phanerozoic/Coq-MathComp · Datasets at Hugging Face

statement stringlengths 1 4.33k	proof stringlengths 0 37.9k	type stringclasses 25 values	symbolic_name stringlengths 1 67	library stringclasses 10 values	filename stringclasses 112 values	imports listlengths 2 138	deps listlengths 0 64	source_url stringclasses 1 value	commit stringclasses 1 value
fmorphV : {morph f: x / x^-1}.	Proof. move=> x; have [-> \| nz_x] := eqVneq x 0; first by rewrite !(invr0, rmorph0). by rewrite rmorphV ?unitfE. Qed.	Lemma	fmorphV	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "eqVneq", "invr0", "rmorph0", "rmorphV", "unitfE" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
fmorph_div : {morph f : x y / x / y}.	Proof. by move=> x y; rewrite rmorphM /= fmorphV. Qed.	Lemma	fmorph_div	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "fmorphV", "rmorphM" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
scalerK a : a != 0 -> cancel ( :%R a : V -> V) ( :%R a^-1).	Proof. by move=> nz_a v; rewrite scalerA mulVf // scale1r. Qed.	Lemma	scalerK	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "mulVf", "scale1r", "scalerA" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
scalerKV a : a != 0 -> cancel ( :%R a^-1 : V -> V) ( :%R a).	Proof. by rewrite -invr_eq0 -{3}[a]invrK; apply: scalerK. Qed.	Lemma	scalerKV	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "invrK", "invr_eq0", "scalerK" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
scalerI a : a != 0 -> injective ( *:%R a : V -> V).	Proof. by move=> nz_a; apply: can_inj (scalerK nz_a). Qed.	Lemma	scalerI	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "scalerK" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
scaler_eq0 a v : (a *: v == 0) = (a == 0) \|\| (v == 0).	Proof. have [-> \| nz_a] := eqVneq a; first by rewrite scale0r eqxx. by rewrite (can2_eq (scalerK nz_a) (scalerKV nz_a)) scaler0. Qed.	Lemma	scaler_eq0	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "can2_eq", "eqVneq", "eqxx", "scale0r", "scaler0", "scalerK", "scalerKV" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pchar_lalg (A : nzLalgType F) : pchar A =i pchar F.	Proof. by move=> p; rewrite inE -scaler_nat scaler_eq0 oner_eq0 orbF. Qed.	Lemma	pchar_lalg	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "inE", "oner_eq0", "pchar", "scaler_eq0", "scaler_nat" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rpredV x : (x^-1 \in S) = (x \in S).	Proof. by apply/idP/idP=> /rpredVr; rewrite ?invrK. Qed.	Lemma	rpredV	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "invrK", "rpredVr" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rpred_div : {in S &, forall x y, x / y \in S}.	Proof. by move=> x y Sx Sy; rewrite /= rpredM ?rpredV. Qed.	Lemma	rpred_div	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "rpredM", "rpredV" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rpredXN n : {in S, forall x, x ^- n \in S}.	Proof. by move=> x Sx; rewrite /= rpredV rpredX. Qed.	Lemma	rpredXN	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "rpredV", "rpredX" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rpredMl x y : x \in S -> x \is a unit-> (x * y \in S) = (y \in S).	Proof. move=> Sx Ux; apply/idP/idP=> [Sxy \| /(rpredM _ _ Sx)-> //]. by rewrite -(mulKr Ux y); rewrite rpredM ?rpredV. Qed.	Lemma	rpredMl	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "mulKr", "rpredM", "rpredV", "unit" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rpredMr x y : x \in S -> x \is a unit -> (y * x \in S) = (y \in S).	Proof. move=> Sx Ux; apply/idP/idP=> [Sxy \| /rpredM-> //]. by rewrite -(mulrK Ux y); rewrite rpred_div. Qed.	Lemma	rpredMr	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "mulrK", "rpredM", "rpred_div", "unit" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rpred_divr x y : x \in S -> x \is a unit -> (y / x \in S) = (y \in S).	Proof. by rewrite -rpredV -unitrV; apply: rpredMr. Qed.	Lemma	rpred_divr	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "rpredMr", "rpredV", "unit", "unitrV" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rpred_divl x y : x \in S -> x \is a unit -> (x / y \in S) = (y \in S).	Proof. by rewrite -(rpredV y); apply: rpredMl. Qed.	Lemma	rpred_divl	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "rpredMl", "rpredV", "unit" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divringClosedP (divS : divringClosed R) : divring_closed divS.	Proof. split; [ exact: rpred1 \| exact: rpredB \| exact: rpred_div ]. Qed.	Lemma	divringClosedP	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "divring_closed", "rpred1", "rpredB", "rpred_div", "split" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
