Theorem mhpmpl

Description: A homogeneous polynomial is a polynomial. (Contributed by Steven Nguyen, 25-Aug-2023) Remove hypotheses using reverse closure. (Revised by SN, 4-Aug-2025)

	Ref	Expression
Hypotheses	mhpmpl.h	$$⊢H=I\mathrm{mHomP}R$$
	mhpmpl.p	$$⊢P=I\mathrm{mPoly}R$$
	mhpmpl.b	$$⊢B={\mathrm{Base}}_P$$
	mhpmpl.x	$$⊢\phi \to X\in H\left(N\right)$$
Assertion	mhpmpl	$$⊢\phi \to X\in B$$

Proof

Step	Hyp	Ref	Expression
1		mhpmpl.h	$$⊢H=I\mathrm{mHomP}R$$
2		mhpmpl.p	$$⊢P=I\mathrm{mPoly}R$$
3		mhpmpl.b	$$⊢B={\mathrm{Base}}_P$$
4		mhpmpl.x	$$⊢\phi \to X\in H\left(N\right)$$
5		eqid	$$⊢0_R=0_R$$
6		eqid	$$⊢\left\{h\in \left({{\mathbb{N}}_0}^I\right)|h^{-1}\left[\mathbb{N}\right]\in \mathrm{Fin}\right\}=\left\{h\in \left({{\mathbb{N}}_0}^I\right)|h^{-1}\left[\mathbb{N}\right]\in \mathrm{Fin}\right\}$$
7		reldmmhp	$$⊢\mathrm{Rel}\mathrm{dom}\mathrm{mHomP}$$
8	7 1 4	elfvov1	$$⊢\phi \to I\in \mathrm{V}$$
9	7 1 4	elfvov2	$$⊢\phi \to R\in \mathrm{V}$$
10	1 4	mhprcl	$$⊢\phi \to N\in {\mathbb{N}}_0$$
11	1 2 3 5 6 8 9 10	ismhp	$$⊢\phi \to \left(X\in H\left(N\right)\leftrightarrow \left(X\in B\wedge X\mathrm{supp}{0}_R\subseteq \left\{g\in \left\{h\in \left({{\mathbb{N}}_0}^I\right)|h^{-1}\left[\mathbb{N}\right]\in \mathrm{Fin}\right\}|{\sum }_{{ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}{\mathbb{N}}_0}g=N\right\}\right)\right)$$
12	11	simprbda	$$⊢\left(\phi \wedge X\in H\left(N\right)\right)\to X\in B$$
13	4 12	mpdan	$$⊢\phi \to X\in B$$