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Lemma iscomm_CFmult : β x y : X, x * y = y * x. Proof. now apply ringcomm2. Qed.
|
Lemma
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\ConstructiveStructures.v
|
iscomm_CFmult
| 800
|
Lemma islinv_CFinv : β (x : X) (Hx0 : x β 0), (CFinv x Hx0) * x = 1. Proof. exact (islinv_CCDRinv (X := ConstructiveField_ConstructiveCommutativeDivisionRig X)). Qed.
|
Lemma
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\ConstructiveStructures.v
|
islinv_CFinv
| 801
|
Lemma isrinv_CFinv : β (x : X) (Hx0 : x β 0), x * (CFinv x Hx0) = 1. Proof. exact (isrinv_CCDRinv (X := ConstructiveField_ConstructiveCommutativeDivisionRig X)). Qed.
|
Lemma
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\ConstructiveStructures.v
|
isrinv_CFinv
| 802
|
Lemma islabsorb_CFzero_CFmult : β x : X, 0 * x = 0. Proof. now apply ringmult0x. Qed.
|
Lemma
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\ConstructiveStructures.v
|
islabsorb_CFzero_CFmult
| 803
|
Lemma israbsorb_CFzero_CFmult : β x : X, x * 0 = 0. Proof. now apply ringmultx0. Qed.
|
Lemma
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\ConstructiveStructures.v
|
israbsorb_CFzero_CFmult
| 804
|
Lemma isldistr_CFplus_CFmult : β x y z : X, z * (x + y) = z * x + z * y. Proof. now apply ringdistraxs. Qed.
|
Lemma
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\ConstructiveStructures.v
|
isldistr_CFplus_CFmult
| 805
|
Lemma isrdistr_CFplus_CFmult : β x y z : X, (x + y) * z = x * z + y * z. Proof. intros. rewrite !(iscomm_CFmult _ z). now apply isldistr_CFplus_CFmult. Qed.
|
Lemma
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\ConstructiveStructures.v
|
isrdistr_CFplus_CFmult
| 806
|
Lemma apCFplus : β x x' y y' : X, x + y β x' + y' β x β x' β¨ y β y'. Proof. exact (apCCDRplus (X := ConstructiveField_ConstructiveCommutativeDivisionRig X)). Qed.
|
Lemma
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\ConstructiveStructures.v
|
apCFplus
| 807
|
Lemma CFplus_apcompat_l : β x y z : X, y + x β z + x <-> y β z. Proof. intros x y z. split. - exact (CCDRplus_apcompat_l (X := ConstructiveField_ConstructiveCommutativeDivisionRig X) _ _ _). - intros Hap. apply (CCDRplus_apcompat_l (X := ConstructiveField_ConstructiveCommutativeDivisionRig X) (- x)). change ((y + x) + - x β (z + x) + - x). rewrite !isassoc_CFplus, isrinv_CFopp, !isrunit_CFzero_CFplus. exact Hap. Qed.
|
Lemma
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\ConstructiveStructures.v
|
CFplus_apcompat_l
| 808
|
Lemma CFplus_apcompat_r : β x y z : X, x + y β x + z <-> y β z. Proof. intros x y z. rewrite !(iscomm_CFplus x). now apply CFplus_apcompat_l. Qed.
|
Lemma
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\ConstructiveStructures.v
|
CFplus_apcompat_r
| 809
|
Lemma apCFmult : β x x' y y' : X, x * y β x' * y' β x β x' β¨ y β y'. Proof. exact (apCCDRmult (X := ConstructiveField_ConstructiveCommutativeDivisionRig X)). Qed.
|
Lemma
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\ConstructiveStructures.v
|
apCFmult
| 810
|
Lemma CFmult_apcompat_l : β x y z : X, y * x β z * x β y β z. Proof. exact (CCDRmult_apcompat_l (X := ConstructiveField_ConstructiveCommutativeDivisionRig X)). Qed.
|
Lemma
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\ConstructiveStructures.v
|
CFmult_apcompat_l
| 811
|
Lemma CFmult_apcompat_l' : β x y z : X, x β 0 β y β z β y * x β z * x. Proof. exact (CCDRmult_apcompat_l' (X := ConstructiveField_ConstructiveCommutativeDivisionRig X)). Qed.