unitr_sdivr_closed : @sdivr_closed R unit.	Proof. by split=> [\|x y Ux Uy]; rewrite ?unitrN1 // unitrMl ?unitrV. Qed.	Fact	unitr_sdivr_closed	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "sdivr_closed", "split", "unit", "unitrMl", "unitrN1", "unitrV" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
unitrN x : (- x \is a unit) = (x \is a unit).	Proof. exact: rpredN. Qed.	Lemma	unitrN	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "rpredN", "unit" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invrN x : (- x)^-1 = - x^-1.	Proof. have [Ux \| U'x] := boolP (x \is a unit); last by rewrite !invr_out ?unitrN. by rewrite -mulN1r invrM ?unitrN1 // invrN1 mulrN1. Qed.	Lemma	invrN	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "invrM", "invrN1", "invr_out", "last", "mulN1r", "mulrN1", "unit", "unitrN", "unitrN1" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divrNN x y : (- x) / (- y) = x / y.	Proof. by rewrite invrN mulrNN. Qed.	Lemma	divrNN	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "invrN", "mulrNN" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divrN x y : x / (- y) = - (x / y).	Proof. by rewrite invrN mulrN. Qed.	Lemma	divrN	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "invrN", "mulrN" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invr_signM n x : ((-1) ^+ n * x)^-1 = (-1) ^+ n * x^-1.	Proof. by rewrite -signr_odd !mulr_sign; case: ifP => // _; rewrite invrN. Qed.	Lemma	invr_signM	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "invrN", "mulr_sign", "signr_odd" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divr_signM (b1 b2 : bool) x1 x2: ((-1) ^+ b1 * x1) / ((-1) ^+ b2 * x2) = (-1) ^+ (b1 (+) b2) * (x1 / x2).	Proof. by rewrite invr_signM mulr_signM. Qed.	Lemma	divr_signM	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "invr_signM", "mulr_signM" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rpredZeq (S : submodClosed V) a v : (a *: v \in S) = (a == 0) \|\| (v \in S).	Proof. have [-> \| nz_a] := eqVneq; first by rewrite scale0r rpred0. by apply/idP/idP; first rewrite -{2}(scalerK nz_a v); apply: rpredZ. Qed.	Lemma	rpredZeq	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "eqVneq", "rpred0", "rpredZ", "scale0r", "scalerK" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
fpredMl x y : x \in S -> x != 0 -> (x * y \in S) = (y \in S).	Proof. by rewrite -!unitfE; apply: rpredMl. Qed.	Lemma	fpredMl	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "rpredMl", "unitfE" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
fpredMr x y : x \in S -> x != 0 -> (y * x \in S) = (y \in S).	Proof. by rewrite -!unitfE; apply: rpredMr. Qed.	Lemma	fpredMr	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "rpredMr", "unitfE" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
fpred_divl x y : x \in S -> x != 0 -> (x / y \in S) = (y \in S).	Proof. by rewrite -!unitfE; apply: rpred_divl. Qed.	Lemma	fpred_divl	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "rpred_divl", "unitfE" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
fpred_divr x y : x \in S -> x != 0 -> (y / x \in S) = (y \in S).	Proof. by rewrite -!unitfE; apply: rpred_divr. Qed.	Lemma	fpred_divr	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "rpred_divr", "unitfE" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
inU v Sv : U	:= Sub v Sv.	Let	inU	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "Sub", "Sv" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invU (u : U)	:= inU (rpredVr _ (valP u)).	Let	invU	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "inU", "rpredVr", "valP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulVr : {in [pred x \| val x \is a unit], left_inverse 1 invU *%R}.	Proof. by move=> x /[!inE] xu; apply: val_inj; rewrite rmorphM rmorph1 /= SubK mulVr. Qed.	Lemma	mulVr	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "SubK", "apply", "inE", "invU", "rmorph1", "rmorphM", "unit", "val", "val_inj" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divrr : {in [pred x \| val x \is a unit], right_inverse 1 invU *%R}.	by move=> x /[!inE] xu; apply: val_inj; rewrite rmorphM rmorph1 /= SubK mulrV. Qed.	Lemma	divrr	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "SubK", "apply", "inE", "invU", "mulrV", "rmorph1", "rmorphM", "unit", "val", "val_inj" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
