Theorem mhpmulcl 22115

Description: A product of homogeneous polynomials is a homogeneous polynomial whose degree is the sum of the degrees of the factors. Compare mdegmulle2 26044 (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 5090	. . . . . . . . 9 ⊢ (𝑑 = 𝑥 → (𝑐 ∘r ≤ 𝑑 ↔ 𝑐 ∘r ≤ 𝑥))
2	1	rabbidv 3397	. . . . . . . 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 7390	. . . . . . . . 9 ⊢ (𝑑 = 𝑥 → (𝑄‘(𝑑 ∘f − 𝑒)) = (𝑄‘(𝑥 ∘f − 𝑒)))
4	3	oveq2d 7383	. . . . . . . 8 ⊢ (𝑑 = 𝑥 → ((𝑃‘𝑒)(.r‘𝑅)(𝑄‘(𝑑 ∘f − 𝑒))) = ((𝑃‘𝑒)(.r‘𝑅)(𝑄‘(𝑥 ∘f − 𝑒))))
5	2, 4	mpteq12dv 5173	. . . . . . 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 7383	. . . . . 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 2737	. . . . . . . 8 ⊢ (Base‘𝑌) = (Base‘𝑌)
9		eqid 2737	. . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅)
10		mhpmulcl.t	. . . . . . . 8 ⊢ · = (.r‘𝑌)
11		eqid 2737	. . . . . . . 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 22110	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ (Base‘𝑌))
15		mhpmulcl.q	. . . . . . . . 9 ⊢ (𝜑 → 𝑄 ∈ (𝐻‘𝑁))
16	12, 7, 8, 15	mhpmpl 22110	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ (Base‘𝑌))
17	7, 8, 9, 10, 11, 14, 16	mplmul 21989	. . . . . . 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 480	. . . . . 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 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
20		ovexd 7402	. . . . . 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 6973	. . . . 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 2992	. . . 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 783	. . . . . . . . . . . . 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 7375	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = 𝑒 → ((ℂfld ↾s ℕ0) Σg 𝑐) = ((ℂfld ↾s ℕ0) Σg 𝑒))
25	24	eqeq1d 2739	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = 𝑒 → (((ℂfld ↾s ℕ0) Σg 𝑐) = 𝑀 ↔ ((ℂfld ↾s ℕ0) Σg 𝑒) = 𝑀))
26	25	necon3bbid 2970	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = 𝑒 → (¬ ((ℂfld ↾s ℕ0) Σg 𝑐) = 𝑀 ↔ ((ℂfld ↾s ℕ0) Σg 𝑒) ≠ 𝑀))
27		elrabi 3631	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥} → 𝑒 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
28	27	ad2antlr 728	. . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . 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 3637	. . . . . . . . . . . . . 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 4263	. . . . . . . . . . . . . 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 2848	. . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑅) = (Base‘𝑅)
34	7, 33, 8, 11, 14	mplelf 21976	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑃:{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶(Base‘𝑅))
35		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (0g‘𝑅) = (0g‘𝑅)
36	12, 35, 11, 13	mhpdeg 22111	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑃 supp (0g‘𝑅)) ⊆ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑐) = 𝑀})
37		fvexd 6856	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0g‘𝑅) ∈ V)
38	34, 36, 13, 37	suppssrg 8146	. . . . . . . . . . . . 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 585	. . . . . . . . . . . 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 7382	. . . . . . . . . . 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 733	. . . . . . . . . . . 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 733	. . . . . . . . . . . . . 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 21976	. . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . 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 21907	. . . . . . . . . . . . . . 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 761	. . . . . . . . . . . . . 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 3631	. . . . . . . . . . . . . 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 17	. . . . . . . . . . . . 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 7038	. . . . . . . . . . . 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 20276	. . . . . . . . . . 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 2772	. . . . . . . . . 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 783	. . . . . . . . . . . . 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 7375	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = (𝑥 ∘f − 𝑒) → ((ℂfld ↾s ℕ0) Σg 𝑐) = ((ℂfld ↾s ℕ0) Σg (𝑥 ∘f − 𝑒)))
55	54	eqeq1d 2739	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = (𝑥 ∘f − 𝑒) → (((ℂfld ↾s ℕ0) Σg 𝑐) = 𝑁 ↔ ((ℂfld ↾s ℕ0) Σg (𝑥 ∘f − 𝑒)) = 𝑁))
