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
zero_lfunE x : (0 : 'Hom(aT, rT)) x = 0.	Proof. exact: lfunE. Qed.	Lemma	zero_lfunE	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "aT", "lfunE" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
add_lfunE f g x : (f + g) x = f x + g x.	Proof. exact: lfunE. Qed.	Lemma	add_lfunE	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "lfunE" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sum_lfunE I (r : seq I) (P : pred I) (fs : I -> 'Hom(aT, rT)) x : (\sum_(i <- r \| P i) fs i) x = \sum_(i <- r \| P i) fs i x.	Proof. by elim/big_rec2: _ => [\|i _ f _ <-]; rewrite lfunE. Qed.	Lemma	sum_lfunE	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "aT", "big_rec2", "lfunE", "seq" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
zero_lfun	:= @zero_lfun R aT rT.	Let	zero_lfun	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "aT" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
add_lfun	:= @add_lfun R aT rT.	Let	add_lfun	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "aT" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
opp_lfun f	:= linfun (-%R \o f).	Definition	opp_lfun	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "linfun" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lfun_addN : left_inverse zero_lfun opp_lfun add_lfun.	Proof. by move=> f; apply/lfunP=> v; rewrite !lfunE /= lfunE addNr. Qed.	Lemma	lfun_addN	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "addNr", "add_lfun", "apply", "lfunE", "lfunP", "opp_lfun", "zero_lfun" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
opp_lfunE f x : (- f) x = - f x.	Proof. exact: lfunE. Qed.	Lemma	opp_lfunE	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "lfunE" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
scale_lfun k f	:= linfun (k \*: f).	Definition	scale_lfun	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "linfun" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lfun_scaleA k1 k2 f : k1 :l (k2 :l f) = (k1 * k2) *:l f.	Proof. by apply/lfunP=> v; rewrite !lfunE /= lfunE scalerA. Qed.	Fact	lfun_scaleA	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "lfunE", "lfunP", "scalerA" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lfun_scale0 f : 0 *:l f = 0.	Proof. by apply/lfunP=> v; rewrite !lfunE /= scale0r. Qed.	Fact	lfun_scale0	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "lfunE", "lfunP", "scale0r" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lfun_scale1 f : 1 *:l f = f.	Proof. by apply/lfunP=> v; rewrite lfunE /= scale1r. Qed.	Fact	lfun_scale1	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "lfunE", "lfunP", "scale1r" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lfun_scaleDr k f1 f2 : k :l (f1 + f2) = k :l f1 + k *:l f2.	Proof. by apply/lfunP=> v; rewrite !lfunE /= !lfunE scalerDr. Qed.	Fact	lfun_scaleDr	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "f1", "f2", "lfunE", "lfunP", "scalerDr" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lfun_scaleDl f k1 k2 : (k1 + k2) :l f = k1 :l f + k2 *:l f.	Proof. by apply/lfunP=> v; rewrite !lfunE /= !lfunE scalerDl. Qed.	Fact	lfun_scaleDl	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "lfunE", "lfunP", "scalerDl" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
scale_lfunE k f x : (k : f) x = k : f x.	Proof. exact: lfunE. Qed.	Lemma	scale_lfunE	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "lfunE" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lfun_vect_iso : SemiVector.axiom (dim aT * dim rT) 'Hom(aT, rT).	Proof. exists (mxvec \o f2mx). by split => [a f\|f g]; rewrite -semilinearP/=; rewrite -[A in _ = mxvec A]/(f2mx (Hom _)); congr (mxvec (f2mx _)); apply/lfunP=> v; rewrite lfunE/= unlock /= -!semilinearP. apply: Bijective (Hom \o vec_mx) _ _ => [[A]\|A] /=; last exact: vec_mxK. by rewrite mxvecK. Qed.	Fact	lfun_vect_iso	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "aT", "apply", "axiom", "dim", "f2mx", "last", "lfunE", "lfunP", "mxvec", "mxvecK", "semilinearP", "split", "vec_mx", "vec_mxK" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
id_lfunE u: \1%VF u = u :> aT.	Proof. exact: lfunE. Qed.	Lemma	id_lfunE	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "aT", "lfunE" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_lfunE f g u : (f \o g)%VF u = f (g u).	Proof. exact: lfunE. Qed.	Lemma	comp_lfunE	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "lfunE" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_lfunA f g h : (f \o (g \o h) = (f \o g) \o h)%VF.	Proof. by apply/lfunP=> u; do !rewrite lfunE /=. Qed.	Lemma	comp_lfunA	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "lfunE", "lfunP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_lfun1l f : (\1 \o f)%VF = f.	Proof. by apply/lfunP=> u; do !rewrite lfunE /=. Qed.	Lemma	comp_lfun1l	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "lfunE", "lfunP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_lfun1r f : (f \o \1)%VF = f.	Proof. by apply/lfunP=> u; do !rewrite lfunE /=. Qed.	Lemma	comp_lfun1r	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "lfunE", "lfunP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_lfun0l g : (0 \o g)%VF = 0 :> 'Hom(aT, rT).	Proof. by apply/lfunP=> u; do !rewrite lfunE /=. Qed.	Lemma	comp_lfun0l	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "aT", "apply", "lfunE", "lfunP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_lfun0r f : (f \o 0)%VF = 0 :> 'Hom(aT, rT).	Proof. by apply/lfunP=> u; do !rewrite lfunE /=; rewrite linear0. Qed.	Lemma	comp_lfun0r	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "aT", "apply", "lfunE", "lfunP", "linear0" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_lfunDl f1 f2 g : ((f1 + f2) \o g = (f1 \o g) + (f2 \o g))%VF.	Proof. by apply/lfunP=> u; do !rewrite lfunE /=. Qed.	Lemma	comp_lfunDl	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "f1", "f2", "lfunE", "lfunP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_lfunDr f g1 g2 : (f \o (g1 + g2) = (f \o g1) + (f \o g2))%VF.	Proof. by apply/lfunP=> u; do !rewrite lfunE /=; rewrite linearD. Qed.	Lemma	comp_lfunDr	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "lfunE", "lfunP", "linearD" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_lfunNl f g : ((- f) \o g = - (f \o g))%VF.	Proof. by apply/lfunP=> u; do !rewrite lfunE /=. Qed.	Lemma	comp_lfunNl	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "lfunE", "lfunP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_lfunNr f g : (f \o (- g) = - (f \o g))%VF.	Proof. by apply/lfunP=> u; do !rewrite lfunE /=; rewrite linearN. Qed.	Lemma	comp_lfunNr	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "lfunE", "lfunP", "linearN" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lfun_simp	:= (comp_lfunE, scale_lfunE, opp_lfunE, add_lfunE, sum_lfunE, lfunE).	Definition	lfun_simp	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "add_lfunE", "comp_lfunE", "lfunE", "opp_lfunE", "scale_lfunE", "sum_lfunE" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_lfunZl k f g : (k : (f \o g) = (k : f) \o g)%VF.	Proof. by apply/lfunP=> u; do !rewrite lfunE /=. Qed.	Lemma	comp_lfunZl	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "lfunE", "lfunP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_lfunZr k f g : (k : (f \o g) = f \o (k : g))%VF.	Proof. by apply/lfunP=> u; do !rewrite lfunE /=; rewrite linearZ. Qed.	Lemma	comp_lfunZr	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "lfunE", "lfunP", "linearZ" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