unitrP (x y : U) : y * x = 1 /\ x * y = 1 -> val x \is a unit.	Proof. move=> -[/(congr1 val) yx1 /(congr1 val) xy1]. by apply: rev_unitrP (val y) _; rewrite !rmorphM rmorph1 /= in yx1 xy1. Qed.	Lemma	unitrP	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "rev_unitrP", "rmorph1", "rmorphM", "unit", "val" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invr_out : {in [pred x \| val x \isn't a unit], invU =1 id}.	Proof. by move=> x /[!inE] xNU; apply: val_inj; rewrite SubK invr_out. Qed.	Lemma	invr_out	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "SubK", "apply", "id", "inE", "invU", "unit", "val", "val_inj" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
id : IntegralDomain.axiom U.	Proof. move=> x y /(congr1 val)/eqP; rewrite rmorphM /=. by rewrite -!(inj_eq val_inj) rmorph0 -mulf_eq0. Qed.	Lemma	id	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "axiom", "inj_eq", "mulf_eq0", "rmorph0", "rmorphM", "val", "val_inj" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
fieldP : Field.axiom U.	Proof. by move=> u; rewrite -(inj_eq val_inj) rmorph0 -unitfE subfield_subproof. Qed.	Lemma	fieldP	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "axiom", "inj_eq", "rmorph0", "unitfE", "val_inj" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'SubNzRing_isSubUnitRing' 'of' U 'by' <: ]"	:= (SubNzRing_isSubUnitRing.Build _ _ U (divringClosedP _)) (format "[ 'SubNzRing_isSubUnitRing' 'of' U 'by' <: ]") : form_scope.	Notation	[ 'SubNzRing_isSubUnitRing' 'of' U 'by' <: ]	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "Build", "divringClosedP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'SubChoice_isSubUnitRing' 'of' U 'by' <: ]"	:= (SubChoice_isSubUnitRing.Build _ _ U (divringClosedP _)) (format "[ 'SubChoice_isSubUnitRing' 'of' U 'by' <: ]") : form_scope.	Notation	[ 'SubChoice_isSubUnitRing' 'of' U 'by' <: ]	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "Build", "divringClosedP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'SubChoice_isSubComUnitRing' 'of' U 'by' <: ]"	:= (SubChoice_isSubComUnitRing.Build _ _ U (divringClosedP _)) (format "[ 'SubChoice_isSubComUnitRing' 'of' U 'by' <: ]") : form_scope.	Notation	[ 'SubChoice_isSubComUnitRing' 'of' U 'by' <: ]	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "Build", "divringClosedP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'SubComUnitRing_isSubIntegralDomain' 'of' U 'by' <: ]"	:= (SubComUnitRing_isSubIntegralDomain.Build _ _ U) (format "[ 'SubComUnitRing_isSubIntegralDomain' 'of' U 'by' <: ]") : form_scope.	Notation	[ 'SubComUnitRing_isSubIntegralDomain' 'of' U 'by' <: ]	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "Build" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'SubChoice_isSubIntegralDomain' 'of' U 'by' <: ]"	:= (SubChoice_isSubIntegralDomain.Build _ _ U (divringClosedP _)) (format "[ 'SubChoice_isSubIntegralDomain' 'of' U 'by' <: ]") : form_scope.	Notation	[ 'SubChoice_isSubIntegralDomain' 'of' U 'by' <: ]	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "Build", "divringClosedP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'SubIntegralDomain_isSubField' 'of' U 'by' <: ]"	:= (SubIntegralDomain_isSubField.Build _ _ U (frefl _)) (format "[ 'SubIntegralDomain_isSubField' 'of' U 'by' <: ]") : form_scope.	Notation	[ 'SubIntegralDomain_isSubField' 'of' U 'by' <: ]	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "Build" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lregP {R x}	:= @lregP R x.	Definition	lregP	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rregP {R x}	:= @rregP R x.	Definition	rregP	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulrV	:= mulrV.	Definition	mulrV	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divrr	:= divrr.	Definition	divrr	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulVr	:= mulVr.	Definition	mulVr	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invr_out	:= invr_out.	Definition	invr_out	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
unitrP {R x}	:= @unitrP R x.	Definition	unitrP	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulKr	:= mulKr.	Definition	mulKr	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulVKr	:= mulVKr.	Definition	mulVKr	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulrK	:= mulrK.	Definition	mulrK	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulrVK	:= mulrVK.	Definition	mulrVK	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divrK	:= divrK.	Definition	divrK	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulrI	:= mulrI.	Definition	mulrI	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulIr	:= mulIr.	Definition	mulIr	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divrI	:= divrI.	Definition	divrI	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divIr	:= divIr.	Definition	divIr	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
