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

Theorem ghminv 19016
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 19011 . . . . . 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 18802 . . . . . 6 ((𝑆 ∈ Grp ∧ 𝑋𝐵) → (𝑋(+g𝑆)(𝑀𝑋)) = (0g𝑆))
71, 6sylan 581 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → (𝑋(+g𝑆)(𝑀𝑋)) = (0g𝑆))
87fveq2d 6847 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → (𝐹‘(𝑋(+g𝑆)(𝑀𝑋))) = (𝐹‘(0g𝑆)))
92, 5grpinvcl 18799 . . . . . 6 ((𝑆 ∈ Grp ∧ 𝑋𝐵) → (𝑀𝑋) ∈ 𝐵)
101, 9sylan 581 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → (𝑀𝑋) ∈ 𝐵)
11 eqid 2737 . . . . . 6 (+g𝑇) = (+g𝑇)
122, 3, 11ghmlin 19014 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝐵) → (𝐹‘(𝑋(+g𝑆)(𝑀𝑋))) = ((𝐹𝑋)(+g𝑇)(𝐹‘(𝑀𝑋))))
1310, 12mpd3an3 1463 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → (𝐹‘(𝑋(+g𝑆)(𝑀𝑋))) = ((𝐹𝑋)(+g𝑇)(𝐹‘(𝑀𝑋))))
14 eqid 2737 . . . . . 6 (0g𝑇) = (0g𝑇)
154, 14ghmid 19015 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(0g𝑆)) = (0g𝑇))
1615adantr 482 . . . 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 19012 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
1918adantr 482 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → 𝑇 ∈ Grp)
20 eqid 2737 . . . . . 6 (Base‘𝑇) = (Base‘𝑇)
212, 20ghmf 19013 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝐵⟶(Base‘𝑇))
2221ffvelcdmda 7036 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → (𝐹𝑋) ∈ (Base‘𝑇))
2321adantr 482 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → 𝐹:𝐵⟶(Base‘𝑇))
2423, 10ffvelcdmd 7037 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → (𝐹‘(𝑀𝑋)) ∈ (Base‘𝑇))
25 ghminv.z . . . . 5 𝑁 = (invg𝑇)
2620, 11, 14, 25grpinvid1 18803 . . . 4 ((𝑇 ∈ Grp ∧ (𝐹𝑋) ∈ (Base‘𝑇) ∧ (𝐹‘(𝑀𝑋)) ∈ (Base‘𝑇)) → ((𝑁‘(𝐹𝑋)) = (𝐹‘(𝑀𝑋)) ↔ ((𝐹𝑋)(+g𝑇)(𝐹‘(𝑀𝑋))) = (0g𝑇)))
2719, 22, 24, 26syl3anc 1372 . . 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 205  wa 397   = wceq 1542  wcel 2107  wf 6493  cfv 6497  (class class class)co 7358  Basecbs 17084  +gcplusg 17134  0gc0g 17322  Grpcgrp 18749  invgcminusg 18750   GrpHom cghm 19006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-0g 17324  df-mgm 18498  df-sgrp 18547  df-mnd 18558  df-grp 18752  df-minusg 18753  df-ghm 19007
This theorem is referenced by:  ghmsub  19017  ghmmulg  19021  ghmrn  19022  ghmpreima  19031  ghmeql  19032  frgpup3lem  19560  psgninv  20989  zrhpsgnodpm  20999  asclinvg  21295  mplind  21481  cpmatinvcl  22069  sum2dchr  26625  ghmquskerlem1  32198  fldhmf1  40550
  Copyright terms: Public domain W3C validator