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Lemma isapropisabsorb {X : hSet} (opp1 opp2 : binop X) : isaprop (isabsorb opp1 opp2). Proof. apply impred_isaprop ; intros x. apply impred_isaprop ; intros y. apply (setproperty X). Defined.
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Lemma
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Algebra
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Algebra\BinaryOperations.v
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isapropisabsorb
| 400 |
Definition isrigops {X : UU} (opp1 opp2 : binop X) : UU := (β axs : (isabmonoidop opp1) Γ (ismonoidop opp2), (β x : X, opp2 (unel_is (pr1 axs)) x = unel_is (pr1 axs)) Γ (β x : X, opp2 x (unel_is (pr1 axs)) = unel_is (pr1 axs))) Γ (isdistr opp1 opp2).
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Definition
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Algebra
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Algebra\BinaryOperations.v
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isrigops
| 401 |
Definition make_isrigops {X : UU} {opp1 opp2 : binop X} (H1 : isabmonoidop opp1) (H2 : ismonoidop opp2) (H3 : β x : X, (opp2 (unel_is H1) x) = (unel_is H1)) (H4 : β x : X, (opp2 x (unel_is H1)) = (unel_is H1)) (H5 : isdistr opp1 opp2) : isrigops opp1 opp2 := ((H1 ,, H2) ,, H3 ,, H4) ,, H5.
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Definition
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Algebra
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Algebra\BinaryOperations.v
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make_isrigops
| 402 |
Definition rigop1axs_is {X : UU} {opp1 opp2 : binop X} : isrigops opp1 opp2 β isabmonoidop opp1 := Ξ» is, pr1 (pr1 (pr1 is)).
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Definition
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Algebra
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Algebra\BinaryOperations.v
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rigop1axs_is
| 403 |
Definition rigop2axs_is {X : UU} {opp1 opp2 : binop X} : isrigops opp1 opp2 β ismonoidop opp2 := Ξ» is, pr2 (pr1 (pr1 is)).
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Definition
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Algebra
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Algebra\BinaryOperations.v
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rigop2axs_is
| 404 |
Definition rigdistraxs_is {X : UU} {opp1 opp2 : binop X} : isrigops opp1 opp2 β isdistr opp1 opp2 := Ξ» is, pr2 is.
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Definition
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Algebra
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Algebra\BinaryOperations.v
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rigdistraxs_is
| 405 |
Definition rigldistrax_is {X : UU} {opp1 opp2 : binop X} : isrigops opp1 opp2 β isldistr opp1 opp2 := Ξ» is, pr1 (pr2 is).
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Definition
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Algebra
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Algebra\BinaryOperations.v
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rigldistrax_is
| 406 |
Definition rigrdistrax_is {X : UU} {opp1 opp2 : binop X} : isrigops opp1 opp2 β isrdistr opp1 opp2 := Ξ» is, pr2 (pr2 is).
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Definition
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Algebra
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Algebra\BinaryOperations.v
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rigrdistrax_is
| 407 |
Definition rigunel1_is {X : UU} {opp1 opp2 : binop X} (is : isrigops opp1 opp2) : X := pr1 (pr2 (pr1 (rigop1axs_is is))).
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Definition
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Algebra
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Algebra\BinaryOperations.v
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rigunel1_is
| 408 |
Definition rigunel2_is {X : UU} {opp1 opp2 : binop X} (is : isrigops opp1 opp2) : X := (pr1 (pr2 (rigop2axs_is is))).
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Definition
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Algebra
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Algebra\BinaryOperations.v
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rigunel2_is
| 409 |
Definition rigmult0x_is {X : UU} {opp1 opp2 : binop X} (is : isrigops opp1 opp2) (x : X) : opp2 (rigunel1_is is) x = rigunel1_is is := pr1 (pr2 (pr1 is)) x.
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Definition
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Algebra
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Algebra\BinaryOperations.v
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rigmult0x_is
| 410 |
Definition rigmultx0_is {X : UU} {opp1 opp2 : binop X} (is : isrigops opp1 opp2) (x : X) : opp2 x (rigunel1_is is) = rigunel1_is is := pr2 (pr2 (pr1 is)) x.
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Definition
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Algebra
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Algebra\BinaryOperations.v
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rigmultx0_is
| 411 |
Lemma isapropisrigops {X : hSet} (opp1 opp2 : binop X) : isaprop (isrigops opp1 opp2). Proof. apply (isofhleveldirprod 1). - apply (isofhleveltotal2 1). + apply (isofhleveldirprod 1). * apply isapropisabmonoidop. * apply isapropismonoidop. + intro x. apply (isofhleveldirprod 1). * apply impred. intro x'. apply (setproperty X). * apply impred. intro x'. apply (setproperty X). - apply isapropisdistr. Defined.
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Lemma
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Algebra
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Algebra\BinaryOperations.v
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isapropisrigops
| 412 |
Definition isringops {X : UU} (opp1 opp2 : binop X) : UU := (isabgrop opp1 Γ ismonoidop opp2) Γ isdistr opp1 opp2.
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Definition
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Algebra
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Algebra\BinaryOperations.v
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isringops
| 413 |
Definition make_isringops {X : UU} {opp1 opp2 : binop X} (H1 : isabgrop opp1) (H2 : ismonoidop opp2) (H3 : isdistr opp1 opp2) : isringops opp1 opp2 := (H1 ,, H2) ,, H3.
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Definition
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Algebra
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Algebra\BinaryOperations.v
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make_isringops
| 414 |
Definition ringop1axs_is {X : UU} {opp1 opp2 : binop X} : isringops opp1 opp2 β isabgrop opp1 := Ξ» is, pr1 (pr1 is).