|
Lemma
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\ConstructiveStructures.v
|
CFmult_apcompat_l'
| 812
|
Lemma CFmult_apcompat_r : β x y z : X, x * y β x * z β y β z. Proof. exact (CCDRmult_apcompat_r (X := ConstructiveField_ConstructiveCommutativeDivisionRig X)). Qed.
|
Lemma
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\ConstructiveStructures.v
|
CFmult_apcompat_r
| 813
|
Lemma CFmult_apcompat_r' : β x y z : X, x β 0 β y β z β x * y β x * z. Proof. exact (CCDRmult_apcompat_r' (X := ConstructiveField_ConstructiveCommutativeDivisionRig X)). Qed.
|
Lemma
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\ConstructiveStructures.v
|
CFmult_apcompat_r'
| 814
|
Lemma CFmultapCFzero : β x y : X, x * y β 0 β x β 0 β§ y β 0. Proof. exact (CCDRmultapCCDRzero (X := ConstructiveField_ConstructiveCommutativeDivisionRig X)). Qed.
|
Lemma
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\ConstructiveStructures.v
|
CFmultapCFzero
| 815
|
Definition isnonzerorig (X : rig) : UU := (1%rig : X) != 0%rig.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
isnonzerorig
| 816
|
Definition isDivRig (X : rig) : UU := isnonzerorig X Γ (β x : X, x != 0%rig β multinvpair X x).
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
isDivRig
| 817
|
Lemma isaprop_isDivRig (X : rig) : isaprop (isDivRig X). Proof. apply isofhleveldirprod. - now apply isapropneg. - apply impred_isaprop ; intro. apply isapropimpl. now apply isapropinvpair. Qed.
|
Lemma
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
isaprop_isDivRig
| 818
|
Definition isDivRig_zero {X : rig} (is : isDivRig X) : X := 0%rig.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
isDivRig_zero
| 819
|
Definition isDivRig_one {X : rig} (is : isDivRig X) : X := 1%rig.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
isDivRig_one
| 820
|
Definition isDivRig_plus {X : rig} (is : isDivRig X) : binop X := Ξ» x y : X, (x + y)%rig.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
isDivRig_plus
| 821
|
Definition isDivRig_mult {X : rig} (is : isDivRig X) : binop X := Ξ» x y : X, (x * y)%rig.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
isDivRig_mult
| 822
|
Definition isDivRig_inv {X : rig} (is : isDivRig X) : (β x : X, x != isDivRig_zero is) β X := Ξ» x, pr1 ((pr2 is) (pr1 x) (pr2 x)).
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
isDivRig_inv
| 823
|
Definition isDivRig_isassoc_plus {X : rig} (is : isDivRig X) : isassoc (isDivRig_plus is) := rigassoc1 X.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
isDivRig_isassoc_plus
| 824
|
Definition isDivRig_islunit_x0 {X : rig} (is : isDivRig X) : islunit (isDivRig_plus is) (isDivRig_zero is) := riglunax1 X.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
isDivRig_islunit_x0
| 825
|
Definition isDivRig_isrunit_x0 {X : rig} (is : isDivRig X) : isrunit (isDivRig_plus is) (isDivRig_zero is) := rigrunax1 X.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
isDivRig_isrunit_x0
| 826
|
Definition isDivRig_iscomm_plus {X : rig} (is : isDivRig X) : iscomm (isDivRig_plus is) := rigcomm1 X.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
isDivRig_iscomm_plus
| 827
|
Definition isDivRig_isassoc_mult {X : rig} (is : isDivRig X) : isassoc (isDivRig_mult is) := rigassoc2 X.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
isDivRig_isassoc_mult
| 828
|
Definition isDivRig_islunit_x1 {X : rig} (is : isDivRig X) : islunit (isDivRig_mult is) (isDivRig_one is) := riglunax2 X.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
isDivRig_islunit_x1
| 829
|
Definition isDivRig_isrunit_x1 {X : rig} (is : isDivRig X) : isrunit (isDivRig_mult is) (isDivRig_one is) := rigrunax2 X.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
isDivRig_isrunit_x1
| 830
|
Definition isDivRig_islinv {X : rig} (is : isDivRig X) : β (x : X) (Hx : x != isDivRig_zero is), isDivRig_mult is (isDivRig_inv is (x,, Hx)) x = isDivRig_one is := Ξ» (x : X) (Hx : x != isDivRig_zero is), pr1 (pr2 (pr2 is x Hx)).
