Theorem isghmd

Description: Deduction for a group homomorphism. (Contributed by Stefan O'Rear, 4-Feb-2015)

	Ref	Expression
Hypotheses	isghmd.x	⊢ 𝑋 = ( Base ‘ 𝑆 )
	isghmd.y	⊢ 𝑌 = ( Base ‘ 𝑇 )
	isghmd.a	⊢ + = ( +g ‘ 𝑆 )
	isghmd.b	⊢ ⨣ = ( +g ‘ 𝑇 )
	isghmd.s	⊢ ( 𝜑 → 𝑆 ∈ Grp )
	isghmd.t	⊢ ( 𝜑 → 𝑇 ∈ Grp )
	isghmd.f	⊢ ( 𝜑 → 𝐹 : 𝑋 ⟶ 𝑌 )
	isghmd.l	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ) ) → ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ⨣ ( 𝐹 ‘ 𝑦 ) ) )
Assertion	isghmd	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) )

Proof

Step	Hyp	Ref	Expression
1		isghmd.x	⊢ 𝑋 = ( Base ‘ 𝑆 )
2		isghmd.y	⊢ 𝑌 = ( Base ‘ 𝑇 )
3		isghmd.a	⊢ + = ( +g ‘ 𝑆 )
4		isghmd.b	⊢ ⨣ = ( +g ‘ 𝑇 )
5		isghmd.s	⊢ ( 𝜑 → 𝑆 ∈ Grp )
6		isghmd.t	⊢ ( 𝜑 → 𝑇 ∈ Grp )
7		isghmd.f	⊢ ( 𝜑 → 𝐹 : 𝑋 ⟶ 𝑌 )
8		isghmd.l	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ) ) → ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ⨣ ( 𝐹 ‘ 𝑦 ) ) )
9	8	ralrimivva	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ⨣ ( 𝐹 ‘ 𝑦 ) ) )
10	7 9	jca	⊢ ( 𝜑 → ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ⨣ ( 𝐹 ‘ 𝑦 ) ) ) )
11	1 2 3 4	isghm	⊢ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ↔ ( ( 𝑆 ∈ Grp ∧ 𝑇 ∈ Grp ) ∧ ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ⨣ ( 𝐹 ‘ 𝑦 ) ) ) ) )
12	5 6 10 11	syl21anbrc	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) )