Theorem mhpmulcl 22291

Description: A product of homogeneous polynomials is a homogeneous polynomial whose degree is the sum of the degrees of the factors. Compare mdegmulle2 26215 (which shows less-than-or-equal instead of equal). (Contributed by SN, 22-Jul-2024.) Remove closure hypotheses. (Revised by SN, 4-Sep-2025.)

Hypotheses

Ref	Expression
mhpmulcl.h	⊢ 𝐻 = (𝐼 mHomP 𝑅)
mhpmulcl.y	⊢ 𝑌 = (𝐼 mPoly 𝑅)
mhpmulcl.t	⊢ · = (.r‘𝑌)
mhpmulcl.r	⊢ (𝜑 → 𝑅 ∈ Ring)
mhpmulcl.p	⊢ (𝜑 → 𝑃 ∈ (𝐻‘𝑀))
mhpmulcl.q	⊢ (𝜑 → 𝑄 ∈ (𝐻‘𝑁))

Assertion

Ref	Expression
mhpmulcl	⊢ (𝜑 → (𝑃 · 𝑄) ∈ (𝐻‘(𝑀 + 𝑁)))

Proof of Theorem mhpmulcl

Dummy variables 𝑏 𝑑 𝑒 𝑖 𝑥 𝑐 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 5112	. . . . . . . . 9 ⊢ (𝑑 = 𝑥 → (𝑐 ∘r ≤ 𝑑 ↔ 𝑐 ∘r ≤ 𝑥))
2	1	rabbidv 3421	. . . . . . . 8 ⊢ (𝑑 = 𝑥 → {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑑} = {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥})
3		fvoveq1 7433	. . . . . . . . 9 ⊢ (𝑑 = 𝑥 → (𝑄‘(𝑑 ∘f − 𝑒)) = (𝑄‘(𝑥 ∘f − 𝑒)))
4	3	oveq2d 7426	. . . . . . . 8 ⊢ (𝑑 = 𝑥 → ((𝑃‘𝑒)(.r‘𝑅)(𝑄‘(𝑑 ∘f − 𝑒))) = ((𝑃‘𝑒)(.r‘𝑅)(𝑄‘(𝑥 ∘f − 𝑒))))
5	2, 4	mpteq12dv 5197	. . . . . . 7 ⊢ (𝑑 = 𝑥 → (𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑑} ↦ ((𝑃‘𝑒)(.r‘𝑅)(𝑄‘(𝑑 ∘f − 𝑒)))) = (𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥} ↦ ((𝑃‘𝑒)(.r‘𝑅)(𝑄‘(𝑥 ∘f − 𝑒)))))
6	5	oveq2d 7426	. . . . . 6 ⊢ (𝑑 = 𝑥 → (𝑅 Σg (𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑑} ↦ ((𝑃‘𝑒)(.r‘𝑅)(𝑄‘(𝑑 ∘f − 𝑒))))) = (𝑅 Σg (𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥} ↦ ((𝑃‘𝑒)(.r‘𝑅)(𝑄‘(𝑥 ∘f − 𝑒))))))
7		mhpmulcl.y	. . . . . . . 8 ⊢ 𝑌 = (𝐼 mPoly 𝑅)
8		eqid 2761	. . . . . . . 8 ⊢ (Base‘𝑌) = (Base‘𝑌)
9		eqid 2761	. . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅)
10		mhpmulcl.t	. . . . . . . 8 ⊢ · = (.r‘𝑌)
11		eqid 2761	. . . . . . . 8 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
12		mhpmulcl.h	. . . . . . . . 9 ⊢ 𝐻 = (𝐼 mHomP 𝑅)
13		mhpmulcl.p	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ (𝐻‘𝑀))
14	12, 7, 8, 13	mhpmpl 22286	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ (Base‘𝑌))
15		mhpmulcl.q	. . . . . . . . 9 ⊢ (𝜑 → 𝑄 ∈ (𝐻‘𝑁))
16	12, 7, 8, 15	mhpmpl 22286	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ (Base‘𝑌))
17	7, 8, 9, 10, 11, 14, 16	mplmul 22139	. . . . . . 7 ⊢ (𝜑 → (𝑃 · 𝑄) = (𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑑} ↦ ((𝑃‘𝑒)(.r‘𝑅)(𝑄‘(𝑑 ∘f − 𝑒)))))))
18	17	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑃 · 𝑄) = (𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑑} ↦ ((𝑃‘𝑒)(.r‘𝑅)(𝑄‘(𝑑 ∘f − 𝑒)))))))
19		simpr 489	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
20		ovexd 7445	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑅 Σg (𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥} ↦ ((𝑃‘𝑒)(.r‘𝑅)(𝑄‘(𝑥 ∘f − 𝑒))))) ∈ V)
21	6, 18, 19, 20	fvmptd4 7014	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑃 · 𝑄)‘𝑥) = (𝑅 Σg (𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥} ↦ ((𝑃‘𝑒)(.r‘𝑅)(𝑄‘(𝑥 ∘f − 𝑒))))))
22	21	neeq1d 3015	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((𝑃 · 𝑄)‘𝑥) ≠ (0g‘𝑅) ↔ (𝑅 Σg (𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥} ↦ ((𝑃‘𝑒)(.r‘𝑅)(𝑄‘(𝑥 ∘f − 𝑒))))) ≠ (0g‘𝑅)))
23		simp-4l 794	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑒) ≠ 𝑀) → 𝜑)