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Definition
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Algebra
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Algebra\BinaryOperations.v
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ringop1axs_is
| 415 |
Definition ringop2axs_is {X : UU} {opp1 opp2 : binop X} : isringops opp1 opp2 β ismonoidop opp2 := Ξ» is, pr2 (pr1 is).
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Definition
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Algebra
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Algebra\BinaryOperations.v
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ringop2axs_is
| 416 |
Definition ringdistraxs_is {X : UU} {opp1 opp2 : binop X} : isringops opp1 opp2 β isdistr opp1 opp2 := Ξ» is, pr2 is.
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Definition
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Algebra
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Algebra\BinaryOperations.v
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ringdistraxs_is
| 417 |
Definition ringldistrax_is {X : UU} {opp1 opp2 : binop X} : isringops opp1 opp2 β isldistr opp1 opp2 := Ξ» is, pr1 (pr2 is).
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Definition
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Algebra
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Algebra\BinaryOperations.v
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ringldistrax_is
| 418 |
Definition ringrdistrax_is {X : UU} {opp1 opp2 : binop X} : isringops opp1 opp2 β isrdistr opp1 opp2 := Ξ» is, pr2 (pr2 is).
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Definition
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Algebra
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Algebra\BinaryOperations.v
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ringrdistrax_is
| 419 |
Definition ringunel1_is {X : UU} {opp1 opp2 : binop X} (is : isringops opp1 opp2) : X := unel_is (pr1 (pr1 is)).
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Definition
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Algebra
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Algebra\BinaryOperations.v
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ringunel1_is
| 420 |
Definition ringunel2_is {X : UU} {opp1 opp2 : binop X} (is : isringops opp1 opp2) : X := unel_is (pr2 (pr1 is)).
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Definition
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Algebra
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Algebra\BinaryOperations.v
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ringunel2_is
| 421 |
Lemma isapropisringops {X : hSet} (opp1 opp2 : binop X) : isaprop (isringops opp1 opp2). Proof. apply (isofhleveldirprod 1). - apply (isofhleveldirprod 1). + apply isapropisabgrop. + apply isapropismonoidop. - apply isapropisdistr. Defined.
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Lemma
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Algebra
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Algebra\BinaryOperations.v
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isapropisringops
| 422 |
Lemma multx0_is_l {X : UU} {opp1 opp2 : binop X} (is1 : isgrop opp1) (is2 : ismonoidop opp2) (is12 : isdistr opp1 opp2) (x : X) : opp2 x (unel_is (pr1 is1)) = unel_is (pr1 is1). Proof. induction is12 as [ ldistr0 rdistr0 ]. induction is2 as [ assoc2 [ un2 [ lun2 run2 ] ] ]. simpl in *. apply (invmaponpathsweq (make_weq _ (isweqrmultingr_is is1 (opp2 x un2)))). simpl. induction is1 as [ [ assoc1 [ un1 [ lun1 run1 ] ] ] [ inv0 [ linv0 rinv0 ] ] ]. unfold unel_is. simpl in *. rewrite (lun1 (opp2 x un2)). induction (ldistr0 un1 un2 x). rewrite (run2 x). rewrite (lun1 un2). rewrite (run2 x). apply idpath. Defined.
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Lemma
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Algebra
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Algebra\BinaryOperations.v
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multx0_is_l
| 423 |
Lemma mult0x_is_l {X : UU} {opp1 opp2 : binop X} (is1 : isgrop opp1) (is2 : ismonoidop opp2) (is12 : isdistr opp1 opp2) (x : X) : opp2 (unel_is (pr1 is1)) x = unel_is (pr1 is1). Proof. induction is12 as [ ldistr0 rdistr0 ]. induction is2 as [ assoc2 [ un2 [ lun2 run2 ] ] ]. simpl in *. apply (invmaponpathsweq (make_weq _ (isweqrmultingr_is is1 (opp2 un2 x)))). simpl. induction is1 as [ [ assoc1 [ un1 [ lun1 run1 ] ] ] [ inv0 [ linv0 rinv0 ] ] ]. unfold unel_is. simpl in *. rewrite (lun1 (opp2 un2 x)). induction (rdistr0 un1 un2 x). rewrite (lun2 x). rewrite (lun1 un2). rewrite (lun2 x). apply idpath. Defined.
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Lemma
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Algebra
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Algebra\BinaryOperations.v
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mult0x_is_l
| 424 |
Definition minus1_is_l {X : UU} {opp1 opp2 : binop X} (is1 : isgrop opp1) (is2 : ismonoidop opp2) := (grinv_is is1) (unel_is is2).
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Definition
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Algebra
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Algebra\BinaryOperations.v
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minus1_is_l
| 425 |
Lemma islinvmultwithminus1_is_l {X : UU} {opp1 opp2 : binop X} (is1 : isgrop opp1) (is2 : ismonoidop opp2) (is12 : isdistr opp1 opp2) (x : X) : opp1 (opp2 (minus1_is_l is1 is2) x) x = unel_is (pr1 is1). Proof. set (xinv := opp2 (minus1_is_l is1 is2) x). rewrite <- (lunax_is is2 x). unfold xinv. rewrite <- (pr2 is12 _ _ x). unfold minus1_is_l. unfold grinv_is. rewrite (grlinvax_is is1 _). apply mult0x_is_l. - apply is2. - apply is12. Defined.
