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Theorem mhpmulcl 22170

Description: A product of homogeneous polynomials is a homogeneous polynomial whose degree is the sum of the degrees of the factors. Compare mdegmulle2 26132 (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 5151	. . . . . . . . 9 ⊢ (𝑑 = 𝑥 → (𝑐 ∘_r ≤ 𝑑 ↔ 𝑐 ∘_r ≤ 𝑥))
2	1	rabbidv 3440	. . . . . . . 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 7453	. . . . . . . . 9 ⊢ (𝑑 = 𝑥 → (𝑄‘(𝑑 ∘_f − 𝑒)) = (𝑄‘(𝑥 ∘_f − 𝑒)))
4	3	oveq2d 7446	. . . . . . . 8 ⊢ (𝑑 = 𝑥 → ((𝑃‘𝑒)(._r‘𝑅)(𝑄‘(𝑑 ∘_f − 𝑒))) = ((𝑃‘𝑒)(._r‘𝑅)(𝑄‘(𝑥 ∘_f − 𝑒))))
5	2, 4	mpteq12dv 5238	. . . . . . 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 7446	. . . . . 6 ⊢ (𝑑 = 𝑥 → (𝑅 Σ_g (𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑑} ↦ ((𝑃‘𝑒)(._r‘𝑅)(𝑄‘(𝑑 ∘_f − 𝑒))))) = (𝑅 Σ_g (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 2734	. . . . . . . 8 ⊢ (Base‘𝑌) = (Base‘𝑌)
9		eqid 2734	. . . . . . . 8 ⊢ (._r‘𝑅) = (._r‘𝑅)
10		mhpmulcl.t	. . . . . . . 8 ⊢ · = (._r‘𝑌)
11		eqid 2734	. . . . . . . 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 22165	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ (Base‘𝑌))
15		mhpmulcl.q	. . . . . . . . 9 ⊢ (𝜑 → 𝑄 ∈ (𝐻‘𝑁))
16	12, 7, 8, 15	mhpmpl 22165	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ (Base‘𝑌))
17	7, 8, 9, 10, 11, 14, 16	mplmul 22048	. . . . . . 7 ⊢ (𝜑 → (𝑃 · 𝑄) = (𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ↦ (𝑅 Σ_g (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 Σ_g (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 7465	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → (𝑅 Σ_g (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 7039	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → ((𝑃 · 𝑄)‘𝑥) = (𝑅 Σ_g (𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥} ↦ ((𝑃‘𝑒)(._r‘𝑅)(𝑄‘(𝑥 ∘_f − 𝑒))))))
22	21	neeq1d 2997	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → (((𝑃 · 𝑄)‘𝑥) ≠ (0_g‘𝑅) ↔ (𝑅 Σ_g (𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥} ↦ ((𝑃‘𝑒)(._r‘𝑅)(𝑄‘(𝑥 ∘_f − 𝑒))))) ≠ (0_g‘𝑅)))
23		simp-4l 783	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑒) ≠ 𝑀) → 𝜑)
24		oveq2 7438	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = 𝑒 → ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑐) = ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑒))
25	24	eqeq1d 2736	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = 𝑒 → (((ℂ_fld ↾_s ℕ₀) Σ_g 𝑐) = 𝑀 ↔ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑒) = 𝑀))
26	25	necon3bbid 2975	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = 𝑒 → (¬ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑐) = 𝑀 ↔ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑒) ≠ 𝑀))
27		elrabi 3689	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥} → 𝑒 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin})
28	27	ad2antlr 727	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑒) ≠ 𝑀) → 𝑒 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin})
29		simpr 484	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑒) ≠ 𝑀) → ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑒) ≠ 𝑀)
30	26, 28, 29	elrabd 3696	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑒) ≠ 𝑀) → 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ¬ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑐) = 𝑀})
31		notrab 4327	. . . . . . . . . . . . . 14 ⊢ ({ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∖ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑐) = 𝑀}) = {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ¬ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑐) = 𝑀}
32	30, 31	eleqtrrdi 2849	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑒) ≠ 𝑀) → 𝑒 ∈ ({ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∖ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑐) = 𝑀}))
33		eqid 2734	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑅) = (Base‘𝑅)
34	7, 33, 8, 11, 14	mplelf 22035	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑃:{ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}⟶(Base‘𝑅))