telescope_prodr	:= telescope_prodr.	Definition	telescope_prodr	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
telescope_prodr_eq	:= @telescope_prodr_eq.	Definition	telescope_prodr_eq	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
commrV	:= commrV.	Definition	commrV	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
unitrE	:= unitrE.	Definition	unitrE	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invrK	:= @invrK.	Definition	invrK	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invr_inj	:= @invr_inj.	Definition	invr_inj	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
unitrV	:= unitrV.	Definition	unitrV	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
unitr1	:= unitr1.	Definition	unitr1	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invr1	:= invr1.	Definition	invr1	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divr1	:= divr1.	Definition	divr1	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
div1r	:= div1r.	Definition	div1r	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
natr_div	:= natr_div.	Definition	natr_div	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
unitr0	:= unitr0.	Definition	unitr0	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invr0	:= invr0.	Definition	invr0	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
unitrN1	:= unitrN1.	Definition	unitrN1	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
unitrN	:= unitrN.	Definition	unitrN	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invrN1	:= invrN1.	Definition	invrN1	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invrN	:= invrN.	Definition	invrN	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divrNN	:= divrNN.	Definition	divrNN	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divrN	:= divrN.	Definition	divrN	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invr_sign	:= invr_sign.	Definition	invr_sign	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
unitrMl	:= unitrMl.	Definition	unitrMl	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
unitrMr	:= unitrMr.	Definition	unitrMr	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invrM	:= invrM.	Definition	invrM	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
unitr_prod	:= unitr_prod.	Definition	unitr_prod	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
unitr_prod_in	:= unitr_prod_in.	Definition	unitr_prod_in	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invr_eq0	:= invr_eq0.	Definition	invr_eq0	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invr_eq1	:= invr_eq1.	Definition	invr_eq1	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invr_neq0	:= invr_neq0.	Definition	invr_neq0	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rev_unitrP	:= rev_unitrP.	Definition	rev_unitrP	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rev_prodrV	:= rev_prodrV.	Definition	rev_prodrV	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
unitrM_comm	:= unitrM_comm.	Definition	unitrM_comm	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
unitrX	:= unitrX.	Definition	unitrX	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
unitrX_pos	:= unitrX_pos.	Definition	unitrX_pos	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
exprVn	:= exprVn.	Definition	exprVn	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
exprB	:= exprB.	Definition	exprB	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invr_signM	:= invr_signM.	Definition	invr_signM	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divr_signM	:= divr_signM.	Definition	divr_signM	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rpredVr	:= @rpredVr.	Definition	rpredVr	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rpredV	:= rpredV.	Definition	rpredV	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rpred_div	:= rpred_div.	Definition	rpred_div	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rpredXN	:= rpredXN.	Definition	rpredXN	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rpredZeq	:= rpredZeq.	Definition	rpredZeq	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