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Lemma
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Algebra
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Algebra\BinaryOperations.v
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islinvmultwithminus1_is_l
| 426 |
Lemma isrinvmultwithminus1_is_l {X : UU} {opp1 opp2 : binop X} (is1 : isgrop opp1) (is2 : ismonoidop opp2) (is12 : isdistr opp1 opp2) (x : X) : opp1 x (opp2 (minus1_is_l is1 is2) x) = unel_is (pr1 is1). Proof. set (xinv := opp2 (minus1_is_l is1 is2) x). rewrite <- (lunax_is is2 x). unfold xinv. rewrite <- (pr2 is12 _ _ x). unfold minus1_is_l. unfold grinv_is. rewrite (grrinvax_is is1 _). apply mult0x_is_l. apply is2. apply is12. Defined.
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Lemma
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Algebra
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Algebra\BinaryOperations.v
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isrinvmultwithminus1_is_l
| 427 |
Lemma isminusmultwithminus1_is_l {X : UU} {opp1 opp2 : binop X} (is1 : isgrop opp1) (is2 : ismonoidop opp2) (is12 : isdistr opp1 opp2) (x : X) : opp2 (minus1_is_l is1 is2) x = grinv_is is1 x. Proof. apply (invmaponpathsweq (make_weq _ (isweqrmultingr_is is1 x))). simpl. rewrite (islinvmultwithminus1_is_l is1 is2 is12 x). unfold grinv_is. rewrite (grlinvax_is is1 x). apply idpath. Defined.
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Lemma
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Algebra
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Algebra\BinaryOperations.v
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isminusmultwithminus1_is_l
| 428 |
Lemma isringopsif {X : UU} {opp1 opp2 : binop X} (is1 : isgrop opp1) (is2 : ismonoidop opp2) (is12 : isdistr opp1 opp2) : isringops opp1 opp2. Proof. set (assoc1 := pr1 (pr1 is1)). split. - split. + exists is1. intros x y. apply (invmaponpathsweq (make_weq _ (isweqrmultingr_is is1 (opp2 (minus1_is_l is1 is2) (opp1 x y))))). simpl. rewrite (isrinvmultwithminus1_is_l is1 is2 is12 (opp1 x y)). rewrite (pr1 is12 x y _). induction (assoc1 (opp1 y x) (opp2 (minus1_is_l is1 is2) x) (opp2 (minus1_is_l is1 is2) y)). rewrite (assoc1 y x _). induction (!isrinvmultwithminus1_is_l is1 is2 is12 x). unfold unel_is. rewrite (runax_is (pr1 is1) y). rewrite (isrinvmultwithminus1_is_l is1 is2 is12 y). apply idpath. + apply is2. - apply is12. Defined.
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Lemma
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Algebra
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Algebra\BinaryOperations.v
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isringopsif
| 429 |
Definition ringmultx0_is {X : UU} {opp1 opp2 : binop X} (is : isringops opp1 opp2) : β (x : X), opp2 x (unel_is (pr1 (ringop1axs_is is))) = unel_is (pr1 (ringop1axs_is is)) := multx0_is_l (ringop1axs_is is) (ringop2axs_is is) (ringdistraxs_is is).
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Definition
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Algebra
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Algebra\BinaryOperations.v
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ringmultx0_is
| 430 |
Definition ringmult0x_is {X : UU} {opp1 opp2 : binop X} (is : isringops opp1 opp2) : β (x : X), opp2 (unel_is (pr1 (ringop1axs_is is))) x = unel_is (pr1 (ringop1axs_is is)) := mult0x_is_l (ringop1axs_is is) (ringop2axs_is is) (ringdistraxs_is is).
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Definition
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Algebra
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Algebra\BinaryOperations.v
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ringmult0x_is
| 431 |
Definition ringminus1_is {X : UU} {opp1 opp2 : binop X} (is : isringops opp1 opp2) : X := minus1_is_l (ringop1axs_is is) (ringop2axs_is is).
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Definition
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Algebra
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Algebra\BinaryOperations.v
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ringminus1_is
| 432 |
Definition ringmultwithminus1_is {X : UU} {opp1 opp2 : binop X} (is : isringops opp1 opp2) : β (x : X), opp2 (minus1_is_l (ringop1axs_is is) (ringop2axs_is is)) x = grinv_is (ringop1axs_is is) x := isminusmultwithminus1_is_l (ringop1axs_is is) (ringop2axs_is is) (ringdistraxs_is is).
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Definition
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Algebra
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Algebra\BinaryOperations.v
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ringmultwithminus1_is
| 433 |
Definition isringopstoisrigops (X : UU) (opp1 opp2 : binop X) (is : isringops opp1 opp2) : isrigops opp1 opp2. Proof. split. - exists (isabgroptoisabmonoidop _ _ (ringop1axs_is is) ,, ringop2axs_is is). split. + simpl. apply (ringmult0x_is). + simpl. apply (ringmultx0_is). - apply (ringdistraxs_is is). Defined.
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Definition
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Algebra
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Algebra\BinaryOperations.v
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isringopstoisrigops
| 434 |
Definition iscommrigops {X : UU} (opp1 opp2 : binop X) : UU := (isrigops opp1 opp2) Γ (iscomm opp2).
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Definition
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Algebra
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Algebra\BinaryOperations.v
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iscommrigops
| 435 |
Definition pr1iscommrigops (X : UU) (opp1 opp2 : binop X) : iscommrigops opp1 opp2 β isrigops opp1 opp2 := @pr1 _ _.
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Definition
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Algebra
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Algebra\BinaryOperations.v
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pr1iscommrigops
| 436 |
Definition rigiscommop2_is {X : UU} {opp1 opp2 : binop X} (is : iscommrigops opp1 opp2) : iscomm opp2 := pr2 is.
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Definition
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Algebra
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Algebra\BinaryOperations.v
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rigiscommop2_is
| 437 |
Lemma isapropiscommrig {X : hSet} (opp1 opp2 : binop X) : isaprop (iscommrigops opp1 opp2). Proof. apply (isofhleveldirprod 1). - apply isapropisrigops. - apply isapropiscomm. Defined.