limgS f U V : (U <= V)%VS -> (f @: U <= f @: V)%VS.	Proof. by rewrite unlock /subsetv !genmxE; apply: submxMr. Qed.	Lemma	limgS	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "genmxE", "submxMr", "subsetv" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
limg_line f v : (f @: <[v]> = <[f v]>)%VS.	Proof. apply/eqP; rewrite 2!unlock eqEsubv /subsetv /= r2vK !genmxE. by rewrite !(eqmxMr _ (genmxE _)) submx_refl. Qed.	Lemma	limg_line	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "eqEsubv", "eqmxMr", "genmxE", "r2vK", "submx_refl", "subsetv" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
limg0 f : (f @: 0 = 0)%VS.	Proof. by rewrite limg_line linear0. Qed.	Lemma	limg0	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "limg_line", "linear0" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
memv_img f v U : v \in U -> f v \in (f @: U)%VS.	Proof. by move=> Uv; rewrite memvE -limg_line limgS. Qed.	Lemma	memv_img	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "limgS", "limg_line", "memvE" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
memv_imgP f w U : reflect (exists2 u, u \in U & w = f u) (w \in f @: U)%VS.	Proof. apply: (iffP idP) => [\|[u Uu ->]]; last exact: memv_img. rewrite 2!unlock memvE /subsetv !genmxE => /submxP[ku Drw]. exists (r2v (ku *m vs2mx U)); last by rewrite /= r2vK -mulmxA -Drw v2rK. by rewrite memvE /subsetv !genmxE r2vK submxMl. Qed.	Lemma	memv_imgP	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "Uu", "apply", "genmxE", "last", "memvE", "memv_img", "mulmxA", "r2v", "r2vK", "submxMl", "submxP", "subsetv", "v2rK", "vs2mx" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lim0g U : (0 @: U = 0 :> {vspace rT})%VS.	Proof. apply/eqP; rewrite -subv0; apply/subvP=> _ /memv_imgP[u _ ->]. by rewrite lfunE rpred0. Qed.	Lemma	lim0g	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "lfunE", "memv_imgP", "rpred0", "subv0", "subvP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_in_limg V f g : {in V, f =1 g} -> (f @: V = g @: V)%VS.	Proof. move=> eq_fg; apply/vspaceP=> y. by apply/memv_imgP/memv_imgP=> [][x Vx ->]; exists x; rewrite ?eq_fg. Qed.	Lemma	eq_in_limg	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "memv_imgP", "vspaceP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
limgD f : {morph lfun_img f : U V / U + V}%VS.	Proof. move=> U V; apply/eqP; rewrite unlock eqEsubv /subsetv /= -genmx_adds. by rewrite !genmxE !(eqmxMr _ (genmxE _)) !addsmxMr submx_refl. Qed.	Lemma	limgD	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "addsmxMr", "apply", "eqEsubv", "eqmxMr", "genmxE", "genmx_adds", "lfun_img", "submx_refl", "subsetv" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
limg_sum f I r (P : pred I) Us : (f @: (\sum_(i <- r \| P i) Us i) = \sum_(i <- r \| P i) f @: Us i)%VS.	Proof. exact: (big_morph _ (limgD f) (limg0 f)). Qed.	Lemma	limg_sum	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "big_morph", "limg0", "limgD" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
limg_cap f U V : (f @: (U :&: V) <= f @: U :&: f @: V)%VS.	Proof. by rewrite subv_cap !limgS ?capvSl ?capvSr. Qed.	Lemma	limg_cap	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "capvSl", "capvSr", "limgS", "subv_cap" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
limg_bigcap f I r (P : pred I) Us : (f @: (\bigcap_(i <- r \| P i) Us i) <= \bigcap_(i <- r \| P i) f @: Us i)%VS.	Proof. elim/big_rec2: _ => [\|i V U _ sUV]; first exact: subvf. by rewrite (subv_trans (limg_cap f _ U)) ?capvS. Qed.	Lemma	limg_bigcap	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "big_rec2", "capvS", "limg_cap", "subv_trans", "subvf" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
limg_span f X : (f @: <<X>> = <<map f X>>)%VS.	Proof. by rewrite !span_def big_map limg_sum; apply: eq_bigr => x _; rewrite limg_line. Qed.	Lemma	limg_span	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "big_map", "eq_bigr", "limg_line", "limg_sum", "map", "span_def" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subset_limgP f U (r : seq rT) : {subset r <= (f @: U)%VS} <-> (exists2 a, all (mem U) a & r = map f a).	Proof. split => [\|[{}r /allP/= rE ->] _ /mapP[x xr ->]]; last by rewrite memv_img ?rE. move=> /(_ _ _)/memv_imgP/sig2_eqW-/(all_sig_cond (0 : aT))[f' f'P]. exists (map f' r); first by apply/allP => _ /mapP [x /f'P[? ?] ->]. by symmetry; rewrite -map_comp; apply: map_id_in => x /f'P[]. Qed.	Lemma	subset_limgP	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "aT", "all", "allP", "apply", "last", "map", "mapP", "map_comp", "map_id_in", "memv_img", "memv_imgP", "seq", "sig2_eqW", "split" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lfunPn f g : reflect (exists u, f u != g u) (f != g).	Proof. apply: (iffP idP) => [f'g\|[x]]; last by apply: contraNneq => /lfunP->. suffices /subvPn[_ /memv_imgP[u _ ->]]: ~~ (limg (f - g) <= 0)%VS. by rewrite lfunE /= lfunE /= memv0 subr_eq0; exists u. apply: contra f'g => /subvP fg0; apply/eqP/lfunP=> u; apply/eqP. by rewrite -subr_eq0 -opp_lfunE -add_lfunE -memv0 fg0...	Lemma	lfunPn	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "add_lfunE", "apply", "contraNneq", "last", "lfunE", "lfunP", "limg", "memv0", "memv_img", "memv_imgP", "memvf", "opp_lfunE", "subr_eq0", "subvP", "subvPn" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
inv_lfun_def f : (f \o f^-1 \o f)%VF = f.	Proof. apply/lfunP=> u; do !rewrite lfunE /=; rewrite unlock /= !r2vK. by rewrite mulmxKpV ?submxMl. Qed.	Lemma	inv_lfun_def	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "lfunE", "lfunP", "mulmxKpV", "r2vK", "submxMl" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
limg_lfunVK f : {in limg f, cancel f^-1%VF f}.	Proof. by move=> _ /memv_imgP[u _ ->]; rewrite -!comp_lfunE inv_lfun_def. Qed.	Lemma	limg_lfunVK	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "comp_lfunE", "inv_lfun_def", "limg", "memv_imgP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lkerE f U : (U <= lker f)%VS = (f @: U == 0)%VS.	Proof. rewrite unlock -dimv_eq0 /dimv /subsetv !genmxE mxrank_eq0. by rewrite (sameP sub_kermxP eqP). Qed.	Lemma	lkerE	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "dimv", "dimv_eq0", "genmxE", "lker", "mxrank_eq0", "sub_kermxP", "subsetv" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
memv_ker f v : (v \in lker f) = (f v == 0).	Proof. by rewrite -memv0 !memvE subv0 lkerE limg_line. Qed.	Lemma	memv_ker	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "limg_line", "lker", "lkerE", "memv0", "memvE", "subv0" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eqlfunP f g v : reflect (f v = g v) (v \in lker (f - g)).	Proof. by rewrite memv_ker !lfun_simp subr_eq0; apply: eqP. Qed.	Lemma	eqlfunP	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "lfun_simp", "lker", "memv_ker", "subr_eq0" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eqlfun_inP V f g : reflect {in V, f =1 g} (V <= lker (f - g))%VS.	Proof. by apply: (iffP subvP) => E x /E/eqlfunP. Qed.	Lemma	eqlfun_inP	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "eqlfunP", "lker", "subvP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
limg_ker_compl f U : (f @: (U :\: lker f) = f @: U)%VS.	Proof. rewrite -{2}(addv_diff_cap U (lker f)) limgD; apply/esym/addv_idPl. by rewrite (subv_trans _ (sub0v _)) // subv0 -lkerE capvSr. Qed.	Lemma	limg_ker_compl	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "addv_diff_cap", "addv_idPl", "apply", "capvSr", "limgD", "lker", "lkerE", "sub0v", "subv0", "subv_trans" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
