Theorem mhpaddcl

Description: Homogeneous polynomials are closed under addition. (Contributed by SN, 26-Aug-2023) Remove sethood hypothesis. (Revised by SN, 18-May-2025)

	Ref	Expression
Hypotheses	mhpaddcl.h	⊢ 𝐻 = ( 𝐼 mHomP 𝑅 )
	mhpaddcl.p	⊢ 𝑃 = ( 𝐼 mPoly 𝑅 )
	mhpaddcl.a	⊢ + = ( +g ‘ 𝑃 )
	mhpaddcl.r	⊢ ( 𝜑 → 𝑅 ∈ Grp )
	mhpaddcl.n	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
	mhpaddcl.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝐻 ‘ 𝑁 ) )
	mhpaddcl.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝐻 ‘ 𝑁 ) )
Assertion	mhpaddcl	⊢ ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ ( 𝐻 ‘ 𝑁 ) )

Proof

Step	Hyp	Ref	Expression
1		mhpaddcl.h	⊢ 𝐻 = ( 𝐼 mHomP 𝑅 )
2		mhpaddcl.p	⊢ 𝑃 = ( 𝐼 mPoly 𝑅 )
3		mhpaddcl.a	⊢ + = ( +g ‘ 𝑃 )
4		mhpaddcl.r	⊢ ( 𝜑 → 𝑅 ∈ Grp )
5		mhpaddcl.n	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
6		mhpaddcl.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝐻 ‘ 𝑁 ) )
7		mhpaddcl.y	⊢ ( 𝜑 → 𝑌 ∈ ( 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		reldmmhp	⊢ Rel dom mHomP
12	11 1 6	elfvov1	⊢ ( 𝜑 → 𝐼 ∈ V )
13	2	mplgrp	⊢ ( ( 𝐼 ∈ V ∧ 𝑅 ∈ Grp ) → 𝑃 ∈ Grp )
14	12 4 13	syl2anc	⊢ ( 𝜑 → 𝑃 ∈ Grp )
15	1 2 8 12 4 5 6	mhpmpl	⊢ ( 𝜑 → 𝑋 ∈ ( Base ‘ 𝑃 ) )
16	1 2 8 12 4 5 7	mhpmpl	⊢ ( 𝜑 → 𝑌 ∈ ( Base ‘ 𝑃 ) )
17	8 3	grpcl	⊢ ( ( 𝑃 ∈ Grp ∧ 𝑋 ∈ ( Base ‘ 𝑃 ) ∧ 𝑌 ∈ ( Base ‘ 𝑃 ) ) → ( 𝑋 + 𝑌 ) ∈ ( Base ‘ 𝑃 ) )
18	14 15 16 17	syl3anc	⊢ ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ ( Base ‘ 𝑃 ) )
19		eqid	⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 )
20	2 8 19 3 15 16	mpladd	⊢ ( 𝜑 → ( 𝑋 + 𝑌 ) = ( 𝑋 ∘f ( +g ‘ 𝑅 ) 𝑌 ) )
21	20	oveq1d	⊢ ( 𝜑 → ( ( 𝑋 + 𝑌 ) supp ( 0g ‘ 𝑅 ) ) = ( ( 𝑋 ∘f ( +g ‘ 𝑅 ) 𝑌 ) supp ( 0g ‘ 𝑅 ) ) )
22		ovexd	⊢ ( 𝜑 → ( ℕ0 ↑m 𝐼 ) ∈ V )
23	10 22	rabexd	⊢ ( 𝜑 → { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∈ V )
24		eqid	⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
25	24 9	grpidcl	⊢ ( 𝑅 ∈ Grp → ( 0g ‘ 𝑅 ) ∈ ( Base ‘ 𝑅 ) )
26	4 25	syl	⊢ ( 𝜑 → ( 0g ‘ 𝑅 ) ∈ ( Base ‘ 𝑅 ) )
27	2 24 8 10 15	mplelf	⊢ ( 𝜑 → 𝑋 : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
28	2 24 8 10 16	mplelf	⊢ ( 𝜑 → 𝑌 : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
29	24 19 9	grplid	⊢ ( ( 𝑅 ∈ Grp ∧ ( 0g ‘ 𝑅 ) ∈ ( Base ‘ 𝑅 ) ) → ( ( 0g ‘ 𝑅 ) ( +g ‘ 𝑅 ) ( 0g ‘ 𝑅 ) ) = ( 0g ‘ 𝑅 ) )
30	4 26 29	syl2anc	⊢ ( 𝜑 → ( ( 0g ‘ 𝑅 ) ( +g ‘ 𝑅 ) ( 0g ‘ 𝑅 ) ) = ( 0g ‘ 𝑅 ) )
31	23 26 27 28 30	suppofssd	⊢ ( 𝜑 → ( ( 𝑋 ∘f ( +g ‘ 𝑅 ) 𝑌 ) supp ( 0g ‘ 𝑅 ) ) ⊆ ( ( 𝑋 supp ( 0g ‘ 𝑅 ) ) ∪ ( 𝑌 supp ( 0g ‘ 𝑅 ) ) ) )
32	21 31	eqsstrd	⊢ ( 𝜑 → ( ( 𝑋 + 𝑌 ) supp ( 0g ‘ 𝑅 ) ) ⊆ ( ( 𝑋 supp ( 0g ‘ 𝑅 ) ) ∪ ( 𝑌 supp ( 0g ‘ 𝑅 ) ) ) )
33	1 9 10 12 4 5 6	mhpdeg	⊢ ( 𝜑 → ( 𝑋 supp ( 0g ‘ 𝑅 ) ) ⊆ { 𝑔 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } )
34	1 9 10 12 4 5 7	mhpdeg	⊢ ( 𝜑 → ( 𝑌 supp ( 0g ‘ 𝑅 ) ) ⊆ { 𝑔 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } )
35	33 34	unssd	⊢ ( 𝜑 → ( ( 𝑋 supp ( 0g ‘ 𝑅 ) ) ∪ ( 𝑌 supp ( 0g ‘ 𝑅 ) ) ) ⊆ { 𝑔 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } )
36	32 35	sstrd	⊢ ( 𝜑 → ( ( 𝑋 + 𝑌 ) supp ( 0g ‘ 𝑅 ) ) ⊆ { 𝑔 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } )
37	1 2 8 9 10 12 4 5 18 36	ismhp2	⊢ ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ ( 𝐻 ‘ 𝑁 ) )