56	55	necon3bbid 2970	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = (𝑥 ∘f − 𝑒) → (¬ ((ℂfld ↾s ℕ0) Σg 𝑐) = 𝑁 ↔ ((ℂfld ↾s ℕ0) Σg (𝑥 ∘f − 𝑒)) ≠ 𝑁))
57	46	ad5ant24 761	. . . . . . . . . . . . . . . 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 17	. . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . 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 3637	. . . . . . . . . . . . . 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 4263	. . . . . . . . . . . . . 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 2848	. . . . . . . . . . . . 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 21976	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑄:{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶(Base‘𝑅))
64	12, 35, 11, 15	mhpdeg 22111	. . . . . . . . . . . . . 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 8146	. . . . . . . . . . . . 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 585	. . . . . . . . . . . 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 7383	. . . . . . . . . . 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 733	. . . . . . . . . . . 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 733	. . . . . . . . . . . . . 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 21976	. . . . . . . . . . . . 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 728	. . . . . . . . . . . . 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 7038	. . . . . . . . . . . 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 20277	. . . . . . . . . . 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 2772	. . . . . . . . . 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 21402	. . . . . . . . . . . . . . . 16 ⊢ ℕ0 ∈ (SubMnd‘ℂfld)
76		eqid 2737	. . . . . . . . . . . . . . . . 17 ⊢ (ℂfld ↾s ℕ0) = (ℂfld ↾s ℕ0)
77	76	submbas 18782	. . . . . . . . . . . . . . . 16 ⊢ (ℕ0 ∈ (SubMnd‘ℂfld) → ℕ0 = (Base‘(ℂfld ↾s ℕ0)))
78	75, 77	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ℕ0 = (Base‘(ℂfld ↾s ℕ0))
79		cnfld0 21376	. . . . . . . . . . . . . . . . 17 ⊢ 0 = (0g‘ℂfld)
80	76, 79	subm0 18783	. . . . . . . . . . . . . . . 16 ⊢ (ℕ0 ∈ (SubMnd‘ℂfld) → 0 = (0g‘(ℂfld ↾s ℕ0)))
81	75, 80	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ 0 = (0g‘(ℂfld ↾s ℕ0))
82		nn0ex 12443	. . . . . . . . . . . . . . . 16 ⊢ ℕ0 ∈ V
83		cnfldadd 21358	. . . . . . . . . . . . . . . . 17 ⊢ + = (+g‘ℂfld)
84	76, 83	ressplusg 17254	. . . . . . . . . . . . . . . 16 ⊢ (ℕ0 ∈ V → + = (+g‘(ℂfld ↾s ℕ0)))
85	82, 84	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ + = (+g‘(ℂfld ↾s ℕ0))
86		cnring 21374	. . . . . . . . . . . . . . . . . 18 ⊢ ℂfld ∈ Ring
87		ringcmn 20263	. . . . . . . . . . . . . . . . . 18 ⊢ (ℂfld ∈ Ring → ℂfld ∈ CMnd)
88	86, 87	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ℂfld ∈ CMnd
89	76	submcmn 19813	. . . . . . . . . . . . . . . . 17 ⊢ ((ℂfld ∈ CMnd ∧ ℕ0 ∈ (SubMnd‘ℂfld)) → (ℂfld ↾s ℕ0) ∈ CMnd)
90	88, 75, 89	mp2an 693	. . . . . . . . . . . . . . . 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 22103	. . . . . . . . . . . . . . . . 17 ⊢ Rel dom mHomP
93	92, 12, 13	elfvov1 7409	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐼 ∈ V)
94	93	ad3antrrr 731	. . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . 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 21898	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → 𝑒:𝐼⟶ℕ0)
97	95, 96	syl 17	. . . . . . . . . . . . . . 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 21898	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → 𝑥:𝐼⟶ℕ0)
99	98	ad3antlr 732	. . . . . . . . . . . . . . . . . 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 6670	. . . . . . . . . . . . . . . . 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 6670	. . . . . . . . . . . . . . . . 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 4168	. . . . . . . . . . . . . . . . 17 ⊢ (𝐼 ∩ 𝐼) = 𝐼
103		eqidd 2738	. . . . . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . . . . 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 7640	. . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . . 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 5089	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 = 𝑒 → (𝑐 ∘r ≤ 𝑥 ↔ 𝑒 ∘r ≤ 𝑥))
108	107	elrab 3635	. . . . . . . . . . . . . . . . . . . 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 497	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥} → 𝑒 ∘r ≤ 𝑥)