limg_ker_dim f U : (\dim (U :&: lker f) + \dim (f @: U) = \dim U)%N.	Proof. rewrite unlock /dimv /= genmx_cap genmx_id -genmx_cap !genmxE. by rewrite addnC mxrank_mul_ker. Qed.	Lemma	limg_ker_dim	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "addnC", "dim", "dimv", "genmxE", "genmx_cap", "genmx_id", "lker", "mxrank_mul_ker" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
limg_dim_eq f U : (U :&: lker f = 0)%VS -> \dim (f @: U) = \dim U.	Proof. by rewrite -(limg_ker_dim f U) => ->; rewrite dimv0. Qed.	Lemma	limg_dim_eq	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "dim", "dimv0", "limg_ker_dim", "lker" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
limg_basis_of f U X : (U :&: lker f = 0)%VS -> basis_of U X -> basis_of (f @: U) (map f X).	Proof. move=> injUf /andP[/eqP defU /eqnP freeX]. by rewrite /basis_of /free size_map -limg_span -freeX defU limg_dim_eq ?eqxx. Qed.	Lemma	limg_basis_of	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "basis_of", "defU", "eqnP", "eqxx", "free", "limg_dim_eq", "limg_span", "lker", "map", "size_map" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lker0P f : reflect (injective f) (lker f == 0%VS).	Proof. rewrite -subv0; apply: (iffP subvP) => [injf u v eq_fuv \| injf u]. apply/eqP; rewrite -subr_eq0 -memv0 injf //. by rewrite memv_ker linearB /= eq_fuv subrr. by rewrite memv_ker memv0 -(inj_eq injf) linear0. Qed.	Lemma	lker0P	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "inj_eq", "injf", "linear0", "linearB", "lker", "memv0", "memv_ker", "subr_eq0", "subrr", "subv0", "subvP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
limg_ker0 f U V : lker f == 0%VS -> (f @: U <= f @: V)%VS = (U <= V)%VS.	Proof. move/lker0P=> injf; apply/idP/idP=> [/subvP sfUV \| ]; last exact: limgS. by apply/subvP=> u Uu; have /memv_imgP[v Vv /injf->] := sfUV _ (memv_img f Uu). Qed.	Lemma	limg_ker0	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "Uu", "apply", "injf", "last", "limgS", "lker", "lker0P", "memv_img", "memv_imgP", "subvP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_limg_ker0 f U V : lker f == 0%VS -> (f @: U == f @: V)%VS = (U == V).	Proof. by move=> injf; rewrite !eqEsubv !limg_ker0. Qed.	Lemma	eq_limg_ker0	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "eqEsubv", "injf", "limg_ker0", "lker" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lker0_lfunK f : lker f == 0%VS -> cancel f f^-1%VF.	Proof. by move/lker0P=> injf u; apply: injf; rewrite limg_lfunVK ?memv_img ?memvf. Qed.	Lemma	lker0_lfunK	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "injf", "limg_lfunVK", "lker", "lker0P", "memv_img", "memvf" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lker0_compVf f : lker f == 0%VS -> (f^-1 \o f = \1)%VF.	Proof. by move/lker0_lfunK=> fK; apply/lfunP=> u; rewrite !lfunE /= fK. Qed.	Lemma	lker0_compVf	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "fK", "lfunE", "lfunP", "lker", "lker0_lfunK" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lker0_img_cap f U V : lker f == 0%VS -> (f @: (U :&: V) = f @: U :&: f @: V)%VS.	Proof. move=> kf0; apply/eqP; rewrite eqEsubv limg_cap/=; apply/subvP => x. rewrite memv_cap => /andP[/memv_imgP[u uU ->]] /memv_imgP[v vV]. by move=> /(lker0P _ kf0) eq_uv; rewrite memv_img// memv_cap uU eq_uv vV. Qed.	Lemma	lker0_img_cap	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "eqEsubv", "limg_cap", "lker", "lker0P", "memv_cap", "memv_img", "memv_imgP", "subvP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
fixedSpace f : {vspace vT}	:= lker (f - \1%VF).	Definition	fixedSpace	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "lker", "vT" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
fixedSpaceP f a : reflect (f a = a) (a \in fixedSpace f).	Proof. by rewrite memv_ker add_lfunE opp_lfunE id_lfunE subr_eq0; apply: eqP. Qed.	Lemma	fixedSpaceP	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "add_lfunE", "apply", "fixedSpace", "id_lfunE", "memv_ker", "opp_lfunE", "subr_eq0" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
fixedSpacesP f U : reflect {in U, f =1 id} (U <= fixedSpace f)%VS.	Proof. by apply: (iffP subvP) => cUf x /cUf/fixedSpaceP. Qed.	Lemma	fixedSpacesP	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "fixedSpace", "fixedSpaceP", "id", "subvP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
fixedSpace_limg f U : (U <= fixedSpace f -> f @: U = U)%VS.	Proof. move/fixedSpacesP=> cUf; apply/vspaceP=> x. by apply/memv_imgP/idP=> [[{}x Ux ->] \| Ux]; last exists x; rewrite ?cUf. Qed.	Lemma	fixedSpace_limg	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "fixedSpace", "fixedSpacesP", "last", "memv_imgP", "vspaceP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
fixedSpace_id : fixedSpace \1 = {:vT}%VS.	Proof. by apply/vspaceP=> x; rewrite memvf; apply/fixedSpaceP; rewrite lfunE. Qed.	Lemma	fixedSpace_id	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "fixedSpace", "fixedSpaceP", "lfunE", "memvf", "vT", "vspaceP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
kerf0 : lker f == 0%VS.		Hypothesis	kerf0	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "lker" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lker0_limgf : limg f = fullv.	Proof. by apply/eqP; rewrite eqEdim subvf limg_dim_eq //= (eqP kerf0) capv0. Qed.	Lemma	lker0_limgf	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "capv0", "eqEdim", "fullv", "kerf0", "limg", "limg_dim_eq", "subvf" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lker0_lfunVK : cancel f^-1%VF f.	Proof. by move=> u; rewrite limg_lfunVK // lker0_limgf memvf. Qed.	Lemma	lker0_lfunVK	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "limg_lfunVK", "lker0_limgf", "memvf" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lker0_compfV : (f \o f^-1 = \1)%VF.	Proof. by apply/lfunP=> u; rewrite !lfunE /= lker0_lfunVK. Qed.	Lemma	lker0_compfV	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "lfunE", "lfunP", "lker0_lfunVK" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lker0_compVKf aT g : (f \o (f^-1 \o g))%VF = g :> 'Hom(aT, vT).	Proof. by rewrite comp_lfunA lker0_compfV comp_lfun1l. Qed.	Lemma	lker0_compVKf	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "aT", "comp_lfun1l", "comp_lfunA", "lker0_compfV", "vT" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lker0_compKf aT g : (f^-1 \o (f \o g))%VF = g :> 'Hom(aT, vT).	Proof. by rewrite comp_lfunA lker0_compVf ?comp_lfun1l. Qed.	Lemma	lker0_compKf	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "aT", "comp_lfun1l", "comp_lfunA", "lker0_compVf", "vT" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lker0_compfK rT h : ((h \o f) \o f^-1)%VF = h :> 'Hom(vT, rT).	Proof. by rewrite -comp_lfunA lker0_compfV comp_lfun1r. Qed.	Lemma	lker0_compfK	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "comp_lfun1r", "comp_lfunA", "lker0_compfV", "vT" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lker0_compfVK rT h : ((h \o f^-1) \o f)%VF = h :> 'Hom(vT, rT).	Proof. by rewrite -comp_lfunA lker0_compVf ?comp_lfun1r. Qed.	Lemma	lker0_compfVK	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "comp_lfun1r", "comp_lfunA", "lker0_compVf", "vT" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lim1g U : (\1 @: U)%VS = U.	Proof. have /andP[/eqP <- _] := vbasisP U; rewrite limg_span map_id_in // => u _. by rewrite lfunE. Qed.	Lemma	lim1g	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "lfunE", "limg_span", "map_id_in", "vbasisP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