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
isDivRig_islinv
| 831
|
Definition isDivRig_isrinv {X : rig} (is : isDivRig X) : β (x : X) (Hx : x != isDivRig_zero is), isDivRig_mult is x (isDivRig_inv is (x,, Hx)) = isDivRig_one is := Ξ» (x : X) (Hx : x != isDivRig_zero is), pr2 (pr2 (pr2 is x Hx)).
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
isDivRig_isrinv
| 832
|
Definition isDivRig_isldistr {X : rig} (is : isDivRig X) : isldistr (isDivRig_plus is) (isDivRig_mult is) := rigldistr X.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
isDivRig_isldistr
| 833
|
Definition isDivRig_isrdistr {X : rig} (is : isDivRig X) : isrdistr (isDivRig_plus is) (isDivRig_mult is) := rigrdistr X.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
isDivRig_isrdistr
| 834
|
Definition DivRig : UU := β (X : rig), isDivRig X.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
DivRig
| 835
|
Definition pr1DivRig (F : DivRig) : hSet := pr1 F.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
pr1DivRig
| 836
|
Definition zeroDivRig {F : DivRig} : F := isDivRig_zero (pr2 F).
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
zeroDivRig
| 837
|
Definition oneDivRig {F : DivRig} : F := isDivRig_one (pr2 F).
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
oneDivRig
| 838
|
Definition plusDivRig {F : DivRig} : binop F := isDivRig_plus (pr2 F).
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
plusDivRig
| 839
|
Definition multDivRig {F : DivRig} : binop F := isDivRig_mult (pr2 F).
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
multDivRig
| 840
|
Definition invDivRig {F : DivRig} : (β x : F, x != zeroDivRig) β F := isDivRig_inv (pr2 F).
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
invDivRig
| 841
|
Definition divDivRig {F : DivRig} : F β (β x : F, x != zeroDivRig) β F := Ξ» x y, multDivRig x (invDivRig y).
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
divDivRig
| 842
|
Definition DivRig_isDivRig (F : DivRig) : isDivRig (pr1 F) := (pr2 F).
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
DivRig_isDivRig
| 843
|
Definition isDivRig_DivRig {X : rig} : isDivRig X β DivRig := Ξ» is : isDivRig X, X ,, is.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
isDivRig_DivRig
| 844
|
Definition DivRig_isassoc_plus: β x y z : F, x + y + z = x + (y + z) := isDivRig_isassoc_plus (DivRig_isDivRig F).
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
DivRig_isassoc_plus
| 845
|
Definition DivRig_islunit_zero: β x : F, 0 + x = x := isDivRig_islunit_x0 (DivRig_isDivRig F).
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
DivRig_islunit_zero
| 846
|
Definition DivRig_isrunit_zero: β x : F, x + 0 = x := isDivRig_isrunit_x0 (DivRig_isDivRig F).
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
DivRig_isrunit_zero
| 847
|
Definition DivRig_iscomm_plus: β x y : F, x + y = y + x := isDivRig_iscomm_plus (DivRig_isDivRig F).
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
DivRig_iscomm_plus
| 848
|
Definition DivRig_isassoc_mult: β x y z : F, x * y * z = x * (y * z) := isDivRig_isassoc_mult (DivRig_isDivRig F).
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
DivRig_isassoc_mult
| 849
|
Definition DivRig_islunit_one: β x : F, 1 * x = x := isDivRig_islunit_x1 (DivRig_isDivRig F).
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
DivRig_islunit_one
| 850
|
Definition DivRig_isrunit_one: β x : F, x * 1 = x := isDivRig_isrunit_x1 (DivRig_isDivRig F).
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
DivRig_isrunit_one
| 851
|
Definition DivRig_islinv: β (x : F) (Hx : x != 0), / (x,, Hx) * x = 1 := isDivRig_islinv (DivRig_isDivRig F).