24		oveq2 7418	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = 𝑒 → ((ℂfld ↾s ℕ0) Σg 𝑐) = ((ℂfld ↾s ℕ0) Σg 𝑒))
25	24	eqeq1d 2763	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = 𝑒 → (((ℂfld ↾s ℕ0) Σg 𝑐) = 𝑀 ↔ ((ℂfld ↾s ℕ0) Σg 𝑒) = 𝑀))
26	25	necon3bbid 2993	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = 𝑒 → (¬ ((ℂfld ↾s ℕ0) Σg 𝑐) = 𝑀 ↔ ((ℂfld ↾s ℕ0) Σg 𝑒) ≠ 𝑀))
27		elrabi 3645	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥} → 𝑒 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
28	27	ad2antlr 739	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑒) ≠ 𝑀) → 𝑒 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
29		simpr 489	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑒) ≠ 𝑀) → ((ℂfld ↾s ℕ0) Σg 𝑒) ≠ 𝑀)
30	26, 28, 29	elrabd 3651	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑒) ≠ 𝑀) → 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ¬ ((ℂfld ↾s ℕ0) Σg 𝑐) = 𝑀})
31		notrab 4274	. . . . . . . . . . . . . 14 ⊢ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∖ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑐) = 𝑀}) = {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ¬ ((ℂfld ↾s ℕ0) Σg 𝑐) = 𝑀}
32	30, 31	eleqtrrdi 2872	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑒) ≠ 𝑀) → 𝑒 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∖ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑐) = 𝑀}))
33		eqid 2761	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑅) = (Base‘𝑅)
34	7, 33, 8, 11, 14	mplelf 22126	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑃:{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶(Base‘𝑅))
35		eqid 2761	. . . . . . . . . . . . . . 15 ⊢ (0g‘𝑅) = (0g‘𝑅)
36	12, 35, 11, 13	mhpdeg 22287	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑃 supp (0g‘𝑅)) ⊆ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑐) = 𝑀})
37		fvexd 6896	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0g‘𝑅) ∈ V)
38	34, 36, 13, 37	suppssrg 8191	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑒 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∖ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑐) = 𝑀})) → (𝑃‘𝑒) = (0g‘𝑅))
39	23, 32, 38	syl2anc 595	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑒) ≠ 𝑀) → (𝑃‘𝑒) = (0g‘𝑅))
40	39	oveq1d 7425	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑒) ≠ 𝑀) → ((𝑃‘𝑒)(.r‘𝑅)(𝑄‘(𝑥 ∘f − 𝑒))) = ((0g‘𝑅)(.r‘𝑅)(𝑄‘(𝑥 ∘f − 𝑒))))
41		mhpmulcl.r	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ Ring)
42	41	ad4antr 744	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑒) ≠ 𝑀) → 𝑅 ∈ Ring)
43	16	ad4antr 744	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑒) ≠ 𝑀) → 𝑄 ∈ (Base‘𝑌))
44	7, 33, 8, 11, 43	mplelf 22126	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑒) ≠ 𝑀) → 𝑄:{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶(Base‘𝑅))
45		eqid 2761	. . . . . . . . . . . . . . . 16 ⊢ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥} = {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}
46	11, 45	psrbagconcl 22056	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) → (𝑥 ∘f − 𝑒) ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥})
47	46	ad5ant24 772	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑒) ≠ 𝑀) → (𝑥 ∘f − 𝑒) ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥})
