Theorem rhmcl

Description: Closure law for a ring homomorphism. (Contributed by Jeff Madsen, 3-Jan-2011) (Revised by AV, 10-Jan-2025)

	Ref	Expression
Hypotheses	rhmf.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	rhmf.c	⊢ 𝐶 = ( Base ‘ 𝑆 )
Assertion	rhmcl	⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ Ring ∧ 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ) ∧ 𝐴 ∈ 𝐵 ) → ( 𝐹 ‘ 𝐴 ) ∈ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		rhmf.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		rhmf.c	⊢ 𝐶 = ( Base ‘ 𝑆 )
3	1 2	rhmf	⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → 𝐹 : 𝐵 ⟶ 𝐶 )
4	3	3ad2ant3	⊢ ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ Ring ∧ 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ) → 𝐹 : 𝐵 ⟶ 𝐶 )
5	4	ffvelcdmda	⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ Ring ∧ 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ) ∧ 𝐴 ∈ 𝐵 ) → ( 𝐹 ‘ 𝐴 ) ∈ 𝐶 )