110	109	ad2antlr 728	. . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . 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 7643	. . . . . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . . . . 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 7037	. . . . . . . . . . . . . . . . . 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 7037	. . . . . . . . . . . . . . . . . 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 12487	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑒‘𝑖) ∈ ℕ0 ∧ (𝑥‘𝑖) ∈ ℕ0) → ((𝑒‘𝑖) ≤ (𝑥‘𝑖) ↔ ((𝑥‘𝑖) − (𝑒‘𝑖)) ∈ ℕ0))
117	114, 115, 116	syl2anc 585	. . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . 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 7069	. . . . . . . . . . . . . . 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 6673	. . . . . . . . . . . . . . . 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 11138	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0 ∈ V
122	94, 121	jctir 520	. . . . . . . . . . . . . . . . . . 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 8125	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐼 ∈ V ∧ 0 ∈ V) → (𝑒:𝐼⟶ℕ0 → (𝑒 supp 0) = (◡𝑒 “ (ℕ0 ∖ {0}))))
124	122, 97, 123	sylc 65	. . . . . . . . . . . . . . . . . 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 12450	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ = (ℕ0 ∖ {0})
126	125	imaeq2i 6024	. . . . . . . . . . . . . . . . . 18 ⊢ (◡𝑒 “ ℕ) = (◡𝑒 “ (ℕ0 ∖ {0}))
127	124, 126	eqtr4di 2790	. . . . . . . . . . . . . . . . 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 21897	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐼 ∈ V → (𝑒 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↔ (𝑒:𝐼⟶ℕ0 ∧ (◡𝑒 “ ℕ) ∈ Fin)))
129	94, 128	syl 17	. . . . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . . . 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 495	. . . . . . . . . . . . . . . . 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 2837	. . . . . . . . . . . . . . . 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 3454	. . . . . . . . . . . . . . . . 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 9278	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑒 ∈ V ∧ 0 ∈ V) → (𝑒 finSupp 0 ↔ (Fun 𝑒 ∧ (𝑒 supp 0) ∈ Fin)))
135	133, 121, 134	sylancl 587	. . . . . . . . . . . . . . . 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 714	. . . . . . . . . . . . . . 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 7402	. . . . . . . . . . . . . . . 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 12452	. . . . . . . . . . . . . . . . 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 7645	. . . . . . . . . . . . . . . 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 21899	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → 𝑥 finSupp 0)
142	141	ad3antlr 732	. . . . . . . . . . . . . . . . . 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 9301	. . . . . . . . . . . . . . . . 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 12296	. . . . . . . . . . . . . . . . . . 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 8153	. . . . . . . . . . . . . . . . 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 9179	. . . . . . . . . . . . . . . 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 9279	. . . . . . . . . . . . . . 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 19898	. . . . . . . . . . . . . 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 7037	. . . . . . . . . . . . . . . . . . 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 12500	. . . . . . . . . . . . . . . . . 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 7037	. . . . . . . . . . . . . . . . . . 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 12500	. . . . . . . . . . . . . . . . . 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 11508	. . . . . . . . . . . . . . . . 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 5179	. . . . . . . . . . . . . . . 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 6856	. . . . . . . . . . . . . . . . 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 7402	. . . . . . . . . . . . . . . . 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 6909	. . . . . . . . . . . . . . . . 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 6909	. . . . . . . . . . . . . . . . . 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 7651	. . . . . . . . . . . . . . . . 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 7651	. . . . . . . . . . . . . . . 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 2782	. . . . . . . . . . . . . . 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 7383	. . