Theorem ringvcl

Description: Tuple-wise multiplication closure in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015)

	Ref	Expression
Hypotheses	ringvcl.b	$$⊢B={\mathrm{Base}}_R$$
	ringvcl.t	$$⊢\underset{\dot{}}{·}={\cdot }_R$$
Assertion	ringvcl	$$⊢\left(R\in \mathrm{Ring}\wedge X\in \left(B^I\right)\wedge Y\in \left(B^I\right)\right)\to X{\underset{\dot{}}{·}}_{f}Y\in \left(B^I\right)$$

Proof

Step	Hyp	Ref	Expression
1		ringvcl.b	$$⊢B={\mathrm{Base}}_R$$
2		ringvcl.t	$$⊢\underset{\dot{}}{·}={\cdot }_R$$
3		eqid	$$⊢{\mathrm{mulGrp}}_R={\mathrm{mulGrp}}_R$$
4	3	ringmgp	$$⊢R\in \mathrm{Ring}\to {\mathrm{mulGrp}}_R\in \mathrm{Mnd}$$
5	3 1	mgpbas	$$⊢B={\mathrm{Base}}_{{\mathrm{mulGrp}}_R}$$
6	3 2	mgpplusg	$$⊢\underset{\dot{}}{·}={+}_{{\mathrm{mulGrp}}_R}$$
7	5 6	mndvcl	$$⊢\left({\mathrm{mulGrp}}_R\in \mathrm{Mnd}\wedge X\in \left(B^I\right)\wedge Y\in \left(B^I\right)\right)\to X{\underset{\dot{}}{·}}_{f}Y\in \left(B^I\right)$$
8	4 7	syl3an1	$$⊢\left(R\in \mathrm{Ring}\wedge X\in \left(B^I\right)\wedge Y\in \left(B^I\right)\right)\to X{\underset{\dot{}}{·}}_{f}Y\in \left(B^I\right)$$