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Lemma
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Algebra
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Algebra\BinaryOperations.v
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isapropiscommrig
| 438 |
Definition iscommringops {X : UU} (opp1 opp2 : binop X) : UU := (isringops opp1 opp2) Γ (iscomm opp2).
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Definition
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Algebra
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Algebra\BinaryOperations.v
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iscommringops
| 439 |
Definition pr1iscommringops (X : UU) (opp1 opp2 : binop X) : iscommringops opp1 opp2 β isringops opp1 opp2 := @pr1 _ _.
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Definition
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Algebra
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Algebra\BinaryOperations.v
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pr1iscommringops
| 440 |
Definition ringiscommop2_is {X : UU} {opp1 opp2 : binop X} (is : iscommringops opp1 opp2) : iscomm opp2 := pr2 is.
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Definition
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Algebra
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Algebra\BinaryOperations.v
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ringiscommop2_is
| 441 |
Lemma isapropiscommring {X : hSet} (opp1 opp2 : binop X) : isaprop (iscommringops opp1 opp2). Proof. apply (isofhleveldirprod 1). - apply isapropisringops. - apply isapropiscomm. Defined.
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Lemma
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Algebra
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Algebra\BinaryOperations.v
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isapropiscommring
| 442 |
Definition iscommringopstoiscommrigops (X : UU) (opp1 opp2 : binop X) (is : iscommringops opp1 opp2) : iscommrigops opp1 opp2 := isringopstoisrigops _ _ _ (pr1 is) ,, pr2 is.
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Definition
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Algebra
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Algebra\BinaryOperations.v
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iscommringopstoiscommrigops
| 443 |
Lemma isassoc_weq_fwd {X Y : UU} (H : X β Y) (opp : binop X) : isassoc opp β isassoc (binop_weq_fwd H opp). Proof. intros is x y z. apply (maponpaths H). refine (_ @ is _ _ _ @ _). - apply (maponpaths (Ξ» x, opp x _)). apply homotinvweqweq. - apply maponpaths. apply homotinvweqweq0. Defined.
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Lemma
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Algebra
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Algebra\BinaryOperations.v
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isassoc_weq_fwd
| 444 |
Lemma islunit_weq_fwd {X Y : UU} (H : X β Y) (opp : binop X) (x0 : X) : islunit opp x0 β islunit (binop_weq_fwd H opp) (H x0). Proof. intros is y. unfold binop_weq_fwd. refine (maponpaths _ _ @ _). - refine (maponpaths (Ξ» x, opp x _) _ @ _). + apply homotinvweqweq. + apply is. - apply homotweqinvweq. Defined.
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Lemma
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Algebra
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Algebra\BinaryOperations.v
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islunit_weq_fwd
| 445 |
Lemma isrunit_weq_fwd {X Y : UU} (H : X β Y) (opp : binop X) (x0 : X) : isrunit opp x0 β isrunit (binop_weq_fwd H opp) (H x0). Proof. intros is y. unfold binop_weq_fwd. refine (maponpaths _ _ @ _). - refine (maponpaths (opp _) _ @ _). + apply homotinvweqweq. + apply is. - apply homotweqinvweq. Defined.
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Lemma
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Algebra
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Algebra\BinaryOperations.v
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isrunit_weq_fwd
| 446 |
Lemma isunit_weq_fwd {X Y : UU} (H : X β Y) (opp : binop X) (x0 : X) : isunit opp x0 β isunit (binop_weq_fwd H opp) (H x0). Proof. intro is. split. apply islunit_weq_fwd, (pr1 is). apply isrunit_weq_fwd, (pr2 is). Defined.
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Lemma
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Algebra
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Algebra\BinaryOperations.v
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isunit_weq_fwd
| 447 |
Lemma isunital_weq_fwd {X Y : UU} (H : X β Y) (opp : binop X) : isunital opp β isunital (binop_weq_fwd H opp). Proof. intro is. exists (H (pr1 is)). apply isunit_weq_fwd, (pr2 is). Defined.
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Lemma
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Algebra
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Algebra\BinaryOperations.v
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isunital_weq_fwd
| 448 |
Lemma ismonoidop_weq_fwd {X Y : UU} (H : X β Y) (opp : binop X) : ismonoidop opp β ismonoidop (binop_weq_fwd H opp). Proof. intro is. split. apply isassoc_weq_fwd, (pr1 is). apply isunital_weq_fwd, (pr2 is). Defined.
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Lemma
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Algebra
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Algebra\BinaryOperations.v
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ismonoidop_weq_fwd
| 449 |
Lemma islinv_weq_fwd {X Y : UU} (H : X β Y) (opp : binop X) (x0 : X) (inv : X β X) : islinv opp x0 inv β islinv (binop_weq_fwd H opp) (H x0) (Ξ» y : Y, H (inv (invmap H y))). Proof. intros is y. unfold binop_weq_fwd. apply maponpaths. refine (_ @ is _). apply (maponpaths (Ξ» x, opp x _)). apply homotinvweqweq. Defined.
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Lemma
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Algebra
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Algebra\BinaryOperations.v
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islinv_weq_fwd
| 450 |
Lemma isrinv_weq_fwd {X Y : UU} (H : X β Y) (opp : binop X) (x0 : X) (inv : X β X) : isrinv opp x0 inv β isrinv (binop_weq_fwd H opp) (H x0) (Ξ» y : Y, H (inv (invmap H y))). Proof. intros is y. unfold binop_weq_fwd. apply maponpaths. refine (_ @ is _). apply (maponpaths (opp _)). apply homotinvweqweq. Defined.