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
DivRig_islinv
| 852
|
Definition DivRig_isrinv: β (x : F) (Hx : x != 0), x * / (x,, Hx) = 1 := isDivRig_isrinv (DivRig_isDivRig F).
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
DivRig_isrinv
| 853
|
Definition DivRig_isldistr: β x y z : F, z * (x + y) = z * x + z * y := isDivRig_isldistr (DivRig_isDivRig F).
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
DivRig_isldistr
| 854
|
Definition DivRig_isrdistr: β x y z : F, (x + y) * z = x * z + y * z := isDivRig_isrdistr (DivRig_isDivRig F).
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
DivRig_isrdistr
| 855
|
Definition CommDivRig : UU := β (X : commrig), isDivRig X.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
CommDivRig
| 856
|
Definition CommDivRig_DivRig (F : CommDivRig) : DivRig := commrigtorig (pr1 F) ,, pr2 F.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
CommDivRig_DivRig
| 857
|
Definition CommDivRig_isassoc_plus: β x y z : F, x + y + z = x + (y + z) := DivRig_isassoc_plus.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
CommDivRig_isassoc_plus
| 858
|
Definition CommDivRig_islunit_zero: β x : F, 0 + x = x := DivRig_islunit_zero.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
CommDivRig_islunit_zero
| 859
|
Definition CommDivRig_isrunit_zero: β x : F, x + 0 = x := DivRig_isrunit_zero.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
CommDivRig_isrunit_zero
| 860
|
Definition CommDivRig_iscomm_plus: β x y : F, x + y = y + x := DivRig_iscomm_plus.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
CommDivRig_iscomm_plus
| 861
|
Definition CommDivRig_isassoc_mult: β x y z : F, x * y * z = x * (y * z) := DivRig_isassoc_mult.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
CommDivRig_isassoc_mult
| 862
|
Definition CommDivRig_islunit_one: β x : F, 1 * x = x := DivRig_islunit_one.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
CommDivRig_islunit_one
| 863
|
Definition CommDivRig_isrunit_one: β x : F, x * 1 = x := DivRig_isrunit_one.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
CommDivRig_isrunit_one
| 864
|
Definition CommDivRig_iscomm_mult: β x y : F, x * y = y * x := rigcomm2 (pr1 F).
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
CommDivRig_iscomm_mult
| 865
|
Definition CommDivRig_islinv: β (x : F) (Hx : x != 0), / (x,, Hx) * x = 1 := DivRig_islinv.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
CommDivRig_islinv
| 866
|
Definition CommDivRig_isrinv: β (x : F) (Hx : x != 0), x * / (x,, Hx) = 1 := DivRig_isrinv.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
CommDivRig_isrinv
| 867
|
Definition CommDivRig_isldistr: β x y z : F, z * (x + y) = z * x + z * y := DivRig_isldistr.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
CommDivRig_isldistr
| 868
|
Definition CommDivRig_isrdistr: β x y z : F, (x + y) * z = x * z + y * z := DivRig_isrdistr.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Tactics.
|
Algebra\DivisionRig.v
|
CommDivRig_isrdistr
| 869
|
Lemma islcancelableif {X : hSet} (opp : binop X) (x : X) (is : β a b : X, opp x a = opp x b β a = b) : islcancelable opp x. Proof. intros. apply isinclbetweensets. - apply (setproperty X). - apply (setproperty X). - apply is. Defined.
|
Lemma
|
Algebra
|
Require Import UniMath.Algebra.Groups.
|
Algebra\Domains_and_Fields.v
|
islcancelableif
| 870
|
Lemma isrcancelableif {X : hSet} (opp : binop X) (x : X) (is : β a b : X, opp a x = opp b x β a = b) : isrcancelable opp x. Proof. intros. apply isinclbetweensets. - apply (setproperty X). - apply (setproperty X). - apply is. Defined.
|
Lemma
|
Algebra
|
Require Import UniMath.Algebra.Groups.
|
Algebra\Domains_and_Fields.v
|
isrcancelableif
| 871
|
Definition iscancelableif {X : hSet} (opp : binop X) (x : X) (isl : β a b : X, opp x a = opp x b β a = b) (isr : β a b : X, opp a x = opp b x β a = b) : iscancelable opp x := islcancelableif opp x isl ,, isrcancelableif opp x isr.