48		elrabi 3645	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∘f − 𝑒) ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥} → (𝑥 ∘f − 𝑒) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
49	47, 48	syl 18	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑒) ≠ 𝑀) → (𝑥 ∘f − 𝑒) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
50	44, 49	ffvelcdmd 7080	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑒) ≠ 𝑀) → (𝑄‘(𝑥 ∘f − 𝑒)) ∈ (Base‘𝑅))
51	33, 9, 35, 42, 50	ringlzd 20377	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑒) ≠ 𝑀) → ((0g‘𝑅)(.r‘𝑅)(𝑄‘(𝑥 ∘f − 𝑒))) = (0g‘𝑅))
52	40, 51	eqtrd 2796	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑒) ≠ 𝑀) → ((𝑃‘𝑒)(.r‘𝑅)(𝑄‘(𝑥 ∘f − 𝑒))) = (0g‘𝑅))
53		simp-4l 794	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ ((ℂfld ↾s ℕ0) Σg (𝑥 ∘f − 𝑒)) ≠ 𝑁) → 𝜑)
54		oveq2 7418	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = (𝑥 ∘f − 𝑒) → ((ℂfld ↾s ℕ0) Σg 𝑐) = ((ℂfld ↾s ℕ0) Σg (𝑥 ∘f − 𝑒)))
55	54	eqeq1d 2763	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = (𝑥 ∘f − 𝑒) → (((ℂfld ↾s ℕ0) Σg 𝑐) = 𝑁 ↔ ((ℂfld ↾s ℕ0) Σg (𝑥 ∘f − 𝑒)) = 𝑁))
56	55	necon3bbid 2993	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = (𝑥 ∘f − 𝑒) → (¬ ((ℂfld ↾s ℕ0) Σg 𝑐) = 𝑁 ↔ ((ℂfld ↾s ℕ0) Σg (𝑥 ∘f − 𝑒)) ≠ 𝑁))
57	46	ad5ant24 772	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ ((ℂfld ↾s ℕ0) Σg (𝑥 ∘f − 𝑒)) ≠ 𝑁) → (𝑥 ∘f − 𝑒) ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥})
58	57, 48	syl 18	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ ((ℂfld ↾s ℕ0) Σg (𝑥 ∘f − 𝑒)) ≠ 𝑁) → (𝑥 ∘f − 𝑒) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
59		simpr 489	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ ((ℂfld ↾s ℕ0) Σg (𝑥 ∘f − 𝑒)) ≠ 𝑁) → ((ℂfld ↾s ℕ0) Σg (𝑥 ∘f − 𝑒)) ≠ 𝑁)
60	56, 58, 59	elrabd 3651	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ ((ℂfld ↾s ℕ0) Σg (𝑥 ∘f − 𝑒)) ≠ 𝑁) → (𝑥 ∘f − 𝑒) ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ¬ ((ℂfld ↾s ℕ0) Σg 𝑐) = 𝑁})
61		notrab 4274	. . . . . . . . . . . . . 14 ⊢ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∖ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑐) = 𝑁}) = {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ¬ ((ℂfld ↾s ℕ0) Σg 𝑐) = 𝑁}
62	60, 61	eleqtrrdi 2872	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ ((ℂfld ↾s ℕ0) Σg (𝑥 ∘f − 𝑒)) ≠ 𝑁) → (𝑥 ∘f − 𝑒) ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∖ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑐) = 𝑁}))
63	7, 33, 8, 11, 16	mplelf 22126	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑄:{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶(Base‘𝑅))
64	12, 35, 11, 15	mhpdeg 22287	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑄 supp (0g‘𝑅)) ⊆ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑐) = 𝑁})
65	63, 64, 15, 37	suppssrg 8191	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∘f − 𝑒) ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∖ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑐) = 𝑁})) → (𝑄‘(𝑥 ∘f − 𝑒)) = (0g‘𝑅))
66	53, 62, 65	syl2anc 595	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ ((ℂfld ↾s ℕ0) Σg (𝑥 ∘f − 𝑒)) ≠ 𝑁) → (𝑄‘(𝑥 ∘f − 𝑒)) = (0g‘𝑅))
67	66	oveq2d 7426	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ ((ℂfld ↾s ℕ0) Σg (𝑥 ∘f − 𝑒)) ≠ 𝑁) → ((𝑃‘𝑒)(.r‘𝑅)(𝑄‘(𝑥 ∘f − 𝑒))) = ((𝑃‘𝑒)(.r‘𝑅)(0g‘𝑅)))
68	41	ad4antr 744	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ ((ℂfld ↾s ℕ0) Σg (𝑥 ∘f − 𝑒)) ≠ 𝑁) → 𝑅 ∈ Ring)
