Theorem ghmgrp2 19160

Description: A group homomorphism is only defined when the codomain is a group. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Assertion

Ref	Expression
ghmgrp2	⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)

Proof of Theorem ghmgrp2

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2737	. . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆)
2		eqid 2737	. . . 4 ⊢ (Base‘𝑇) = (Base‘𝑇)
3		eqid 2737	. . . 4 ⊢ (+g‘𝑆) = (+g‘𝑆)
4		eqid 2737	. . . 4 ⊢ (+g‘𝑇) = (+g‘𝑇)
5	1, 2, 3, 4	isghm 19156	. . 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	simprd 495	1 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  +_gcplusg 17189  Grpcgrp 18875   GrpHom cghm 19153

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 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-map 8777  df-ghm 19154

This theorem is referenced by:  ghmid  19163  ghminv  19164  ghmmhm  19167  ghmmulg  19169  ghmrn  19170  resghm  19173  ghmco  19177  ghmker  19183  ghmeqker  19184  ghmf1  19187  ghmf1o  19189  ghmpropd  19197  isgim  19203  gicrcl  19215  ghmqusnsglem1  19221  ghmquskerlem1  19224  lactghmga  19346  ghmplusg  19787  ghmcyg  19837  ghmcnp  24071  abliso  33128  gicabl  43450