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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.

Lemma

Algebra

Algebra\BinaryOperations.v

isapropisabsorb

Definition isrigops {X : UU} (opp1 opp2 : binop X) : UU := (∑ axs : (isabmonoidop opp1) × (ismonoidop opp2), (∏ x : X, opp2 (unel_is (pr1 axs)) x = unel_is (pr1 axs)) × (∏ x : X, opp2 x (unel_is (pr1 axs)) = unel_is (pr1 axs))) × (isdistr opp1 opp2).

Definition

Algebra

Algebra\BinaryOperations.v

isrigops

Definition make_isrigops {X : UU} {opp1 opp2 : binop X} (H1 : isabmonoidop opp1) (H2 : ismonoidop opp2) (H3 : ∏ x : X, (opp2 (unel_is H1) x) = (unel_is H1)) (H4 : ∏ x : X, (opp2 x (unel_is H1)) = (unel_is H1)) (H5 : isdistr opp1 opp2) : isrigops opp1 opp2 := ((H1 ,, H2) ,, H3 ,, H4) ,, H5.

Definition

Algebra

Algebra\BinaryOperations.v

make_isrigops

Definition rigop1axs_is {X : UU} {opp1 opp2 : binop X} : isrigops opp1 opp2 → isabmonoidop opp1 := λ is, pr1 (pr1 (pr1 is)).

Definition

Algebra

Algebra\BinaryOperations.v

rigop1axs_is

Definition rigop2axs_is {X : UU} {opp1 opp2 : binop X} : isrigops opp1 opp2 → ismonoidop opp2 := λ is, pr2 (pr1 (pr1 is)).

Definition

Algebra

Algebra\BinaryOperations.v

rigop2axs_is

Definition rigdistraxs_is {X : UU} {opp1 opp2 : binop X} : isrigops opp1 opp2 → isdistr opp1 opp2 := λ is, pr2 is.

Definition

Algebra

Algebra\BinaryOperations.v

rigdistraxs_is

Definition rigldistrax_is {X : UU} {opp1 opp2 : binop X} : isrigops opp1 opp2 → isldistr opp1 opp2 := λ is, pr1 (pr2 is).

Definition

Algebra

Algebra\BinaryOperations.v

rigldistrax_is

Definition rigrdistrax_is {X : UU} {opp1 opp2 : binop X} : isrigops opp1 opp2 → isrdistr opp1 opp2 := λ is, pr2 (pr2 is).

Definition

Algebra

Algebra\BinaryOperations.v

rigrdistrax_is

Definition rigunel1_is {X : UU} {opp1 opp2 : binop X} (is : isrigops opp1 opp2) : X := pr1 (pr2 (pr1 (rigop1axs_is is))).

Definition

Algebra

Algebra\BinaryOperations.v

rigunel1_is

Definition rigunel2_is {X : UU} {opp1 opp2 : binop X} (is : isrigops opp1 opp2) : X := (pr1 (pr2 (rigop2axs_is is))).

Definition

Algebra

Algebra\BinaryOperations.v

rigunel2_is

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.

Definition

Algebra

Algebra\BinaryOperations.v

rigmult0x_is

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.

Definition

Algebra

Algebra\BinaryOperations.v

rigmultx0_is

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.

Lemma

Algebra

Algebra\BinaryOperations.v

isapropisrigops

Definition isringops {X : UU} (opp1 opp2 : binop X) : UU := (isabgrop opp1 × ismonoidop opp2) × isdistr opp1 opp2.

Definition

Algebra

Algebra\BinaryOperations.v

isringops

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.

Definition

Algebra

Algebra\BinaryOperations.v

make_isringops

Definition ringop1axs_is {X : UU} {opp1 opp2 : binop X} : isringops opp1 opp2 → isabgrop opp1 := λ is, pr1 (pr1 is).

Definition

Algebra

Algebra\BinaryOperations.v

ringop1axs_is

Definition ringop2axs_is {X : UU} {opp1 opp2 : binop X} : isringops opp1 opp2 → ismonoidop opp2 := λ is, pr2 (pr1 is).

Definition

Algebra

Algebra\BinaryOperations.v

ringop2axs_is

Definition ringdistraxs_is {X : UU} {opp1 opp2 : binop X} : isringops opp1 opp2 → isdistr opp1 opp2 := λ is, pr2 is.

