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Theorem ghmgrp1 18305
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 2826 . . . 4 (Base‘𝑆) = (Base‘𝑆)
2 eqid 2826 . . . 4 (Base‘𝑇) = (Base‘𝑇)
3 eqid 2826 . . . 4 (+g𝑆) = (+g𝑆)
4 eqid 2826 . . . 4 (+g𝑇) = (+g𝑇)
51, 2, 3, 4isghm 18303 . . 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 498 . 2 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝑆 ∈ Grp ∧ 𝑇 ∈ Grp))
76simpld 495 1 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  wral 3143  wf 6350  cfv 6354  (class class class)co 7150  Basecbs 16478  +gcplusg 16560  Grpcgrp 18048   GrpHom cghm 18300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7153  df-oprab 7154  df-mpo 7155  df-ghm 18301
This theorem is referenced by:  ghmid  18309  ghminv  18310  ghmsub  18311  ghmmhm  18313  ghmmulg  18315  ghmrn  18316  resghm2  18320  resghm2b  18321  ghmco  18323  ghmpreima  18325  ghmeql  18326  ghmnsgima  18327  ghmnsgpreima  18328  ghmeqker  18330  ghmf1  18332  ghmf1o  18333  ghmpropd  18341  isgim  18347  giclcl  18357  lactghmga  18469  invghm  18890  ghmplusg  18902  f1ghm0to0  19428  kerf1ghm  19433  evl1addd  20439  evl1subd  20440  ghmcnp  22657  gicabl  39583
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