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Theorem ghmeql 18127
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 18114 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 ∈ (𝑆 MndHom 𝑇))
2 ghmmhm 18114 . . 3 (𝐺 ∈ (𝑆 GrpHom 𝑇) → 𝐺 ∈ (𝑆 MndHom 𝑇))
3 mhmeql 17808 . . 3 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → dom (𝐹𝐺) ∈ (SubMnd‘𝑆))
41, 2, 3syl2an 595 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → dom (𝐹𝐺) ∈ (SubMnd‘𝑆))
5 fveq2 6543 . . . . . . . 8 (𝑦 = ((invg𝑆)‘𝑥) → (𝐹𝑦) = (𝐹‘((invg𝑆)‘𝑥)))
6 fveq2 6543 . . . . . . . 8 (𝑦 = ((invg𝑆)‘𝑥) → (𝐺𝑦) = (𝐺‘((invg𝑆)‘𝑥)))
75, 6eqeq12d 2810 . . . . . . 7 (𝑦 = ((invg𝑆)‘𝑥) → ((𝐹𝑦) = (𝐺𝑦) ↔ (𝐹‘((invg𝑆)‘𝑥)) = (𝐺‘((invg𝑆)‘𝑥))))
8 ghmgrp1 18106 . . . . . . . . . 10 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
98adantr 481 . . . . . . . . 9 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → 𝑆 ∈ Grp)
109adantr 481 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) → 𝑆 ∈ Grp)
11 simprl 767 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) → 𝑥 ∈ (Base‘𝑆))
12 eqid 2795 . . . . . . . . 9 (Base‘𝑆) = (Base‘𝑆)
13 eqid 2795 . . . . . . . . 9 (invg𝑆) = (invg𝑆)
1412, 13grpinvcl 17913 . . . . . . . 8 ((𝑆 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑆)) → ((invg𝑆)‘𝑥) ∈ (Base‘𝑆))
1510, 11, 14syl2anc 584 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) → ((invg𝑆)‘𝑥) ∈ (Base‘𝑆))
16 simprr 769 . . . . . . . . 9 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) → (𝐹𝑥) = (𝐺𝑥))
1716fveq2d 6547 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) → ((invg𝑇)‘(𝐹𝑥)) = ((invg𝑇)‘(𝐺𝑥)))
18 eqid 2795 . . . . . . . . . 10 (invg𝑇) = (invg𝑇)
1912, 13, 18ghminv 18111 . . . . . . . . 9 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → (𝐹‘((invg𝑆)‘𝑥)) = ((invg𝑇)‘(𝐹𝑥)))
2019ad2ant2r 743 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) → (𝐹‘((invg𝑆)‘𝑥)) = ((invg𝑇)‘(𝐹𝑥)))
2112, 13, 18ghminv 18111 . . . . . . . . 9 ((𝐺 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → (𝐺‘((invg𝑆)‘𝑥)) = ((invg𝑇)‘(𝐺𝑥)))
2221ad2ant2lr 744 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) → (𝐺‘((invg𝑆)‘𝑥)) = ((invg𝑇)‘(𝐺𝑥)))
2317, 20, 223eqtr4d 2841 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) → (𝐹‘((invg𝑆)‘𝑥)) = (𝐺‘((invg𝑆)‘𝑥)))
247, 15, 23elrabd 3621 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) → ((invg𝑆)‘𝑥) ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹𝑦) = (𝐺𝑦)})
2524expr 457 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑥 ∈ (Base‘𝑆)) → ((𝐹𝑥) = (𝐺𝑥) → ((invg𝑆)‘𝑥) ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹𝑦) = (𝐺𝑦)}))
2625ralrimiva 3149 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → ∀𝑥 ∈ (Base‘𝑆)((𝐹𝑥) = (𝐺𝑥) → ((invg𝑆)‘𝑥) ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹𝑦) = (𝐺𝑦)}))
27 fveq2 6543 . . . . . 6 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
28 fveq2 6543 . . . . . 6 (𝑦 = 𝑥 → (𝐺𝑦) = (𝐺𝑥))
2927, 28eqeq12d 2810 . . . . 5 (𝑦 = 𝑥 → ((𝐹𝑦) = (𝐺𝑦) ↔ (𝐹𝑥) = (𝐺𝑥)))
3029ralrab 3624 . . . 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)}))
3126, 30sylibr 235 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → ∀𝑥 ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹𝑦) = (𝐺𝑦)} ((invg𝑆)‘𝑥) ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹𝑦) = (𝐺𝑦)})
32 eqid 2795 . . . . . . . 8 (Base‘𝑇) = (Base‘𝑇)
3312, 32ghmf 18108 . . . . . . 7 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
