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Theorem lmhmeql 19268
Description: The equalizer of two module homomorphisms is a subspace. (Contributed by Stefan O'Rear, 7-Mar-2015.)
Hypothesis
Ref Expression
lmhmeql.u 𝑈 = (LSubSp‘𝑆)
Assertion
Ref Expression
lmhmeql ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) → dom (𝐹𝐺) ∈ 𝑈)

Proof of Theorem lmhmeql
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmghm 19244 . . 3 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
2 lmghm 19244 . . 3 (𝐺 ∈ (𝑆 LMHom 𝑇) → 𝐺 ∈ (𝑆 GrpHom 𝑇))
3 ghmeql 17891 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → dom (𝐹𝐺) ∈ (SubGrp‘𝑆))
41, 2, 3syl2an 583 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) → dom (𝐹𝐺) ∈ (SubGrp‘𝑆))
5 lmhmlmod1 19246 . . . . . . . . . 10 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
65adantr 466 . . . . . . . . 9 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) → 𝑆 ∈ LMod)
76ad2antrr 705 . . . . . . . 8 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → 𝑆 ∈ LMod)
8 simplr 752 . . . . . . . 8 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → 𝑥 ∈ (Base‘(Scalar‘𝑆)))
9 simprl 754 . . . . . . . 8 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → 𝑦 ∈ (Base‘𝑆))
10 eqid 2771 . . . . . . . . 9 (Base‘𝑆) = (Base‘𝑆)
11 eqid 2771 . . . . . . . . 9 (Scalar‘𝑆) = (Scalar‘𝑆)
12 eqid 2771 . . . . . . . . 9 ( ·𝑠𝑆) = ( ·𝑠𝑆)
13 eqid 2771 . . . . . . . . 9 (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
1410, 11, 12, 13lmodvscl 19090 . . . . . . . 8 ((𝑆 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥( ·𝑠𝑆)𝑦) ∈ (Base‘𝑆))
157, 8, 9, 14syl3anc 1476 . . . . . . 7 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → (𝑥( ·𝑠𝑆)𝑦) ∈ (Base‘𝑆))
16 oveq2 6801 . . . . . . . . 9 ((𝐹𝑦) = (𝐺𝑦) → (𝑥( ·𝑠𝑇)(𝐹𝑦)) = (𝑥( ·𝑠𝑇)(𝐺𝑦)))
1716ad2antll 708 . . . . . . . 8 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → (𝑥( ·𝑠𝑇)(𝐹𝑦)) = (𝑥( ·𝑠𝑇)(𝐺𝑦)))
18 simplll 758 . . . . . . . . 9 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → 𝐹 ∈ (𝑆 LMHom 𝑇))
19 eqid 2771 . . . . . . . . . 10 ( ·𝑠𝑇) = ( ·𝑠𝑇)
2011, 13, 10, 12, 19lmhmlin 19248 . . . . . . . . 9 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥( ·𝑠𝑆)𝑦)) = (𝑥( ·𝑠𝑇)(𝐹𝑦)))
2118, 8, 9, 20syl3anc 1476 . . . . . . . 8 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → (𝐹‘(𝑥( ·𝑠𝑆)𝑦)) = (𝑥( ·𝑠𝑇)(𝐹𝑦)))
22 simpllr 760 . . . . . . . . 9 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → 𝐺 ∈ (𝑆 LMHom 𝑇))
2311, 13, 10, 12, 19lmhmlin 19248 . . . . . . . . 9 ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐺‘(𝑥( ·𝑠𝑆)𝑦)) = (𝑥( ·𝑠𝑇)(𝐺𝑦)))
2422, 8, 9, 23syl3anc 1476 . . . . . . . 8 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → (𝐺‘(𝑥( ·𝑠𝑆)𝑦)) = (𝑥( ·𝑠𝑇)(𝐺𝑦)))
2517, 21, 243eqtr4d 2815 . . . . . . 7 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → (𝐹‘(𝑥( ·𝑠𝑆)𝑦)) = (𝐺‘(𝑥( ·𝑠𝑆)𝑦)))
26 fveq2 6332 . . . . . . . . 9 (𝑧 = (𝑥( ·𝑠𝑆)𝑦) → (𝐹𝑧) = (𝐹‘(𝑥( ·𝑠𝑆)𝑦)))
27 fveq2 6332 . . . . . . . . 9 (𝑧 = (𝑥( ·𝑠𝑆)𝑦) → (𝐺𝑧) = (𝐺‘(𝑥( ·𝑠𝑆)𝑦)))
2826, 27eqeq12d 2786 . . . . . . . 8 (𝑧 = (𝑥( ·𝑠𝑆)𝑦) → ((𝐹𝑧) = (𝐺𝑧) ↔ (𝐹‘(𝑥( ·𝑠𝑆)𝑦)) = (𝐺‘(𝑥( ·𝑠𝑆)𝑦))))
2928elrab 3515 . . . . . . 7 ((𝑥( ·𝑠𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} ↔ ((𝑥( ·𝑠𝑆)𝑦) ∈ (Base‘𝑆) ∧ (𝐹‘(𝑥( ·𝑠𝑆)𝑦)) = (𝐺‘(𝑥( ·𝑠𝑆)𝑦))))
3015, 25, 29sylanbrc 572 . . . . . 6 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → (𝑥( ·𝑠𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)})
3130expr 444 . . . . 5 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆))) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹𝑦) = (𝐺𝑦) → (𝑥( ·𝑠𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)}))
3231ralrimiva 3115 . . . 4 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆))) → ∀𝑦 ∈ (Base‘𝑆)((𝐹𝑦) = (𝐺𝑦) → (𝑥( ·𝑠𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)}))
33 eqid 2771 . . . . . . . . 9 (Base‘𝑇) = (Base‘𝑇)
3410, 33lmhmf 19247 . . . . . . . 8 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