fmorphV : {morph f: x / x^-1}.

Proof. move=> x; have [-> | nz_x] := eqVneq x 0; first by rewrite !(invr0, rmorph0). by rewrite rmorphV ?unitfE. Qed.

Lemma

fmorphV

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "eqVneq", "invr0", "rmorph0", "rmorphV", "unitfE" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

fmorph_div : {morph f : x y / x / y}.

Proof. by move=> x y; rewrite rmorphM /= fmorphV. Qed.

Lemma

fmorph_div

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "fmorphV", "rmorphM" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

scalerK a : a != 0 -> cancel ( *:%R a : V -> V) ( *:%R a^-1).

Proof. by move=> nz_a v; rewrite scalerA mulVf // scale1r. Qed.

Lemma

scalerK

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "mulVf", "scale1r", "scalerA" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

scalerKV a : a != 0 -> cancel ( *:%R a^-1 : V -> V) ( *:%R a).

Proof. by rewrite -invr_eq0 -{3}[a]invrK; apply: scalerK. Qed.

Lemma

scalerKV

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "invrK", "invr_eq0", "scalerK" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

scalerI a : a != 0 -> injective ( *:%R a : V -> V).

Proof. by move=> nz_a; apply: can_inj (scalerK nz_a). Qed.

Lemma

scalerI

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "scalerK" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

scaler_eq0 a v : (a *: v == 0) = (a == 0) || (v == 0).

Proof. have [-> | nz_a] := eqVneq a; first by rewrite scale0r eqxx. by rewrite (can2_eq (scalerK nz_a) (scalerKV nz_a)) scaler0. Qed.

Lemma

scaler_eq0

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "can2_eq", "eqVneq", "eqxx", "scale0r", "scaler0", "scalerK", "scalerKV" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pchar_lalg (A : nzLalgType F) : pchar A =i pchar F.

Proof. by move=> p; rewrite inE -scaler_nat scaler_eq0 oner_eq0 orbF. Qed.

Lemma

pchar_lalg

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "inE", "oner_eq0", "pchar", "scaler_eq0", "scaler_nat" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rpredV x : (x^-1 \in S) = (x \in S).

Proof. by apply/idP/idP=> /rpredVr; rewrite ?invrK. Qed.

Lemma

rpredV

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "invrK", "rpredVr" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rpred_div : {in S &, forall x y, x / y \in S}.

Proof. by move=> x y Sx Sy; rewrite /= rpredM ?rpredV. Qed.

Lemma

rpred_div

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "rpredM", "rpredV" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rpredXN n : {in S, forall x, x ^- n \in S}.

Proof. by move=> x Sx; rewrite /= rpredV rpredX. Qed.

Lemma

rpredXN

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "rpredV", "rpredX" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rpredMl x y : x \in S -> x \is a unit-> (x * y \in S) = (y \in S).

Proof. move=> Sx Ux; apply/idP/idP=> [Sxy | /(rpredM _ _ Sx)-> //]. by rewrite -(mulKr Ux y); rewrite rpredM ?rpredV. Qed.

Lemma

rpredMl

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "mulKr", "rpredM", "rpredV", "unit" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rpredMr x y : x \in S -> x \is a unit -> (y * x \in S) = (y \in S).

Proof. move=> Sx Ux; apply/idP/idP=> [Sxy | /rpredM-> //]. by rewrite -(mulrK Ux y); rewrite rpred_div. Qed.

Lemma

rpredMr

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "mulrK", "rpredM", "rpred_div", "unit" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rpred_divr x y : x \in S -> x \is a unit -> (y / x \in S) = (y \in S).

Proof. by rewrite -rpredV -unitrV; apply: rpredMr. Qed.

Lemma

rpred_divr

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "rpredMr", "rpredV", "unit", "unitrV" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rpred_divl x y : x \in S -> x \is a unit -> (x / y \in S) = (y \in S).

