Metamath Proof Explorer

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	$⊢ ((R \in Ring \land S \in Ring \land F \in (R RingHom S)) \land A \in B) \to F (A) \in C$

Step	Hyp	Ref	Expression
1		rhmf.b	$⊢ B = {Base}_{R}$
2		rhmf.c	$⊢ C = {Base}_{S}$
3	1 2	rhmf	$⊢ F \in (R RingHom S) \to F : B ⟶ C$
4	3	3ad2ant3	$⊢ (R \in Ring \land S \in Ring \land F \in (R RingHom S)) \to F : B ⟶ C$
5	4	ffvelcdmda	$⊢ ((R \in Ring \land S \in Ring \land F \in (R RingHom S)) \land A \in B) \to F (A) \in C$