Theorem mhpfval

Description: Value of the "homogeneous polynomial" function. (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	$$⊢\phi \to R\in W$$
Assertion	mhpfval	$$⊢\phi \to H=\left(n\in {\mathbb{N}}_0⟼\left\{f\in B|f\mathrm{supp}\underset{\dot{}}{0}\subseteq \left\{g\in D|\sum _{j\in {\mathbb{N}}_0}g\left(j\right)=n\right\}\right\}\right)$$

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	6	elexd	$$⊢\phi \to I\in \mathrm{V}$$
9	7	elexd	$$⊢\phi \to R\in \mathrm{V}$$
10		nn0ex	$$⊢{\mathbb{N}}_0\in \mathrm{V}$$
11	10	mptex	$$⊢\left(n\in {\mathbb{N}}_0⟼\left\{f\in B|f\mathrm{supp}\underset{\dot{}}{0}\subseteq \left\{g\in D|\sum _{j\in {\mathbb{N}}_0}g\left(j\right)=n\right\}\right\}\right)\in \mathrm{V}$$
12	11	a1i	$$⊢\phi \to \left(n\in {\mathbb{N}}_0⟼\left\{f\in B|f\mathrm{supp}\underset{\dot{}}{0}\subseteq \left\{g\in D|\sum _{j\in {\mathbb{N}}_0}g\left(j\right)=n\right\}\right\}\right)\in \mathrm{V}$$
13		oveq12	$$⊢\left(i=I\wedge r=R\right)\to i\mathrm{mPoly}r=I\mathrm{mPoly}R$$
14	13 2	syl6eqr	$$⊢\left(i=I\wedge r=R\right)\to i\mathrm{mPoly}r=P$$
15	14	fveq2d	$$⊢\left(i=I\wedge r=R\right)\to {\mathrm{Base}}_{\left(i\mathrm{mPoly}r\right)}={\mathrm{Base}}_P$$
16	15 3	syl6eqr	$$⊢\left(i=I\wedge r=R\right)\to {\mathrm{Base}}_{\left(i\mathrm{mPoly}r\right)}=B$$
17		fveq2	$$⊢r=R\to 0_r=0_R$$
18	17 4	syl6eqr	$$⊢r=R\to 0_r=\underset{\dot{}}{0}$$
19	18	oveq2d	$$⊢r=R\to f\mathrm{supp}{0}_r=f\mathrm{supp}\underset{\dot{}}{0}$$
20	19	adantl	$$⊢\left(i=I\wedge r=R\right)\to f\mathrm{supp}{0}_r=f\mathrm{supp}\underset{\dot{}}{0}$$
21		oveq2	$$⊢i=I\to {{\mathbb{N}}_0}^i={{\mathbb{N}}_0}^I$$
22	21	rabeqdv	$$⊢i=I\to \left\{h\in \left({{\mathbb{N}}_0}^i\right)|h^{-1}\left[\mathbb{N}\right]\in \mathrm{Fin}\right\}=\left\{h\in \left({{\mathbb{N}}_0}^I\right)|h^{-1}\left[\mathbb{N}\right]\in \mathrm{Fin}\right\}$$
23	22 5	syl6eqr	$$⊢i=I\to \left\{h\in \left({{\mathbb{N}}_0}^i\right)|h^{-1}\left[\mathbb{N}\right]\in \mathrm{Fin}\right\}=D$$
24	23	rabeqdv	$$⊢i=I\to \left\{g\in \left\{h\in \left({{\mathbb{N}}_0}^i\right)|h^{-1}\left[\mathbb{N}\right]\in \mathrm{Fin}\right\}|\sum _{j\in {\mathbb{N}}_0}g\left(j\right)=n\right\}=\left\{g\in D|\sum _{j\in {\mathbb{N}}_0}g\left(j\right)=n\right\}$$
25	24	adantr	$$⊢\left(i=I\wedge r=R\right)\to \left\{g\in \left\{h\in \left({{\mathbb{N}}_0}^i\right)|h^{-1}\left[\mathbb{N}\right]\in \mathrm{Fin}\right\}|\sum _{j\in {\mathbb{N}}_0}g\left(j\right)=n\right\}=\left\{g\in D|\sum _{j\in {\mathbb{N}}_0}g\left(j\right)=n\right\}$$
