Theorem ghmgrp1 19193

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.

Step	Hyp	Ref	Expression
1		eqid 2736	. . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆)
2		eqid 2736	. . . 4 ⊢ (Base‘𝑇) = (Base‘𝑇)
3		eqid 2736	. . . 4 ⊢ (+g‘𝑆) = (+g‘𝑆)
4		eqid 2736	. . . 4 ⊢ (+g‘𝑇) = (+g‘𝑇)
5	1, 2, 3, 4	isghm 19190	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) ∧ ∀𝑦 ∈ (Base‘𝑆)∀𝑥 ∈ (Base‘𝑆)(𝐹‘(𝑦(+g‘𝑆)𝑥)) = ((𝐹‘𝑦)(+g‘𝑇)(𝐹‘𝑥)))))
6	5	simplbi 496	. 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝑆 ∈ Grp ∧ 𝑇 ∈ Grp))
7	6	simpld 494	1 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  +_gcplusg 17220  Grpcgrp 18909   GrpHom cghm 19187

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-map 8775  df-ghm 19188

This theorem is referenced by:  ghmid  19197  ghminv  19198  ghmsub  19199  ghmmhm  19201  ghmmulg  19203  ghmrn  19204  resghm2  19208  resghm2b  19209  ghmco  19211  ghmpreima  19213  ghmeql  19214  ghmnsgima  19215  ghmnsgpreima  19216  ghmeqker  19218  f1ghm0to0  19220  ghmf1  19221  kerf1ghm  19222  ghmf1o  19223  ghmpropd  19231  isgim  19237  giclcl  19248  ghmqusnsglem1  19255  ghmqusnsglem2  19256  ghmqusnsg  19257  ghmquskerlem1  19258  ghmquskerlem2  19260  ghmquskerlem3  19261  ghmqusker  19262  lactghmga  19380  invghm  19808  ghmplusg  19821  evladdval  22081  evl1addd  22306  evl1subd  22307  ghmcnp  24080  fxpsubg  33234  evlsaddval  43004  gicabl  43527