limg_comp f g U : ((g \o f) @: U = g @: (f @: U))%VS.	Proof. have /andP[/eqP <- _] := vbasisP U; rewrite !limg_span; congr (span _). by rewrite -map_comp; apply/eq_map => u; rewrite lfunE. Qed.	Lemma	limg_comp	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "eq_map", "lfunE", "limg_span", "map_comp", "span", "vbasisP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lpreim_cap_limg f W : (f @^-1: (W :&: limg f))%VS = (f @^-1: W)%VS.	Proof. by rewrite /lfun_preim -capvA capvv. Qed.	Lemma	lpreim_cap_limg	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "capvA", "capvv", "lfun_preim", "limg" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lpreim0 f : (f @^-1: 0)%VS = lker f.	Proof. by rewrite /lfun_preim cap0v limg0 add0v. Qed.	Lemma	lpreim0	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "add0v", "cap0v", "lfun_preim", "limg0", "lker" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lpreimS f V W : (V <= W)%VS-> (f @^-1: V <= f @^-1: W)%VS.	Proof. by move=> sVW; rewrite addvS // limgS // capvS. Qed.	Lemma	lpreimS	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "addvS", "capvS", "limgS" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lpreimK f W : (W <= limg f)%VS -> (f @: (f @^-1: W))%VS = W.	Proof. move=> sWf; rewrite limgD (capv_idPl sWf) // -limg_comp. have /eqP->: (f @: lker f == 0)%VS by rewrite -lkerE. have /andP[/eqP defW _] := vbasisP W; rewrite addv0 -defW limg_span. rewrite map_id_in // => x Xx; rewrite lfunE /= limg_lfunVK //. by apply: span_subvP Xx; rewrite defW. Qed.	Lemma	lpreimK	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "addv0", "apply", "capv_idPl", "lfunE", "limg", "limgD", "limg_comp", "limg_lfunVK", "limg_span", "lker", "lkerE", "map_id_in", "span_subvP", "vbasisP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
memv_preim f u W : (f u \in W) = (u \in f @^-1: W)%VS.	Proof. apply/idP/idP=> [Wfu \| /(memv_img f)]; last first. by rewrite -lpreim_cap_limg lpreimK ?capvSr // => /memv_capP[]. rewrite -[u](addNKr (f^-1%VF (f u))) memv_add ?memv_img //. by rewrite memv_cap Wfu memv_img ?memvf. by rewrite memv_ker addrC linearB /= subr_eq0 limg_lfunVK ?memv_img ?memvf. Qed.	Lemma	memv_preim	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "addNKr", "addrC", "apply", "capvSr", "last", "limg_lfunVK", "linearB", "lpreimK", "lpreim_cap_limg", "memv_add", "memv_cap", "memv_capP", "memv_img", "memv_ker", "memvf", "subr_eq0" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lfun_comp_pzSemiRingType : pzSemiRingType	:= HB.pack 'End(vT) (GRing.Nmodule_isPzSemiRing.Build 'End(vT) comp_lfunA comp_lfun1l comp_lfun1r comp_lfunDl comp_lfunDr (comp_lfun0l _) (@comp_lfun0r _ _ _ _)).	Definition	lfun_comp_pzSemiRingType	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "Build", "comp_lfun0l", "comp_lfun0r", "comp_lfun1l", "comp_lfun1r", "comp_lfunA", "comp_lfunDl", "comp_lfunDr", "vT" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lfun1_neq0 : \1%VF != 0 :> 'End(vT).	Proof. apply/eqP=> /lfunP/(_ (r2v (const_mx 1))); rewrite !lfunE /= => /(canRL r2vK). by move=> /rowP/(_ (Ordinal dim_gt0))/eqP; rewrite linear0 !mxE oner_eq0. Qed.	Fact	lfun1_neq0	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "const_mx", "dim_gt0", "lfunE", "lfunP", "linear0", "mxE", "oner_eq0", "r2v", "r2vK", "rowP", "vT" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vT_proper : dim vT > 0.		Hypothesis	vT_proper	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "dim", "vT" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lfun_comp_nzRingType : nzRingType	:= 'End(vT)^c.	Definition	lfun_comp_nzRingType	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "vT" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lfun_nzRingType : nzRingType	:= 'End(vT).	Definition	lfun_nzRingType	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "vT" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lfun_lalgType : nzLalgType R	:= 'End(vT).	Definition	lfun_lalgType	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "vT" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lfun_algType : nzAlgType R	:= 'End(vT).	Definition	lfun_algType	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "vT" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
daddv_pi U V	:= Hom (proj_mx (vs2mx U) (vs2mx V)).	Definition	daddv_pi	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "proj_mx", "vs2mx" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
projv U	:= daddv_pi U U^C.	Definition	projv	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "daddv_pi" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
addv_pi1 U V	:= daddv_pi (U :\: V) V.	Definition	addv_pi1	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "daddv_pi" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
addv_pi2 U V	:= daddv_pi V (U :\: V).	Definition	addv_pi2	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "daddv_pi" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
memv_pi U V w : (daddv_pi U V) w \in U.	Proof. by rewrite unlock memvE /subsetv genmxE /= r2vK proj_mx_sub. Qed.	Lemma	memv_pi	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "daddv_pi", "genmxE", "memvE", "proj_mx_sub", "r2vK", "subsetv" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
memv_proj U w : projv U w \in U.	Proof. exact: memv_pi. Qed.	Lemma	memv_proj	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "memv_pi", "projv" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
memv_pi1 U V w : (addv_pi1 U V) w \in U.	Proof. by rewrite (subvP (diffvSl U V)) ?memv_pi. Qed.	Lemma	memv_pi1	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "addv_pi1", "diffvSl", "memv_pi", "subvP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
memv_pi2 U V w : (addv_pi2 U V) w \in V.	Proof. exact: memv_pi. Qed.	Lemma	memv_pi2	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "addv_pi2", "memv_pi" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
daddv_pi_id U V u : (U :&: V = 0)%VS -> u \in U -> daddv_pi U V u = u.	Proof. move/eqP; rewrite -dimv_eq0 memvE /subsetv /dimv !genmxE mxrank_eq0 => /eqP. by move=> dxUV Uu; rewrite unlock /= proj_mx_id ?v2rK. Qed.	Lemma	daddv_pi_id	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "Uu", "daddv_pi", "dimv", "dimv_eq0", "genmxE", "memvE", "mxrank_eq0", "proj_mx_id", "subsetv", "v2rK" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
daddv_pi_proj U V w (pi := daddv_pi U V) : (U :&: V = 0)%VS -> pi (pi w) = pi w.	Proof. by move/daddv_pi_id=> -> //; apply: memv_pi. Qed.	Lemma	daddv_pi_proj	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "daddv_pi", "daddv_pi_id", "memv_pi", "pi" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
daddv_pi_add U V w : (U :&: V = 0)%VS -> (w \in U + V)%VS -> daddv_pi U V w + daddv_pi V U w = w.	Proof. move/eqP; rewrite -dimv_eq0 memvE /subsetv /dimv !genmxE mxrank_eq0 => /eqP. by move=> dxUW UVw; rewrite unlock /= -linearD /= add_proj_mx ?v2rK. Qed.	Lemma	daddv_pi_add	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "add_proj_mx", "daddv_pi", "dimv", "dimv_eq0", "genmxE", "linearD", "memvE", "mxrank_eq0", "subsetv", "v2rK" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
projv_id U u : u \in U -> projv U u = u.	Proof. exact: daddv_pi_id (capv_compl _). Qed.	Lemma	projv_id	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "capv_compl", "daddv_pi_id", "projv" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
projv_proj U w : projv U (projv U w) = projv U w.	Proof. exact: daddv_pi_proj (capv_compl _). Qed.	Lemma	projv_proj	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "capv_compl", "daddv_pi_proj", "projv" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