Definition

Algebra

Algebra\BinaryOperations.v

ringdistraxs_is

Definition ringldistrax_is {X : UU} {opp1 opp2 : binop X} : isringops opp1 opp2 → isldistr opp1 opp2 := λ is, pr1 (pr2 is).

Definition

Algebra

Algebra\BinaryOperations.v

ringldistrax_is

Definition ringrdistrax_is {X : UU} {opp1 opp2 : binop X} : isringops opp1 opp2 → isrdistr opp1 opp2 := λ is, pr2 (pr2 is).

Definition

Algebra

Algebra\BinaryOperations.v

ringrdistrax_is

Definition ringunel1_is {X : UU} {opp1 opp2 : binop X} (is : isringops opp1 opp2) : X := unel_is (pr1 (pr1 is)).

Definition

Algebra

Algebra\BinaryOperations.v

ringunel1_is

Definition ringunel2_is {X : UU} {opp1 opp2 : binop X} (is : isringops opp1 opp2) : X := unel_is (pr2 (pr1 is)).

Definition

Algebra

Algebra\BinaryOperations.v

ringunel2_is

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.

Lemma

Algebra

Algebra\BinaryOperations.v

isapropisringops

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.

Lemma

Algebra

Algebra\BinaryOperations.v

multx0_is_l

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.

Lemma

Algebra

Algebra\BinaryOperations.v

mult0x_is_l

Definition minus1_is_l {X : UU} {opp1 opp2 : binop X} (is1 : isgrop opp1) (is2 : ismonoidop opp2) := (grinv_is is1) (unel_is is2).

Definition

Algebra

Algebra\BinaryOperations.v

minus1_is_l

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.

Lemma

Algebra

Algebra\BinaryOperations.v

islinvmultwithminus1_is_l

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.

Lemma

Algebra

Algebra\BinaryOperations.v

isrinvmultwithminus1_is_l

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.

Lemma

Algebra

Algebra\BinaryOperations.v

isminusmultwithminus1_is_l

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.

Lemma

Algebra

Algebra\BinaryOperations.v

isringopsif

Definition ringmultx0_is {X : UU} {opp1 opp2 : binop X} (is : isringops opp1 opp2) : ∏ (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).

Definition

Algebra

Algebra\BinaryOperations.v

ringmultx0_is

Definition ringmult0x_is {X : UU} {opp1 opp2 : binop X} (is : isringops opp1 opp2) : ∏ (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).

Definition

Algebra

Algebra\BinaryOperations.v

ringmult0x_is

Definition ringminus1_is {X : UU} {opp1 opp2 : binop X} (is : isringops opp1 opp2) : X := minus1_is_l (ringop1axs_is is) (ringop2axs_is is).

Definition

Algebra

Algebra\BinaryOperations.v

ringminus1_is

Definition ringmultwithminus1_is {X : UU} {opp1 opp2 : binop X} (is : isringops opp1 opp2) : ∏ (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).

Definition

Algebra

Algebra\BinaryOperations.v

ringmultwithminus1_is

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.

Definition

Algebra

Algebra\BinaryOperations.v

isringopstoisrigops

Definition iscommrigops {X : UU} (opp1 opp2 : binop X) : UU := (isrigops opp1 opp2) × (iscomm opp2).

Definition

Algebra

Algebra\BinaryOperations.v

iscommrigops

Definition pr1iscommrigops (X : UU) (opp1 opp2 : binop X) : iscommrigops opp1 opp2 → isrigops opp1 opp2 := @pr1 _ _.

Definition

Algebra

Algebra\BinaryOperations.v

pr1iscommrigops

Definition rigiscommop2_is {X : UU} {opp1 opp2 : binop X} (is : iscommrigops opp1 opp2) : iscomm opp2 := pr2 is.

Definition

Algebra

Algebra\BinaryOperations.v

rigiscommop2_is

Lemma isapropiscommrig {X : hSet} (opp1 opp2 : binop X) : isaprop (iscommrigops opp1 opp2). Proof. apply (isofhleveldirprod 1). - apply isapropisrigops. - apply isapropiscomm. Defined.

Lemma

Algebra

Algebra\BinaryOperations.v

isapropiscommrig

Definition iscommringops {X : UU} (opp1 opp2 : binop X) : UU := (isringops opp1 opp2) × (iscomm opp2).

Definition

Algebra

Algebra\BinaryOperations.v

iscommringops

Definition pr1iscommringops (X : UU) (opp1 opp2 : binop X) : iscommringops opp1 opp2 → isringops opp1 opp2 := @pr1 _ _.

