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Theorem ghmeql 17891
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 17878 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 ∈ (𝑆 MndHom 𝑇))
2 ghmmhm 17878 . . 3 (𝐺 ∈ (𝑆 GrpHom 𝑇) → 𝐺 ∈ (𝑆 MndHom 𝑇))
3 mhmeql 17572 . . 3 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → dom (𝐹𝐺) ∈ (SubMnd‘𝑆))
41, 2, 3syl2an 583 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → dom (𝐹𝐺) ∈ (SubMnd‘𝑆))
5 ghmgrp1 17870 . . . . . . . . . 10 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
65adantr 466 . . . . . . . . 9 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → 𝑆 ∈ Grp)
76adantr 466 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) → 𝑆 ∈ Grp)
8 simprl 754 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) → 𝑥 ∈ (Base‘𝑆))
9 eqid 2771 . . . . . . . . 9 (Base‘𝑆) = (Base‘𝑆)
10 eqid 2771 . . . . . . . . 9 (invg𝑆) = (invg𝑆)
119, 10grpinvcl 17675 . . . . . . . 8 ((𝑆 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑆)) → ((invg𝑆)‘𝑥) ∈ (Base‘𝑆))
127, 8, 11syl2anc 573 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) → ((invg𝑆)‘𝑥) ∈ (Base‘𝑆))
13 simprr 756 . . . . . . . . 9 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) → (𝐹𝑥) = (𝐺𝑥))
1413fveq2d 6337 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) → ((invg𝑇)‘(𝐹𝑥)) = ((invg𝑇)‘(𝐺𝑥)))
15 eqid 2771 . . . . . . . . . 10 (invg𝑇) = (invg𝑇)
169, 10, 15ghminv 17875 . . . . . . . . 9 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → (𝐹‘((invg𝑆)‘𝑥)) = ((invg𝑇)‘(𝐹𝑥)))
1716ad2ant2r 741 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) → (𝐹‘((invg𝑆)‘𝑥)) = ((invg𝑇)‘(𝐹𝑥)))
189, 10, 15ghminv 17875 . . . . . . . . 9 ((𝐺 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → (𝐺‘((invg𝑆)‘𝑥)) = ((invg𝑇)‘(𝐺𝑥)))
1918ad2ant2lr 742 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) → (𝐺‘((invg𝑆)‘𝑥)) = ((invg𝑇)‘(𝐺𝑥)))
2014, 17, 193eqtr4d 2815 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) → (𝐹‘((invg𝑆)‘𝑥)) = (𝐺‘((invg𝑆)‘𝑥)))
21 fveq2 6333 . . . . . . . . 9 (𝑦 = ((invg𝑆)‘𝑥) → (𝐹𝑦) = (𝐹‘((invg𝑆)‘𝑥)))
22 fveq2 6333 . . . . . . . . 9 (𝑦 = ((invg𝑆)‘𝑥) → (𝐺𝑦) = (𝐺‘((invg𝑆)‘𝑥)))
2321, 22eqeq12d 2786 . . . . . . . 8 (𝑦 = ((invg𝑆)‘𝑥) → ((𝐹𝑦) = (𝐺𝑦) ↔ (𝐹‘((invg𝑆)‘𝑥)) = (𝐺‘((invg𝑆)‘𝑥))))
2423elrab 3515 . . . . . . 7 (((invg𝑆)‘𝑥) ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹𝑦) = (𝐺𝑦)} ↔ (((invg𝑆)‘𝑥) ∈ (Base‘𝑆) ∧ (𝐹‘((invg𝑆)‘𝑥)) = (𝐺‘((invg𝑆)‘𝑥))))
2512, 20, 24sylanbrc 572 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) → ((invg𝑆)‘𝑥) ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹𝑦) = (𝐺𝑦)})
2625expr 444 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑥 ∈ (Base‘𝑆)) → ((𝐹𝑥) = (𝐺𝑥) → ((invg𝑆)‘𝑥) ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹𝑦) = (𝐺𝑦)}))
2726ralrimiva 3115 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → ∀𝑥 ∈ (Base‘𝑆)((𝐹𝑥) = (𝐺𝑥) → ((invg𝑆)‘𝑥) ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹𝑦) = (𝐺𝑦)}))
