Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ghmeql Structured version   Visualization version   GIF version

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
 Copyright terms: Public domain W3C validator