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Lemma
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Algebra
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Algebra\BinaryOperations.v
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isrinv_weq_fwd
| 451 |
Lemma isinv_weq_fwd {X Y : UU} (H : X β Y) (opp : binop X) (x0 : X) (inv : X β X) : isinv opp x0 inv β isinv (binop_weq_fwd H opp) (H x0) (Ξ» y : Y, H (inv (invmap H y))). Proof. intro is. split. apply islinv_weq_fwd, (pr1 is). apply isrinv_weq_fwd, (pr2 is). Defined.
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Lemma
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Algebra
| null |
Algebra\BinaryOperations.v
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isinv_weq_fwd
| 452 |
Lemma invstruct_weq_fwd {X Y : UU} (H : X β Y) (opp : binop X) (is : ismonoidop opp) : invstruct opp is β invstruct (binop_weq_fwd H opp) (ismonoidop_weq_fwd H opp is). Proof. intro inv. exists (Ξ» y : Y, H (pr1 inv (invmap H y))). apply isinv_weq_fwd, (pr2 inv). Defined.
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Lemma
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Algebra
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Algebra\BinaryOperations.v
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invstruct_weq_fwd
| 453 |
Lemma isgrop_weq_fwd {X Y : UU} (H : X β Y) (opp : binop X) : isgrop opp β isgrop (binop_weq_fwd H opp). Proof. intro is. use tpair. - apply ismonoidop_weq_fwd, (pr1 is). - apply invstruct_weq_fwd, (pr2 is). Defined.
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Lemma
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Algebra
| null |
Algebra\BinaryOperations.v
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isgrop_weq_fwd
| 454 |
Lemma iscomm_weq_fwd {X Y : UU} (H : X β Y) (opp : binop X) : iscomm opp β iscomm (binop_weq_fwd H opp). Proof. intros is x y. unfold binop_weq_fwd. apply maponpaths, is. Defined.
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Lemma
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Algebra
| null |
Algebra\BinaryOperations.v
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iscomm_weq_fwd
| 455 |
Lemma isabmonoidop_weq_fwd {X Y : UU} (H : X β Y) (opp : binop X) : isabmonoidop opp β isabmonoidop (binop_weq_fwd H opp). Proof. intro is. split. apply ismonoidop_weq_fwd, (pr1 is). apply iscomm_weq_fwd, (pr2 is). Defined.
|
Lemma
|
Algebra
| null |
Algebra\BinaryOperations.v
|
isabmonoidop_weq_fwd
| 456 |
Lemma isabgrop_weq_fwd {X Y : UU} (H : X β Y) (opp : binop X) : isabgrop opp β isabgrop (binop_weq_fwd H opp). Proof. intro is. split. apply isgrop_weq_fwd, (pr1 is). apply iscomm_weq_fwd, (pr2 is). Defined.
|
Lemma
|
Algebra
| null |
Algebra\BinaryOperations.v
|
isabgrop_weq_fwd
| 457 |
Lemma isldistr_weq_fwd {X Y : UU} (H : X β Y) (op1 op2 : binop X) : isldistr op1 op2 β isldistr (binop_weq_fwd H op1) (binop_weq_fwd H op2). Proof. intros is x y z. unfold binop_weq_fwd. apply maponpaths. refine (_ @ is _ _ _ @ _). - apply maponpaths. apply homotinvweqweq. - apply map_on_two_paths ; apply homotinvweqweq0. Defined.
|
Lemma
|
Algebra
| null |
Algebra\BinaryOperations.v
|
isldistr_weq_fwd
| 458 |
Lemma isrdistr_weq_fwd {X Y : UU} (H : X β Y) (op1 op2 : binop X) : isrdistr op1 op2 β isrdistr (binop_weq_fwd H op1) (binop_weq_fwd H op2). Proof. intros is x y z. unfold binop_weq_fwd. apply maponpaths. refine (_ @ is _ _ _ @ _). - apply (maponpaths (Ξ» x, op2 x _)). apply homotinvweqweq. - apply map_on_two_paths ; apply homotinvweqweq0. Defined.
|
Lemma
|
Algebra
| null |
Algebra\BinaryOperations.v
|
isrdistr_weq_fwd
| 459 |
Lemma isdistr_weq_fwd {X Y : UU} (H : X β Y) (op1 op2 : binop X) : isdistr op1 op2 β isdistr (binop_weq_fwd H op1) (binop_weq_fwd H op2). Proof. intro is. split. apply isldistr_weq_fwd, (pr1 is). apply isrdistr_weq_fwd, (pr2 is). Defined.
|
Lemma
|
Algebra
| null |
Algebra\BinaryOperations.v
|
isdistr_weq_fwd
| 460 |
Lemma isabsorb_weq_fwd {X Y : UU} (H : X β Y) (op1 op2 : binop X) : isabsorb op1 op2 β isabsorb (binop_weq_fwd H op1) (binop_weq_fwd H op2). Proof. intros is x y. unfold binop_weq_fwd. refine (_ @ homotweqinvweq H _). apply maponpaths. refine (_ @ is _ _). apply maponpaths. apply (homotinvweqweq H). Defined.
|
Lemma
|
Algebra
| null |
Algebra\BinaryOperations.v
|
isabsorb_weq_fwd
| 461 |
Lemma isrigops_weq_fwd {X Y : UU} (H : X β Y) (op1 op2 : binop X) : isrigops op1 op2 β isrigops (binop_weq_fwd H op1) (binop_weq_fwd H op2). Proof. intro is. split. - use tpair. + split. apply isabmonoidop_weq_fwd, (pr1 (pr1 (pr1 is))). apply ismonoidop_weq_fwd, (pr2 (pr1 (pr1 is))). + split ; simpl. * intros x. apply (maponpaths H). refine (_ @ pr1 (pr2 (pr1 is)) _). apply (maponpaths (Ξ» x, op2 x _)). apply homotinvweqweq. * intros x. apply (maponpaths H). refine (_ @ pr2 (pr2 (pr1 is)) _). apply (maponpaths (op2 _)). apply homotinvweqweq. - apply isdistr_weq_fwd, (pr2 is). Defined.
