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Theorem isghmd 19255
Description: Deduction for a group homomorphism. (Contributed by Stefan O'Rear, 4-Feb-2015.)
Hypotheses
Ref Expression
isghmd.x 𝑋 = (Base‘𝑆)
isghmd.y 𝑌 = (Base‘𝑇)
isghmd.a + = (+g𝑆)
isghmd.b = (+g𝑇)
isghmd.s (𝜑𝑆 ∈ Grp)
isghmd.t (𝜑𝑇 ∈ Grp)
isghmd.f (𝜑𝐹:𝑋𝑌)
isghmd.l ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
Assertion
Ref Expression
isghmd (𝜑𝐹 ∈ (𝑆 GrpHom 𝑇))
Distinct variable groups:   𝜑,𝑥,𝑦   𝑥,𝐹,𝑦   𝑥,𝑆,𝑦   𝑥,𝑇,𝑦   𝑥, + ,𝑦   𝑥, ,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦

Proof of Theorem isghmd
StepHypRef Expression
1 isghmd.s . 2 (𝜑𝑆 ∈ Grp)
2 isghmd.t . 2 (𝜑𝑇 ∈ Grp)
3 isghmd.f . . 3 (𝜑𝐹:𝑋𝑌)
4 isghmd.l . . . 4 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
54ralrimivva 3199 . . 3 (𝜑 → ∀𝑥𝑋𝑦𝑋 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
63, 5jca 511 . 2 (𝜑 → (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))))
7 isghmd.x . . 3 𝑋 = (Base‘𝑆)
8 isghmd.y . . 3 𝑌 = (Base‘𝑇)
9 isghmd.a . . 3 + = (+g𝑆)
10 isghmd.b . . 3 = (+g𝑇)
117, 8, 9, 10isghm 19245 . 2 (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))))
121, 2, 6, 11syl21anbrc 1343 1 (𝜑𝐹 ∈ (𝑆 GrpHom 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  wral 3058  wf 6558  cfv 6562  (class class class)co 7430  Basecbs 17244  +gcplusg 17297  Grpcgrp 18963   GrpHom cghm 19242
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-map 8866  df-ghm 19243
This theorem is referenced by:  ghmmhmb  19257  resghm  19262  conjghm  19279  qusghm  19285  ghmqusnsg  19312  ghmquskerlem3  19316  invoppggim  19393  galactghm  19436  pj1ghm  19735  frgpup1  19807  mulgghm  19860  ghmfghm  19862  invghm  19865  ghmplusg  19878  ringlghm  20325  ringrghm  20326  isrnghmd  20467  isrhmd  20504  lmodvsghm  20937  pwssplit2  21076  rngqiprngghm  21326  cygznlem3  21605  psgnghm  21615  frlmup1  21835  asclghm  21920  evlslem1  22123  mat1ghm  22504  scmatghm  22554  mat2pmatghm  22751  pm2mpghm  22837  reefgim  26508  lmodvslmhm  33035  imasghm  33362  qqhghm  33950  aks6d1c6isolem2  42156  frlmsnic  42526  mplmapghm  42542  imasgim  43088  amgmlemALT  49033
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