Theorem rhmghm

Description: A ring homomorphism is an additive group homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015)

	Ref	Expression
Assertion	rhmghm	⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( mulGrp ‘ 𝑅 ) = ( mulGrp ‘ 𝑅 )
2		eqid	⊢ ( mulGrp ‘ 𝑆 ) = ( mulGrp ‘ 𝑆 )
3	1 2	isrhm	⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ↔ ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ Ring ) ∧ ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ 𝐹 ∈ ( ( mulGrp ‘ 𝑅 ) MndHom ( mulGrp ‘ 𝑆 ) ) ) ) )
4	3	simprbi	⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ 𝐹 ∈ ( ( mulGrp ‘ 𝑅 ) MndHom ( mulGrp ‘ 𝑆 ) ) ) )
5	4	simpld	⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) )