|
Definition
|
Algebra
|
Require Import UniMath.Algebra.Groups.
|
Algebra\Domains_and_Fields.v
|
iscancelableif
| 872
|
Definition linvpair (X : monoid) (x : X) : UU := β (x' : X), x' * x = 1.
|
Definition
|
Algebra
|
Require Import UniMath.Algebra.Groups.
|
Algebra\Domains_and_Fields.v
|
linvpair
| 873
|
Definition pr1linvpair (X : monoid) (x : X) : linvpair X x β X := @pr1 _ _.
|
Definition
|
Algebra
|
Require Import UniMath.Algebra.Groups.
|
Algebra\Domains_and_Fields.v
|
pr1linvpair
| 874
|
Definition linvpairxy (X : monoid) (x y : X) (x' : linvpair X x) (y' : linvpair X y) : linvpair X (x * y). Proof. intros. exists (pr1 y' * pr1 x'). rewrite (assocax _ _ _ (x * y)). rewrite <- (assocax _ _ x y). rewrite (pr2 x'). rewrite (lunax _ y). rewrite (pr2 y'). apply idpath. Defined.
|
Definition
|
Algebra
|
Require Import UniMath.Algebra.Groups.
|
Algebra\Domains_and_Fields.v
|
linvpairxy
| 875
|
Definition lcanfromlinv (X : monoid) (a b c : X) (c' : linvpair X c) (e : c * a = c * b) : a = b. Proof. intros. set (e' := maponpaths (Ξ» x : X, (pr1 c') * x) e). simpl in e'. rewrite <- (assocax X _ _ _) in e'. rewrite <- (assocax X _ _ _) in e'. rewrite (pr2 c') in e'. rewrite (lunax X a) in e'. rewrite (lunax X b) in e'. apply e'. Defined.
|
Definition
|
Algebra
|
Require Import UniMath.Algebra.Groups.
|
Algebra\Domains_and_Fields.v
|
lcanfromlinv
| 876
|
Definition rinvpair (X : monoid) (x : X) : UU := β (x' : X), x * x' = 1.
|
Definition
|
Algebra
|
Require Import UniMath.Algebra.Groups.
|
Algebra\Domains_and_Fields.v
|
rinvpair
| 877
|
Definition pr1rinvpair (X : monoid) (x : X) : rinvpair X x β X := @pr1 _ _.
|
Definition
|
Algebra
|
Require Import UniMath.Algebra.Groups.
|
Algebra\Domains_and_Fields.v
|
pr1rinvpair
| 878
|
Definition rinvpairxy (X : monoid) (x y : X) (x' : rinvpair X x) (y' : rinvpair X y) : rinvpair X (x * y). Proof. intros. exists (pr1 y' * pr1 x'). rewrite (assocax _ x y _). rewrite <- (assocax _ y _ _). rewrite (pr2 y'). rewrite (lunax _ _). rewrite (pr2 x'). apply idpath. Defined.
|
Definition
|
Algebra
|
Require Import UniMath.Algebra.Groups.
|
Algebra\Domains_and_Fields.v
|
rinvpairxy
| 879
|
Definition rcanfromrinv (X : monoid) (a b c : X) (c' : rinvpair X c) (e : a * c = b * c) : a = b. Proof. intros. set (e' := maponpaths (Ξ» x : X, x * (pr1 c')) e). simpl in e'. rewrite (assocax X _ _ _) in e'. rewrite (assocax X _ _ _) in e'. rewrite (pr2 c') in e'. rewrite (runax X a) in e'. rewrite (runax X b) in e'. apply e'. Defined.
|
Definition
|
Algebra
|
Require Import UniMath.Algebra.Groups.
|
Algebra\Domains_and_Fields.v
|
rcanfromrinv
| 880
|
Lemma pathslinvtorinv (X : monoid) (x : X) (x' : linvpair X x) (x'' : rinvpair X x) : pr1 x' = pr1 x''. Proof. intros. induction (runax X (pr1 x')). induction (pr2 x''). set (int := x * pr1 x''). rewrite <- (lunax X (pr1 x'')). induction (pr2 x'). unfold int. apply (!assocax X _ _ _). Defined.
|
Lemma
|
Algebra
|
Require Import UniMath.Algebra.Groups.
