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Theorem isghmd 19243
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 3202 . . 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 19233 . 2 (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))))
121, 2, 6, 11syl21anbrc 1345 1 (𝜑𝐹 ∈ (𝑆 GrpHom 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wral 3061  wf 6557  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  Grpcgrp 18951   GrpHom cghm 19230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868  df-ghm 19231
This theorem is referenced by:  ghmmhmb  19245  resghm  19250  conjghm  19267  qusghm  19273  ghmqusnsg  19300  ghmquskerlem3  19304  invoppggim  19379  galactghm  19422  pj1ghm  19721  frgpup1  19793  mulgghm  19846  ghmfghm  19848  invghm  19851  ghmplusg  19864  ringlghm  20309  ringrghm  20310  isrnghmd  20451  isrhmd  20488  lmodvsghm  20921  pwssplit2  21059  rngqiprngghm  21309  cygznlem3  21588  psgnghm  21598  frlmup1  21818  asclghm  21903  evlslem1  22106  mat1ghm  22489  scmatghm  22539  mat2pmatghm  22736  pm2mpghm  22822  reefgim  26494  lmodvslmhm  33053  imasghm  33383  qqhghm  33989  aks6d1c6isolem2  42176  frlmsnic  42550  mplmapghm  42566  imasgim  43112  amgmlemALT  49322
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