zero_lfunE x : (0 : 'Hom(aT, rT)) x = 0.

Proof. exact: lfunE. Qed.

Lemma

zero_lfunE

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "aT", "lfunE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

add_lfunE f g x : (f + g) x = f x + g x.

Proof. exact: lfunE. Qed.

Lemma

add_lfunE

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "lfunE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sum_lfunE I (r : seq I) (P : pred I) (fs : I -> 'Hom(aT, rT)) x : (\sum_(i <- r | P i) fs i) x = \sum_(i <- r | P i) fs i x.

Proof. by elim/big_rec2: _ => [|i _ f _ <-]; rewrite lfunE. Qed.

Lemma

sum_lfunE

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "aT", "big_rec2", "lfunE", "seq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

zero_lfun

:= @zero_lfun R aT rT.

Let

zero_lfun

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "aT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

add_lfun

:= @add_lfun R aT rT.

Let

add_lfun

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "aT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

opp_lfun f

:= linfun (-%R \o f).

Definition

opp_lfun

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "linfun" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lfun_addN : left_inverse zero_lfun opp_lfun add_lfun.

Proof. by move=> f; apply/lfunP=> v; rewrite !lfunE /= lfunE addNr. Qed.

Lemma

lfun_addN

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "addNr", "add_lfun", "apply", "lfunE", "lfunP", "opp_lfun", "zero_lfun" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

opp_lfunE f x : (- f) x = - f x.

Proof. exact: lfunE. Qed.

Lemma

opp_lfunE

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "lfunE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

scale_lfun k f

:= linfun (k \*: f).

Definition

scale_lfun

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "linfun" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lfun_scaleA k1 k2 f : k1 *:l (k2 *:l f) = (k1 * k2) *:l f.

Proof. by apply/lfunP=> v; rewrite !lfunE /= lfunE scalerA. Qed.

Fact

lfun_scaleA

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "lfunE", "lfunP", "scalerA" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lfun_scale0 f : 0 *:l f = 0.

Proof. by apply/lfunP=> v; rewrite !lfunE /= scale0r. Qed.

Fact

lfun_scale0

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "lfunE", "lfunP", "scale0r" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lfun_scale1 f : 1 *:l f = f.

Proof. by apply/lfunP=> v; rewrite lfunE /= scale1r. Qed.

Fact

lfun_scale1

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "lfunE", "lfunP", "scale1r" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lfun_scaleDr k f1 f2 : k *:l (f1 + f2) = k *:l f1 + k *:l f2.

Proof. by apply/lfunP=> v; rewrite !lfunE /= !lfunE scalerDr. Qed.

Fact

lfun_scaleDr

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "f1", "f2", "lfunE", "lfunP", "scalerDr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lfun_scaleDl f k1 k2 : (k1 + k2) *:l f = k1 *:l f + k2 *:l f.

Proof. by apply/lfunP=> v; rewrite !lfunE /= !lfunE scalerDl. Qed.

Fact

lfun_scaleDl

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "lfunE", "lfunP", "scalerDl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

scale_lfunE k f x : (k *: f) x = k *: f x.

Proof. exact: lfunE. Qed.

Lemma

scale_lfunE

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "lfunE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lfun_vect_iso : SemiVector.axiom (dim aT * dim rT) 'Hom(aT, rT).

Proof. exists (mxvec \o f2mx). by split => [a f|f g]; rewrite -semilinearP/=; rewrite -[A in _ = mxvec A]/(f2mx (Hom _)); congr (mxvec (f2mx _)); apply/lfunP=> v; rewrite lfunE/= unlock /= -!semilinearP. apply: Bijective (Hom \o vec_mx) _ _ => [[A]|A] /=; last exact: vec_mxK. by rewrite mxvecK. Qed.

Fact

lfun_vect_iso

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "aT", "apply", "axiom", "dim", "f2mx", "last", "lfunE", "lfunP", "mxvec", "mxvecK", "semilinearP", "split", "vec_mx", "vec_mxK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

id_lfunE u: \1%VF u = u :> aT.

Proof. exact: lfunE. Qed.

Lemma

id_lfunE

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "aT", "lfunE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comp_lfunE f g u : (f \o g)%VF u = f (g u).

Proof. exact: lfunE. Qed.

Lemma

comp_lfunE

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "lfunE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comp_lfunA f g h : (f \o (g \o h) = (f \o g) \o h)%VF.

Proof. by apply/lfunP=> u; do !rewrite lfunE /=. Qed.

Lemma

comp_lfunA

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "lfunE", "lfunP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comp_lfun1l f : (\1 \o f)%VF = f.

Proof. by apply/lfunP=> u; do !rewrite lfunE /=. Qed.

Lemma

comp_lfun1l

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "lfunE", "lfunP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comp_lfun1r f : (f \o \1)%VF = f.

Proof. by apply/lfunP=> u; do !rewrite lfunE /=. Qed.

Lemma

comp_lfun1r

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "lfunE", "lfunP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comp_lfun0l g : (0 \o g)%VF = 0 :> 'Hom(aT, rT).

Proof. by apply/lfunP=> u; do !rewrite lfunE /=. Qed.

Lemma

comp_lfun0l

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "aT", "apply", "lfunE", "lfunP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comp_lfun0r f : (f \o 0)%VF = 0 :> 'Hom(aT, rT).

Proof. by apply/lfunP=> u; do !rewrite lfunE /=; rewrite linear0. Qed.

Lemma

comp_lfun0r

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "aT", "apply", "lfunE", "lfunP", "linear0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comp_lfunDl f1 f2 g : ((f1 + f2) \o g = (f1 \o g) + (f2 \o g))%VF.

Proof. by apply/lfunP=> u; do !rewrite lfunE /=. Qed.

Lemma

comp_lfunDl

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "f1", "f2", "lfunE", "lfunP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comp_lfunDr f g1 g2 : (f \o (g1 + g2) = (f \o g1) + (f \o g2))%VF.

Proof. by apply/lfunP=> u; do !rewrite lfunE /=; rewrite linearD. Qed.

Lemma

comp_lfunDr

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "lfunE", "lfunP", "linearD" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comp_lfunNl f g : ((- f) \o g = - (f \o g))%VF.

Proof. by apply/lfunP=> u; do !rewrite lfunE /=. Qed.

Lemma

comp_lfunNl

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "lfunE", "lfunP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comp_lfunNr f g : (f \o (- g) = - (f \o g))%VF.

Proof. by apply/lfunP=> u; do !rewrite lfunE /=; rewrite linearN. Qed.

Lemma

comp_lfunNr

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "lfunE", "lfunP", "linearN" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lfun_simp

:= (comp_lfunE, scale_lfunE, opp_lfunE, add_lfunE, sum_lfunE, lfunE).

Definition

lfun_simp

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "add_lfunE", "comp_lfunE", "lfunE", "opp_lfunE", "scale_lfunE", "sum_lfunE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comp_lfunZl k f g : (k *: (f \o g) = (k *: f) \o g)%VF.

Proof. by apply/lfunP=> u; do !rewrite lfunE /=. Qed.

Lemma

comp_lfunZl

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "lfunE", "lfunP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comp_lfunZr k f g : (k *: (f \o g) = f \o (k *: g))%VF.

Proof. by apply/lfunP=> u; do !rewrite lfunE /=; rewrite linearZ. Qed.

Lemma

comp_lfunZr

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "lfunE", "lfunP", "linearZ" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

limgS f U V : (U <= V)%VS -> (f @: U <= f @: V)%VS.

Proof. by rewrite unlock /subsetv !genmxE; apply: submxMr. Qed.

Lemma

limgS

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "genmxE", "submxMr", "subsetv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

limg_line f v : (f @: <[v]> = <[f v]>)%VS.