Definition

Algebra

Algebra\BinaryOperations.v

pr1iscommringops

Definition ringiscommop2_is {X : UU} {opp1 opp2 : binop X} (is : iscommringops opp1 opp2) : iscomm opp2 := pr2 is.

Definition

Algebra

Algebra\BinaryOperations.v

ringiscommop2_is

Lemma isapropiscommring {X : hSet} (opp1 opp2 : binop X) : isaprop (iscommringops opp1 opp2). Proof. apply (isofhleveldirprod 1). - apply isapropisringops. - apply isapropiscomm. Defined.

Lemma

Algebra

Algebra\BinaryOperations.v

isapropiscommring

Definition iscommringopstoiscommrigops (X : UU) (opp1 opp2 : binop X) (is : iscommringops opp1 opp2) : iscommrigops opp1 opp2 := isringopstoisrigops _ _ _ (pr1 is) ,, pr2 is.

Definition

Algebra

Algebra\BinaryOperations.v

iscommringopstoiscommrigops

Lemma isassoc_weq_fwd {X Y : UU} (H : X ≃ Y) (opp : binop X) : isassoc opp → isassoc (binop_weq_fwd H opp). Proof. intros is x y z. apply (maponpaths H). refine (_ @ is _ _ _ @ _). - apply (maponpaths (λ x, opp x _)). apply homotinvweqweq. - apply maponpaths. apply homotinvweqweq0. Defined.

Lemma

Algebra

Algebra\BinaryOperations.v

isassoc_weq_fwd

Lemma islunit_weq_fwd {X Y : UU} (H : X ≃ Y) (opp : binop X) (x0 : X) : islunit opp x0 → islunit (binop_weq_fwd H opp) (H x0). Proof. intros is y. unfold binop_weq_fwd. refine (maponpaths _ _ @ _). - refine (maponpaths (λ x, opp x _) _ @ _). + apply homotinvweqweq. + apply is. - apply homotweqinvweq. Defined.

Lemma

Algebra

Algebra\BinaryOperations.v

islunit_weq_fwd

Lemma isrunit_weq_fwd {X Y : UU} (H : X ≃ Y) (opp : binop X) (x0 : X) : isrunit opp x0 → 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.

Lemma

Algebra

Algebra\BinaryOperations.v

isrunit_weq_fwd

Lemma isunit_weq_fwd {X Y : UU} (H : X ≃ Y) (opp : binop X) (x0 : X) : isunit opp x0 → 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.

Lemma

Algebra

Algebra\BinaryOperations.v

isunit_weq_fwd

Lemma isunital_weq_fwd {X Y : UU} (H : X ≃ Y) (opp : binop X) : isunital opp → isunital (binop_weq_fwd H opp). Proof. intro is. exists (H (pr1 is)). apply isunit_weq_fwd, (pr2 is). Defined.

Lemma

Algebra

Algebra\BinaryOperations.v

isunital_weq_fwd

Lemma ismonoidop_weq_fwd {X Y : UU} (H : X ≃ Y) (opp : binop X) : ismonoidop opp → ismonoidop (binop_weq_fwd H opp). Proof. intro is. split. apply isassoc_weq_fwd, (pr1 is). apply isunital_weq_fwd, (pr2 is). Defined.

Lemma

Algebra

Algebra\BinaryOperations.v

ismonoidop_weq_fwd

Lemma islinv_weq_fwd {X Y : UU} (H : X ≃ Y) (opp : binop X) (x0 : X) (inv : X → X) : islinv opp x0 inv → islinv (binop_weq_fwd H opp) (H x0) (λ y : Y, H (inv (invmap H y))). Proof. intros is y. unfold binop_weq_fwd. apply maponpaths. refine (_ @ is _). apply (maponpaths (λ x, opp x _)). apply homotinvweqweq. Defined.

Lemma

Algebra

Algebra\BinaryOperations.v

islinv_weq_fwd

Lemma isrinv_weq_fwd {X Y : UU} (H : X ≃ Y) (opp : binop X) (x0 : X) (inv : X → X) : isrinv opp x0 inv → isrinv (binop_weq_fwd H opp) (H x0) (λ 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.

Lemma

Algebra

Algebra\BinaryOperations.v

isrinv_weq_fwd

Lemma isinv_weq_fwd {X Y : UU} (H : X ≃ Y) (opp : binop X) (x0 : X) (inv : X → X) : isinv opp x0 inv → isinv (binop_weq_fwd H opp) (H x0) (λ y : Y, H (inv (invmap H y))). Proof. intro is. split. apply islinv_weq_fwd, (pr1 is). apply isrinv_weq_fwd, (pr2 is). Defined.

