Theorem ringcld

Description: Closure of the multiplication operation of a ring. (Contributed by SN, 29-Jul-2024)

	Ref	Expression
Hypotheses	ringcld.b	$$⊢B={\mathrm{Base}}_R$$
	ringcld.t	$$⊢\underset{\dot{}}{·}={\cdot }_R$$
	ringcld.r	$$⊢\phi \to R\in \mathrm{Ring}$$
	ringcld.x	$$⊢\phi \to X\in B$$
	ringcld.y	$$⊢\phi \to Y\in B$$
Assertion	ringcld	$$⊢\phi \to X\underset{\dot{}}{·}Y\in B$$

Proof

Step	Hyp	Ref	Expression
1		ringcld.b	$$⊢B={\mathrm{Base}}_R$$
2		ringcld.t	$$⊢\underset{\dot{}}{·}={\cdot }_R$$
3		ringcld.r	$$⊢\phi \to R\in \mathrm{Ring}$$
4		ringcld.x	$$⊢\phi \to X\in B$$
5		ringcld.y	$$⊢\phi \to Y\in B$$
6	1 2	ringcl	$$⊢\left(R\in \mathrm{Ring}\wedge X\in B\wedge Y\in B\right)\to X\underset{\dot{}}{·}Y\in B$$
7	3 4 5 6	syl3anc	$$⊢\phi \to X\underset{\dot{}}{·}Y\in B$$