Theorem mhpinvcl

Description: Homogeneous polynomials are closed under taking the opposite. (Contributed by SN, 12-Sep-2023) Remove closure hypotheses. (Revised by SN, 4-Sep-2025)

	Ref	Expression
Hypotheses	mhpinvcl.h	⊢ 𝐻 = ( 𝐼 mHomP 𝑅 )
	mhpinvcl.p	⊢ 𝑃 = ( 𝐼 mPoly 𝑅 )
	mhpinvcl.m	⊢ 𝑀 = ( invg ‘ 𝑃 )
	mhpinvcl.r	⊢ ( 𝜑 → 𝑅 ∈ Grp )
	mhpinvcl.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝐻 ‘ 𝑁 ) )
Assertion	mhpinvcl	⊢ ( 𝜑 → ( 𝑀 ‘ 𝑋 ) ∈ ( 𝐻 ‘ 𝑁 ) )

Proof

Step	Hyp	Ref	Expression
1		mhpinvcl.h	⊢ 𝐻 = ( 𝐼 mHomP 𝑅 )
2		mhpinvcl.p	⊢ 𝑃 = ( 𝐼 mPoly 𝑅 )
3		mhpinvcl.m	⊢ 𝑀 = ( invg ‘ 𝑃 )
4		mhpinvcl.r	⊢ ( 𝜑 → 𝑅 ∈ Grp )
5		mhpinvcl.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝐻 ‘ 𝑁 ) )
6		eqid	⊢ ( Base ‘ 𝑃 ) = ( Base ‘ 𝑃 )
7		eqid	⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 )
8		eqid	⊢ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin }
9	1 5	mhprcl	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
10		reldmmhp	⊢ Rel dom mHomP
11	10 1 5	elfvov1	⊢ ( 𝜑 → 𝐼 ∈ V )
12	2	mplgrp	⊢ ( ( 𝐼 ∈ V ∧ 𝑅 ∈ Grp ) → 𝑃 ∈ Grp )
13	11 4 12	syl2anc	⊢ ( 𝜑 → 𝑃 ∈ Grp )
14	1 2 6 5	mhpmpl	⊢ ( 𝜑 → 𝑋 ∈ ( Base ‘ 𝑃 ) )
15	6 3 13 14	grpinvcld	⊢ ( 𝜑 → ( 𝑀 ‘ 𝑋 ) ∈ ( Base ‘ 𝑃 ) )
16		eqid	⊢ ( invg ‘ 𝑅 ) = ( invg ‘ 𝑅 )
17	2 6 16 3 11 4 14	mplneg	⊢ ( 𝜑 → ( 𝑀 ‘ 𝑋 ) = ( ( invg ‘ 𝑅 ) ∘ 𝑋 ) )
18	17	oveq1d	⊢ ( 𝜑 → ( ( 𝑀 ‘ 𝑋 ) supp ( 0g ‘ 𝑅 ) ) = ( ( ( invg ‘ 𝑅 ) ∘ 𝑋 ) supp ( 0g ‘ 𝑅 ) ) )
19		eqid	⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
20	19 16	grpinvfn	⊢ ( invg ‘ 𝑅 ) Fn ( Base ‘ 𝑅 )
21	20	a1i	⊢ ( 𝜑 → ( invg ‘ 𝑅 ) Fn ( Base ‘ 𝑅 ) )
22	2 19 6 8 14	mplelf	⊢ ( 𝜑 → 𝑋 : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
23		ovex	⊢ ( ℕ0 ↑m 𝐼 ) ∈ V
24	23	rabex	⊢ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∈ V
25	24	a1i	⊢ ( 𝜑 → { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∈ V )
26		fvexd	⊢ ( 𝜑 → ( 0g ‘ 𝑅 ) ∈ V )
27	7 16	grpinvid	⊢ ( 𝑅 ∈ Grp → ( ( invg ‘ 𝑅 ) ‘ ( 0g ‘ 𝑅 ) ) = ( 0g ‘ 𝑅 ) )
28	4 27	syl	⊢ ( 𝜑 → ( ( invg ‘ 𝑅 ) ‘ ( 0g ‘ 𝑅 ) ) = ( 0g ‘ 𝑅 ) )
29	21 22 25 26 28	suppcoss	⊢ ( 𝜑 → ( ( ( invg ‘ 𝑅 ) ∘ 𝑋 ) supp ( 0g ‘ 𝑅 ) ) ⊆ ( 𝑋 supp ( 0g ‘ 𝑅 ) ) )
30	18 29	eqsstrd	⊢ ( 𝜑 → ( ( 𝑀 ‘ 𝑋 ) supp ( 0g ‘ 𝑅 ) ) ⊆ ( 𝑋 supp ( 0g ‘ 𝑅 ) ) )
31	1 7 8 5	mhpdeg	⊢ ( 𝜑 → ( 𝑋 supp ( 0g ‘ 𝑅 ) ) ⊆ { 𝑔 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } )
32	30 31	sstrd	⊢ ( 𝜑 → ( ( 𝑀 ‘ 𝑋 ) supp ( 0g ‘ 𝑅 ) ) ⊆ { 𝑔 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } )
33	1 2 6 7 8 9 15 32	ismhp2	⊢ ( 𝜑 → ( 𝑀 ‘ 𝑋 ) ∈ ( 𝐻 ‘ 𝑁 ) )