26	20 25	sseq12d	$$⊢\left(i=I\wedge r=R\right)\to \left(f\mathrm{supp}{0}_r\subseteq \left\{g\in \left\{h\in \left({{\mathbb{N}}_0}^i\right)|h^{-1}\left[\mathbb{N}\right]\in \mathrm{Fin}\right\}|\sum _{j\in {\mathbb{N}}_0}g\left(j\right)=n\right\}\leftrightarrow f\mathrm{supp}\underset{\dot{}}{0}\subseteq \left\{g\in D|\sum _{j\in {\mathbb{N}}_0}g\left(j\right)=n\right\}\right)$$
27	16 26	rabeqbidv	$$⊢\left(i=I\wedge r=R\right)\to \left\{f\in {\mathrm{Base}}_{\left(i\mathrm{mPoly}r\right)}|f\mathrm{supp}{0}_r\subseteq \left\{g\in \left\{h\in \left({{\mathbb{N}}_0}^i\right)|h^{-1}\left[\mathbb{N}\right]\in \mathrm{Fin}\right\}|\sum _{j\in {\mathbb{N}}_0}g\left(j\right)=n\right\}\right\}=\left\{f\in B|f\mathrm{supp}\underset{\dot{}}{0}\subseteq \left\{g\in D|\sum _{j\in {\mathbb{N}}_0}g\left(j\right)=n\right\}\right\}$$
28	27	mpteq2dv	$$⊢\left(i=I\wedge r=R\right)\to \left(n\in {\mathbb{N}}_0⟼\left\{f\in {\mathrm{Base}}_{\left(i\mathrm{mPoly}r\right)}|f\mathrm{supp}{0}_r\subseteq \left\{g\in \left\{h\in \left({{\mathbb{N}}_0}^i\right)|h^{-1}\left[\mathbb{N}\right]\in \mathrm{Fin}\right\}|\sum _{j\in {\mathbb{N}}_0}g\left(j\right)=n\right\}\right\}\right)=\left(n\in {\mathbb{N}}_0⟼\left\{f\in B|f\mathrm{supp}\underset{\dot{}}{0}\subseteq \left\{g\in D|\sum _{j\in {\mathbb{N}}_0}g\left(j\right)=n\right\}\right\}\right)$$
29		df-mhp	$$⊢\mathrm{mHomP}=\left(i\in \mathrm{V},r\in \mathrm{V}⟼\left(n\in {\mathbb{N}}_0⟼\left\{f\in {\mathrm{Base}}_{\left(i\mathrm{mPoly}r\right)}|f\mathrm{supp}{0}_r\subseteq \left\{g\in \left\{h\in \left({{\mathbb{N}}_0}^i\right)|h^{-1}\left[\mathbb{N}\right]\in \mathrm{Fin}\right\}|\sum _{j\in {\mathbb{N}}_0}g\left(j\right)=n\right\}\right\}\right)\right)$$
30	28 29	ovmpoga	$$⊢\left(I\in \mathrm{V}\wedge R\in \mathrm{V}\wedge \left(n\in {\mathbb{N}}_0⟼\left\{f\in B|f\mathrm{supp}\underset{\dot{}}{0}\subseteq \left\{g\in D|\sum _{j\in {\mathbb{N}}_0}g\left(j\right)=n\right\}\right\}\right)\in \mathrm{V}\right)\to I\mathrm{mHomP}R=\left(n\in {\mathbb{N}}_0⟼\left\{f\in B|f\mathrm{supp}\underset{\dot{}}{0}\subseteq \left\{g\in D|\sum _{j\in {\mathbb{N}}_0}g\left(j\right)=n\right\}\right\}\right)$$
31	8 9 12 30	syl3anc	$$⊢\phi \to I\mathrm{mHomP}R=\left(n\in {\mathbb{N}}_0⟼\left\{f\in B|f\mathrm{supp}\underset{\dot{}}{0}\subseteq \left\{g\in D|\sum _{j\in {\mathbb{N}}_0}g\left(j\right)=n\right\}\right\}\right)$$
32	1 31	syl5eq	$$⊢\phi \to H=\left(n\in {\mathbb{N}}_0⟼\left\{f\in B|f\mathrm{supp}\underset{\dot{}}{0}\subseteq \left\{g\in D|\sum _{j\in {\mathbb{N}}_0}g\left(j\right)=n\right\}\right\}\right)$$