Theorem mhpvscacl

Description: Homogeneous polynomials are closed under scalar multiplication. (Contributed by SN, 25-Sep-2023) Remove closure hypotheses. (Revised by SN, 4-Sep-2025)

	Ref	Expression
Hypotheses	mhpvscacl.h	⊢ 𝐻 = ( 𝐼 mHomP 𝑅 )
	mhpvscacl.p	⊢ 𝑃 = ( 𝐼 mPoly 𝑅 )
	mhpvscacl.t	⊢ · = ( ·𝑠 ‘ 𝑃 )
	mhpvscacl.k	⊢ 𝐾 = ( Base ‘ 𝑅 )
	mhpvscacl.r	⊢ ( 𝜑 → 𝑅 ∈ Ring )
	mhpvscacl.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐾 )
	mhpvscacl.f	⊢ ( 𝜑 → 𝐹 ∈ ( 𝐻 ‘ 𝑁 ) )
Assertion	mhpvscacl	⊢ ( 𝜑 → ( 𝑋 · 𝐹 ) ∈ ( 𝐻 ‘ 𝑁 ) )

Proof

Step	Hyp	Ref	Expression
1		mhpvscacl.h	⊢ 𝐻 = ( 𝐼 mHomP 𝑅 )
2		mhpvscacl.p	⊢ 𝑃 = ( 𝐼 mPoly 𝑅 )
3		mhpvscacl.t	⊢ · = ( ·𝑠 ‘ 𝑃 )
4		mhpvscacl.k	⊢ 𝐾 = ( Base ‘ 𝑅 )
5		mhpvscacl.r	⊢ ( 𝜑 → 𝑅 ∈ Ring )
6		mhpvscacl.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐾 )
7		mhpvscacl.f	⊢ ( 𝜑 → 𝐹 ∈ ( 𝐻 ‘ 𝑁 ) )
8		eqid	⊢ ( Base ‘ 𝑃 ) = ( Base ‘ 𝑃 )
9		eqid	⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 )
10		eqid	⊢ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin }
11	1 7	mhprcl	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
12		eqid	⊢ ( Scalar ‘ 𝑃 ) = ( Scalar ‘ 𝑃 )
13		eqid	⊢ ( Base ‘ ( Scalar ‘ 𝑃 ) ) = ( Base ‘ ( Scalar ‘ 𝑃 ) )
14		reldmmhp	⊢ Rel dom mHomP
15	14 1 7	elfvov1	⊢ ( 𝜑 → 𝐼 ∈ V )
16	2 15 5	mpllmodd	⊢ ( 𝜑 → 𝑃 ∈ LMod )
17	6 4	eleqtrdi	⊢ ( 𝜑 → 𝑋 ∈ ( Base ‘ 𝑅 ) )
18	2 15 5	mplsca	⊢ ( 𝜑 → 𝑅 = ( Scalar ‘ 𝑃 ) )
19	18	fveq2d	⊢ ( 𝜑 → ( Base ‘ 𝑅 ) = ( Base ‘ ( Scalar ‘ 𝑃 ) ) )
20	17 19	eleqtrd	⊢ ( 𝜑 → 𝑋 ∈ ( Base ‘ ( Scalar ‘ 𝑃 ) ) )
21	1 2 8 7	mhpmpl	⊢ ( 𝜑 → 𝐹 ∈ ( Base ‘ 𝑃 ) )
22	8 12 3 13 16 20 21	lmodvscld	⊢ ( 𝜑 → ( 𝑋 · 𝐹 ) ∈ ( Base ‘ 𝑃 ) )
23	2 4 8 10 22	mplelf	⊢ ( 𝜑 → ( 𝑋 · 𝐹 ) : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ 𝐾 )
24		eqid	⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 )
25	6	adantr	⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∖ ( 𝐹 supp ( 0g ‘ 𝑅 ) ) ) ) → 𝑋 ∈ 𝐾 )
26	21	adantr	⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∖ ( 𝐹 supp ( 0g ‘ 𝑅 ) ) ) ) → 𝐹 ∈ ( Base ‘ 𝑃 ) )
27		eldifi	⊢ ( 𝑘 ∈ ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∖ ( 𝐹 supp ( 0g ‘ 𝑅 ) ) ) → 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
28	27	adantl	⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∖ ( 𝐹 supp ( 0g ‘ 𝑅 ) ) ) ) → 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
29	2 3 4 8 24 10 25 26 28	mplvscaval	⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∖ ( 𝐹 supp ( 0g ‘ 𝑅 ) ) ) ) → ( ( 𝑋 · 𝐹 ) ‘ 𝑘 ) = ( 𝑋 ( .r ‘ 𝑅 ) ( 𝐹 ‘ 𝑘 ) ) )
30	2 4 8 10 21	mplelf	⊢ ( 𝜑 → 𝐹 : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ 𝐾 )
31		ssidd	⊢ ( 𝜑 → ( 𝐹 supp ( 0g ‘ 𝑅 ) ) ⊆ ( 𝐹 supp ( 0g ‘ 𝑅 ) ) )
32		fvexd	⊢ ( 𝜑 → ( 0g ‘ 𝑅 ) ∈ V )
33	30 31 7 32	suppssrg	⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∖ ( 𝐹 supp ( 0g ‘ 𝑅 ) ) ) ) → ( 𝐹 ‘ 𝑘 ) = ( 0g ‘ 𝑅 ) )
34	33	oveq2d	⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∖ ( 𝐹 supp ( 0g ‘ 𝑅 ) ) ) ) → ( 𝑋 ( .r ‘ 𝑅 ) ( 𝐹 ‘ 𝑘 ) ) = ( 𝑋 ( .r ‘ 𝑅 ) ( 0g ‘ 𝑅 ) ) )
35	4 24 9 5 6	ringrzd	⊢ ( 𝜑 → ( 𝑋 ( .r ‘ 𝑅 ) ( 0g ‘ 𝑅 ) ) = ( 0g ‘ 𝑅 ) )
36	35	adantr	⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∖ ( 𝐹 supp ( 0g ‘ 𝑅 ) ) ) ) → ( 𝑋 ( .r ‘ 𝑅 ) ( 0g ‘ 𝑅 ) ) = ( 0g ‘ 𝑅 ) )
37	29 34 36	3eqtrd	⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∖ ( 𝐹 supp ( 0g ‘ 𝑅 ) ) ) ) → ( ( 𝑋 · 𝐹 ) ‘ 𝑘 ) = ( 0g ‘ 𝑅 ) )
38	23 37	suppss	⊢ ( 𝜑 → ( ( 𝑋 · 𝐹 ) supp ( 0g ‘ 𝑅 ) ) ⊆ ( 𝐹 supp ( 0g ‘ 𝑅 ) ) )
39	1 9 10 7	mhpdeg	⊢ ( 𝜑 → ( 𝐹 supp ( 0g ‘ 𝑅 ) ) ⊆ { 𝑔 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } )
40	38 39	sstrd	⊢ ( 𝜑 → ( ( 𝑋 · 𝐹 ) supp ( 0g ‘ 𝑅 ) ) ⊆ { 𝑔 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } )
41	1 2 8 9 10 11 22 40	ismhp2	⊢ ( 𝜑 → ( 𝑋 · 𝐹 ) ∈ ( 𝐻 ‘ 𝑁 ) )