Theorem 0mplric

Description: Multivariate polynomials with no variables are isomorphic with the underlying ring. (Contributed by Thierry Arnoux, 4-May-2026)

	Ref	Expression
Hypotheses	0mplric.b	$$⊢B={\mathrm{Base}}_P$$
	0mplric.p	$$⊢P=\varnothing \mathrm{mPoly}R$$
	0mplric.r	$$⊢\phi \to R\in \mathrm{Ring}$$
Assertion	0mplric	$$⊢\phi \to P{\simeq }_{𝑟}R$$

Proof

Step	Hyp	Ref	Expression
1		0mplric.b	$$⊢B={\mathrm{Base}}_P$$
2		0mplric.p	$$⊢P=\varnothing \mathrm{mPoly}R$$
3		0mplric.r	$$⊢\phi \to R\in \mathrm{Ring}$$
4		fveq1	$$⊢q=p\to q\left(\varnothing \right)=p\left(\varnothing \right)$$
5	4	cbvmptv	$$⊢\left(q\in B⟼q\left(\varnothing \right)\right)=\left(p\in B⟼p\left(\varnothing \right)\right)$$
6	1 2 3 5	0mplrim	$$⊢\phi \to \left(q\in B⟼q\left(\varnothing \right)\right)\in \left(P\mathrm{RingIso}R\right)$$
7		brrici	$$⊢\left(q\in B⟼q\left(\varnothing \right)\right)\in \left(P\mathrm{RingIso}R\right)\to P{\simeq }_{𝑟}R$$
8	6 7	syl	$$⊢\phi \to P{\simeq }_{𝑟}R$$