|
Lemma
|
Algebra
| null |
Algebra\BinaryOperations.v
|
isrigops_weq_fwd
| 462 |
Lemma isringops_weq_fwd {X Y : UU} (H : X β Y) (op1 op2 : binop X) : isringops op1 op2 β isringops (binop_weq_fwd H op1) (binop_weq_fwd H op2). Proof. intro is. split. - split. + apply isabgrop_weq_fwd, (pr1 (pr1 is)). + apply ismonoidop_weq_fwd, (pr2 (pr1 is)). - apply isdistr_weq_fwd, (pr2 is). Defined.
|
Lemma
|
Algebra
| null |
Algebra\BinaryOperations.v
|
isringops_weq_fwd
| 463 |
Lemma iscommrigops_weq_fwd {X Y : UU} (H : X β Y) (op1 op2 : binop X) : iscommrigops op1 op2 β iscommrigops (binop_weq_fwd H op1) (binop_weq_fwd H op2). Proof. intro is. split. - apply isrigops_weq_fwd, (pr1 is). - apply iscomm_weq_fwd, (pr2 is). Defined.
|
Lemma
|
Algebra
| null |
Algebra\BinaryOperations.v
|
iscommrigops_weq_fwd
| 464 |
Lemma iscommringops_weq_fwd {X Y : UU} (H : X β Y) (op1 op2 : binop X) : iscommringops op1 op2 β iscommringops (binop_weq_fwd H op1) (binop_weq_fwd H op2). Proof. intro is. split. - apply isringops_weq_fwd, (pr1 is). - apply iscomm_weq_fwd, (pr2 is). Defined.
|
Lemma
|
Algebra
| null |
Algebra\BinaryOperations.v
|
iscommringops_weq_fwd
| 465 |
Lemma isassoc_weq_bck {X Y : UU} (H : X β Y) (opp : binop Y) : isassoc opp β isassoc (binop_weq_bck H opp). Proof. intros is x y z. apply (maponpaths (invmap H)). refine (_ @ is _ _ _ @ _). - apply (maponpaths (Ξ» x, opp x _)). apply homotweqinvweq. - apply maponpaths. symmetry. apply homotweqinvweq. Defined.
|
Lemma
|
Algebra
| null |
Algebra\BinaryOperations.v
|
isassoc_weq_bck
| 466 |
Lemma islunit_weq_bck {X Y : UU} (H : X β Y) (opp : binop Y) (x0 : Y) : islunit opp x0 β islunit (binop_weq_bck H opp) (invmap H x0). Proof. intros is y. unfold binop_weq_bck. refine (maponpaths _ _ @ _). - refine (maponpaths (Ξ» x, opp x _) _ @ _). + apply homotweqinvweq. + apply is. - apply homotinvweqweq. Defined.
|
Lemma
|
Algebra
| null |
Algebra\BinaryOperations.v
|
islunit_weq_bck
| 467 |
Lemma isrunit_weq_bck {X Y : UU} (H : X β Y) (opp : binop Y) (x0 : Y) : isrunit opp x0 β isrunit (binop_weq_bck H opp) (invmap H x0). Proof. intros is y. unfold binop_weq_bck. refine (maponpaths _ _ @ _). - refine (maponpaths (opp _) _ @ _). + apply homotweqinvweq. + apply is. - apply homotinvweqweq. Defined.
|
Lemma
|
Algebra
| null |
Algebra\BinaryOperations.v
|
isrunit_weq_bck
| 468 |
Lemma isunit_weq_bck {X Y : UU} (H : X β Y) (opp : binop Y) (x0 : Y) : isunit opp x0 β isunit (binop_weq_bck H opp) (invmap H x0). Proof. intro is. split. apply islunit_weq_bck, (pr1 is). apply isrunit_weq_bck, (pr2 is). Defined.
|
Lemma
|
Algebra
| null |
Algebra\BinaryOperations.v
|
isunit_weq_bck
| 469 |
Lemma isunital_weq_bck {X Y : UU} (H : X β Y) (opp : binop Y) : isunital opp β isunital (binop_weq_bck H opp). Proof. intro is. exists (invmap H (pr1 is)). apply isunit_weq_bck, (pr2 is). Defined.
|
Lemma
|
Algebra
| null |
Algebra\BinaryOperations.v
|
isunital_weq_bck
| 470 |
Lemma ismonoidop_weq_bck {X Y : UU} (H : X β Y) (opp : binop Y) : ismonoidop opp β ismonoidop (binop_weq_bck H opp). Proof. intro is. split. apply isassoc_weq_bck, (pr1 is). apply isunital_weq_bck, (pr2 is). Defined.
|
Lemma
|
Algebra
| null |
Algebra\BinaryOperations.v
|
ismonoidop_weq_bck
| 471 |
Lemma islinv_weq_bck {X Y : UU} (H : X β Y) (opp : binop Y) (x0 : Y) (inv : Y β Y) : islinv opp x0 inv β islinv (binop_weq_bck H opp) (invmap H x0) (Ξ» y : X, invmap H (inv (H y))). Proof. intros is y. unfold binop_weq_bck. apply maponpaths. refine (_ @ is _). apply (maponpaths (Ξ» x, opp x _)). apply homotweqinvweq. Defined.