Lemma

Algebra

Algebra\BinaryOperations.v

isinv_weq_fwd

Lemma invstruct_weq_fwd {X Y : UU} (H : X ≃ Y) (opp : binop X) (is : ismonoidop opp) : invstruct opp is → invstruct (binop_weq_fwd H opp) (ismonoidop_weq_fwd H opp is). Proof. intro inv. exists (λ y : Y, H (pr1 inv (invmap H y))). apply isinv_weq_fwd, (pr2 inv). Defined.

Lemma

Algebra

Algebra\BinaryOperations.v

invstruct_weq_fwd

Lemma isgrop_weq_fwd {X Y : UU} (H : X ≃ Y) (opp : binop X) : isgrop opp → isgrop (binop_weq_fwd H opp). Proof. intro is. use tpair. - apply ismonoidop_weq_fwd, (pr1 is). - apply invstruct_weq_fwd, (pr2 is). Defined.

Lemma

Algebra

Algebra\BinaryOperations.v

isgrop_weq_fwd

Lemma iscomm_weq_fwd {X Y : UU} (H : X ≃ Y) (opp : binop X) : iscomm opp → iscomm (binop_weq_fwd H opp). Proof. intros is x y. unfold binop_weq_fwd. apply maponpaths, is. Defined.

Lemma

Algebra

Algebra\BinaryOperations.v

iscomm_weq_fwd

Lemma isabmonoidop_weq_fwd {X Y : UU} (H : X ≃ Y) (opp : binop X) : isabmonoidop opp → 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

Algebra\BinaryOperations.v

isabmonoidop_weq_fwd

Lemma isabgrop_weq_fwd {X Y : UU} (H : X ≃ Y) (opp : binop X) : isabgrop opp → 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

Algebra\BinaryOperations.v

isabgrop_weq_fwd

Lemma isldistr_weq_fwd {X Y : UU} (H : X ≃ Y) (op1 op2 : binop X) : isldistr op1 op2 → 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

Algebra\BinaryOperations.v

isldistr_weq_fwd

Lemma isrdistr_weq_fwd {X Y : UU} (H : X ≃ Y) (op1 op2 : binop X) : isrdistr op1 op2 → 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 (λ x, op2 x _)). apply homotinvweqweq. - apply map_on_two_paths ; apply homotinvweqweq0. Defined.

Lemma

Algebra

Algebra\BinaryOperations.v

isrdistr_weq_fwd

Lemma isdistr_weq_fwd {X Y : UU} (H : X ≃ Y) (op1 op2 : binop X) : isdistr op1 op2 → 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

Algebra\BinaryOperations.v

isdistr_weq_fwd

Lemma isabsorb_weq_fwd {X Y : UU} (H : X ≃ Y) (op1 op2 : binop X) : isabsorb op1 op2 → 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

Algebra\BinaryOperations.v

isabsorb_weq_fwd

Lemma isrigops_weq_fwd {X Y : UU} (H : X ≃ Y) (op1 op2 : binop X) : isrigops op1 op2 → 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 (λ 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

Algebra\BinaryOperations.v

isrigops_weq_fwd

Lemma isringops_weq_fwd {X Y : UU} (H : X ≃ Y) (op1 op2 : binop X) : isringops op1 op2 → 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

Algebra\BinaryOperations.v

isringops_weq_fwd

Lemma iscommrigops_weq_fwd {X Y : UU} (H : X ≃ Y) (op1 op2 : binop X) : iscommrigops op1 op2 → 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

Algebra\BinaryOperations.v

iscommrigops_weq_fwd

Lemma iscommringops_weq_fwd {X Y : UU} (H : X ≃ Y) (op1 op2 : binop X) : iscommringops op1 op2 → 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

Algebra\BinaryOperations.v

iscommringops_weq_fwd

Lemma isassoc_weq_bck {X Y : UU} (H : X ≃ Y) (opp : binop Y) : isassoc opp → isassoc (binop_weq_bck H opp). Proof. intros is x y z. apply (maponpaths (invmap H)). refine (_ @ is _ _ _ @ _). - apply (maponpaths (λ x, opp x _)). apply homotweqinvweq. - apply maponpaths. symmetry. apply homotweqinvweq. Defined.