35		eqid 2734	. . . . . . . . . . . . . . 15 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
36	12, 35, 11, 13	mhpdeg 22166	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑃 supp (0_g‘𝑅)) ⊆ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑐) = 𝑀})
37		fvexd 6921	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0_g‘𝑅) ∈ V)
38	34, 36, 13, 37	suppssrg 8219	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑒 ∈ ({ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∖ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑐) = 𝑀})) → (𝑃‘𝑒) = (0_g‘𝑅))
39	23, 32, 38	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑒) ≠ 𝑀) → (𝑃‘𝑒) = (0_g‘𝑅))
40	39	oveq1d 7445	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑒) ≠ 𝑀) → ((𝑃‘𝑒)(._r‘𝑅)(𝑄‘(𝑥 ∘_f − 𝑒))) = ((0_g‘𝑅)(._r‘𝑅)(𝑄‘(𝑥 ∘_f − 𝑒))))
41		mhpmulcl.r	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ Ring)
42	41	ad4antr 732	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑒) ≠ 𝑀) → 𝑅 ∈ Ring)
43	16	ad4antr 732	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑒) ≠ 𝑀) → 𝑄 ∈ (Base‘𝑌))
44	7, 33, 8, 11, 43	mplelf 22035	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑒) ≠ 𝑀) → 𝑄:{ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}⟶(Base‘𝑅))
45		eqid 2734	. . . . . . . . . . . . . . . 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 21964	. . . . . . . . . . . . . . 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}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑒) ≠ 𝑀) → (𝑥 ∘_f − 𝑒) ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥})
48		elrabi 3689	. . . . . . . . . . . . . 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}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑒) ≠ 𝑀) → (𝑥 ∘_f − 𝑒) ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin})
50	44, 49	ffvelcdmd 7104	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑒) ≠ 𝑀) → (𝑄‘(𝑥 ∘_f − 𝑒)) ∈ (Base‘𝑅))
51	33, 9, 35, 42, 50	ringlzd 20308	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑒) ≠ 𝑀) → ((0_g‘𝑅)(._r‘𝑅)(𝑄‘(𝑥 ∘_f − 𝑒))) = (0_g‘𝑅))
52	40, 51	eqtrd 2774	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑒) ≠ 𝑀) → ((𝑃‘𝑒)(._r‘𝑅)(𝑄‘(𝑥 ∘_f − 𝑒))) = (0_g‘𝑅))
53		simp-4l 783	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g (𝑥 ∘_f − 𝑒)) ≠ 𝑁) → 𝜑)
54		oveq2 7438	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = (𝑥 ∘_f − 𝑒) → ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑐) = ((ℂ_fld ↾_s ℕ₀) Σ_g (𝑥 ∘_f − 𝑒)))
55	54	eqeq1d 2736	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = (𝑥 ∘_f − 𝑒) → (((ℂ_fld ↾_s ℕ₀) Σ_g 𝑐) = 𝑁 ↔ ((ℂ_fld ↾_s ℕ₀) Σ_g (𝑥 ∘_f − 𝑒)) = 𝑁))
56	55	necon3bbid 2975	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = (𝑥 ∘_f − 𝑒) → (¬ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑐) = 𝑁 ↔ ((ℂ_fld ↾_s ℕ₀) Σ_g (𝑥 ∘_f − 𝑒)) ≠ 𝑁))
57	46	ad5ant24 761	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g (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}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g (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}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g (𝑥 ∘_f − 𝑒)) ≠ 𝑁) → ((ℂ_fld ↾_s ℕ₀) Σ_g (𝑥 ∘_f − 𝑒)) ≠ 𝑁)
60	56, 58, 59	elrabd 3696	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g (𝑥 ∘_f − 𝑒)) ≠ 𝑁) → (𝑥 ∘_f − 𝑒) ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ¬ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑐) = 𝑁})
61		notrab 4327	. . . . . . . . . . . . . 14 ⊢ ({ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∖ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑐) = 𝑁}) = {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ¬ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑐) = 𝑁}
62	60, 61	eleqtrrdi 2849	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g (𝑥 ∘_f − 𝑒)) ≠ 𝑁) → (𝑥 ∘_f − 𝑒) ∈ ({ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∖ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑐) = 𝑁}))
63	7, 33, 8, 11, 16	mplelf 22035	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑄:{ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}⟶(Base‘𝑅))