28 fveq2 6333 . . . . . 6 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
29 fveq2 6333 . . . . . 6 (𝑦 = 𝑥 → (𝐺𝑦) = (𝐺𝑥))
3028, 29eqeq12d 2786 . . . . 5 (𝑦 = 𝑥 → ((𝐹𝑦) = (𝐺𝑦) ↔ (𝐹𝑥) = (𝐺𝑥)))
3130ralrab 3520 . . . 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 2771 . . . . . . . 8 (Base‘𝑇) = (Base‘𝑇)
349, 33ghmf 17872 . . . . . . 7 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
3534adantr 466 . . . . . 6 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
3635ffnd 6185 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → 𝐹 Fn (Base‘𝑆))
379, 33ghmf 17872 . . . . . . 7 (𝐺 ∈ (𝑆 GrpHom 𝑇) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
3837adantl 467 . . . . . 6 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
3938ffnd 6185 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → 𝐺 Fn (Base‘𝑆))
40 fndmin 6469 . . . . 5 ((𝐹 Fn (Base‘𝑆) ∧ 𝐺 Fn (Base‘𝑆)) → dom (𝐹𝐺) = {𝑦 ∈ (Base‘𝑆) ∣ (𝐹𝑦) = (𝐺𝑦)})
4136, 39, 40syl2anc 573 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → dom (𝐹𝐺) = {𝑦 ∈ (Base‘𝑆) ∣ (𝐹𝑦) = (𝐺𝑦)})
42 eleq2 2839 . . . . 5 (dom (𝐹𝐺) = {𝑦 ∈ (Base‘𝑆) ∣ (𝐹𝑦) = (𝐺𝑦)} → (((invg𝑆)‘𝑥) ∈ dom (𝐹𝐺) ↔ ((invg𝑆)‘𝑥) ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹𝑦) = (𝐺𝑦)}))
4342raleqbi1dv 3295 . . . 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)}))
4441, 43syl 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)}))
4532, 44mpbird 247 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → ∀𝑥 ∈ dom (𝐹𝐺)((invg𝑆)‘𝑥) ∈ dom (𝐹𝐺))
4610issubg3 17820 . . 3 (𝑆 ∈ Grp → (dom (𝐹𝐺) ∈ (SubGrp‘𝑆) ↔ (dom (𝐹𝐺) ∈ (SubMnd‘𝑆) ∧ ∀𝑥 ∈ dom (𝐹𝐺)((invg𝑆)‘𝑥) ∈ dom (𝐹𝐺))))
476, 46syl 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))))
484, 45, 47mpbir2and 692 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 382   = wceq 1631  wcel 2145  wral 3061  {crab 3065  cin 3722  dom cdm 5250   Fn wfn 6025  wf 6026  cfv 6030  (class class class)co 6796  Basecbs 16064   MndHom cmhm 17541  SubMndcsubmnd 17542  Grpcgrp 17630  invgcminusg 17631  SubGrpcsubg 17796   GrpHom cghm 17865
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 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-cnex 10198  ax-resscn 10199  ax-1cn 10200  ax-icn 10201  ax-addcl 10202  ax-addrcl 10203  ax-mulcl 10204  ax-mulrcl 10205  ax-mulcom 10206  ax-addass 10207  ax-mulass 10208  ax-distr 10209  ax-i2m1 10210  ax-1ne0 10211  ax-1rid 10212  ax-rnegex 10213  ax-rrecex 10214  ax-cnre 10215  ax-pre-lttri 10216  ax-pre-lttrn 10217  ax-pre-ltadd 10218  ax-pre-mulgt0 10219
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 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-er 7900  df-map 8015  df-en 8114  df-dom 8115  df-sdom 8116  df-pnf 10282  df-mnf 10283  df-xr 10284  df-ltxr 10285  df-le 10286  df-sub 10474  df-neg 10475  df-nn 11227  df-2 11285  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-subg 17799  df-ghm 17866
This theorem is referenced by:  rhmeql  19020  lmhmeql  19268
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