rngisomfv1

Description: If there is a non-unital ring isomorphism between a unital ring and a non-unital ring, then the function value of the ring unity of the unital ring is an element of the base set of the non-unital ring. (Contributed by AV, 27-Feb-2025)

	Ref	Expression
Hypotheses	rngisom1.1	$⊢ \underset{˙}{1} = 1_{R}$
	rngisom1.b	$⊢ B = {Base}_{S}$
Assertion	rngisomfv1	$⊢ (R \in Ring \land F \in (R RngIsom S)) \to F (\underset{˙}{1}) \in B$

Proof

Step	Hyp	Ref	Expression
1		rngisom1.1	$⊢ \underset{˙}{1} = 1_{R}$
2		rngisom1.b	$⊢ B = {Base}_{S}$
3		eqid	$⊢ {Base}_{R} = {Base}_{R}$
4	3 2	rngimf1o	$⊢ F \in (R RngIsom S) \to F : {Base}_{R} \underset{1-1 onto}{⟶} B$
5		f1of	$⊢ F : {Base}_{R} \underset{1-1 onto}{⟶} B \to F : {Base}_{R} ⟶ B$
6	4 5	syl	$⊢ F \in (R RngIsom S) \to F : {Base}_{R} ⟶ B$
7	6	adantl	$⊢ (R \in Ring \land F \in (R RngIsom S)) \to F : {Base}_{R} ⟶ B$
8	3 1	ringidcl	$⊢ R \in Ring \to \underset{˙}{1} \in {Base}_{R}$
9	8	adantr	$⊢ (R \in Ring \land F \in (R RngIsom S)) \to \underset{˙}{1} \in {Base}_{R}$
10	7 9	ffvelcdmd	$⊢ (R \in Ring \land F \in (R RngIsom S)) \to F (\underset{˙}{1}) \in B$

Metamath Proof Explorer

Theorem rngisomfv1

Proof