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Theorem lmhmeql 19909
 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 19885 . . 3 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
2 lmghm 19885 . . 3 (𝐺 ∈ (𝑆 LMHom 𝑇) → 𝐺 ∈ (𝑆 GrpHom 𝑇))
3 ghmeql 18462 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → dom (𝐹𝐺) ∈ (SubGrp‘𝑆))
41, 2, 3syl2an 598 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) → dom (𝐹𝐺) ∈ (SubGrp‘𝑆))
5 fveq2 6663 . . . . . . . 8 (𝑧 = (𝑥( ·𝑠𝑆)𝑦) → (𝐹𝑧) = (𝐹‘(𝑥( ·𝑠𝑆)𝑦)))
6 fveq2 6663 . . . . . . . 8 (𝑧 = (𝑥( ·𝑠𝑆)𝑦) → (𝐺𝑧) = (𝐺‘(𝑥( ·𝑠𝑆)𝑦)))
75, 6eqeq12d 2774 . . . . . . 7 (𝑧 = (𝑥( ·𝑠𝑆)𝑦) → ((𝐹𝑧) = (𝐺𝑧) ↔ (𝐹‘(𝑥( ·𝑠𝑆)𝑦)) = (𝐺‘(𝑥( ·𝑠𝑆)𝑦))))
8 lmhmlmod1 19887 . . . . . . . . . 10 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
98adantr 484 . . . . . . . . 9 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) → 𝑆 ∈ LMod)
109ad2antrr 725 . . . . . . . 8 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → 𝑆 ∈ LMod)
11 simplr 768 . . . . . . . 8 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → 𝑥 ∈ (Base‘(Scalar‘𝑆)))
12 simprl 770 . . . . . . . 8 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → 𝑦 ∈ (Base‘𝑆))
13 eqid 2758 . . . . . . . . 9 (Base‘𝑆) = (Base‘𝑆)
14 eqid 2758 . . . . . . . . 9 (Scalar‘𝑆) = (Scalar‘𝑆)
15 eqid 2758 . . . . . . . . 9 ( ·𝑠𝑆) = ( ·𝑠𝑆)
16 eqid 2758 . . . . . . . . 9 (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
1713, 14, 15, 16lmodvscl 19733 . . . . . . . 8 ((𝑆 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥( ·𝑠𝑆)𝑦) ∈ (Base‘𝑆))
1810, 11, 12, 17syl3anc 1368 . . . . . . 7 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → (𝑥( ·𝑠𝑆)𝑦) ∈ (Base‘𝑆))
19 oveq2 7164 . . . . . . . . 9 ((𝐹𝑦) = (𝐺𝑦) → (𝑥( ·𝑠𝑇)(𝐹𝑦)) = (𝑥( ·𝑠𝑇)(𝐺𝑦)))
2019ad2antll 728 . . . . . . . 8 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → (𝑥( ·𝑠𝑇)(𝐹𝑦)) = (𝑥( ·𝑠𝑇)(𝐺𝑦)))
21 simplll 774 . . . . . . . . 9 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → 𝐹 ∈ (𝑆 LMHom 𝑇))
22 eqid 2758 . . . . . . . . . 10 ( ·𝑠𝑇) = ( ·𝑠𝑇)
2314, 16, 13, 15, 22lmhmlin 19889 . . . . . . . . 9 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥( ·𝑠𝑆)𝑦)) = (𝑥( ·𝑠𝑇)(𝐹𝑦)))
2421, 11, 12, 23syl3anc 1368 . . . . . . . 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)))
25 simpllr 775 . . . . . . . . 9 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → 𝐺 ∈ (𝑆 LMHom 𝑇))
2614, 16, 13, 15, 22lmhmlin 19889 . . . . . . . . 9 ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐺‘(𝑥( ·𝑠𝑆)𝑦)) = (𝑥( ·𝑠𝑇)(𝐺𝑦)))
2725, 11, 12, 26syl3anc 1368 . . . . . . . 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)))
2820, 24, 273eqtr4d 2803 . . . . . . 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)))
297, 18, 28elrabd 3606 . . . . . 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)})
3029expr 460 . . . . 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)}))
3130ralrimiva 3113 . . . 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)}))
32 eqid 2758 . . . . . . . . 9 (Base‘𝑇) = (Base‘𝑇)
3313, 32lmhmf 19888 . . . . . . . 8 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
3433ffnd 6504 . . . . . . 7 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 Fn (Base‘𝑆))
3513, 32lmhmf 19888 . . . . . . . 8 (𝐺 ∈ (𝑆 LMHom 𝑇) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
