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Theorem ghminv 18629
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 18624 . . . . . 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 18417 . . . . . 6 ((𝑆 ∈ Grp ∧ 𝑋𝐵) → (𝑋(+g𝑆)(𝑀𝑋)) = (0g𝑆))
71, 6sylan 583 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → (𝑋(+g𝑆)(𝑀𝑋)) = (0g𝑆))
87fveq2d 6721 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → (𝐹‘(𝑋(+g𝑆)(𝑀𝑋))) = (𝐹‘(0g𝑆)))
92, 5grpinvcl 18415 . . . . . 6 ((𝑆 ∈ Grp ∧ 𝑋𝐵) → (𝑀𝑋) ∈ 𝐵)
101, 9sylan 583 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → (𝑀𝑋) ∈ 𝐵)
11 eqid 2737 . . . . . 6 (+g𝑇) = (+g𝑇)
122, 3, 11ghmlin 18627 . . . . 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 18628 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(0g𝑆)) = (0g𝑇))
1615adantr 484 . . . 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 18625 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
1918adantr 484 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → 𝑇 ∈ Grp)
20 eqid 2737 . . . . . 6 (Base‘𝑇) = (Base‘𝑇)
212, 20ghmf 18626 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝐵⟶(Base‘𝑇))
2221ffvelrnda 6904 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → (𝐹𝑋) ∈ (Base‘𝑇))
2321adantr 484 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → 𝐹:𝐵⟶(Base‘𝑇))
2423, 10ffvelrnd 6905 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → (𝐹‘(𝑀𝑋)) ∈ (Base‘𝑇))
25 ghminv.z . . . . 5 𝑁 = (invg𝑇)
2620, 11, 14, 25grpinvid1 18418 . . . 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 260 . 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 209  wa 399   = wceq 1543  wcel 2110  wf 6376  cfv 6380  (class class class)co 7213  Basecbs 16760  +gcplusg 16802  0gc0g 16944  Grpcgrp 18365  invgcminusg 18366   GrpHom cghm 18619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-0g 16946  df-mgm 18114  df-sgrp 18163  df-mnd 18174  df-grp 18368  df-minusg 18369  df-ghm 18620
This theorem is referenced by:  ghmsub  18630  ghmmulg  18634  ghmrn  18635  ghmpreima  18644  ghmeql  18645  frgpup3lem  19167  psgninv  20544  zrhpsgnodpm  20554  asclinvg  20849  mplind  21028  cpmatinvcl  21614  sum2dchr  26155
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