69	14	ad4antr 744	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ ((ℂfld ↾s ℕ0) Σg (𝑥 ∘f − 𝑒)) ≠ 𝑁) → 𝑃 ∈ (Base‘𝑌))
70	7, 33, 8, 11, 69	mplelf 22126	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ ((ℂfld ↾s ℕ0) Σg (𝑥 ∘f − 𝑒)) ≠ 𝑁) → 𝑃:{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶(Base‘𝑅))
71	27	ad2antlr 739	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ ((ℂfld ↾s ℕ0) Σg (𝑥 ∘f − 𝑒)) ≠ 𝑁) → 𝑒 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
72	70, 71	ffvelcdmd 7080	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ ((ℂfld ↾s ℕ0) Σg (𝑥 ∘f − 𝑒)) ≠ 𝑁) → (𝑃‘𝑒) ∈ (Base‘𝑅))
73	33, 9, 35, 68, 72	ringrzd 20378	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ ((ℂfld ↾s ℕ0) Σg (𝑥 ∘f − 𝑒)) ≠ 𝑁) → ((𝑃‘𝑒)(.r‘𝑅)(0g‘𝑅)) = (0g‘𝑅))
74	67, 73	eqtrd 2796	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ ((ℂfld ↾s ℕ0) Σg (𝑥 ∘f − 𝑒)) ≠ 𝑁) → ((𝑃‘𝑒)(.r‘𝑅)(𝑄‘(𝑥 ∘f − 𝑒))) = (0g‘𝑅))
75		nn0subm 21551	. . . . . . . . . . . . . . . 16 ⊢ ℕ0 ∈ (SubMnd‘ℂfld)
76		eqid 2761	. . . . . . . . . . . . . . . . 17 ⊢ (ℂfld ↾s ℕ0) = (ℂfld ↾s ℕ0)
77	76	submbas 18872	. . . . . . . . . . . . . . . 16 ⊢ (ℕ0 ∈ (SubMnd‘ℂfld) → ℕ0 = (Base‘(ℂfld ↾s ℕ0)))
78	75, 77	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ℕ0 = (Base‘(ℂfld ↾s ℕ0))
79		cnfld0 21525	. . . . . . . . . . . . . . . . 17 ⊢ 0 = (0g‘ℂfld)
80	76, 79	subm0 18873	. . . . . . . . . . . . . . . 16 ⊢ (ℕ0 ∈ (SubMnd‘ℂfld) → 0 = (0g‘(ℂfld ↾s ℕ0)))
81	75, 80	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ 0 = (0g‘(ℂfld ↾s ℕ0))
82		nn0ex 12509	. . . . . . . . . . . . . . . 16 ⊢ ℕ0 ∈ V
83		cnfldadd 21507	. . . . . . . . . . . . . . . . 17 ⊢ + = (+g‘ℂfld)
84	76, 83	ressplusg 17343	. . . . . . . . . . . . . . . 16 ⊢ (ℕ0 ∈ V → + = (+g‘(ℂfld ↾s ℕ0)))
85	82, 84	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ + = (+g‘(ℂfld ↾s ℕ0))
86		cnring 21523	. . . . . . . . . . . . . . . . . 18 ⊢ ℂfld ∈ Ring
87		ringcmn 20364	. . . . . . . . . . . . . . . . . 18 ⊢ (ℂfld ∈ Ring → ℂfld ∈ CMnd)
88	86, 87	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ℂfld ∈ CMnd
89	76	submcmn 19907	. . . . . . . . . . . . . . . . 17 ⊢ ((ℂfld ∈ CMnd ∧ ℕ0 ∈ (SubMnd‘ℂfld)) → (ℂfld ↾s ℕ0) ∈ CMnd)
90	88, 75, 89	mp2an 704	. . . . . . . . . . . . . . . 16 ⊢ (ℂfld ↾s ℕ0) ∈ CMnd
91	90	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) → (ℂfld ↾s ℕ0) ∈ CMnd)
92		reldmmhp 22279	. . . . . . . . . . . . . . . . 17 ⊢ Rel dom mHomP
93	92, 12, 13	elfvov1 7452	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐼 ∈ V)
94	93	ad3antrrr 742	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) → 𝐼 ∈ V)
95	27	adantl 486	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) → 𝑒 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
96	11	psrbagf 22047	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → 𝑒:𝐼⟶ℕ0)
97	95, 96	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) → 𝑒:𝐼⟶ℕ0)
98	11	psrbagf 22047	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → 𝑥:𝐼⟶ℕ0)
99	98	ad3antlr 743	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) → 𝑥:𝐼⟶ℕ0)
100	99	ffnd 6706	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) → 𝑥 Fn 𝐼)
101	97	ffnd 6706	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) → 𝑒 Fn 𝐼)
102		inidm 4178	. . . . . . . . . . . . . . . . 17 ⊢ (𝐼 ∩ 𝐼) = 𝐼
