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Theorem ghmgrp1 19132
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 2730 . . . 4 (Base‘𝑆) = (Base‘𝑆)
2 eqid 2730 . . . 4 (Base‘𝑇) = (Base‘𝑇)
3 eqid 2730 . . . 4 (+g𝑆) = (+g𝑆)
4 eqid 2730 . . . 4 (+g𝑇) = (+g𝑇)
51, 2, 3, 4isghm 19130 . . 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 496 . 2 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝑆 ∈ Grp ∧ 𝑇 ∈ Grp))
76simpld 493 1 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1539  wcel 2104  wral 3059  wf 6538  cfv 6542  (class class class)co 7411  Basecbs 17148  +gcplusg 17201  Grpcgrp 18855   GrpHom cghm 19127
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-ghm 19128
This theorem is referenced by:  ghmid  19136  ghminv  19137  ghmsub  19138  ghmmhm  19140  ghmmulg  19142  ghmrn  19143  resghm2  19147  resghm2b  19148  ghmco  19150  ghmpreima  19152  ghmeql  19153  ghmnsgima  19154  ghmnsgpreima  19155  ghmeqker  19157  f1ghm0to0  19159  ghmf1  19160  kerf1ghm  19161  ghmf1o  19162  ghmpropd  19170  isgim  19176  giclcl  19187  lactghmga  19314  invghm  19742  ghmplusg  19755  evl1addd  22080  evl1subd  22081  ghmcnp  23839  ghmquskerlem1  32802  ghmquskerlem2  32804  ghmquskerlem3  32805  ghmqusker  32806  evlsaddval  41442  evladdval  41449  gicabl  42143
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