3635ffnd 6504 . . . . . . 7 (𝐺 ∈ (𝑆 LMHom 𝑇) → 𝐺 Fn (Base‘𝑆))
37 fndmin 6811 . . . . . . 7 ((𝐹 Fn (Base‘𝑆) ∧ 𝐺 Fn (Base‘𝑆)) → dom (𝐹𝐺) = {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)})
3834, 36, 37syl2an 598 . . . . . 6 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) → dom (𝐹𝐺) = {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)})
3938adantr 484 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆))) → dom (𝐹𝐺) = {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)})
40 eleq2 2840 . . . . . . 7 (dom (𝐹𝐺) = {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} → ((𝑥( ·𝑠𝑆)𝑦) ∈ dom (𝐹𝐺) ↔ (𝑥( ·𝑠𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)}))
4140raleqbi1dv 3321 . . . . . 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)}))
42 fveq2 6663 . . . . . . . 8 (𝑧 = 𝑦 → (𝐹𝑧) = (𝐹𝑦))
43 fveq2 6663 . . . . . . . 8 (𝑧 = 𝑦 → (𝐺𝑧) = (𝐺𝑦))
4442, 43eqeq12d 2774 . . . . . . 7 (𝑧 = 𝑦 → ((𝐹𝑧) = (𝐺𝑧) ↔ (𝐹𝑦) = (𝐺𝑦)))
4544ralrab 3610 . . . . . 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)}))
4641, 45bitrdi 290 . . . . 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)})))
4739, 46syl 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)})))
4831, 47mpbird 260 . . 3 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆))) → ∀𝑦 ∈ dom (𝐹𝐺)(𝑥( ·𝑠𝑆)𝑦) ∈ dom (𝐹𝐺))
4948ralrimiva 3113 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) → ∀𝑥 ∈ (Base‘(Scalar‘𝑆))∀𝑦 ∈ dom (𝐹𝐺)(𝑥( ·𝑠𝑆)𝑦) ∈ dom (𝐹𝐺))
50 lmhmeql.u . . . 4 𝑈 = (LSubSp‘𝑆)
5114, 16, 13, 15, 50islss4 19816 . . 3 (𝑆 ∈ LMod → (dom (𝐹𝐺) ∈ 𝑈 ↔ (dom (𝐹𝐺) ∈ (SubGrp‘𝑆) ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝑆))∀𝑦 ∈ dom (𝐹𝐺)(𝑥( ·𝑠𝑆)𝑦) ∈ dom (𝐹𝐺))))
529, 51syl 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))))
534, 49, 52mpbir2and 712 1 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) → dom (𝐹𝐺) ∈ 𝑈)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  {crab 3074   ∩ cin 3859  dom cdm 5528   Fn wfn 6335  ‘cfv 6340  (class class class)co 7156  Basecbs 16555  Scalarcsca 16640   ·𝑠 cvsca 16641  SubGrpcsubg 18354   GrpHom cghm 18436  LModclmod 19716  LSubSpclss 19785   LMHom clmhm 19873 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465  ax-cnex 10644  ax-resscn 10645  ax-1cn 10646  ax-icn 10647  ax-addcl 10648  ax-addrcl 10649  ax-mulcl 10650  ax-mulrcl 10651  ax-mulcom 10652  ax-addass 10653  ax-mulass 10654  ax-distr 10655  ax-i2m1 10656  ax-1ne0 10657  ax-1rid 10658  ax-rnegex 10659  ax-rrecex 10660  ax-cnre 10661  ax-pre-lttri 10662  ax-pre-lttrn 10663  ax-pre-ltadd 10664  ax-pre-mulgt0 10665 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7586  df-1st 7699  df-2nd 7700  df-wrecs 7963  df-recs 8024  df-rdg 8062  df-er 8305  df-map 8424  df-en 8541  df-dom 8542  df-sdom 8543  df-pnf 10728  df-mnf 10729  df-xr 10730  df-ltxr 10731  df-le 10732  df-sub 10923  df-neg 10924  df-nn 11688  df-2 11750  df-ndx 16558  df-slot 16559  df-base 16561  df-sets 16562  df-ress 16563  df-plusg 16650  df-0g 16787  df-mgm 17932  df-sgrp 17981  df-mnd 17992  df-mhm 18036  df-submnd 18037  df-grp 18186  df-minusg 18187  df-sbg 18188  df-subg 18357  df-ghm 18437  df-mgp 19322  df-ur 19334  df-ring 19381  df-lmod 19718  df-lss 19786  df-lmhm 19876 This theorem is referenced by:  lspextmo  19910
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