103		eqidd 2762	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ 𝑖 ∈ 𝐼) → (𝑥‘𝑖) = (𝑥‘𝑖))
104		eqidd 2762	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ 𝑖 ∈ 𝐼) → (𝑒‘𝑖) = (𝑒‘𝑖))
105	100, 101, 94, 94, 102, 103, 104	offval 7683	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) → (𝑥 ∘f − 𝑒) = (𝑖 ∈ 𝐼 ↦ ((𝑥‘𝑖) − (𝑒‘𝑖))))
106		simpl 487	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ 𝑖 ∈ 𝐼) → (((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}))
107		breq1 5111	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 = 𝑒 → (𝑐 ∘r ≤ 𝑥 ↔ 𝑒 ∘r ≤ 𝑥))
108	107	elrab 3649	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥} ↔ (𝑒 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∧ 𝑒 ∘r ≤ 𝑥))
109	108	simprbi 502	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥} → 𝑒 ∘r ≤ 𝑥)
110	109	ad2antlr 739	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ 𝑖 ∈ 𝐼) → 𝑒 ∘r ≤ 𝑥)
111		simpr 489	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐼)
112	101, 100, 94, 94, 102, 104, 103	ofrval 7686	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ 𝑒 ∘r ≤ 𝑥 ∧ 𝑖 ∈ 𝐼) → (𝑒‘𝑖) ≤ (𝑥‘𝑖))
113	106, 110, 111, 112	syl3anc 1396	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ 𝑖 ∈ 𝐼) → (𝑒‘𝑖) ≤ (𝑥‘𝑖))
114	97	ffvelcdmda 7079	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ 𝑖 ∈ 𝐼) → (𝑒‘𝑖) ∈ ℕ0)
115	99	ffvelcdmda 7079	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ 𝑖 ∈ 𝐼) → (𝑥‘𝑖) ∈ ℕ0)
116		nn0sub 12553	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑒‘𝑖) ∈ ℕ0 ∧ (𝑥‘𝑖) ∈ ℕ0) → ((𝑒‘𝑖) ≤ (𝑥‘𝑖) ↔ ((𝑥‘𝑖) − (𝑒‘𝑖)) ∈ ℕ0))
117	114, 115, 116	syl2anc 595	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ 𝑖 ∈ 𝐼) → ((𝑒‘𝑖) ≤ (𝑥‘𝑖) ↔ ((𝑥‘𝑖) − (𝑒‘𝑖)) ∈ ℕ0))
118	113, 117	mpbid 235	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ 𝑖 ∈ 𝐼) → ((𝑥‘𝑖) − (𝑒‘𝑖)) ∈ ℕ0)
119	105, 118	fmpt3d 7111	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) → (𝑥 ∘f − 𝑒):𝐼⟶ℕ0)
120	97	ffund 6710	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) → Fun 𝑒)
121		c0ex 11199	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0 ∈ V
122	94, 121	jctir 529	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) → (𝐼 ∈ V ∧ 0 ∈ V))
123		fsuppeq 8170	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐼 ∈ V ∧ 0 ∈ V) → (𝑒:𝐼⟶ℕ0 → (𝑒 supp 0) = (◡𝑒 “ (ℕ0 ∖ {0}))))
124	122, 97, 123	sylc 66	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) → (𝑒 supp 0) = (◡𝑒 “ (ℕ0 ∖ {0})))
125		dfn2 12516	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ = (ℕ0 ∖ {0})
126	125	imaeq2i 6060	. . . . . . . . . . . . . . . . . 18 ⊢ (◡𝑒 “ ℕ) = (◡𝑒 “ (ℕ0 ∖ {0}))
127	124, 126	eqtr4di 2814	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) → (𝑒 supp 0) = (◡𝑒 “ ℕ))
128	11	psrbag 22046	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐼 ∈ V → (𝑒 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↔ (𝑒:𝐼⟶ℕ0 ∧ (◡𝑒 “ ℕ) ∈ Fin)))
129	94, 128	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) → (𝑒 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↔ (𝑒:𝐼⟶ℕ0 ∧ (◡𝑒 “ ℕ) ∈ Fin)))
130	95, 129	mpbid 235	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) → (𝑒:𝐼⟶ℕ0 ∧ (◡𝑒 “ ℕ) ∈ Fin))
131	130	simprd 500	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) → (◡𝑒 “ ℕ) ∈ Fin)
132	127, 131	eqeltrd 2861	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) → (𝑒 supp 0) ∈ Fin)
