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Theorem ghminv 19254
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 19249 . . . . . 6 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
2 ghminv.b . . . . . . 7 𝐵 = (Base‘𝑆)
3 eqid 2735 . . . . . . 7 (+g𝑆) = (+g𝑆)
4 eqid 2735 . . . . . . 7 (0g𝑆) = (0g𝑆)
5 ghminv.y . . . . . . 7 𝑀 = (invg𝑆)
62, 3, 4, 5grprinv 19021 . . . . . 6 ((𝑆 ∈ Grp ∧ 𝑋𝐵) → (𝑋(+g𝑆)(𝑀𝑋)) = (0g𝑆))
71, 6sylan 580 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → (𝑋(+g𝑆)(𝑀𝑋)) = (0g𝑆))
87fveq2d 6911 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → (𝐹‘(𝑋(+g𝑆)(𝑀𝑋))) = (𝐹‘(0g𝑆)))
92, 5grpinvcl 19018 . . . . . 6 ((𝑆 ∈ Grp ∧ 𝑋𝐵) → (𝑀𝑋) ∈ 𝐵)
101, 9sylan 580 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → (𝑀𝑋) ∈ 𝐵)
11 eqid 2735 . . . . . 6 (+g𝑇) = (+g𝑇)
122, 3, 11ghmlin 19252 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝐵) → (𝐹‘(𝑋(+g𝑆)(𝑀𝑋))) = ((𝐹𝑋)(+g𝑇)(𝐹‘(𝑀𝑋))))
1310, 12mpd3an3 1461 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → (𝐹‘(𝑋(+g𝑆)(𝑀𝑋))) = ((𝐹𝑋)(+g𝑇)(𝐹‘(𝑀𝑋))))
14 eqid 2735 . . . . . 6 (0g𝑇) = (0g𝑇)
154, 14ghmid 19253 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(0g𝑆)) = (0g𝑇))
1615adantr 480 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → (𝐹‘(0g𝑆)) = (0g𝑇))
178, 13, 163eqtr3d 2783 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → ((𝐹𝑋)(+g𝑇)(𝐹‘(𝑀𝑋))) = (0g𝑇))
18 ghmgrp2 19250 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
1918adantr 480 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → 𝑇 ∈ Grp)
20 eqid 2735 . . . . . 6 (Base‘𝑇) = (Base‘𝑇)
212, 20ghmf 19251 . . . . 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 19022 . . . 4 ((𝑇 ∈ Grp ∧ (𝐹𝑋) ∈ (Base‘𝑇) ∧ (𝐹‘(𝑀𝑋)) ∈ (Base‘𝑇)) → ((𝑁‘(𝐹𝑋)) = (𝐹‘(𝑀𝑋)) ↔ ((𝐹𝑋)(+g𝑇)(𝐹‘(𝑀𝑋))) = (0g𝑇)))
2719, 22, 24, 26syl3anc 1370 . . 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 2741 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 1537  wcel 2106  wf 6559  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  0gc0g 17486  Grpcgrp 18964  invgcminusg 18965   GrpHom cghm 19243
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-ghm 19244
This theorem is referenced by:  ghmsub  19255  ghmmulg  19259  ghmrn  19260  ghmpreima  19269  ghmeql  19270  ghmqusnsglem1  19311  ghmquskerlem1  19314  frgpup3lem  19810  psgninv  21618  zrhpsgnodpm  21628  asclinvg  21927  mplind  22112  cpmatinvcl  22739  sum2dchr  27333  zrhneg  33941  zrhcntr  33942  fldhmf1  42072
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