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Theorem ghmgrp1 19011
Description: A group homomorphism is only defined when the domain is a group. (Contributed by Stefan O'Rear, 31-Dec-2014.)
Assertion
Ref Expression
ghmgrp1 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)

Proof of Theorem ghmgrp1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . . . 4 (Base‘𝑆) = (Base‘𝑆)
2 eqid 2737 . . . 4 (Base‘𝑇) = (Base‘𝑇)
3 eqid 2737 . . . 4 (+g𝑆) = (+g𝑆)
4 eqid 2737 . . . 4 (+g𝑇) = (+g𝑇)
51, 2, 3, 4isghm 19009 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) ∧ ∀𝑦 ∈ (Base‘𝑆)∀𝑥 ∈ (Base‘𝑆)(𝐹‘(𝑦(+g𝑆)𝑥)) = ((𝐹𝑦)(+g𝑇)(𝐹𝑥)))))
65simplbi 499 . 2 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝑆 ∈ Grp ∧ 𝑇 ∈ Grp))
76simpld 496 1 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3065  wf 6493  cfv 6497  (class class class)co 7358  Basecbs 17084  +gcplusg 17134  Grpcgrp 18749   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-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-ov 7361  df-oprab 7362  df-mpo 7363  df-ghm 19007
This theorem is referenced by:  ghmid  19015  ghminv  19016  ghmsub  19017  ghmmhm  19019  ghmmulg  19021  ghmrn  19022  resghm2  19026  resghm2b  19027  ghmco  19029  ghmpreima  19031  ghmeql  19032  ghmnsgima  19033  ghmnsgpreima  19034  ghmeqker  19036  ghmf1  19038  ghmf1o  19039  ghmpropd  19047  isgim  19053  giclcl  19063  lactghmga  19188  invghm  19613  ghmplusg  19625  f1ghm0to0  20175  kerf1ghm  20178  evl1addd  21710  evl1subd  21711  ghmcnp  23469  ghmquskerlem1  32198  ghmquskerlem2  32200  ghmqusker  32201  evlsaddval  40753  evladdval  40756  gicabl  41429
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