133	95	elexd 3476	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) → 𝑒 ∈ V)
134		isfsupp 9324	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑒 ∈ V ∧ 0 ∈ V) → (𝑒 finSupp 0 ↔ (Fun 𝑒 ∧ (𝑒 supp 0) ∈ Fin)))
135	133, 121, 134	sylancl 597	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) → (𝑒 finSupp 0 ↔ (Fun 𝑒 ∧ (𝑒 supp 0) ∈ Fin)))
136	120, 132, 135	mpbir2and 725	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) → 𝑒 finSupp 0)
137		ovexd 7445	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) → (𝑥 ∘f − 𝑒) ∈ V)
138		0nn0 12518	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℕ0
139	138	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) → 0 ∈ ℕ0)
140	100, 101, 94, 94	offun 7688	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) → Fun (𝑥 ∘f − 𝑒))
141	11	psrbagfsupp 22048	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → 𝑥 finSupp 0)
142	141	ad3antlr 743	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) → 𝑥 finSupp 0)
143	142, 136	fsuppunfi 9347	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) → ((𝑥 supp 0) ∪ (𝑒 supp 0)) ∈ Fin)
144		0m0e0 12358	. . . . . . . . . . . . . . . . . . 19 ⊢ (0 − 0) = 0
145	144	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) → (0 − 0) = 0)
146	94, 139, 99, 97, 145	suppofssd 8198	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) → ((𝑥 ∘f − 𝑒) supp 0) ⊆ ((𝑥 supp 0) ∪ (𝑒 supp 0)))
147	143, 146	ssfid 9228	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) → ((𝑥 ∘f − 𝑒) supp 0) ∈ Fin)
148	137, 139, 140, 147	isfsuppd 9325	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) → (𝑥 ∘f − 𝑒) finSupp 0)
149	78, 81, 85, 91, 94, 97, 119, 136, 148	gsumadd 19992	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) → ((ℂfld ↾s ℕ0) Σg (𝑒 ∘f + (𝑥 ∘f − 𝑒))) = (((ℂfld ↾s ℕ0) Σg 𝑒) + ((ℂfld ↾s ℕ0) Σg (𝑥 ∘f − 𝑒))))
150	97	ffvelcdmda 7079	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ 𝑏 ∈ 𝐼) → (𝑒‘𝑏) ∈ ℕ0)
151	150	nn0cnd 12566	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ 𝑏 ∈ 𝐼) → (𝑒‘𝑏) ∈ ℂ)
152	99	ffvelcdmda 7079	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ 𝑏 ∈ 𝐼) → (𝑥‘𝑏) ∈ ℕ0)
153	152	nn0cnd 12566	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ 𝑏 ∈ 𝐼) → (𝑥‘𝑏) ∈ ℂ)
154	151, 153	pncan3d 11571	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ 𝑏 ∈ 𝐼) → ((𝑒‘𝑏) + ((𝑥‘𝑏) − (𝑒‘𝑏))) = (𝑥‘𝑏))
155	154	mpteq2dva 5203	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) → (𝑏 ∈ 𝐼 ↦ ((𝑒‘𝑏) + ((𝑥‘𝑏) − (𝑒‘𝑏)))) = (𝑏 ∈ 𝐼 ↦ (𝑥‘𝑏)))
156		fvexd 6896	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ 𝑏 ∈ 𝐼) → (𝑒‘𝑏) ∈ V)
157		ovexd 7445	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) ∧ 𝑏 ∈ 𝐼) → ((𝑥‘𝑏) − (𝑒‘𝑏)) ∈ V)
158	97	feqmptd 6949	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) → 𝑒 = (𝑏 ∈ 𝐼 ↦ (𝑒‘𝑏)))
159	99	feqmptd 6949	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) → 𝑥 = (𝑏 ∈ 𝐼 ↦ (𝑥‘𝑏)))
160	94, 152, 150, 159, 158	offval2 7694	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) → (𝑥 ∘f − 𝑒) = (𝑏 ∈ 𝐼 ↦ ((𝑥‘𝑏) − (𝑒‘𝑏))))
161	94, 156, 157, 158, 160	offval2 7694	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) → (𝑒 ∘f + (𝑥 ∘f − 𝑒)) = (𝑏 ∈ 𝐼 ↦ ((𝑒‘𝑏) + ((𝑥‘𝑏) − (𝑒‘𝑏)))))
162	155, 161, 159	3eqtr4d 2806	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) → (𝑒 ∘f + (𝑥 ∘f − 𝑒)) = 𝑥)
163	162	oveq2d 7426	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) → ((ℂfld ↾s ℕ0) Σg (𝑒 ∘f + (𝑥 ∘f − 𝑒))) = ((ℂfld ↾s ℕ0) Σg 𝑥))
