Theorem ghmgrp1

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

	Ref	Expression
Assertion	ghmgrp1	⊢ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) → 𝑆 ∈ Grp )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 )
2		eqid	⊢ ( Base ‘ 𝑇 ) = ( Base ‘ 𝑇 )
3		eqid	⊢ ( +g ‘ 𝑆 ) = ( +g ‘ 𝑆 )
4		eqid	⊢ ( +g ‘ 𝑇 ) = ( +g ‘ 𝑇 )
5	1 2 3 4	isghm	⊢ ( 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	⊢ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) → ( 𝑆 ∈ Grp ∧ 𝑇 ∈ Grp ) )
7	6	simpld	⊢ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) → 𝑆 ∈ Grp )