Proof. apply/eqP; rewrite 2!unlock eqEsubv /subsetv /= r2vK !genmxE. by rewrite !(eqmxMr _ (genmxE _)) submx_refl. Qed.

Lemma

limg_line

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "eqEsubv", "eqmxMr", "genmxE", "r2vK", "submx_refl", "subsetv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

limg0 f : (f @: 0 = 0)%VS.

Proof. by rewrite limg_line linear0. Qed.

Lemma

limg0

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "limg_line", "linear0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

memv_img f v U : v \in U -> f v \in (f @: U)%VS.

Proof. by move=> Uv; rewrite memvE -limg_line limgS. Qed.

Lemma

memv_img

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "limgS", "limg_line", "memvE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

memv_imgP f w U : reflect (exists2 u, u \in U & w = f u) (w \in f @: U)%VS.

Proof. apply: (iffP idP) => [|[u Uu ->]]; last exact: memv_img. rewrite 2!unlock memvE /subsetv !genmxE => /submxP[ku Drw]. exists (r2v (ku *m vs2mx U)); last by rewrite /= r2vK -mulmxA -Drw v2rK. by rewrite memvE /subsetv !genmxE r2vK submxMl. Qed.

Lemma

memv_imgP

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "Uu", "apply", "genmxE", "last", "memvE", "memv_img", "mulmxA", "r2v", "r2vK", "submxMl", "submxP", "subsetv", "v2rK", "vs2mx" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lim0g U : (0 @: U = 0 :> {vspace rT})%VS.

Proof. apply/eqP; rewrite -subv0; apply/subvP=> _ /memv_imgP[u _ ->]. by rewrite lfunE rpred0. Qed.

Lemma

lim0g

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "lfunE", "memv_imgP", "rpred0", "subv0", "subvP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_in_limg V f g : {in V, f =1 g} -> (f @: V = g @: V)%VS.

Proof. move=> eq_fg; apply/vspaceP=> y. by apply/memv_imgP/memv_imgP=> [][x Vx ->]; exists x; rewrite ?eq_fg. Qed.

Lemma

eq_in_limg

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "memv_imgP", "vspaceP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

limgD f : {morph lfun_img f : U V / U + V}%VS.

Proof. move=> U V; apply/eqP; rewrite unlock eqEsubv /subsetv /= -genmx_adds. by rewrite !genmxE !(eqmxMr _ (genmxE _)) !addsmxMr submx_refl. Qed.

Lemma

limgD

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "addsmxMr", "apply", "eqEsubv", "eqmxMr", "genmxE", "genmx_adds", "lfun_img", "submx_refl", "subsetv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

limg_sum f I r (P : pred I) Us : (f @: (\sum_(i <- r | P i) Us i) = \sum_(i <- r | P i) f @: Us i)%VS.

Proof. exact: (big_morph _ (limgD f) (limg0 f)). Qed.

Lemma

limg_sum

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "big_morph", "limg0", "limgD" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

limg_cap f U V : (f @: (U :&: V) <= f @: U :&: f @: V)%VS.

Proof. by rewrite subv_cap !limgS ?capvSl ?capvSr. Qed.

Lemma

limg_cap

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "capvSl", "capvSr", "limgS", "subv_cap" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

limg_bigcap f I r (P : pred I) Us : (f @: (\bigcap_(i <- r | P i) Us i) <= \bigcap_(i <- r | P i) f @: Us i)%VS.

Proof. elim/big_rec2: _ => [|i V U _ sUV]; first exact: subvf. by rewrite (subv_trans (limg_cap f _ U)) ?capvS. Qed.

Lemma

limg_bigcap

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "big_rec2", "capvS", "limg_cap", "subv_trans", "subvf" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

limg_span f X : (f @: <<X>> = <<map f X>>)%VS.

Proof. by rewrite !span_def big_map limg_sum; apply: eq_bigr => x _; rewrite limg_line. Qed.

Lemma

limg_span

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "big_map", "eq_bigr", "limg_line", "limg_sum", "map", "span_def" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subset_limgP f U (r : seq rT) : {subset r <= (f @: U)%VS} <-> (exists2 a, all (mem U) a & r = map f a).

Proof. split => [|[{}r /allP/= rE ->] _ /mapP[x xr ->]]; last by rewrite memv_img ?rE. move=> /(_ _ _)/memv_imgP/sig2_eqW-/(all_sig_cond (0 : aT))[f' f'P]. exists (map f' r); first by apply/allP => _ /mapP [x /f'P[? ?] ->]. by symmetry; rewrite -map_comp; apply: map_id_in => x /f'P[]. Qed.

Lemma

subset_limgP

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "aT", "all", "allP", "apply", "last", "map", "mapP", "map_comp", "map_id_in", "memv_img", "memv_imgP", "seq", "sig2_eqW", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lfunPn f g : reflect (exists u, f u != g u) (f != g).

Proof. apply: (iffP idP) => [f'g|[x]]; last by apply: contraNneq => /lfunP->. suffices /subvPn[_ /memv_imgP[u _ ->]]: ~~ (limg (f - g) <= 0)%VS. by rewrite lfunE /= lfunE /= memv0 subr_eq0; exists u. apply: contra f'g => /subvP fg0; apply/eqP/lfunP=> u; apply/eqP. by rewrite -subr_eq0 -opp_lfunE -add_lfunE -memv0 fg0...

Lemma

lfunPn

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "add_lfunE", "apply", "contraNneq", "last", "lfunE", "lfunP", "limg", "memv0", "memv_img", "memv_imgP", "memvf", "opp_lfunE", "subr_eq0", "subvP", "subvPn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

inv_lfun_def f : (f \o f^-1 \o f)%VF = f.

Proof. apply/lfunP=> u; do !rewrite lfunE /=; rewrite unlock /= !r2vK. by rewrite mulmxKpV ?submxMl. Qed.

Lemma

inv_lfun_def

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "lfunE", "lfunP", "mulmxKpV", "r2vK", "submxMl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

limg_lfunVK f : {in limg f, cancel f^-1%VF f}.

Proof. by move=> _ /memv_imgP[u _ ->]; rewrite -!comp_lfunE inv_lfun_def. Qed.

Lemma

limg_lfunVK

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "comp_lfunE", "inv_lfun_def", "limg", "memv_imgP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lkerE f U : (U <= lker f)%VS = (f @: U == 0)%VS.

Proof. rewrite unlock -dimv_eq0 /dimv /subsetv !genmxE mxrank_eq0. by rewrite (sameP sub_kermxP eqP). Qed.

Lemma

lkerE

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "dimv", "dimv_eq0", "genmxE", "lker", "mxrank_eq0", "sub_kermxP", "subsetv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

memv_ker f v : (v \in lker f) = (f v == 0).

Proof. by rewrite -memv0 !memvE subv0 lkerE limg_line. Qed.

Lemma

memv_ker

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "limg_line", "lker", "lkerE", "memv0", "memvE", "subv0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eqlfunP f g v : reflect (f v = g v) (v \in lker (f - g)).

Proof. by rewrite memv_ker !lfun_simp subr_eq0; apply: eqP. Qed.

Lemma

eqlfunP

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "lfun_simp", "lker", "memv_ker", "subr_eq0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eqlfun_inP V f g : reflect {in V, f =1 g} (V <= lker (f - g))%VS.

Proof. by apply: (iffP subvP) => E x /E/eqlfunP. Qed.

Lemma

eqlfun_inP

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "eqlfunP", "lker", "subvP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

limg_ker_compl f U : (f @: (U :\: lker f) = f @: U)%VS.

Proof. rewrite -{2}(addv_diff_cap U (lker f)) limgD; apply/esym/addv_idPl. by rewrite (subv_trans _ (sub0v _)) // subv0 -lkerE capvSr. Qed.

Lemma

limg_ker_compl

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "addv_diff_cap", "addv_idPl", "apply", "capvSr", "limgD", "lker", "lkerE", "sub0v", "subv0", "subv_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

limg_ker_dim f U : (\dim (U :&: lker f) + \dim (f @: U) = \dim U)%N.