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . 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 769	. . . . . . . . . . . . 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 3000	. . . . . . . . . . . 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 7376	. . . . . . . . . . . . . 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 2946	. . . . . . . . . . . 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 15	. . . . . . . . . . 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 3028	. . . . . . . . . . 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 234	. . . . . . . . . 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 961	. . . . . . . . 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 5179	. . . . . . . 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 7383	. . . . . . 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 20224	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
177	41, 176	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ Mnd)
178	177	ad2antrr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂfld ↾s ℕ0) Σg 𝑥) ≠ (𝑀 + 𝑁)) → 𝑅 ∈ Mnd)
179		ovex 7400	. . . . . . . . . 10 ⊢ (ℕ0 ↑m 𝐼) ∈ V
180	179	rabex 5281	. . . . . . . . 9 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∈ V
181	180	rabex 5281	. . . . . . . 8 ⊢ {𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘r ≤ 𝑥} ∈ V
182	35	gsumz 18804	. . . . . . . 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 587	. . . . . . 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 2772	. . . . . 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 412	. . . . 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 2955	. . . 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 240	. . 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 3130	. 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 22109	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ0)
190	12, 15	mhprcl 22109	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
191	189, 190	nn0addcld 12502	. . 3 ⊢ (𝜑 → (𝑀 + 𝑁) ∈ ℕ0)
192	7, 93, 41	mplringd 22001	. . . 4 ⊢ (𝜑 → 𝑌 ∈ Ring)
193	8, 10, 192, 14, 16	ringcld 20241	. . 3 ⊢ (𝜑 → (𝑃 · 𝑄) ∈ (Base‘𝑌))
194	12, 7, 8, 35, 11, 191, 193	ismhp3 22108	. 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 257	1 ⊢ (𝜑 → (𝑃 · 𝑄) ∈ (𝐻‘(𝑀 + 𝑁)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  {crab 3390  Vcvv 3430   ∖ cdif 3887   ∪ cun 3888  {csn 4568   class class class wbr 5086   ↦ cmpt 5167  ^◡ccnv 5630   “ cima 5634  Fun wfun 6493  ⟶wf 6495  ‘cfv 6499  (class class class)co 7367   ∘_f cof 7629   ∘_r cofr 7630   supp csupp 8110   ↑_m cmap 8773  Fincfn 8893   finSupp cfsupp 9274  0cc0 11038   + caddc 11041   ≤ cle 11180   − cmin 11377  ℕcn 12174  ℕ₀cn0 12437  Basecbs 17179   ↾_s cress 17200  +_gcplusg 17220  ._rcmulr 17221  0_gc0g 17402   Σ_g cgsu 17403  Mndcmnd 18702  SubMndcsubmnd 18750  CMndccmn 19755  Ringcrg 20214  ℂ_fldccnfld 21352   mPoly cmpl 21886   mHomP cmhp 22095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5308  ax-pr 5376  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-addf 11117

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6266  df-ord 6327  df-on 6328  df-lim 6329  df-suc 6330  df-iota 6455  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-isom 6508  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-of 7631  df-ofr 7632  df-om 7818  df-1st 7942  df-2nd 7943  df-supp 8111  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-er 8643  df-map 8775  df-pm 8776  df-ixp 8846  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-fsupp 9275  df-sup 9355  df-oi 9425  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-7 12249  df-8 12250  df-9 12251  df-n0 12438  df-z 12525  df-dec 12645  df-uz 12789  df-fz 13462  df-fzo 13609  df-seq 13964  df-hash 14293  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-starv 17235  df-sca 17236  df-vsca 17237  df-ip 17238  df-tset 17239  df-ple 17240  df-ds 17242  df-unif 17243  df-hom 17244  df-cco 17245  df-0g 17404  df-gsum 17405  df-prds 17410  df-pws 17412  df-mre 17548  df-mrc 17549  df-acs 17551  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-mhm 18751  df-submnd 18752  df-grp 18912  df-minusg 18913  df-mulg 19044  df-subg 19099  df-ghm 19188  df-cntz 19292  df-cmn 19757  df-abl 19758  df-mgp 20122  df-rng 20134  df-ur 20163  df-ring 20216  df-cring 20217  df-subrng 20523  df-subrg 20547  df-cnfld 21353  df-psr 21889  df-mpl 21891  df-mhp 22102

This theorem is referenced by:  mhppwdeg  22116