Theorem mhp0cl

Description: The zero polynomial is homogeneous. Under df-mhp , it has any (nonnegative integer) degree which loosely corresponds to the value "undefined". The values -oo and 0 are also used in Metamath (by df-mdeg and df-dgr respectively) and the literature: https://math.stackexchange.com/a/1796314/593843 . (Contributed by SN, 12-Sep-2023)

	Ref	Expression
Hypotheses	mhp0cl.h	⊢ 𝐻 = ( 𝐼 mHomP 𝑅 )
	mhp0cl.0	⊢ 0 = ( 0g ‘ 𝑅 )
	mhp0cl.d	⊢ 𝐷 = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin }
	mhp0cl.i	⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
	mhp0cl.r	⊢ ( 𝜑 → 𝑅 ∈ Grp )
	mhp0cl.n	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
Assertion	mhp0cl	⊢ ( 𝜑 → ( 𝐷 × { 0 } ) ∈ ( 𝐻 ‘ 𝑁 ) )

Proof

Step	Hyp	Ref	Expression
1		mhp0cl.h	⊢ 𝐻 = ( 𝐼 mHomP 𝑅 )
2		mhp0cl.0	⊢ 0 = ( 0g ‘ 𝑅 )
3		mhp0cl.d	⊢ 𝐷 = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin }
4		mhp0cl.i	⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
5		mhp0cl.r	⊢ ( 𝜑 → 𝑅 ∈ Grp )
6		mhp0cl.n	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
7		eqid	⊢ ( 𝐼 mPoly 𝑅 ) = ( 𝐼 mPoly 𝑅 )
8		eqid	⊢ ( Base ‘ ( 𝐼 mPoly 𝑅 ) ) = ( Base ‘ ( 𝐼 mPoly 𝑅 ) )
9		eqid	⊢ ( 0g ‘ ( 𝐼 mPoly 𝑅 ) ) = ( 0g ‘ ( 𝐼 mPoly 𝑅 ) )
10	7 3 2 9 4 5	mpl0	⊢ ( 𝜑 → ( 0g ‘ ( 𝐼 mPoly 𝑅 ) ) = ( 𝐷 × { 0 } ) )
11	7	mplgrp	⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Grp ) → ( 𝐼 mPoly 𝑅 ) ∈ Grp )
12	4 5 11	syl2anc	⊢ ( 𝜑 → ( 𝐼 mPoly 𝑅 ) ∈ Grp )
13	8 9	grpidcl	⊢ ( ( 𝐼 mPoly 𝑅 ) ∈ Grp → ( 0g ‘ ( 𝐼 mPoly 𝑅 ) ) ∈ ( Base ‘ ( 𝐼 mPoly 𝑅 ) ) )
14	12 13	syl	⊢ ( 𝜑 → ( 0g ‘ ( 𝐼 mPoly 𝑅 ) ) ∈ ( Base ‘ ( 𝐼 mPoly 𝑅 ) ) )
15	10 14	eqeltrrd	⊢ ( 𝜑 → ( 𝐷 × { 0 } ) ∈ ( Base ‘ ( 𝐼 mPoly 𝑅 ) ) )
16		fczsupp0	⊢ ( ( 𝐷 × { 0 } ) supp 0 ) = ∅
17		0ss	⊢ ∅ ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 }
18	16 17	eqsstri	⊢ ( ( 𝐷 × { 0 } ) supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 }
19	18	a1i	⊢ ( 𝜑 → ( ( 𝐷 × { 0 } ) supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } )
20	1 7 8 2 3 6 15 19	ismhp2	⊢ ( 𝜑 → ( 𝐷 × { 0 } ) ∈ ( 𝐻 ‘ 𝑁 ) )