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Theorem ghmgrp1 18440
 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 2758 . . . 4 (Base‘𝑆) = (Base‘𝑆)
2 eqid 2758 . . . 4 (Base‘𝑇) = (Base‘𝑇)
3 eqid 2758 . . . 4 (+g𝑆) = (+g𝑆)
4 eqid 2758 . . . 4 (+g𝑇) = (+g𝑇)
51, 2, 3, 4isghm 18438 . . 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 501 . 2 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝑆 ∈ Grp ∧ 𝑇 ∈ Grp))
76simpld 498 1 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  ⟶wf 6336  ‘cfv 6340  (class class class)co 7156  Basecbs 16554  +gcplusg 16636  Grpcgrp 18182   GrpHom cghm 18435 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7159  df-oprab 7160  df-mpo 7161  df-ghm 18436 This theorem is referenced by:  ghmid  18444  ghminv  18445  ghmsub  18446  ghmmhm  18448  ghmmulg  18450  ghmrn  18451  resghm2  18455  resghm2b  18456  ghmco  18458  ghmpreima  18460  ghmeql  18461  ghmnsgima  18462  ghmnsgpreima  18463  ghmeqker  18465  ghmf1  18467  ghmf1o  18468  ghmpropd  18476  isgim  18482  giclcl  18492  lactghmga  18613  invghm  19035  ghmplusg  19047  f1ghm0to0  19576  kerf1ghm  19579  evl1addd  21073  evl1subd  21074  ghmcnp  22828  evlsaddval  39817  gicabl  40451
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