Proof. by rewrite -(rpredV y); apply: rpredMl. Qed.

Lemma

rpred_divl

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "rpredMl", "rpredV", "unit" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divringClosedP (divS : divringClosed R) : divring_closed divS.

Proof. split; [ exact: rpred1 | exact: rpredB | exact: rpred_div ]. Qed.

Lemma

divringClosedP

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "divring_closed", "rpred1", "rpredB", "rpred_div", "split" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

unitr_sdivr_closed : @sdivr_closed R unit.

Proof. by split=> [|x y Ux Uy]; rewrite ?unitrN1 // unitrMl ?unitrV. Qed.

Fact

unitr_sdivr_closed

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "sdivr_closed", "split", "unit", "unitrMl", "unitrN1", "unitrV" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

unitrN x : (- x \is a unit) = (x \is a unit).

Proof. exact: rpredN. Qed.

Lemma

unitrN

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "rpredN", "unit" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invrN x : (- x)^-1 = - x^-1.

Proof. have [Ux | U'x] := boolP (x \is a unit); last by rewrite !invr_out ?unitrN. by rewrite -mulN1r invrM ?unitrN1 // invrN1 mulrN1. Qed.

Lemma

invrN

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "invrM", "invrN1", "invr_out", "last", "mulN1r", "mulrN1", "unit", "unitrN", "unitrN1" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divrNN x y : (- x) / (- y) = x / y.

Proof. by rewrite invrN mulrNN. Qed.

Lemma

divrNN

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "invrN", "mulrNN" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divrN x y : x / (- y) = - (x / y).

Proof. by rewrite invrN mulrN. Qed.

Lemma

divrN

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "invrN", "mulrN" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invr_signM n x : ((-1) ^+ n * x)^-1 = (-1) ^+ n * x^-1.

Proof. by rewrite -signr_odd !mulr_sign; case: ifP => // _; rewrite invrN. Qed.

Lemma

invr_signM

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "invrN", "mulr_sign", "signr_odd" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divr_signM (b1 b2 : bool) x1 x2: ((-1) ^+ b1 * x1) / ((-1) ^+ b2 * x2) = (-1) ^+ (b1 (+) b2) * (x1 / x2).

Proof. by rewrite invr_signM mulr_signM. Qed.

Lemma

divr_signM

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "invr_signM", "mulr_signM" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rpredZeq (S : submodClosed V) a v : (a *: v \in S) = (a == 0) || (v \in S).

Proof. have [-> | nz_a] := eqVneq; first by rewrite scale0r rpred0. by apply/idP/idP; first rewrite -{2}(scalerK nz_a v); apply: rpredZ. Qed.

Lemma

rpredZeq

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "eqVneq", "rpred0", "rpredZ", "scale0r", "scalerK" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

fpredMl x y : x \in S -> x != 0 -> (x * y \in S) = (y \in S).

Proof. by rewrite -!unitfE; apply: rpredMl. Qed.

Lemma

fpredMl

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "rpredMl", "unitfE" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

fpredMr x y : x \in S -> x != 0 -> (y * x \in S) = (y \in S).

Proof. by rewrite -!unitfE; apply: rpredMr. Qed.

Lemma

fpredMr

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "rpredMr", "unitfE" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

fpred_divl x y : x \in S -> x != 0 -> (x / y \in S) = (y \in S).

Proof. by rewrite -!unitfE; apply: rpred_divl. Qed.

Lemma

fpred_divl

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "rpred_divl", "unitfE" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

fpred_divr x y : x \in S -> x != 0 -> (y / x \in S) = (y \in S).

Proof. by rewrite -!unitfE; apply: rpred_divr. Qed.

Lemma

fpred_divr

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "rpred_divr", "unitfE" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

inU v Sv : U

:= Sub v Sv.

Let

inU

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "Sub", "Sv" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invU (u : U)

:= inU (rpredVr _ (valP u)).

Let

invU

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "inU", "rpredVr", "valP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulVr : {in [pred x | val x \is a unit], left_inverse 1 invU *%R}.

