Theorem rnghmghm

Description: A non-unital ring homomorphism is an additive group homomorphism. (Contributed by AV, 23-Feb-2020)

	Ref	Expression
Assertion	rnghmghm	⊢ ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) → 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
2		eqid	⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 )
3		eqid	⊢ ( .r ‘ 𝑆 ) = ( .r ‘ 𝑆 )
4	1 2 3	isrnghm	⊢ ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) ↔ ( ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) ∧ ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ∀ 𝑦 ∈ ( Base ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ) ) )
5		simprl	⊢ ( ( ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) ∧ ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ∀ 𝑦 ∈ ( Base ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ) ) → 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) )
6	4 5	sylbi	⊢ ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) → 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) )