35 ffn 6185 . . . . . . . 8 (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → 𝐹 Fn (Base‘𝑆))
3634, 35syl 17 . . . . . . 7 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 Fn (Base‘𝑆))
3710, 33lmhmf 19247 . . . . . . . 8 (𝐺 ∈ (𝑆 LMHom 𝑇) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
38 ffn 6185 . . . . . . . 8 (𝐺:(Base‘𝑆)⟶(Base‘𝑇) → 𝐺 Fn (Base‘𝑆))
3937, 38syl 17 . . . . . . 7 (𝐺 ∈ (𝑆 LMHom 𝑇) → 𝐺 Fn (Base‘𝑆))
40 fndmin 6467 . . . . . . 7 ((𝐹 Fn (Base‘𝑆) ∧ 𝐺 Fn (Base‘𝑆)) → dom (𝐹𝐺) = {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)})
4136, 39, 40syl2an 583 . . . . . 6 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) → dom (𝐹𝐺) = {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)})
4241adantr 466 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆))) → dom (𝐹𝐺) = {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)})
43 eleq2 2839 . . . . . . 7 (dom (𝐹𝐺) = {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} → ((𝑥( ·𝑠𝑆)𝑦) ∈ dom (𝐹𝐺) ↔ (𝑥( ·𝑠𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)}))
4443raleqbi1dv 3295 . . . . . 6 (dom (𝐹𝐺) = {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} → (∀𝑦 ∈ dom (𝐹𝐺)(𝑥( ·𝑠𝑆)𝑦) ∈ dom (𝐹𝐺) ↔ ∀𝑦 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} (𝑥( ·𝑠𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)}))
45 fveq2 6332 . . . . . . . 8 (𝑧 = 𝑦 → (𝐹𝑧) = (𝐹𝑦))
46 fveq2 6332 . . . . . . . 8 (𝑧 = 𝑦 → (𝐺𝑧) = (𝐺𝑦))
4745, 46eqeq12d 2786 . . . . . . 7 (𝑧 = 𝑦 → ((𝐹𝑧) = (𝐺𝑧) ↔ (𝐹𝑦) = (𝐺𝑦)))
4847ralrab 3520 . . . . . 6 (∀𝑦 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} (𝑥( ·𝑠𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} ↔ ∀𝑦 ∈ (Base‘𝑆)((𝐹𝑦) = (𝐺𝑦) → (𝑥( ·𝑠𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)}))
4944, 48syl6bb 276 . . . . 5 (dom (𝐹𝐺) = {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} → (∀𝑦 ∈ dom (𝐹𝐺)(𝑥( ·𝑠𝑆)𝑦) ∈ dom (𝐹𝐺) ↔ ∀𝑦 ∈ (Base‘𝑆)((𝐹𝑦) = (𝐺𝑦) → (𝑥( ·𝑠𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)})))
5042, 49syl 17 . . . 4 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆))) → (∀𝑦 ∈ dom (𝐹𝐺)(𝑥( ·𝑠𝑆)𝑦) ∈ dom (𝐹𝐺) ↔ ∀𝑦 ∈ (Base‘𝑆)((𝐹𝑦) = (𝐺𝑦) → (𝑥( ·𝑠𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)})))
5132, 50mpbird 247 . . 3 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆))) → ∀𝑦 ∈ dom (𝐹𝐺)(𝑥( ·𝑠𝑆)𝑦) ∈ dom (𝐹𝐺))
5251ralrimiva 3115 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) → ∀𝑥 ∈ (Base‘(Scalar‘𝑆))∀𝑦 ∈ dom (𝐹𝐺)(𝑥( ·𝑠𝑆)𝑦) ∈ dom (𝐹𝐺))
53 lmhmeql.u . . . 4 𝑈 = (LSubSp‘𝑆)
5411, 13, 10, 12, 53islss4 19175 . . 3 (𝑆 ∈ LMod → (dom (𝐹𝐺) ∈ 𝑈 ↔ (dom (𝐹𝐺) ∈ (SubGrp‘𝑆) ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝑆))∀𝑦 ∈ dom (𝐹𝐺)(𝑥( ·𝑠𝑆)𝑦) ∈ dom (𝐹𝐺))))
556, 54syl 17 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) → (dom (𝐹𝐺) ∈ 𝑈 ↔ (dom (𝐹𝐺) ∈ (SubGrp‘𝑆) ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝑆))∀𝑦 ∈ dom (𝐹𝐺)(𝑥( ·𝑠𝑆)𝑦) ∈ dom (𝐹𝐺))))
564, 52, 55mpbir2and 692 1 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) → dom (𝐹𝐺) ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061  {crab 3065  cin 3722  dom cdm 5249   Fn wfn 6026  wf 6027  cfv 6031  (class class class)co 6793  Basecbs 16064  Scalarcsca 16152   ·𝑠 cvsca 16153  SubGrpcsubg 17796   GrpHom cghm 17865  LModclmod 19073  LSubSpclss 19142   LMHom clmhm 19232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-er 7896  df-map 8011  df-en 8110  df-dom 8111  df-sdom 8112  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-2 11281  df-ndx 16067  df-slot 16068  df-base 16070  df-sets 16071  df-ress 16072  df-plusg 16162  df-0g 16310  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-mhm 17543  df-submnd 17544  df-grp 17633  df-minusg 17634  df-sbg 17635  df-subg 17799  df-ghm 17866  df-mgp 18698  df-ur 18710  df-ring 18757  df-lmod 19075  df-lss 19143  df-lmhm 19235
This theorem is referenced by:  lspextmo  19269
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