|
Lemma
|
Algebra
| null |
Algebra\BinaryOperations.v
|
islinv_weq_bck
| 472 |
Lemma isrinv_weq_bck {X Y : UU} (H : X β Y) (opp : binop Y) (x0 : Y) (inv : Y β Y) : isrinv opp x0 inv β isrinv (binop_weq_bck H opp) (invmap H x0) (Ξ» y : X, invmap H (inv (H y))). Proof. intros is y. unfold binop_weq_bck. apply maponpaths. refine (_ @ is _). apply (maponpaths (opp _)). apply homotweqinvweq. Defined.
|
Lemma
|
Algebra
| null |
Algebra\BinaryOperations.v
|
isrinv_weq_bck
| 473 |
Lemma isinv_weq_bck {X Y : UU} (H : X β Y) (opp : binop Y) (x0 : Y) (inv : Y β Y) : isinv opp x0 inv β isinv (binop_weq_bck H opp) (invmap H x0) (Ξ» y : X, invmap H (inv (H y))). Proof. intro is. split. apply islinv_weq_bck, (pr1 is). apply isrinv_weq_bck, (pr2 is). Defined.
|
Lemma
|
Algebra
| null |
Algebra\BinaryOperations.v
|
isinv_weq_bck
| 474 |
Lemma invstruct_weq_bck {X Y : UU} (H : X β Y) (opp : binop Y) (is : ismonoidop opp) : invstruct opp is β invstruct (binop_weq_bck H opp) (ismonoidop_weq_bck H opp is). Proof. intro inv. exists (Ξ» y : X, invmap H (pr1 inv (H y))). apply isinv_weq_bck, (pr2 inv). Defined.
|
Lemma
|
Algebra
| null |
Algebra\BinaryOperations.v
|
invstruct_weq_bck
| 475 |
Lemma isgrop_weq_bck {X Y : UU} (H : X β Y) (opp : binop Y) : isgrop opp β isgrop (binop_weq_bck H opp). Proof. intro is. use tpair. apply ismonoidop_weq_bck, (pr1 is). apply invstruct_weq_bck, (pr2 is). Defined.
|
Lemma
|
Algebra
| null |
Algebra\BinaryOperations.v
|
isgrop_weq_bck
| 476 |
Lemma iscomm_weq_bck {X Y : UU} (H : X β Y) (opp : binop Y) : iscomm opp β iscomm (binop_weq_bck H opp). Proof. intros is x y. unfold binop_weq_bck. apply maponpaths, is. Defined.
|
Lemma
|
Algebra
| null |
Algebra\BinaryOperations.v
|
iscomm_weq_bck
| 477 |
Lemma isabmonoidop_weq_bck {X Y : UU} (H : X β Y) (opp : binop Y) : isabmonoidop opp β isabmonoidop (binop_weq_bck H opp). Proof. intro is. split. apply ismonoidop_weq_bck, (pr1 is). apply iscomm_weq_bck, (pr2 is). Defined.
|
Lemma
|
Algebra
| null |
Algebra\BinaryOperations.v
|
isabmonoidop_weq_bck
| 478 |
Lemma isabgrop_weq_bck {X Y : UU} (H : X β Y) (opp : binop Y) : isabgrop opp β isabgrop (binop_weq_bck H opp). Proof. intro is. split. apply isgrop_weq_bck, (pr1 is). apply iscomm_weq_bck, (pr2 is). Defined.
|
Lemma
|
Algebra
| null |
Algebra\BinaryOperations.v
|
isabgrop_weq_bck
| 479 |
Lemma isldistr_weq_bck {X Y : UU} (H : X β Y) (op1 op2 : binop Y) : isldistr op1 op2 β isldistr (binop_weq_bck H op1) (binop_weq_bck H op2). Proof. intros is x y z. unfold binop_weq_bck. apply maponpaths. refine (_ @ is _ _ _ @ _). - apply maponpaths. apply homotweqinvweq. - apply map_on_two_paths; exact (!homotweqinvweq _ _). Defined.
|
Lemma
|
Algebra
| null |
Algebra\BinaryOperations.v
|
isldistr_weq_bck
| 480 |
Lemma isrdistr_weq_bck {X Y : UU} (H : X β Y) (op1 op2 : binop Y) : isrdistr op1 op2 β isrdistr (binop_weq_bck H op1) (binop_weq_bck H op2). Proof. intros is x y z. unfold binop_weq_bck. apply maponpaths. refine (_ @ is _ _ _ @ _). - apply (maponpaths (Ξ» x, op2 x _)). apply homotweqinvweq. - apply map_on_two_paths; exact (!homotweqinvweq _ _). Defined.
|
Lemma
|
Algebra
| null |
Algebra\BinaryOperations.v
|
isrdistr_weq_bck
| 481 |
Lemma isdistr_weq_bck {X Y : UU} (H : X β Y) (op1 op2 : binop Y) : isdistr op1 op2 β isdistr (binop_weq_bck H op1) (binop_weq_bck H op2). Proof. intro is. split. apply isldistr_weq_bck, (pr1 is). apply isrdistr_weq_bck, (pr2 is). Defined.
|
Lemma
|
Algebra
| null |
Algebra\BinaryOperations.v
|
isdistr_weq_bck
| 482 |
Lemma isabsorb_weq_bck {X Y : UU} (H : X β Y) (op1 op2 : binop Y) : isabsorb op1 op2 β isabsorb (binop_weq_bck H op1) (binop_weq_bck H op2). Proof. intros is x y. unfold binop_weq_bck. refine (_ @ homotinvweqweq H _). apply maponpaths. refine (_ @ is _ _). apply maponpaths. apply (homotweqinvweq H). Defined.