|
Algebra\Domains_and_Fields.v
|
pathslinvtorinv
| 881
|
Definition invpair (X : monoid) (x : X) : UU := β (x' : X), (x' * x = 1) Γ (x * x' = 1).
|
Definition
|
Algebra
|
Require Import UniMath.Algebra.Groups.
|
Algebra\Domains_and_Fields.v
|
invpair
| 882
|
Definition pr1invpair (X : monoid) (x : X) : invpair X x β X := @pr1 _ _.
|
Definition
|
Algebra
|
Require Import UniMath.Algebra.Groups.
|
Algebra\Domains_and_Fields.v
|
pr1invpair
| 883
|
Definition invtolinv (X : monoid) (x : X) (x' : invpair X x) : linvpair X x := pr1 x' ,, pr12 x'.
|
Definition
|
Algebra
|
Require Import UniMath.Algebra.Groups.
|
Algebra\Domains_and_Fields.v
|
invtolinv
| 884
|
Definition invtorinv (X : monoid) (x : X) (x' : invpair X x) : rinvpair X x := pr1 x' ,, pr22 x'.
|
Definition
|
Algebra
|
Require Import UniMath.Algebra.Groups.
|
Algebra\Domains_and_Fields.v
|
invtorinv
| 885
|
Lemma isapropinvpair (X : monoid) (x : X) : isaprop (invpair X x). Proof. intros. apply invproofirrelevance. intros x' x''. apply (invmaponpathsincl _ (isinclpr1 _ (Ξ» a, isapropdirprod _ _ (setproperty X _ _) (setproperty X _ _)))). apply (pathslinvtorinv X x (invtolinv X x x') (invtorinv X x x'')). Defined.
|
Lemma
|
Algebra
|
Require Import UniMath.Algebra.Groups.
|
Algebra\Domains_and_Fields.v
|
isapropinvpair
| 886
|
Definition invpairxy (X : monoid) (x y : X) (x' : invpair X x) (y' : invpair X y) : invpair X (x * y). Proof. intros. exists (pr1 y' * pr1 x'). split. - apply (pr2 (linvpairxy _ x y (invtolinv _ x x') (invtolinv _ y y'))). - apply (pr2 (rinvpairxy _ x y (invtorinv _ x x') (invtorinv _ y y'))). Defined.
|
Definition
|
Algebra
|
Require Import UniMath.Algebra.Groups.
|
Algebra\Domains_and_Fields.v
|
invpairxy
| 887
|
Lemma grfrompathsxy (X : gr) {a b : X} (e : a = b) : a * grinv X b = 1. Proof. intros. set (e' := maponpaths (Ξ» x : X, x * grinv X b) e). simpl in e'. rewrite (grrinvax X _) in e'. apply e'. Defined.
|
Lemma
|
Algebra
|
Require Import UniMath.Algebra.Groups.
|
Algebra\Domains_and_Fields.v
|
grfrompathsxy
| 888
|
Lemma grtopathsxy (X : gr) {a b : X} (e : a * grinv X b = 1) : a = b . Proof. intros. set (e' := maponpaths (Ξ» x, x * b) e). simpl in e'. rewrite (assocax X) in e'. rewrite (grlinvax X) in e'. rewrite (lunax X) in e'. rewrite (runax X) in e'. apply e'. Defined.
|
Lemma
|
Algebra
|
Require Import UniMath.Algebra.Groups.
|
Algebra\Domains_and_Fields.v
|
grtopathsxy
| 889
|
Definition multlinvpair (X : rig) (x : X) : UU := linvpair (rigmultmonoid X) x.
|
Definition
|
Algebra
|
Require Import UniMath.Algebra.Groups.
|
Algebra\Domains_and_Fields.v
|
multlinvpair
| 890
|
Definition multrinvpair (X : rig) (x : X) : UU := rinvpair (rigmultmonoid X) x.
|
Definition
|
Algebra
|
Require Import UniMath.Algebra.Groups.
|
Algebra\Domains_and_Fields.v
|
multrinvpair
| 891
|
Definition multinvpair (X : rig) (x : X) : UU := invpair (rigmultmonoid X) x.
|
Definition
|
Algebra
|
Require Import UniMath.Algebra.Groups.
