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Theorem ghminv 19241
Description: A homomorphism of groups preserves inverses. (Contributed by Stefan O'Rear, 31-Dec-2014.)
Hypotheses
Ref Expression
ghminv.b 𝐵 = (Base‘𝑆)
ghminv.y 𝑀 = (invg𝑆)
ghminv.z 𝑁 = (invg𝑇)
Assertion
Ref Expression
ghminv ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → (𝐹‘(𝑀𝑋)) = (𝑁‘(𝐹𝑋)))

Proof of Theorem ghminv
StepHypRef Expression
1 ghmgrp1 19236 . . . . . 6 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
2 ghminv.b . . . . . . 7 𝐵 = (Base‘𝑆)
3 eqid 2737 . . . . . . 7 (+g𝑆) = (+g𝑆)
4 eqid 2737 . . . . . . 7 (0g𝑆) = (0g𝑆)
5 ghminv.y . . . . . . 7 𝑀 = (invg𝑆)
62, 3, 4, 5grprinv 19008 . . . . . 6 ((𝑆 ∈ Grp ∧ 𝑋𝐵) → (𝑋(+g𝑆)(𝑀𝑋)) = (0g𝑆))
71, 6sylan 580 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → (𝑋(+g𝑆)(𝑀𝑋)) = (0g𝑆))
87fveq2d 6910 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → (𝐹‘(𝑋(+g𝑆)(𝑀𝑋))) = (𝐹‘(0g𝑆)))
92, 5grpinvcl 19005 . . . . . 6 ((𝑆 ∈ Grp ∧ 𝑋𝐵) → (𝑀𝑋) ∈ 𝐵)
101, 9sylan 580 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → (𝑀𝑋) ∈ 𝐵)
11 eqid 2737 . . . . . 6 (+g𝑇) = (+g𝑇)
122, 3, 11ghmlin 19239 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝐵) → (𝐹‘(𝑋(+g𝑆)(𝑀𝑋))) = ((𝐹𝑋)(+g𝑇)(𝐹‘(𝑀𝑋))))
1310, 12mpd3an3 1464 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → (𝐹‘(𝑋(+g𝑆)(𝑀𝑋))) = ((𝐹𝑋)(+g𝑇)(𝐹‘(𝑀𝑋))))
14 eqid 2737 . . . . . 6 (0g𝑇) = (0g𝑇)
154, 14ghmid 19240 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(0g𝑆)) = (0g𝑇))
1615adantr 480 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → (𝐹‘(0g𝑆)) = (0g𝑇))
178, 13, 163eqtr3d 2785 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → ((𝐹𝑋)(+g𝑇)(𝐹‘(𝑀𝑋))) = (0g𝑇))
18 ghmgrp2 19237 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
1918adantr 480 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → 𝑇 ∈ Grp)
20 eqid 2737 . . . . . 6 (Base‘𝑇) = (Base‘𝑇)
212, 20ghmf 19238 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝐵⟶(Base‘𝑇))
2221ffvelcdmda 7104 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → (𝐹𝑋) ∈ (Base‘𝑇))
2321adantr 480 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → 𝐹:𝐵⟶(Base‘𝑇))
2423, 10ffvelcdmd 7105 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → (𝐹‘(𝑀𝑋)) ∈ (Base‘𝑇))
25 ghminv.z . . . . 5 𝑁 = (invg𝑇)
2620, 11, 14, 25grpinvid1 19009 . . . 4 ((𝑇 ∈ Grp ∧ (𝐹𝑋) ∈ (Base‘𝑇) ∧ (𝐹‘(𝑀𝑋)) ∈ (Base‘𝑇)) → ((𝑁‘(𝐹𝑋)) = (𝐹‘(𝑀𝑋)) ↔ ((𝐹𝑋)(+g𝑇)(𝐹‘(𝑀𝑋))) = (0g𝑇)))
2719, 22, 24, 26syl3anc 1373 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → ((𝑁‘(𝐹𝑋)) = (𝐹‘(𝑀𝑋)) ↔ ((𝐹𝑋)(+g𝑇)(𝐹‘(𝑀𝑋))) = (0g𝑇)))
2817, 27mpbird 257 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → (𝑁‘(𝐹𝑋)) = (𝐹‘(𝑀𝑋)))
2928eqcomd 2743 1 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → (𝐹‘(𝑀𝑋)) = (𝑁‘(𝐹𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wf 6557  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  0gc0g 17484  Grpcgrp 18951  invgcminusg 18952   GrpHom cghm 19230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-ghm 19231
This theorem is referenced by:  ghmsub  19242  ghmmulg  19246  ghmrn  19247  ghmpreima  19256  ghmeql  19257  ghmqusnsglem1  19298  ghmquskerlem1  19301  frgpup3lem  19795  psgninv  21600  zrhpsgnodpm  21610  asclinvg  21909  mplind  22094  cpmatinvcl  22723  sum2dchr  27318  zrhneg  33979  zrhcntr  33980  fldhmf1  42091
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