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

Theorem ghminv 17490
 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 17485 . . . . . 6 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
2 ghminv.b . . . . . . 7 𝐵 = (Base‘𝑆)
3 eqid 2610 . . . . . . 7 (+g𝑆) = (+g𝑆)
4 eqid 2610 . . . . . . 7 (0g𝑆) = (0g𝑆)
5 ghminv.y . . . . . . 7 𝑀 = (invg𝑆)
62, 3, 4, 5grprinv 17292 . . . . . 6 ((𝑆 ∈ Grp ∧ 𝑋𝐵) → (𝑋(+g𝑆)(𝑀𝑋)) = (0g𝑆))
71, 6sylan 487 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → (𝑋(+g𝑆)(𝑀𝑋)) = (0g𝑆))
87fveq2d 6107 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → (𝐹‘(𝑋(+g𝑆)(𝑀𝑋))) = (𝐹‘(0g𝑆)))
92, 5grpinvcl 17290 . . . . . 6 ((𝑆 ∈ Grp ∧ 𝑋𝐵) → (𝑀𝑋) ∈ 𝐵)
101, 9sylan 487 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → (𝑀𝑋) ∈ 𝐵)
11 eqid 2610 . . . . . 6 (+g𝑇) = (+g𝑇)
122, 3, 11ghmlin 17488 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝐵) → (𝐹‘(𝑋(+g𝑆)(𝑀𝑋))) = ((𝐹𝑋)(+g𝑇)(𝐹‘(𝑀𝑋))))
1310, 12mpd3an3 1417 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → (𝐹‘(𝑋(+g𝑆)(𝑀𝑋))) = ((𝐹𝑋)(+g𝑇)(𝐹‘(𝑀𝑋))))
14 eqid 2610 . . . . . 6 (0g𝑇) = (0g𝑇)
154, 14ghmid 17489 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(0g𝑆)) = (0g𝑇))
1615adantr 480 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → (𝐹‘(0g𝑆)) = (0g𝑇))
178, 13, 163eqtr3d 2652 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → ((𝐹𝑋)(+g𝑇)(𝐹‘(𝑀𝑋))) = (0g𝑇))
18 ghmgrp2 17486 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
1918adantr 480 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → 𝑇 ∈ Grp)
20 eqid 2610 . . . . . 6 (Base‘𝑇) = (Base‘𝑇)
212, 20ghmf 17487 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝐵⟶(Base‘𝑇))
2221ffvelrnda 6267 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → (𝐹𝑋) ∈ (Base‘𝑇))
2321adantr 480 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → 𝐹:𝐵⟶(Base‘𝑇))
2423, 10ffvelrnd 6268 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → (𝐹‘(𝑀𝑋)) ∈ (Base‘𝑇))
25 ghminv.z . . . . 5 𝑁 = (invg𝑇)
2620, 11, 14, 25grpinvid1 17293 . . . 4 ((𝑇 ∈ Grp ∧ (𝐹𝑋) ∈ (Base‘𝑇) ∧ (𝐹‘(𝑀𝑋)) ∈ (Base‘𝑇)) → ((𝑁‘(𝐹𝑋)) = (𝐹‘(𝑀𝑋)) ↔ ((𝐹𝑋)(+g𝑇)(𝐹‘(𝑀𝑋))) = (0g𝑇)))
2719, 22, 24, 26syl3anc 1318 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → ((𝑁‘(𝐹𝑋)) = (𝐹‘(𝑀𝑋)) ↔ ((𝐹𝑋)(+g𝑇)(𝐹‘(𝑀𝑋))) = (0g𝑇)))
2817, 27mpbird 246 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → (𝑁‘(𝐹𝑋)) = (𝐹‘(𝑀𝑋)))
2928eqcomd 2616 1 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → (𝐹‘(𝑀𝑋)) = (𝑁‘(𝐹𝑋)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  0gc0g 15923  Grpcgrp 17245  invgcminusg 17246   GrpHom cghm 17480 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-ghm 17481 This theorem is referenced by:  ghmsub  17491  ghmmulg  17495  ghmrn  17496  ghmpreima  17505  ghmeql  17506  frgpup3lem  18013  asclinvg  19162  mplind  19323  psgninv  19747  zrhpsgnodpm  19757  cpmatinvcl  20341  sum2dchr  24799
 Copyright terms: Public domain W3C validator