Theorem isghm3 19257

Description: Property of a group homomorphism, similar to ismhm 18819. (Contributed by Mario Carneiro, 7-Mar-2015.)

Hypotheses

Ref	Expression
isghm.w	⊢ 𝑋 = (Base‘𝑆)
isghm.x	⊢ 𝑌 = (Base‘𝑇)
isghm.a	⊢ + = (+g‘𝑆)
isghm.b	⊢ ⨣ = (+g‘𝑇)

Assertion

Ref	Expression
isghm3	⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹‘𝑢) ⨣ (𝐹‘𝑣)))))

Distinct variable groups:   𝑣,𝑢,𝑆   𝑢,𝑇,𝑣   𝑢,𝑋,𝑣   𝑢, + ,𝑣   𝑢,𝑌,𝑣   𝑢, ⨣ ,𝑣   𝑢,𝐹,𝑣

Proof of Theorem isghm3

Step	Hyp	Ref	Expression
1		isghm.w	. . 3 ⊢ 𝑋 = (Base‘𝑆)
2		isghm.x	. . 3 ⊢ 𝑌 = (Base‘𝑇)
3		isghm.a	. . 3 ⊢ + = (+g‘𝑆)
4		isghm.b	. . 3 ⊢ ⨣ = (+g‘𝑇)
5	1, 2, 3, 4	isghm 19256	. 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹‘𝑢) ⨣ (𝐹‘𝑣)))))
6	5	baib 543	1 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹‘𝑢) ⨣ (𝐹‘𝑣)))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ∀wral 3076  ⟶wf 6517  ‘cfv 6521  (class class class)co 7396  Basecbs 17245  +_gcplusg 17286  Grpcgrp 18975   GrpHom cghm 19253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1st 7970  df-2nd 7971  df-map 8810  df-ghm 19254

This theorem is referenced by:  dfrhm2  20523