Proof. rewrite unlock /dimv /= genmx_cap genmx_id -genmx_cap !genmxE. by rewrite addnC mxrank_mul_ker. Qed.

Lemma

limg_ker_dim

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "addnC", "dim", "dimv", "genmxE", "genmx_cap", "genmx_id", "lker", "mxrank_mul_ker" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

limg_dim_eq f U : (U :&: lker f = 0)%VS -> \dim (f @: U) = \dim U.

Proof. by rewrite -(limg_ker_dim f U) => ->; rewrite dimv0. Qed.

Lemma

limg_dim_eq

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "dim", "dimv0", "limg_ker_dim", "lker" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

limg_basis_of f U X : (U :&: lker f = 0)%VS -> basis_of U X -> basis_of (f @: U) (map f X).

Proof. move=> injUf /andP[/eqP defU /eqnP freeX]. by rewrite /basis_of /free size_map -limg_span -freeX defU limg_dim_eq ?eqxx. Qed.

Lemma

limg_basis_of

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "basis_of", "defU", "eqnP", "eqxx", "free", "limg_dim_eq", "limg_span", "lker", "map", "size_map" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lker0P f : reflect (injective f) (lker f == 0%VS).

Proof. rewrite -subv0; apply: (iffP subvP) => [injf u v eq_fuv | injf u]. apply/eqP; rewrite -subr_eq0 -memv0 injf //. by rewrite memv_ker linearB /= eq_fuv subrr. by rewrite memv_ker memv0 -(inj_eq injf) linear0. Qed.

Lemma

lker0P

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "inj_eq", "injf", "linear0", "linearB", "lker", "memv0", "memv_ker", "subr_eq0", "subrr", "subv0", "subvP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

limg_ker0 f U V : lker f == 0%VS -> (f @: U <= f @: V)%VS = (U <= V)%VS.

Proof. move/lker0P=> injf; apply/idP/idP=> [/subvP sfUV | ]; last exact: limgS. by apply/subvP=> u Uu; have /memv_imgP[v Vv /injf->] := sfUV _ (memv_img f Uu). Qed.

Lemma

limg_ker0

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "Uu", "apply", "injf", "last", "limgS", "lker", "lker0P", "memv_img", "memv_imgP", "subvP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_limg_ker0 f U V : lker f == 0%VS -> (f @: U == f @: V)%VS = (U == V).

Proof. by move=> injf; rewrite !eqEsubv !limg_ker0. Qed.

Lemma

eq_limg_ker0

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "eqEsubv", "injf", "limg_ker0", "lker" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lker0_lfunK f : lker f == 0%VS -> cancel f f^-1%VF.

Proof. by move/lker0P=> injf u; apply: injf; rewrite limg_lfunVK ?memv_img ?memvf. Qed.

Lemma

lker0_lfunK

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "injf", "limg_lfunVK", "lker", "lker0P", "memv_img", "memvf" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lker0_compVf f : lker f == 0%VS -> (f^-1 \o f = \1)%VF.

Proof. by move/lker0_lfunK=> fK; apply/lfunP=> u; rewrite !lfunE /= fK. Qed.

Lemma

lker0_compVf

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "fK", "lfunE", "lfunP", "lker", "lker0_lfunK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lker0_img_cap f U V : lker f == 0%VS -> (f @: (U :&: V) = f @: U :&: f @: V)%VS.

Proof. move=> kf0; apply/eqP; rewrite eqEsubv limg_cap/=; apply/subvP => x. rewrite memv_cap => /andP[/memv_imgP[u uU ->]] /memv_imgP[v vV]. by move=> /(lker0P _ kf0) eq_uv; rewrite memv_img// memv_cap uU eq_uv vV. Qed.

Lemma

lker0_img_cap

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "eqEsubv", "limg_cap", "lker", "lker0P", "memv_cap", "memv_img", "memv_imgP", "subvP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

fixedSpace f : {vspace vT}

:= lker (f - \1%VF).

Definition

fixedSpace

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "lker", "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

fixedSpaceP f a : reflect (f a = a) (a \in fixedSpace f).

Proof. by rewrite memv_ker add_lfunE opp_lfunE id_lfunE subr_eq0; apply: eqP. Qed.

Lemma

fixedSpaceP

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "add_lfunE", "apply", "fixedSpace", "id_lfunE", "memv_ker", "opp_lfunE", "subr_eq0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

fixedSpacesP f U : reflect {in U, f =1 id} (U <= fixedSpace f)%VS.

Proof. by apply: (iffP subvP) => cUf x /cUf/fixedSpaceP. Qed.

Lemma

fixedSpacesP

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "fixedSpace", "fixedSpaceP", "id", "subvP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

fixedSpace_limg f U : (U <= fixedSpace f -> f @: U = U)%VS.

Proof. move/fixedSpacesP=> cUf; apply/vspaceP=> x. by apply/memv_imgP/idP=> [[{}x Ux ->] | Ux]; last exists x; rewrite ?cUf. Qed.

Lemma

fixedSpace_limg

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "fixedSpace", "fixedSpacesP", "last", "memv_imgP", "vspaceP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

fixedSpace_id : fixedSpace \1 = {:vT}%VS.

Proof. by apply/vspaceP=> x; rewrite memvf; apply/fixedSpaceP; rewrite lfunE. Qed.

Lemma

fixedSpace_id

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "fixedSpace", "fixedSpaceP", "lfunE", "memvf", "vT", "vspaceP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

kerf0 : lker f == 0%VS.

Hypothesis

kerf0

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "lker" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lker0_limgf : limg f = fullv.

Proof. by apply/eqP; rewrite eqEdim subvf limg_dim_eq //= (eqP kerf0) capv0. Qed.

Lemma

lker0_limgf

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "capv0", "eqEdim", "fullv", "kerf0", "limg", "limg_dim_eq", "subvf" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lker0_lfunVK : cancel f^-1%VF f.

Proof. by move=> u; rewrite limg_lfunVK // lker0_limgf memvf. Qed.

Lemma

lker0_lfunVK

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "limg_lfunVK", "lker0_limgf", "memvf" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lker0_compfV : (f \o f^-1 = \1)%VF.

Proof. by apply/lfunP=> u; rewrite !lfunE /= lker0_lfunVK. Qed.

Lemma

lker0_compfV

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "lfunE", "lfunP", "lker0_lfunVK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lker0_compVKf aT g : (f \o (f^-1 \o g))%VF = g :> 'Hom(aT, vT).

Proof. by rewrite comp_lfunA lker0_compfV comp_lfun1l. Qed.

Lemma

lker0_compVKf

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "aT", "comp_lfun1l", "comp_lfunA", "lker0_compfV", "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lker0_compKf aT g : (f^-1 \o (f \o g))%VF = g :> 'Hom(aT, vT).

Proof. by rewrite comp_lfunA lker0_compVf ?comp_lfun1l. Qed.

Lemma

lker0_compKf

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "aT", "comp_lfun1l", "comp_lfunA", "lker0_compVf", "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lker0_compfK rT h : ((h \o f) \o f^-1)%VF = h :> 'Hom(vT, rT).

Proof. by rewrite -comp_lfunA lker0_compfV comp_lfun1r. Qed.

Lemma

lker0_compfK

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "comp_lfun1r", "comp_lfunA", "lker0_compfV", "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lker0_compfVK rT h : ((h \o f^-1) \o f)%VF = h :> 'Hom(vT, rT).

Proof. by rewrite -comp_lfunA lker0_compVf ?comp_lfun1r. Qed.

Lemma

lker0_compfVK

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "comp_lfun1r", "comp_lfunA", "lker0_compVf", "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lim1g U : (\1 @: U)%VS = U.

