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Theorem ghmeql 17599
 Description: The equalizer of two group homomorphisms is a subgroup. (Contributed by Stefan O'Rear, 7-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
Assertion
Ref Expression
ghmeql ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → dom (𝐹𝐺) ∈ (SubGrp‘𝑆))

Proof of Theorem ghmeql
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ghmmhm 17586 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 ∈ (𝑆 MndHom 𝑇))
2 ghmmhm 17586 . . 3 (𝐺 ∈ (𝑆 GrpHom 𝑇) → 𝐺 ∈ (𝑆 MndHom 𝑇))
3 mhmeql 17280 . . 3 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → dom (𝐹𝐺) ∈ (SubMnd‘𝑆))
41, 2, 3syl2an 494 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → dom (𝐹𝐺) ∈ (SubMnd‘𝑆))
5 ghmgrp1 17578 . . . . . . . . . 10 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
65adantr 481 . . . . . . . . 9 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → 𝑆 ∈ Grp)
76adantr 481 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) → 𝑆 ∈ Grp)
8 simprl 793 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) → 𝑥 ∈ (Base‘𝑆))
9 eqid 2626 . . . . . . . . 9 (Base‘𝑆) = (Base‘𝑆)
10 eqid 2626 . . . . . . . . 9 (invg𝑆) = (invg𝑆)
119, 10grpinvcl 17383 . . . . . . . 8 ((𝑆 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑆)) → ((invg𝑆)‘𝑥) ∈ (Base‘𝑆))
127, 8, 11syl2anc 692 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) → ((invg𝑆)‘𝑥) ∈ (Base‘𝑆))
13 simprr 795 . . . . . . . . 9 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) → (𝐹𝑥) = (𝐺𝑥))
1413fveq2d 6154 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) → ((invg𝑇)‘(𝐹𝑥)) = ((invg𝑇)‘(𝐺𝑥)))
15 eqid 2626 . . . . . . . . . 10 (invg𝑇) = (invg𝑇)
169, 10, 15ghminv 17583 . . . . . . . . 9 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → (𝐹‘((invg𝑆)‘𝑥)) = ((invg𝑇)‘(𝐹𝑥)))
1716ad2ant2r 782 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) → (𝐹‘((invg𝑆)‘𝑥)) = ((invg𝑇)‘(𝐹𝑥)))
189, 10, 15ghminv 17583 . . . . . . . . 9 ((𝐺 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → (𝐺‘((invg𝑆)‘𝑥)) = ((invg𝑇)‘(𝐺𝑥)))
1918ad2ant2lr 783 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) → (𝐺‘((invg𝑆)‘𝑥)) = ((invg𝑇)‘(𝐺𝑥)))
2014, 17, 193eqtr4d 2670 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) → (𝐹‘((invg𝑆)‘𝑥)) = (𝐺‘((invg𝑆)‘𝑥)))
21 fveq2 6150 . . . . . . . . 9 (𝑦 = ((invg𝑆)‘𝑥) → (𝐹𝑦) = (𝐹‘((invg𝑆)‘𝑥)))
22 fveq2 6150 . . . . . . . . 9 (𝑦 = ((invg𝑆)‘𝑥) → (𝐺𝑦) = (𝐺‘((invg𝑆)‘𝑥)))
2321, 22eqeq12d 2641 . . . . . . . 8 (𝑦 = ((invg𝑆)‘𝑥) → ((𝐹𝑦) = (𝐺𝑦) ↔ (𝐹‘((invg𝑆)‘𝑥)) = (𝐺‘((invg𝑆)‘𝑥))))
2423elrab 3351 . . . . . . 7 (((invg𝑆)‘𝑥) ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹𝑦) = (𝐺𝑦)} ↔ (((invg𝑆)‘𝑥) ∈ (Base‘𝑆) ∧ (𝐹‘((invg𝑆)‘𝑥)) = (𝐺‘((invg𝑆)‘𝑥))))
2512, 20, 24sylanbrc 697 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) → ((invg𝑆)‘𝑥) ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹𝑦) = (𝐺𝑦)})
2625expr 642 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑥 ∈ (Base‘𝑆)) → ((𝐹𝑥) = (𝐺𝑥) → ((invg𝑆)‘𝑥) ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹𝑦) = (𝐺𝑦)}))
2726ralrimiva 2965 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → ∀𝑥 ∈ (Base‘𝑆)((𝐹𝑥) = (𝐺𝑥) → ((invg𝑆)‘𝑥) ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹𝑦) = (𝐺𝑦)}))
28 fveq2 6150 . . . . . 6 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
29 fveq2 6150 . . . . . 6 (𝑦 = 𝑥 → (𝐺𝑦) = (𝐺𝑥))
3028, 29eqeq12d 2641 . . . . 5 (𝑦 = 𝑥 → ((𝐹𝑦) = (𝐺𝑦) ↔ (𝐹𝑥) = (𝐺𝑥)))