164	149, 163	eqtr3d 2798	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) → (((ℂfld ↾s ℕ0) Σg 𝑒) + ((ℂfld ↾s ℕ0) Σg (𝑥 ∘f − 𝑒))) = ((ℂfld ↾s ℕ0) Σg 𝑥))
165		simplr 780	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) → ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁))
166	164, 165	eqnetrd 3023	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) → (((ℂfld ↾s ℕ0) Σg 𝑒) + ((ℂfld ↾s ℕ0) Σg (𝑥 ∘f − 𝑒))) ≠ (𝑀 + 𝑁))
167		oveq12 7419	. . . . . . . . . . . . . 14 ⊢ ((((ℂfld ↾s ℕ0) Σg 𝑒) = 𝑀 ∧ ((ℂfld ↾s ℕ0) Σg (𝑥 ∘f − 𝑒)) = 𝑁) → (((ℂfld ↾s ℕ0) Σg 𝑒) + ((ℂfld ↾s ℕ0) Σg (𝑥 ∘f − 𝑒))) = (𝑀 + 𝑁))
168	167	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) → ((((ℂfld ↾s ℕ0) Σg 𝑒) = 𝑀 ∧ ((ℂfld ↾s ℕ0) Σg (𝑥 ∘f − 𝑒)) = 𝑁) → (((ℂfld ↾s ℕ0) Σg 𝑒) + ((ℂfld ↾s ℕ0) Σg (𝑥 ∘f − 𝑒))) = (𝑀 + 𝑁)))
169	168	necon3ad 2969	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) → ((((ℂfld ↾s ℕ0) Σg 𝑒) + ((ℂfld ↾s ℕ0) Σg (𝑥 ∘f − 𝑒))) ≠ (𝑀 + 𝑁) → ¬ (((ℂfld ↾s ℕ0) Σg 𝑒) = 𝑀 ∧ ((ℂfld ↾s ℕ0) Σg (𝑥 ∘f − 𝑒)) = 𝑁)))
170	166, 169	mpd 16	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) → ¬ (((ℂfld ↾s ℕ0) Σg 𝑒) = 𝑀 ∧ ((ℂfld ↾s ℕ0) Σg (𝑥 ∘f − 𝑒)) = 𝑁))
171		neorian 3051	. . . . . . . . . . 11 ⊢ ((((ℂfld ↾s ℕ0) Σg 𝑒) ≠ 𝑀 ∨ ((ℂfld ↾s ℕ0) Σg (𝑥 ∘f − 𝑒)) ≠ 𝑁) ↔ ¬ (((ℂfld ↾s ℕ0) Σg 𝑒) = 𝑀 ∧ ((ℂfld ↾s ℕ0) Σg (𝑥 ∘f − 𝑒)) = 𝑁))
172	170, 171	sylibr 237	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) → (((ℂfld ↾s ℕ0) Σg 𝑒) ≠ 𝑀 ∨ ((ℂfld ↾s ℕ0) Σg (𝑥 ∘f − 𝑒)) ≠ 𝑁))
173	52, 74, 172	mpjaodan 973	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥}) → ((𝑃‘𝑒)(.r‘𝑅)(𝑄‘(𝑥 ∘f − 𝑒))) = (0g‘𝑅))
174	173	mpteq2dva 5203	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) → (𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥} ↦ ((𝑃‘𝑒)(.r‘𝑅)(𝑄‘(𝑥 ∘f − 𝑒)))) = (𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥} ↦ (0g‘𝑅)))
175	174	oveq2d 7426	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) → (𝑅 Σg (𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥} ↦ ((𝑃‘𝑒)(.r‘𝑅)(𝑄‘(𝑥 ∘f − 𝑒))))) = (𝑅 Σg (𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥} ↦ (0g‘𝑅))))
176		ringmnd 20324	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
177	41, 176	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ Mnd)
178	177	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) → 𝑅 ∈ Mnd)
179		ovex 7443	. . . . . . . . . 10 ⊢ (ℕ0 ↑m 𝐼) ∈ V
180	179	rabex 5309	. . . . . . . . 9 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∈ V
181	180	rabex 5309	. . . . . . . 8 ⊢ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥} ∈ V
182	35	gsumz 18894	. . . . . . . 8 ⊢ ((𝑅 ∈ Mnd ∧ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥} ∈ V) → (𝑅 Σg (𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥} ↦ (0g‘𝑅))) = (0g‘𝑅))
183	178, 181, 182	sylancl 597	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) → (𝑅 Σg (𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥} ↦ (0g‘𝑅))) = (0g‘𝑅))
184	175, 183	eqtrd 2796	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) → (𝑅 Σg (𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥} ↦ ((𝑃‘𝑒)(.r‘𝑅)(𝑄‘(𝑥 ∘f − 𝑒))))) = (0g‘𝑅))
185	184	ex 417	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁) → (𝑅 Σg (𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥} ↦ ((𝑃‘𝑒)(.r‘𝑅)(𝑄‘(𝑥 ∘f − 𝑒))))) = (0g‘𝑅)))