64	12, 35, 11, 15	mhpdeg 22166	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑄 supp (0_g‘𝑅)) ⊆ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑐) = 𝑁})
65	63, 64, 15, 37	suppssrg 8219	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∘_f − 𝑒) ∈ ({ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∖ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑐) = 𝑁})) → (𝑄‘(𝑥 ∘_f − 𝑒)) = (0_g‘𝑅))
66	53, 62, 65	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g (𝑥 ∘_f − 𝑒)) ≠ 𝑁) → (𝑄‘(𝑥 ∘_f − 𝑒)) = (0_g‘𝑅))
67	66	oveq2d 7446	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g (𝑥 ∘_f − 𝑒)) ≠ 𝑁) → ((𝑃‘𝑒)(._r‘𝑅)(𝑄‘(𝑥 ∘_f − 𝑒))) = ((𝑃‘𝑒)(._r‘𝑅)(0_g‘𝑅)))
68	41	ad4antr 732	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g (𝑥 ∘_f − 𝑒)) ≠ 𝑁) → 𝑅 ∈ Ring)
69	14	ad4antr 732	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g (𝑥 ∘_f − 𝑒)) ≠ 𝑁) → 𝑃 ∈ (Base‘𝑌))
70	7, 33, 8, 11, 69	mplelf 22035	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g (𝑥 ∘_f − 𝑒)) ≠ 𝑁) → 𝑃:{ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}⟶(Base‘𝑅))
71	27	ad2antlr 727	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g (𝑥 ∘_f − 𝑒)) ≠ 𝑁) → 𝑒 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin})
72	70, 71	ffvelcdmd 7104	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g (𝑥 ∘_f − 𝑒)) ≠ 𝑁) → (𝑃‘𝑒) ∈ (Base‘𝑅))
73	33, 9, 35, 68, 72	ringrzd 20309	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g (𝑥 ∘_f − 𝑒)) ≠ 𝑁) → ((𝑃‘𝑒)(._r‘𝑅)(0_g‘𝑅)) = (0_g‘𝑅))
74	67, 73	eqtrd 2774	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g (𝑥 ∘_f − 𝑒)) ≠ 𝑁) → ((𝑃‘𝑒)(._r‘𝑅)(𝑄‘(𝑥 ∘_f − 𝑒))) = (0_g‘𝑅))
75		nn0subm 21457	. . . . . . . . . . . . . . . 16 ⊢ ℕ₀ ∈ (SubMnd‘ℂ_fld)
76		eqid 2734	. . . . . . . . . . . . . . . . 17 ⊢ (ℂ_fld ↾_s ℕ₀) = (ℂ_fld ↾_s ℕ₀)
77	76	submbas 18839	. . . . . . . . . . . . . . . 16 ⊢ (ℕ₀ ∈ (SubMnd‘ℂ_fld) → ℕ₀ = (Base‘(ℂ_fld ↾_s ℕ₀)))
78	75, 77	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ℕ₀ = (Base‘(ℂ_fld ↾_s ℕ₀))
79		cnfld0 21422	. . . . . . . . . . . . . . . . 17 ⊢ 0 = (0_g‘ℂ_fld)
80	76, 79	subm0 18840	. . . . . . . . . . . . . . . 16 ⊢ (ℕ₀ ∈ (SubMnd‘ℂ_fld) → 0 = (0_g‘(ℂ_fld ↾_s ℕ₀)))
81	75, 80	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ 0 = (0_g‘(ℂ_fld ↾_s ℕ₀))
82		nn0ex 12529	. . . . . . . . . . . . . . . 16 ⊢ ℕ₀ ∈ V
83		cnfldadd 21387	. . . . . . . . . . . . . . . . 17 ⊢ + = (+_g‘ℂ_fld)
84	76, 83	ressplusg 17335	. . . . . . . . . . . . . . . 16 ⊢ (ℕ₀ ∈ V → + = (+_g‘(ℂ_fld ↾_s ℕ₀)))
85	82, 84	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ + = (+_g‘(ℂ_fld ↾_s ℕ₀))
86		cnring 21420	. . . . . . . . . . . . . . . . . 18 ⊢ ℂ_fld ∈ Ring
87		ringcmn 20295	. . . . . . . . . . . . . . . . . 18 ⊢ (ℂ_fld ∈ Ring → ℂ_fld ∈ CMnd)
88	86, 87	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ℂ_fld ∈ CMnd
89	76	submcmn 19870	. . . . . . . . . . . . . . . . 17 ⊢ ((ℂ_fld ∈ CMnd ∧ ℕ₀ ∈ (SubMnd‘ℂ_fld)) → (ℂ_fld ↾_s ℕ₀) ∈ CMnd)
90	88, 75, 89	mp2an 692	. . . . . . . . . . . . . . . 16 ⊢ (ℂ_fld ↾_s ℕ₀) ∈ CMnd
91	90	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) → (ℂ_fld ↾_s ℕ₀) ∈ CMnd)
92		reldmmhp 22158	. . . . . . . . . . . . . . . . 17 ⊢ Rel dom mHomP
93	92, 12, 13	elfvov1 7472	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐼 ∈ V)
94	93	ad3antrrr 730	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 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}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 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 21955	. . . . . . . . . . . . . . . 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}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) → 𝑒:𝐼⟶ℕ₀)
98	11	psrbagf 21955	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} → 𝑥:𝐼⟶ℕ₀)
99	98	ad3antlr 731	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) → 𝑥:𝐼⟶ℕ₀)
100	99	ffnd 6737	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) → 𝑥 Fn 𝐼)
101	97	ffnd 6737	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) → 𝑒 Fn 𝐼)
102		inidm 4234	. . . . . . . . . . . . . . . . 17 ⊢ (𝐼 ∩ 𝐼) = 𝐼
103		eqidd 2735	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) ∧ 𝑖 ∈ 𝐼) → (𝑥‘𝑖) = (𝑥‘𝑖))