Lemma

Algebra

Algebra\BinaryOperations.v

isassoc_weq_bck

Lemma islunit_weq_bck {X Y : UU} (H : X ≃ Y) (opp : binop Y) (x0 : Y) : islunit opp x0 → islunit (binop_weq_bck H opp) (invmap H x0). Proof. intros is y. unfold binop_weq_bck. refine (maponpaths _ _ @ _). - refine (maponpaths (λ x, opp x _) _ @ _). + apply homotweqinvweq. + apply is. - apply homotinvweqweq. Defined.

Lemma

Algebra

Algebra\BinaryOperations.v

islunit_weq_bck

Lemma isrunit_weq_bck {X Y : UU} (H : X ≃ Y) (opp : binop Y) (x0 : Y) : isrunit opp x0 → 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

Algebra\BinaryOperations.v

isrunit_weq_bck

Lemma isunit_weq_bck {X Y : UU} (H : X ≃ Y) (opp : binop Y) (x0 : Y) : isunit opp x0 → 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

Algebra\BinaryOperations.v

isunit_weq_bck

Lemma isunital_weq_bck {X Y : UU} (H : X ≃ Y) (opp : binop Y) : isunital opp → isunital (binop_weq_bck H opp). Proof. intro is. exists (invmap H (pr1 is)). apply isunit_weq_bck, (pr2 is). Defined.

Lemma

Algebra

Algebra\BinaryOperations.v

isunital_weq_bck

Lemma ismonoidop_weq_bck {X Y : UU} (H : X ≃ Y) (opp : binop Y) : ismonoidop opp → 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

Algebra\BinaryOperations.v

ismonoidop_weq_bck

Lemma islinv_weq_bck {X Y : UU} (H : X ≃ Y) (opp : binop Y) (x0 : Y) (inv : Y → Y) : islinv opp x0 inv → islinv (binop_weq_bck H opp) (invmap H x0) (λ y : X, invmap H (inv (H y))). Proof. intros is y. unfold binop_weq_bck. apply maponpaths. refine (_ @ is _). apply (maponpaths (λ x, opp x _)). apply homotweqinvweq. Defined.

Lemma

Algebra

Algebra\BinaryOperations.v

islinv_weq_bck

Lemma isrinv_weq_bck {X Y : UU} (H : X ≃ Y) (opp : binop Y) (x0 : Y) (inv : Y → Y) : isrinv opp x0 inv → isrinv (binop_weq_bck H opp) (invmap H x0) (λ 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

Algebra\BinaryOperations.v

isrinv_weq_bck

Lemma isinv_weq_bck {X Y : UU} (H : X ≃ Y) (opp : binop Y) (x0 : Y) (inv : Y → Y) : isinv opp x0 inv → isinv (binop_weq_bck H opp) (invmap H x0) (λ 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

Algebra\BinaryOperations.v

isinv_weq_bck

Lemma invstruct_weq_bck {X Y : UU} (H : X ≃ Y) (opp : binop Y) (is : ismonoidop opp) : invstruct opp is → invstruct (binop_weq_bck H opp) (ismonoidop_weq_bck H opp is). Proof. intro inv. exists (λ y : X, invmap H (pr1 inv (H y))). apply isinv_weq_bck, (pr2 inv). Defined.

Lemma

Algebra

Algebra\BinaryOperations.v

invstruct_weq_bck

Lemma isgrop_weq_bck {X Y : UU} (H : X ≃ Y) (opp : binop Y) : isgrop opp → 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

Algebra\BinaryOperations.v

isgrop_weq_bck

Lemma iscomm_weq_bck {X Y : UU} (H : X ≃ Y) (opp : binop Y) : iscomm opp → iscomm (binop_weq_bck H opp). Proof. intros is x y. unfold binop_weq_bck. apply maponpaths, is. Defined.

Lemma

Algebra

Algebra\BinaryOperations.v

iscomm_weq_bck

Lemma isabmonoidop_weq_bck {X Y : UU} (H : X ≃ Y) (opp : binop Y) : isabmonoidop opp → 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

Algebra\BinaryOperations.v

isabmonoidop_weq_bck

Lemma isabgrop_weq_bck {X Y : UU} (H : X ≃ Y) (opp : binop Y) : isabgrop opp → 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

Algebra\BinaryOperations.v

isabgrop_weq_bck

Lemma isldistr_weq_bck {X Y : UU} (H : X ≃ Y) (op1 op2 : binop Y) : isldistr op1 op2 → 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

Algebra\BinaryOperations.v

isldistr_weq_bck

Lemma isrdistr_weq_bck {X Y : UU} (H : X ≃ Y) (op1 op2 : binop Y) : isrdistr op1 op2 → 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 (λ x, op2 x _)). apply homotweqinvweq. - apply map_on_two_paths; exact (!homotweqinvweq _ _). Defined.