186	185	necon1d 2978	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑅 Σg (𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥} ↦ ((𝑃‘𝑒)(.r‘𝑅)(𝑄‘(𝑥 ∘f − 𝑒))))) ≠ (0g‘𝑅) → ((ℂfld ↾s ℕ0) Σg 𝑥) = (𝑀 + 𝑁)))
187	22, 186	sylbid 243	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((𝑃 · 𝑄)‘𝑥) ≠ (0g‘𝑅) → ((ℂfld ↾s ℕ0) Σg 𝑥) = (𝑀 + 𝑁)))
188	187	ralrimiva 3155	. 2 ⊢ (𝜑 → ∀𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} (((𝑃 · 𝑄)‘𝑥) ≠ (0g‘𝑅) → ((ℂfld ↾s ℕ0) Σg 𝑥) = (𝑀 + 𝑁)))
189	12, 13	mhprcl 22285	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ0)
190	12, 15	mhprcl 22285	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
191	189, 190	nn0addcld 12568	. . 3 ⊢ (𝜑 → (𝑀 + 𝑁) ∈ ℕ0)
192	7, 93, 41	mplringd 22151	. . . 4 ⊢ (𝜑 → 𝑌 ∈ Ring)
193	8, 10, 192, 14, 16	ringcld 20341	. . 3 ⊢ (𝜑 → (𝑃 · 𝑄) ∈ (Base‘𝑌))
194	12, 7, 8, 35, 11, 191, 193	ismhp3 22284	. 2 ⊢ (𝜑 → ((𝑃 · 𝑄) ∈ (𝐻‘(𝑀 + 𝑁)) ↔ ∀𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} (((𝑃 · 𝑄)‘𝑥) ≠ (0g‘𝑅) → ((ℂfld ↾s ℕ0) Σg 𝑥) = (𝑀 + 𝑁))))
195	188, 194	mpbird 260	1 ⊢ (𝜑 → (𝑃 · 𝑄) ∈ (𝐻‘(𝑀 + 𝑁)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1568   ∈ wcel 2141   ≠ wne 2956  ∀wral 3077  {crab 3414  Vcvv 3453   ∖ cdif 3901   ∪ cun 3902  {csn 4588   class class class wbr 5108   ↦ cmpt 5191  ^◡ccnv 5660   “ cima 5664  Fun wfun 6530  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410   ∘_f cof 7672   ∘_r cofr 7673   supp csupp 8155   ↑_m cmap 8823  Fincfn 8942   finSupp cfsupp 9320  0cc0 11099   + caddc 11102   ≤ cle 11243   − cmin 11440  ℕcn 12232  ℕ₀cn0 12503  Basecbs 17268   ↾_s cress 17289  +_gcplusg 17309  ._rcmulr 17310  0_gc0g 17491   Σ_g cgsu 17492  Mndcmnd 18791  SubMndcsubmnd 18839  CMndccmn 19849  Ringcrg 20314  ℂ_fldccnfld 21501   mPoly cmpl 22035   mHomP cmhp 22275

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176  ax-addf 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-iin 4958  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-of 7674  df-ofr 7675  df-om 7862  df-1st 7985  df-2nd 7986  df-supp 8156  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-2o 8453  df-er 8693  df-map 8825  df-pm 8826  df-ixp 8895  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-fsupp 9321  df-sup 9401  df-oi 9471  df-card 9924  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-nn 12233  df-2 12302  df-3 12303  df-4 12304  df-5 12305  df-6 12306  df-7 12307  df-8 12308  df-9 12309  df-n0 12504  df-z 12591  df-dec 12711  df-uz 12862  df-fz 13535  df-fzo 13683  df-seq 14038  df-hash 14367  df-struct 17206  df-sets 17223  df-slot 17241  df-ndx 17253  df-base 17269  df-ress 17290  df-plusg 17322  df-mulr 17323  df-starv 17324  df-sca 17325  df-vsca 17326  df-ip 17327  df-tset 17328  df-ple 17329  df-ds 17331  df-unif 17332  df-hom 17333  df-cco 17334  df-0g 17493  df-gsum 17494  df-prds 17499  df-pws 17501  df-mre 17637  df-mrc 17638  df-acs 17640  df-mgm 18697  df-sgrp 18776  df-mnd 18792  df-mhm 18840  df-submnd 18841  df-grp 19002  df-minusg 19003  df-mulg 19133  df-subg 19188  df-ghm 19283  df-cntz 19386  df-cmn 19851  df-abl 19852  df-mgp 20216  df-rng 20230  df-ur 20263  df-ring 20316  df-cring 20317  df-subrng 20630  df-subrg 20654  df-cnfld 21502  df-psr 22038  df-mpl 22040  df-mhp 22278

This theorem is referenced by:  mhppwdeg  22292