|
Algebra\Domains_and_Fields.v
|
multinvpair
| 892
|
Definition rigneq0andmultlinv (X : rig) (n m : X) (isnm : ((n * m) != 0)%rig) : n != 0%rig. Proof. intros. intro e. rewrite e in isnm. rewrite (rigmult0x X) in isnm. induction (isnm (idpath _)). Defined.
|
Definition
|
Algebra
|
Require Import UniMath.Algebra.Groups.
|
Algebra\Domains_and_Fields.v
|
rigneq0andmultlinv
| 893
|
Definition rigneq0andmultrinv (X : rig) (n m : X) (isnm : ((n * m) != 0)%rig) : m != 0%rig. Proof. intros. intro e. rewrite e in isnm. rewrite (rigmultx0 _) in isnm. induction (isnm (idpath _)). Defined.
|
Definition
|
Algebra
|
Require Import UniMath.Algebra.Groups.
|
Algebra\Domains_and_Fields.v
|
rigneq0andmultrinv
| 894
|
Definition ringneq0andmultlinv (X : ring) (n m : X) (isnm : ((n * m) != 0)) : n != 0. Proof. intros. intro e. rewrite e in isnm. rewrite (ringmult0x X) in isnm. induction (isnm (idpath _)). Defined.
|
Definition
|
Algebra
|
Require Import UniMath.Algebra.Groups.
|
Algebra\Domains_and_Fields.v
|
ringneq0andmultlinv
| 895
|
Definition ringneq0andmultrinv (X : ring) (n m : X) (isnm : ((n * m) != 0)) : m != 0. Proof. intros. intro e. rewrite e in isnm. rewrite (ringmultx0 _) in isnm. induction (isnm (idpath _)). Defined.
|
Definition
|
Algebra
|
Require Import UniMath.Algebra.Groups.
|
Algebra\Domains_and_Fields.v
|
ringneq0andmultrinv
| 896
|
Definition ringpossubmonoid (X : ring) {R : hrel X} (is1 : isringmultgt X R) (is2 : R 1 0) : @submonoid (ringmultmonoid X). Proof. intros. exists (Ξ» x, R x 0). split. - intros x1 x2. apply is1. apply (pr2 x1). apply (pr2 x2). - apply is2. Defined.
|
Definition
|
Algebra
|
Require Import UniMath.Algebra.Groups.
|
Algebra\Domains_and_Fields.v
|
ringpossubmonoid
| 897
|
Lemma isinvringmultgtif (X : ring) {R : hrel X} (is0 : @isbinophrel X R) (is1 : isringmultgt X R) (nc : neqchoice R) (isa : isasymm R) : isinvringmultgt X R. Proof. intros. split. - intros a b rab0 ra0. assert (int : b != 0). { intro e. rewrite e in rab0. rewrite (ringmultx0 X _) in rab0. apply (isa _ _ rab0 rab0). } induction (nc _ _ int) as [ g | l ]. + apply g. + set (int' := ringmultgt0lt0 X is0 is1 ra0 l). induction (isa _ _ rab0 int'). - intros a b rab0 rb0. assert (int : a != 0). { intro e. rewrite e in rab0. rewrite (ringmult0x X _) in rab0. apply (isa _ _ rab0 rab0). } induction (nc _ _ int) as [ g | l ]. + apply g. + set (int' := ringmultlt0gt0 X is0 is1 l rb0). induction (isa _ _ rab0 int'). Defined.
|
Lemma
|
Algebra
|
Require Import UniMath.Algebra.Groups.
|
Algebra\Domains_and_Fields.v
|
isinvringmultgtif
| 898
|
Lemma isnonzerolinvel (X : ring) (is : isnonzerorig X) (x : X) (x' : multlinvpair X x) : ((pr1 x') != 0). Proof. intros. apply (negf (maponpaths (Ξ» a : X, a * x))). assert (e := pr2 x'). change (pr1 x' * x = 1) in e. change (pr1 x' * x != 0 * x). rewrite e. rewrite (ringmult0x X _). apply is. Defined.
|
Lemma
|
Algebra
|
Require Import UniMath.Algebra.Groups.
|
Algebra\Domains_and_Fields.v
|
isnonzerolinvel
| 899
|
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