3130ralrab 3355 . . . 4 (∀𝑥 ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹𝑦) = (𝐺𝑦)} ((invg𝑆)‘𝑥) ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹𝑦) = (𝐺𝑦)} ↔ ∀𝑥 ∈ (Base‘𝑆)((𝐹𝑥) = (𝐺𝑥) → ((invg𝑆)‘𝑥) ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹𝑦) = (𝐺𝑦)}))
3227, 31sylibr 224 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → ∀𝑥 ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹𝑦) = (𝐺𝑦)} ((invg𝑆)‘𝑥) ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹𝑦) = (𝐺𝑦)})
33 eqid 2626 . . . . . . . 8 (Base‘𝑇) = (Base‘𝑇)
349, 33ghmf 17580 . . . . . . 7 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
3534adantr 481 . . . . . 6 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
36 ffn 6004 . . . . . 6 (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → 𝐹 Fn (Base‘𝑆))
3735, 36syl 17 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → 𝐹 Fn (Base‘𝑆))
389, 33ghmf 17580 . . . . . . 7 (𝐺 ∈ (𝑆 GrpHom 𝑇) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
3938adantl 482 . . . . . 6 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
40 ffn 6004 . . . . . 6 (𝐺:(Base‘𝑆)⟶(Base‘𝑇) → 𝐺 Fn (Base‘𝑆))
4139, 40syl 17 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → 𝐺 Fn (Base‘𝑆))
42 fndmin 6281 . . . . 5 ((𝐹 Fn (Base‘𝑆) ∧ 𝐺 Fn (Base‘𝑆)) → dom (𝐹𝐺) = {𝑦 ∈ (Base‘𝑆) ∣ (𝐹𝑦) = (𝐺𝑦)})
4337, 41, 42syl2anc 692 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → dom (𝐹𝐺) = {𝑦 ∈ (Base‘𝑆) ∣ (𝐹𝑦) = (𝐺𝑦)})
44 eleq2 2693 . . . . 5 (dom (𝐹𝐺) = {𝑦 ∈ (Base‘𝑆) ∣ (𝐹𝑦) = (𝐺𝑦)} → (((invg𝑆)‘𝑥) ∈ dom (𝐹𝐺) ↔ ((invg𝑆)‘𝑥) ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹𝑦) = (𝐺𝑦)}))
4544raleqbi1dv 3140 . . . 4 (dom (𝐹𝐺) = {𝑦 ∈ (Base‘𝑆) ∣ (𝐹𝑦) = (𝐺𝑦)} → (∀𝑥 ∈ dom (𝐹𝐺)((invg𝑆)‘𝑥) ∈ dom (𝐹𝐺) ↔ ∀𝑥 ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹𝑦) = (𝐺𝑦)} ((invg𝑆)‘𝑥) ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹𝑦) = (𝐺𝑦)}))
4643, 45syl 17 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → (∀𝑥 ∈ dom (𝐹𝐺)((invg𝑆)‘𝑥) ∈ dom (𝐹𝐺) ↔ ∀𝑥 ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹𝑦) = (𝐺𝑦)} ((invg𝑆)‘𝑥) ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹𝑦) = (𝐺𝑦)}))
4732, 46mpbird 247 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → ∀𝑥 ∈ dom (𝐹𝐺)((invg𝑆)‘𝑥) ∈ dom (𝐹𝐺))
4810issubg3 17528 . . 3 (𝑆 ∈ Grp → (dom (𝐹𝐺) ∈ (SubGrp‘𝑆) ↔ (dom (𝐹𝐺) ∈ (SubMnd‘𝑆) ∧ ∀𝑥 ∈ dom (𝐹𝐺)((invg𝑆)‘𝑥) ∈ dom (𝐹𝐺))))
496, 48syl 17 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → (dom (𝐹𝐺) ∈ (SubGrp‘𝑆) ↔ (dom (𝐹𝐺) ∈ (SubMnd‘𝑆) ∧ ∀𝑥 ∈ dom (𝐹𝐺)((invg𝑆)‘𝑥) ∈ dom (𝐹𝐺))))
504, 47, 49mpbir2and 956 1 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → dom (𝐹𝐺) ∈ (SubGrp‘𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1480   ∈ wcel 1992  ∀wral 2912  {crab 2916   ∩ cin 3559  dom cdm 5079   Fn wfn 5845  ⟶wf 5846  ‘cfv 5850  (class class class)co 6605  Basecbs 15776   MndHom cmhm 17249  SubMndcsubmnd 17250  Grpcgrp 17338  invgcminusg 17339  SubGrpcsubg 17504   GrpHom cghm 17573 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-er 7688  df-map 7805  df-en 7901  df-dom 7902  df-sdom 7903  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-2 11024  df-ndx 15779  df-slot 15780  df-base 15781  df-sets 15782  df-ress 15783  df-plusg 15870  df-0g 16018  df-mgm 17158  df-sgrp 17200  df-mnd 17211  df-mhm 17251  df-submnd 17252  df-grp 17341  df-minusg 17342  df-subg 17507  df-ghm 17574 This theorem is referenced by:  rhmeql  18726  lmhmeql  18969
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