Proof. have /andP[/eqP <- _] := vbasisP U; rewrite limg_span map_id_in // => u _. by rewrite lfunE. Qed.

Lemma

lim1g

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "lfunE", "limg_span", "map_id_in", "vbasisP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

limg_comp f g U : ((g \o f) @: U = g @: (f @: U))%VS.

Proof. have /andP[/eqP <- _] := vbasisP U; rewrite !limg_span; congr (span _). by rewrite -map_comp; apply/eq_map => u; rewrite lfunE. Qed.

Lemma

limg_comp

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "eq_map", "lfunE", "limg_span", "map_comp", "span", "vbasisP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lpreim_cap_limg f W : (f @^-1: (W :&: limg f))%VS = (f @^-1: W)%VS.

Proof. by rewrite /lfun_preim -capvA capvv. Qed.

Lemma

lpreim_cap_limg

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "capvA", "capvv", "lfun_preim", "limg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lpreim0 f : (f @^-1: 0)%VS = lker f.

Proof. by rewrite /lfun_preim cap0v limg0 add0v. Qed.

Lemma

lpreim0

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "add0v", "cap0v", "lfun_preim", "limg0", "lker" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lpreimS f V W : (V <= W)%VS-> (f @^-1: V <= f @^-1: W)%VS.

Proof. by move=> sVW; rewrite addvS // limgS // capvS. Qed.

Lemma

lpreimS

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "addvS", "capvS", "limgS" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lpreimK f W : (W <= limg f)%VS -> (f @: (f @^-1: W))%VS = W.

Proof. move=> sWf; rewrite limgD (capv_idPl sWf) // -limg_comp. have /eqP->: (f @: lker f == 0)%VS by rewrite -lkerE. have /andP[/eqP defW _] := vbasisP W; rewrite addv0 -defW limg_span. rewrite map_id_in // => x Xx; rewrite lfunE /= limg_lfunVK //. by apply: span_subvP Xx; rewrite defW. Qed.

Lemma

lpreimK

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "addv0", "apply", "capv_idPl", "lfunE", "limg", "limgD", "limg_comp", "limg_lfunVK", "limg_span", "lker", "lkerE", "map_id_in", "span_subvP", "vbasisP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

memv_preim f u W : (f u \in W) = (u \in f @^-1: W)%VS.

Proof. apply/idP/idP=> [Wfu | /(memv_img f)]; last first. by rewrite -lpreim_cap_limg lpreimK ?capvSr // => /memv_capP[]. rewrite -[u](addNKr (f^-1%VF (f u))) memv_add ?memv_img //. by rewrite memv_cap Wfu memv_img ?memvf. by rewrite memv_ker addrC linearB /= subr_eq0 limg_lfunVK ?memv_img ?memvf. Qed.

Lemma

memv_preim

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "addNKr", "addrC", "apply", "capvSr", "last", "limg_lfunVK", "linearB", "lpreimK", "lpreim_cap_limg", "memv_add", "memv_cap", "memv_capP", "memv_img", "memv_ker", "memvf", "subr_eq0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lfun_comp_pzSemiRingType : pzSemiRingType

:= HB.pack 'End(vT) (GRing.Nmodule_isPzSemiRing.Build 'End(vT) comp_lfunA comp_lfun1l comp_lfun1r comp_lfunDl comp_lfunDr (comp_lfun0l _) (@comp_lfun0r _ _ _ _)).

Definition

lfun_comp_pzSemiRingType

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "Build", "comp_lfun0l", "comp_lfun0r", "comp_lfun1l", "comp_lfun1r", "comp_lfunA", "comp_lfunDl", "comp_lfunDr", "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lfun1_neq0 : \1%VF != 0 :> 'End(vT).

Proof. apply/eqP=> /lfunP/(_ (r2v (const_mx 1))); rewrite !lfunE /= => /(canRL r2vK). by move=> /rowP/(_ (Ordinal dim_gt0))/eqP; rewrite linear0 !mxE oner_eq0. Qed.

Fact

lfun1_neq0

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "const_mx", "dim_gt0", "lfunE", "lfunP", "linear0", "mxE", "oner_eq0", "r2v", "r2vK", "rowP", "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vT_proper : dim vT > 0.

Hypothesis

vT_proper

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "dim", "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lfun_comp_nzRingType : nzRingType

:= 'End(vT)^c.

Definition

lfun_comp_nzRingType

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lfun_nzRingType : nzRingType

:= 'End(vT).

Definition

lfun_nzRingType

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lfun_lalgType : nzLalgType R

:= 'End(vT).

Definition

lfun_lalgType

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lfun_algType : nzAlgType R

:= 'End(vT).

Definition

lfun_algType

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

daddv_pi U V

:= Hom (proj_mx (vs2mx U) (vs2mx V)).

Definition

daddv_pi

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "proj_mx", "vs2mx" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

projv U

:= daddv_pi U U^C.

Definition

projv

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "daddv_pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

addv_pi1 U V

:= daddv_pi (U :\: V) V.

Definition

addv_pi1

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "daddv_pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

addv_pi2 U V

:= daddv_pi V (U :\: V).

Definition

addv_pi2

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "daddv_pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

memv_pi U V w : (daddv_pi U V) w \in U.

Proof. by rewrite unlock memvE /subsetv genmxE /= r2vK proj_mx_sub. Qed.

Lemma

memv_pi

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "daddv_pi", "genmxE", "memvE", "proj_mx_sub", "r2vK", "subsetv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

memv_proj U w : projv U w \in U.

Proof. exact: memv_pi. Qed.

Lemma

memv_proj

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "memv_pi", "projv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

memv_pi1 U V w : (addv_pi1 U V) w \in U.

Proof. by rewrite (subvP (diffvSl U V)) ?memv_pi. Qed.

Lemma

memv_pi1

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "addv_pi1", "diffvSl", "memv_pi", "subvP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

memv_pi2 U V w : (addv_pi2 U V) w \in V.

Proof. exact: memv_pi. Qed.

Lemma

memv_pi2

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "addv_pi2", "memv_pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

daddv_pi_id U V u : (U :&: V = 0)%VS -> u \in U -> daddv_pi U V u = u.

Proof. move/eqP; rewrite -dimv_eq0 memvE /subsetv /dimv !genmxE mxrank_eq0 => /eqP. by move=> dxUV Uu; rewrite unlock /= proj_mx_id ?v2rK. Qed.

Lemma

daddv_pi_id

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "Uu", "daddv_pi", "dimv", "dimv_eq0", "genmxE", "memvE", "mxrank_eq0", "proj_mx_id", "subsetv", "v2rK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

daddv_pi_proj U V w (pi := daddv_pi U V) : (U :&: V = 0)%VS -> pi (pi w) = pi w.

Proof. by move/daddv_pi_id=> -> //; apply: memv_pi. Qed.

Lemma

daddv_pi_proj

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "daddv_pi", "daddv_pi_id", "memv_pi", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

daddv_pi_add U V w : (U :&: V = 0)%VS -> (w \in U + V)%VS -> daddv_pi U V w + daddv_pi V U w = w.

Proof. move/eqP; rewrite -dimv_eq0 memvE /subsetv /dimv !genmxE mxrank_eq0 => /eqP. by move=> dxUW UVw; rewrite unlock /= -linearD /= add_proj_mx ?v2rK. Qed.

Lemma

daddv_pi_add

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "add_proj_mx", "daddv_pi", "dimv", "dimv_eq0", "genmxE", "linearD", "memvE", "mxrank_eq0", "subsetv", "v2rK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

projv_id U u : u \in U -> projv U u = u.

Proof. exact: daddv_pi_id (capv_compl _). Qed.

Lemma

projv_id

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "capv_compl", "daddv_pi_id", "projv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

projv_proj U w : projv U (projv U w) = projv U w.

Proof. exact: daddv_pi_proj (capv_compl _). Qed.

Lemma

projv_proj

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "capv_compl", "daddv_pi_proj", "projv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d