Theorem mhpvscacl

Description: Homogeneous polynomials are closed under scalar multiplication. (Contributed by SN, 25-Sep-2023) Remove sethood hypothesis. (Revised by SN, 18-May-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.n	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
	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.n	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
7		mhpvscacl.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐾 )
8		mhpvscacl.f	⊢ ( 𝜑 → 𝐹 ∈ ( 𝐻 ‘ 𝑁 ) )
9		eqid	⊢ ( Base ‘ 𝑃 ) = ( Base ‘ 𝑃 )
10		eqid	⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 )
11		eqid	⊢ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin }
12		reldmmhp	⊢ Rel dom mHomP
13	12 1 8	elfvov1	⊢ ( 𝜑 → 𝐼 ∈ V )
14	2	mpllmod	⊢ ( ( 𝐼 ∈ V ∧ 𝑅 ∈ Ring ) → 𝑃 ∈ LMod )
15	13 5 14	syl2anc	⊢ ( 𝜑 → 𝑃 ∈ LMod )
16	7 4	eleqtrdi	⊢ ( 𝜑 → 𝑋 ∈ ( Base ‘ 𝑅 ) )
17	2 13 5	mplsca	⊢ ( 𝜑 → 𝑅 = ( Scalar ‘ 𝑃 ) )
18	17	fveq2d	⊢ ( 𝜑 → ( Base ‘ 𝑅 ) = ( Base ‘ ( Scalar ‘ 𝑃 ) ) )
19	16 18	eleqtrd	⊢ ( 𝜑 → 𝑋 ∈ ( Base ‘ ( Scalar ‘ 𝑃 ) ) )
20	1 2 9 13 5 6 8	mhpmpl	⊢ ( 𝜑 → 𝐹 ∈ ( Base ‘ 𝑃 ) )
21		eqid	⊢ ( Scalar ‘ 𝑃 ) = ( Scalar ‘ 𝑃 )
22		eqid	⊢ ( Base ‘ ( Scalar ‘ 𝑃 ) ) = ( Base ‘ ( Scalar ‘ 𝑃 ) )
23	9 21 3 22	lmodvscl	⊢ ( ( 𝑃 ∈ LMod ∧ 𝑋 ∈ ( Base ‘ ( Scalar ‘ 𝑃 ) ) ∧ 𝐹 ∈ ( Base ‘ 𝑃 ) ) → ( 𝑋 · 𝐹 ) ∈ ( Base ‘ 𝑃 ) )
24	15 19 20 23	syl3anc	⊢ ( 𝜑 → ( 𝑋 · 𝐹 ) ∈ ( Base ‘ 𝑃 ) )
25	2 4 9 11 24	mplelf	⊢ ( 𝜑 → ( 𝑋 · 𝐹 ) : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ 𝐾 )
26		eqid	⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 )
27	7	adantr	⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∖ ( 𝐹 supp ( 0g ‘ 𝑅 ) ) ) ) → 𝑋 ∈ 𝐾 )
28	20	adantr	⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∖ ( 𝐹 supp ( 0g ‘ 𝑅 ) ) ) ) → 𝐹 ∈ ( Base ‘ 𝑃 ) )
29		eldifi	⊢ ( 𝑘 ∈ ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∖ ( 𝐹 supp ( 0g ‘ 𝑅 ) ) ) → 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
30	29	adantl	⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∖ ( 𝐹 supp ( 0g ‘ 𝑅 ) ) ) ) → 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
31	2 3 4 9 26 11 27 28 30	mplvscaval	⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∖ ( 𝐹 supp ( 0g ‘ 𝑅 ) ) ) ) → ( ( 𝑋 · 𝐹 ) ‘ 𝑘 ) = ( 𝑋 ( .r ‘ 𝑅 ) ( 𝐹 ‘ 𝑘 ) ) )
32	2 4 9 11 20	mplelf	⊢ ( 𝜑 → 𝐹 : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ 𝐾 )
33		ssidd	⊢ ( 𝜑 → ( 𝐹 supp ( 0g ‘ 𝑅 ) ) ⊆ ( 𝐹 supp ( 0g ‘ 𝑅 ) ) )
34		ovexd	⊢ ( 𝜑 → ( ℕ0 ↑m 𝐼 ) ∈ V )
35	11 34	rabexd	⊢ ( 𝜑 → { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∈ V )
36		fvexd	⊢ ( 𝜑 → ( 0g ‘ 𝑅 ) ∈ V )
37	32 33 35 36	suppssr	⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∖ ( 𝐹 supp ( 0g ‘ 𝑅 ) ) ) ) → ( 𝐹 ‘ 𝑘 ) = ( 0g ‘ 𝑅 ) )
38	37	oveq2d	⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∖ ( 𝐹 supp ( 0g ‘ 𝑅 ) ) ) ) → ( 𝑋 ( .r ‘ 𝑅 ) ( 𝐹 ‘ 𝑘 ) ) = ( 𝑋 ( .r ‘ 𝑅 ) ( 0g ‘ 𝑅 ) ) )
39	4 26 10	ringrz	⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ) → ( 𝑋 ( .r ‘ 𝑅 ) ( 0g ‘ 𝑅 ) ) = ( 0g ‘ 𝑅 ) )
40	5 7 39	syl2anc	⊢ ( 𝜑 → ( 𝑋 ( .r ‘ 𝑅 ) ( 0g ‘ 𝑅 ) ) = ( 0g ‘ 𝑅 ) )
41	40	adantr	⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∖ ( 𝐹 supp ( 0g ‘ 𝑅 ) ) ) ) → ( 𝑋 ( .r ‘ 𝑅 ) ( 0g ‘ 𝑅 ) ) = ( 0g ‘ 𝑅 ) )
42	31 38 41	3eqtrd	⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∖ ( 𝐹 supp ( 0g ‘ 𝑅 ) ) ) ) → ( ( 𝑋 · 𝐹 ) ‘ 𝑘 ) = ( 0g ‘ 𝑅 ) )
43	25 42	suppss	⊢ ( 𝜑 → ( ( 𝑋 · 𝐹 ) supp ( 0g ‘ 𝑅 ) ) ⊆ ( 𝐹 supp ( 0g ‘ 𝑅 ) ) )
44	1 10 11 13 5 6 8	mhpdeg	⊢ ( 𝜑 → ( 𝐹 supp ( 0g ‘ 𝑅 ) ) ⊆ { 𝑔 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } )
45	43 44	sstrd	⊢ ( 𝜑 → ( ( 𝑋 · 𝐹 ) supp ( 0g ‘ 𝑅 ) ) ⊆ { 𝑔 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } )
46	1 2 9 10 11 13 5 6 24 45	ismhp2	⊢ ( 𝜑 → ( 𝑋 · 𝐹 ) ∈ ( 𝐻 ‘ 𝑁 ) )