Theorem gimghm

Description: An isomorphism of groups is a homomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015) (Revised by Mario Carneiro, 6-May-2015)

	Ref	Expression
Assertion	gimghm	⊢ ( 𝐹 ∈ ( 𝑅 GrpIso 𝑆 ) → 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
2		eqid	⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 )
3	1 2	isgim	⊢ ( 𝐹 ∈ ( 𝑅 GrpIso 𝑆 ) ↔ ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ 𝐹 : ( Base ‘ 𝑅 ) –1-1-onto→ ( Base ‘ 𝑆 ) ) )
4	3	simplbi	⊢ ( 𝐹 ∈ ( 𝑅 GrpIso 𝑆 ) → 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) )