3433adantr 481 . . . . . 6 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
3534ffnd 6388 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → 𝐹 Fn (Base‘𝑆))
3612, 32ghmf 18108 . . . . . . 7 (𝐺 ∈ (𝑆 GrpHom 𝑇) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
3736adantl 482 . . . . . 6 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
3837ffnd 6388 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → 𝐺 Fn (Base‘𝑆))
39 fndmin 6685 . . . . 5 ((𝐹 Fn (Base‘𝑆) ∧ 𝐺 Fn (Base‘𝑆)) → dom (𝐹𝐺) = {𝑦 ∈ (Base‘𝑆) ∣ (𝐹𝑦) = (𝐺𝑦)})
4035, 38, 39syl2anc 584 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → dom (𝐹𝐺) = {𝑦 ∈ (Base‘𝑆) ∣ (𝐹𝑦) = (𝐺𝑦)})
41 eleq2 2871 . . . . 5 (dom (𝐹𝐺) = {𝑦 ∈ (Base‘𝑆) ∣ (𝐹𝑦) = (𝐺𝑦)} → (((invg𝑆)‘𝑥) ∈ dom (𝐹𝐺) ↔ ((invg𝑆)‘𝑥) ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹𝑦) = (𝐺𝑦)}))
4241raleqbi1dv 3363 . . . 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)}))
4340, 42syl 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)}))
4431, 43mpbird 258 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → ∀𝑥 ∈ dom (𝐹𝐺)((invg𝑆)‘𝑥) ∈ dom (𝐹𝐺))
4513issubg3 18056 . . 3 (𝑆 ∈ Grp → (dom (𝐹𝐺) ∈ (SubGrp‘𝑆) ↔ (dom (𝐹𝐺) ∈ (SubMnd‘𝑆) ∧ ∀𝑥 ∈ dom (𝐹𝐺)((invg𝑆)‘𝑥) ∈ dom (𝐹𝐺))))
469, 45syl 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))))
474, 44, 46mpbir2and 709 1 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → dom (𝐹𝐺) ∈ (SubGrp‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1522  wcel 2081  wral 3105  {crab 3109  cin 3862  dom cdm 5448   Fn wfn 6225  wf 6226  cfv 6230  (class class class)co 7021  Basecbs 16317   MndHom cmhm 17777  SubMndcsubmnd 17778  Grpcgrp 17866  invgcminusg 17867  SubGrpcsubg 18032   GrpHom cghm 18101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5086  ax-sep 5099  ax-nul 5106  ax-pow 5162  ax-pr 5226  ax-un 7324  ax-cnex 10444  ax-resscn 10445  ax-1cn 10446  ax-icn 10447  ax-addcl 10448  ax-addrcl 10449  ax-mulcl 10450  ax-mulrcl 10451  ax-mulcom 10452  ax-addass 10453  ax-mulass 10454  ax-distr 10455  ax-i2m1 10456  ax-1ne0 10457  ax-1rid 10458  ax-rnegex 10459  ax-rrecex 10460  ax-cnre 10461  ax-pre-lttri 10462  ax-pre-lttrn 10463  ax-pre-ltadd 10464  ax-pre-mulgt0 10465
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3710  df-csb 3816  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-pss 3880  df-nul 4216  df-if 4386  df-pw 4459  df-sn 4477  df-pr 4479  df-tp 4481  df-op 4483  df-uni 4750  df-iun 4831  df-br 4967  df-opab 5029  df-mpt 5046  df-tr 5069  df-id 5353  df-eprel 5358  df-po 5367  df-so 5368  df-fr 5407  df-we 5409  df-xp 5454  df-rel 5455  df-cnv 5456  df-co 5457  df-dm 5458  df-rn 5459  df-res 5460  df-ima 5461  df-pred 6028  df-ord 6074  df-on 6075  df-lim 6076  df-suc 6077  df-iota 6194  df-fun 6232  df-fn 6233  df-f 6234  df-f1 6235  df-fo 6236  df-f1o 6237  df-fv 6238  df-riota 6982  df-ov 7024  df-oprab 7025  df-mpo 7026  df-om 7442  df-wrecs 7803  df-recs 7865  df-rdg 7903  df-er 8144  df-map 8263  df-en 8363  df-dom 8364  df-sdom 8365  df-pnf 10528  df-mnf 10529  df-xr 10530  df-ltxr 10531  df-le 10532  df-sub 10724  df-neg 10725  df-nn 11492  df-2 11553  df-ndx 16320  df-slot 16321  df-base 16323  df-sets 16324  df-ress 16325  df-plusg 16412  df-0g 16549  df-mgm 17686  df-sgrp 17728  df-mnd 17739  df-mhm 17779  df-submnd 17780  df-grp 17869  df-minusg 17870  df-subg 18035  df-ghm 18102
This theorem is referenced by:  rhmeql  19260  lmhmeql  19522
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