|
Lemma
|
Algebra
| null |
Algebra\BinaryOperations.v
|
isabsorb_weq_bck
| 483 |
Lemma isrigops_weq_bck {X Y : UU} (H : X β Y) (op1 op2 : binop Y) : isrigops op1 op2 β isrigops (binop_weq_bck H op1) (binop_weq_bck H op2). Proof. intro is. split. - use tpair. + split. apply isabmonoidop_weq_bck, (pr1 (pr1 (pr1 is))). apply ismonoidop_weq_bck, (pr2 (pr1 (pr1 is))). + split ; simpl. * intros x. apply (maponpaths (invmap H)). refine (_ @ pr1 (pr2 (pr1 is)) _). apply (maponpaths (Ξ» x, op2 x _)). apply homotweqinvweq. * intros x. apply (maponpaths (invmap H)). refine (_ @ pr2 (pr2 (pr1 is)) _). apply (maponpaths (op2 _)). apply homotweqinvweq. - apply isdistr_weq_bck, (pr2 is). Defined.
|
Lemma
|
Algebra
| null |
Algebra\BinaryOperations.v
|
isrigops_weq_bck
| 484 |
Lemma isringops_weq_bck {X Y : UU} (H : X β Y) (op1 op2 : binop Y) : isringops op1 op2 β isringops (binop_weq_bck H op1) (binop_weq_bck H op2). Proof. intro is. split. - split. + apply isabgrop_weq_bck, (pr1 (pr1 is)). + apply ismonoidop_weq_bck, (pr2 (pr1 is)). - apply isdistr_weq_bck, (pr2 is). Defined.
|
Lemma
|
Algebra
| null |
Algebra\BinaryOperations.v
|
isringops_weq_bck
| 485 |
Lemma iscommrigops_weq_bck {X Y : UU} (H : X β Y) (op1 op2 : binop Y) : iscommrigops op1 op2 β iscommrigops (binop_weq_bck H op1) (binop_weq_bck H op2). Proof. intro is. split. - apply isrigops_weq_bck, (pr1 is). - apply iscomm_weq_bck, (pr2 is). Defined.
|
Lemma
|
Algebra
| null |
Algebra\BinaryOperations.v
|
iscommrigops_weq_bck
| 486 |
Lemma iscommringops_weq_bck {X Y : UU} (H : X β Y) (op1 op2 : binop Y) : iscommringops op1 op2 β iscommringops (binop_weq_bck H op1) (binop_weq_bck H op2). Proof. intro is. split. - apply isringops_weq_bck, (pr1 is). - apply iscomm_weq_bck, (pr2 is). Defined.
|
Lemma
|
Algebra
| null |
Algebra\BinaryOperations.v
|
iscommringops_weq_bck
| 487 |
Definition setwithbinop : UU := β (X : hSet), binop X.
|
Definition
|
Algebra
| null |
Algebra\BinaryOperations.v
|
setwithbinop
| 488 |
Definition make_setwithbinop (X : hSet) (opp : binop X) : setwithbinop := X ,, opp.
|
Definition
|
Algebra
| null |
Algebra\BinaryOperations.v
|
make_setwithbinop
| 489 |
Definition pr1setwithbinop : setwithbinop β hSet := @pr1 _ (Ξ» X : hSet, binop X).
|
Definition
|
Algebra
| null |
Algebra\BinaryOperations.v
|
pr1setwithbinop
| 490 |
Definition op {X : setwithbinop} : binop X := pr2 X.
|
Definition
|
Algebra
| null |
Algebra\BinaryOperations.v
|
op
| 491 |
Definition isasetbinoponhSet (X : hSet) : isaset (@binop X).
|
Definition
|
Algebra
| null |
Algebra\BinaryOperations.v
|
isasetbinoponhSet
| 492 |
Definition setwithbinop_rev (X : setwithbinop) : setwithbinop := make_setwithbinop X (Ξ» x y, op y x).
|
Definition
|
Algebra
| null |
Algebra\BinaryOperations.v
|
setwithbinop_rev
| 493 |
Definition isbinopfun {X Y : setwithbinop} (f : X β Y) : UU := β x x' : X, f (op x x') = op (f x) (f x').
|
Definition
|
Algebra
| null |
Algebra\BinaryOperations.v
|
isbinopfun
| 494 |
Definition make_isbinopfun {X Y : setwithbinop} {f : X β Y} (H : β x x' : X, f (op x x') = op (f x) (f x')) : isbinopfun f := H.
|
Definition
|
Algebra
| null |
Algebra\BinaryOperations.v
|
make_isbinopfun
| 495 |
Lemma isapropisbinopfun {X Y : setwithbinop} (f : X β Y) : isaprop (isbinopfun f). Proof. apply impred. intro x. apply impred. intro x'. apply (setproperty Y). Defined.
|
Lemma
|
Algebra
| null |
Algebra\BinaryOperations.v
|
isapropisbinopfun
| 496 |
Definition isbinopfun_twooutof3b {A B C : setwithbinop} (f : A β B) (g : B β C) (H : issurjective f) : isbinopfun (g β f)%functions β isbinopfun f β isbinopfun g.
|
Definition
|
Algebra
| null |
Algebra\BinaryOperations.v
|
isbinopfun_twooutof3b
| 497 |
Definition binopfun (X Y : setwithbinop) : UU := β (f : X β Y), isbinopfun f.
|
Definition
|
Algebra
| null |
Algebra\BinaryOperations.v
|
binopfun
| 498 |
Definition make_binopfun {X Y : setwithbinop} (f : X β Y) (is : isbinopfun f) : binopfun X Y := f ,, is.
|
Definition
|
Algebra
| null |
Algebra\BinaryOperations.v
|
make_binopfun
| 499 |
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