Theorem crngcomd

Description: Multiplication is commutative in a commutative ring. (Contributed by SN, 8-Mar-2025)

	Ref	Expression
Hypotheses	crngcomd.b	$$⊢B={\mathrm{Base}}_R$$
	crngcomd.t	$$⊢\underset{\dot{}}{·}={\cdot }_R$$
	crngcomd.r	$$⊢\phi \to R\in \mathrm{CRing}$$
	crngcomd.1	$$⊢\phi \to X\in B$$
	crngcomd.2	$$⊢\phi \to Y\in B$$
Assertion	crngcomd	$$⊢\phi \to X\underset{\dot{}}{·}Y=Y\underset{\dot{}}{·}X$$

Proof

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