Proof. by move=> x /[!inE] xu; apply: val_inj; rewrite rmorphM rmorph1 /= SubK mulVr. Qed.

Lemma

mulVr

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "SubK", "apply", "inE", "invU", "rmorph1", "rmorphM", "unit", "val", "val_inj" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divrr : {in [pred x | val x \is a unit], right_inverse 1 invU *%R}.

by move=> x /[!inE] xu; apply: val_inj; rewrite rmorphM rmorph1 /= SubK mulrV. Qed.

Lemma

divrr

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "SubK", "apply", "inE", "invU", "mulrV", "rmorph1", "rmorphM", "unit", "val", "val_inj" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

unitrP (x y : U) : y * x = 1 /\ x * y = 1 -> val x \is a unit.

Proof. move=> -[/(congr1 val) yx1 /(congr1 val) xy1]. by apply: rev_unitrP (val y) _; rewrite !rmorphM rmorph1 /= in yx1 xy1. Qed.

Lemma

unitrP

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "rev_unitrP", "rmorph1", "rmorphM", "unit", "val" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invr_out : {in [pred x | val x \isn't a unit], invU =1 id}.

Proof. by move=> x /[!inE] xNU; apply: val_inj; rewrite SubK invr_out. Qed.

Lemma

invr_out

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "SubK", "apply", "id", "inE", "invU", "unit", "val", "val_inj" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

id : IntegralDomain.axiom U.

Proof. move=> x y /(congr1 val)/eqP; rewrite rmorphM /=. by rewrite -!(inj_eq val_inj) rmorph0 -mulf_eq0. Qed.

Lemma

id

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "axiom", "inj_eq", "mulf_eq0", "rmorph0", "rmorphM", "val", "val_inj" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

fieldP : Field.axiom U.

Proof. by move=> u; rewrite -(inj_eq val_inj) rmorph0 -unitfE subfield_subproof. Qed.

Lemma

fieldP

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "axiom", "inj_eq", "rmorph0", "unitfE", "val_inj" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'SubNzRing_isSubUnitRing' 'of' U 'by' <: ]"

:= (SubNzRing_isSubUnitRing.Build _ _ U (divringClosedP _)) (format "[ 'SubNzRing_isSubUnitRing' 'of' U 'by' <: ]") : form_scope.

Notation

[ 'SubNzRing_isSubUnitRing' 'of' U 'by' <: ]

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "Build", "divringClosedP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'SubChoice_isSubUnitRing' 'of' U 'by' <: ]"

:= (SubChoice_isSubUnitRing.Build _ _ U (divringClosedP _)) (format "[ 'SubChoice_isSubUnitRing' 'of' U 'by' <: ]") : form_scope.

Notation

[ 'SubChoice_isSubUnitRing' 'of' U 'by' <: ]

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "Build", "divringClosedP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'SubChoice_isSubComUnitRing' 'of' U 'by' <: ]"

:= (SubChoice_isSubComUnitRing.Build _ _ U (divringClosedP _)) (format "[ 'SubChoice_isSubComUnitRing' 'of' U 'by' <: ]") : form_scope.

Notation

[ 'SubChoice_isSubComUnitRing' 'of' U 'by' <: ]

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "Build", "divringClosedP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'SubComUnitRing_isSubIntegralDomain' 'of' U 'by' <: ]"

:= (SubComUnitRing_isSubIntegralDomain.Build _ _ U) (format "[ 'SubComUnitRing_isSubIntegralDomain' 'of' U 'by' <: ]") : form_scope.

Notation

[ 'SubComUnitRing_isSubIntegralDomain' 'of' U 'by' <: ]

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "Build" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'SubChoice_isSubIntegralDomain' 'of' U 'by' <: ]"

:= (SubChoice_isSubIntegralDomain.Build _ _ U (divringClosedP _)) (format "[ 'SubChoice_isSubIntegralDomain' 'of' U 'by' <: ]") : form_scope.

