Theorem ismhp

Description: Property of being a homogeneous polynomial. (Contributed by Steven Nguyen, 25-Aug-2023)

	Ref	Expression
Hypotheses	mhpfval.h	$$⊢H=I\mathrm{mHomP}R$$
	mhpfval.p	$$⊢P=I\mathrm{mPoly}R$$
	mhpfval.b	$$⊢B={\mathrm{Base}}_P$$
	mhpfval.0	$$⊢\underset{\dot{}}{0}=0_R$$
	mhpfval.d	$$⊢D=\left\{h\in \left({{\mathbb{N}}_0}^I\right)|h^{-1}\left[\mathbb{N}\right]\in \mathrm{Fin}\right\}$$
	mhpfval.i	$$⊢\phi \to I\in V$$
	mhpfval.r	$$⊢\phi \to R\in W$$
	mhpval.n	$$⊢\phi \to N\in {\mathbb{N}}_0$$
Assertion	ismhp	$$⊢\phi \to \left(X\in H\left(N\right)\leftrightarrow \left(X\in B\wedge X\mathrm{supp}\underset{\dot{}}{0}\subseteq \left\{g\in D|{\sum }_{{ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}{\mathbb{N}}_0}g=N\right\}\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		mhpfval.h	$$⊢H=I\mathrm{mHomP}R$$
2		mhpfval.p	$$⊢P=I\mathrm{mPoly}R$$
3		mhpfval.b	$$⊢B={\mathrm{Base}}_P$$
4		mhpfval.0	$$⊢\underset{\dot{}}{0}=0_R$$
5		mhpfval.d	$$⊢D=\left\{h\in \left({{\mathbb{N}}_0}^I\right)|h^{-1}\left[\mathbb{N}\right]\in \mathrm{Fin}\right\}$$
6		mhpfval.i	$$⊢\phi \to I\in V$$
7		mhpfval.r	$$⊢\phi \to R\in W$$
8		mhpval.n	$$⊢\phi \to N\in {\mathbb{N}}_0$$
9	1 2 3 4 5 6 7 8	mhpval	$$⊢\phi \to H\left(N\right)=\left\{f\in B|f\mathrm{supp}\underset{\dot{}}{0}\subseteq \left\{g\in D|{\sum }_{{ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}{\mathbb{N}}_0}g=N\right\}\right\}$$
10	9	eleq2d	$$⊢\phi \to \left(X\in H\left(N\right)\leftrightarrow X\in \left\{f\in B|f\mathrm{supp}\underset{\dot{}}{0}\subseteq \left\{g\in D|{\sum }_{{ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}{\mathbb{N}}_0}g=N\right\}\right\}\right)$$
11		oveq1	$$⊢f=X\to f\mathrm{supp}\underset{\dot{}}{0}=X\mathrm{supp}\underset{\dot{}}{0}$$
12	11	sseq1d	$$⊢f=X\to \left(f\mathrm{supp}\underset{\dot{}}{0}\subseteq \left\{g\in D|{\sum }_{{ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}{\mathbb{N}}_0}g=N\right\}\leftrightarrow X\mathrm{supp}\underset{\dot{}}{0}\subseteq \left\{g\in D|{\sum }_{{ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}{\mathbb{N}}_0}g=N\right\}\right)$$
13	12	elrab	$$⊢X\in \left\{f\in B|f\mathrm{supp}\underset{\dot{}}{0}\subseteq \left\{g\in D|{\sum }_{{ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}{\mathbb{N}}_0}g=N\right\}\right\}\leftrightarrow \left(X\in B\wedge X\mathrm{supp}\underset{\dot{}}{0}\subseteq \left\{g\in D|{\sum }_{{ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}{\mathbb{N}}_0}g=N\right\}\right)$$
14	10 13	bitrdi	$$⊢\phi \to \left(X\in H\left(N\right)\leftrightarrow \left(X\in B\wedge X\mathrm{supp}\underset{\dot{}}{0}\subseteq \left\{g\in D|{\sum }_{{ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}{\mathbb{N}}_0}g=N\right\}\right)\right)$$