Theorem mplringd

Description: The polynomial ring is a ring. (Contributed by SN, 7-Feb-2025)

	Ref	Expression
Hypotheses	mplringd.p	$$⊢P=I\mathrm{mPoly}R$$
	mplringd.i	$$⊢\phi \to I\in V$$
	mplringd.r	$$⊢\phi \to R\in \mathrm{Ring}$$
Assertion	mplringd	$$⊢\phi \to P\in \mathrm{Ring}$$

Proof

Step	Hyp	Ref	Expression
1		mplringd.p	$$⊢P=I\mathrm{mPoly}R$$
2		mplringd.i	$$⊢\phi \to I\in V$$
3		mplringd.r	$$⊢\phi \to R\in \mathrm{Ring}$$
4	1	mplring	$$⊢\left(I\in V\wedge R\in \mathrm{Ring}\right)\to P\in \mathrm{Ring}$$
5	2 3 4	syl2anc	$$⊢\phi \to P\in \mathrm{Ring}$$