Notation

[ 'SubChoice_isSubIntegralDomain' 'of' U 'by' <: ]

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "Build", "divringClosedP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'SubIntegralDomain_isSubField' 'of' U 'by' <: ]"

:= (SubIntegralDomain_isSubField.Build _ _ U (frefl _)) (format "[ 'SubIntegralDomain_isSubField' 'of' U 'by' <: ]") : form_scope.

Notation

[ 'SubIntegralDomain_isSubField' 'of' U 'by' <: ]

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "Build" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lregP {R x}

:= @lregP R x.

Definition

lregP

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rregP {R x}

:= @rregP R x.

Definition

rregP

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulrV

:= mulrV.

Definition

mulrV

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divrr

:= divrr.

Definition

divrr

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulVr

:= mulVr.

Definition

mulVr

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invr_out

:= invr_out.

Definition

invr_out

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

unitrP {R x}

:= @unitrP R x.

Definition

unitrP

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulKr

:= mulKr.

Definition

mulKr

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulVKr

:= mulVKr.

Definition

mulVKr

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulrK

:= mulrK.

Definition

mulrK

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulrVK

:= mulrVK.

Definition

mulrVK

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divrK

:= divrK.

Definition

divrK

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulrI

:= mulrI.

Definition

mulrI

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulIr

:= mulIr.

Definition

mulIr

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divrI

:= divrI.

Definition

divrI

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divIr

:= divIr.

Definition

divIr

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

telescope_prodr

:= telescope_prodr.

Definition

telescope_prodr

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

telescope_prodr_eq

:= @telescope_prodr_eq.

Definition

telescope_prodr_eq

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

commrV

:= commrV.

Definition

commrV

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

unitrE

:= unitrE.

Definition

unitrE

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invrK

:= @invrK.

Definition

invrK

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invr_inj

:= @invr_inj.

Definition

invr_inj

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

unitrV

:= unitrV.

Definition

unitrV

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

unitr1

:= unitr1.

Definition

unitr1

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invr1

:= invr1.

Definition

invr1

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divr1

:= divr1.

Definition

divr1

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

div1r

:= div1r.

Definition

div1r

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

natr_div

:= natr_div.

Definition

natr_div

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

unitr0

:= unitr0.

Definition

unitr0

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invr0

:= invr0.

Definition

invr0

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

unitrN1

:= unitrN1.

Definition

unitrN1

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

unitrN

:= unitrN.

Definition

unitrN

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invrN1

:= invrN1.

Definition

invrN1

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invrN

:= invrN.

Definition

invrN

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divrNN

:= divrNN.

Definition

divrNN

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divrN

:= divrN.

Definition

divrN

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invr_sign

:= invr_sign.

Definition

invr_sign

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

unitrMl

:= unitrMl.

Definition

unitrMl

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

unitrMr

:= unitrMr.

Definition

unitrMr

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invrM

:= invrM.

Definition

invrM

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

unitr_prod

:= unitr_prod.

Definition

unitr_prod

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

unitr_prod_in

:= unitr_prod_in.

Definition

unitr_prod_in

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invr_eq0

:= invr_eq0.

Definition

invr_eq0

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invr_eq1

:= invr_eq1.

Definition

invr_eq1

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invr_neq0

:= invr_neq0.

Definition

invr_neq0

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rev_unitrP

:= rev_unitrP.

Definition

rev_unitrP

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rev_prodrV

:= rev_prodrV.

Definition

rev_prodrV

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

unitrM_comm

:= unitrM_comm.

Definition

unitrM_comm

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

unitrX

:= unitrX.

Definition

unitrX

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

unitrX_pos

:= unitrX_pos.

Definition

unitrX_pos

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

exprVn

:= exprVn.

Definition

exprVn

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

exprB

:= exprB.

Definition

exprB

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invr_signM

:= invr_signM.

Definition

invr_signM

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divr_signM

:= divr_signM.

Definition

divr_signM

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rpredVr

:= @rpredVr.

Definition

rpredVr

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rpredV

:= rpredV.

Definition

rpredV

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rpred_div

:= rpred_div.

Definition

rpred_div

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rpredXN

:= rpredXN.

Definition

rpredXN

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rpredZeq

:= rpredZeq.

Definition

rpredZeq

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d