Lemma

Algebra

Algebra\BinaryOperations.v

isrdistr_weq_bck

Lemma isdistr_weq_bck {X Y : UU} (H : X ≃ Y) (op1 op2 : binop Y) : isdistr op1 op2 → 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

Algebra\BinaryOperations.v

isdistr_weq_bck

Lemma isabsorb_weq_bck {X Y : UU} (H : X ≃ Y) (op1 op2 : binop Y) : isabsorb op1 op2 → 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

Algebra\BinaryOperations.v

isabsorb_weq_bck

Lemma isrigops_weq_bck {X Y : UU} (H : X ≃ Y) (op1 op2 : binop Y) : isrigops op1 op2 → 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 (λ 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

Algebra\BinaryOperations.v

isrigops_weq_bck

Lemma isringops_weq_bck {X Y : UU} (H : X ≃ Y) (op1 op2 : binop Y) : isringops op1 op2 → 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

Algebra\BinaryOperations.v

isringops_weq_bck

Lemma iscommrigops_weq_bck {X Y : UU} (H : X ≃ Y) (op1 op2 : binop Y) : iscommrigops op1 op2 → 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

Algebra\BinaryOperations.v

iscommrigops_weq_bck

Lemma iscommringops_weq_bck {X Y : UU} (H : X ≃ Y) (op1 op2 : binop Y) : iscommringops op1 op2 → 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

Algebra\BinaryOperations.v

iscommringops_weq_bck

Definition setwithbinop : UU := ∑ (X : hSet), binop X.

Definition

Algebra

Algebra\BinaryOperations.v

setwithbinop

Definition make_setwithbinop (X : hSet) (opp : binop X) : setwithbinop := X ,, opp.

Definition

Algebra

Algebra\BinaryOperations.v

make_setwithbinop

Definition pr1setwithbinop : setwithbinop → hSet := @pr1 _ (λ X : hSet, binop X).

Definition

Algebra

Algebra\BinaryOperations.v

pr1setwithbinop

Definition op {X : setwithbinop} : binop X := pr2 X.

Definition

Algebra

Algebra\BinaryOperations.v

op

Definition isasetbinoponhSet (X : hSet) : isaset (@binop X).

Definition

Algebra

Algebra\BinaryOperations.v

isasetbinoponhSet

Definition setwithbinop_rev (X : setwithbinop) : setwithbinop := make_setwithbinop X (λ x y, op y x).

Definition

Algebra

Algebra\BinaryOperations.v

setwithbinop_rev

Definition isbinopfun {X Y : setwithbinop} (f : X → Y) : UU := ∏ x x' : X, f (op x x') = op (f x) (f x').

Definition

Algebra

Algebra\BinaryOperations.v

isbinopfun

Definition make_isbinopfun {X Y : setwithbinop} {f : X → Y} (H : ∏ x x' : X, f (op x x') = op (f x) (f x')) : isbinopfun f := H.

Definition

Algebra

Algebra\BinaryOperations.v

make_isbinopfun

Lemma isapropisbinopfun {X Y : setwithbinop} (f : X → Y) : isaprop (isbinopfun f). Proof. apply impred. intro x. apply impred. intro x'. apply (setproperty Y). Defined.

Lemma

Algebra

Algebra\BinaryOperations.v

isapropisbinopfun

Definition isbinopfun_twooutof3b {A B C : setwithbinop} (f : A → B) (g : B → C) (H : issurjective f) : isbinopfun (g ∘ f)%functions → isbinopfun f → isbinopfun g.

Definition

Algebra

Algebra\BinaryOperations.v

isbinopfun_twooutof3b

Definition binopfun (X Y : setwithbinop) : UU := ∑ (f : X → Y), isbinopfun f.

Definition

Algebra

Algebra\BinaryOperations.v

binopfun

Definition make_binopfun {X Y : setwithbinop} (f : X → Y) (is : isbinopfun f) : binopfun X Y := f ,, is.

Definition

Algebra

Algebra\BinaryOperations.v

make_binopfun