104		eqidd 2735	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 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 7705	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 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}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 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}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}))
107		breq1 5150	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 = 𝑒 → (𝑐 ∘_r ≤ 𝑥 ↔ 𝑒 ∘_r ≤ 𝑥))
108	107	elrab 3694	. . . . . . . . . . . . . . . . . . . 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 496	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥} → 𝑒 ∘_r ≤ 𝑥)
110	109	ad2antlr 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 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}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 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 7708	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 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 1370	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) ∧ 𝑖 ∈ 𝐼) → (𝑒‘𝑖) ≤ (𝑥‘𝑖))
114	97	ffvelcdmda 7103	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) ∧ 𝑖 ∈ 𝐼) → (𝑒‘𝑖) ∈ ℕ₀)
115	99	ffvelcdmda 7103	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) ∧ 𝑖 ∈ 𝐼) → (𝑥‘𝑖) ∈ ℕ₀)
116		nn0sub 12573	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑒‘𝑖) ∈ ℕ₀ ∧ (𝑥‘𝑖) ∈ ℕ₀) → ((𝑒‘𝑖) ≤ (𝑥‘𝑖) ↔ ((𝑥‘𝑖) − (𝑒‘𝑖)) ∈ ℕ₀))
117	114, 115, 116	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 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}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 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 7135	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) → (𝑥 ∘_f − 𝑒):𝐼⟶ℕ₀)
120	97	ffund 6740	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) → Fun 𝑒)
121		c0ex 11252	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0 ∈ V
122	94, 121	jctir 520	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) → (𝐼 ∈ V ∧ 0 ∈ V))
123		fsuppeq 8198	. . . . . . . . . . . . . . . . . . 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}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) → (𝑒 supp 0) = (^◡𝑒 “ (ℕ₀ ∖ {0})))
125		dfn2 12536	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ = (ℕ₀ ∖ {0})
126	125	imaeq2i 6077	. . . . . . . . . . . . . . . . . 18 ⊢ (^◡𝑒 “ ℕ) = (^◡𝑒 “ (ℕ₀ ∖ {0}))
127	124, 126	eqtr4di 2792	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) → (𝑒 supp 0) = (^◡𝑒 “ ℕ))
128	11	psrbag 21954	. . . . . . . . . . . . . . . . . . . 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}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 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}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 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}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) → (^◡𝑒 “ ℕ) ∈ Fin)
132	127, 131	eqeltrd 2838	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) → (𝑒 supp 0) ∈ Fin)
133	95	elexd 3501	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) → 𝑒 ∈ V)
134		isfsupp 9402	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑒 ∈ V ∧ 0 ∈ V) → (𝑒 finSupp 0 ↔ (Fun 𝑒 ∧ (𝑒 supp 0) ∈ Fin)))
135	133, 121, 134	sylancl 586	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 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 713	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) → 𝑒 finSupp 0)
137		ovexd 7465	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) → (𝑥 ∘_f − 𝑒) ∈ V)
138		0nn0 12538	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℕ₀
139	138	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) → 0 ∈ ℕ₀)
140	100, 101, 94, 94	offun 7710	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) → Fun (𝑥 ∘_f − 𝑒))
141	11	psrbagfsupp 21956	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} → 𝑥 finSupp 0)
142	141	ad3antlr 731	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) → 𝑥 finSupp 0)
143	142, 136	fsuppunfi 9425	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) → ((𝑥 supp 0) ∪ (𝑒 supp 0)) ∈ Fin)
144		0m0e0 12383	. . . . . . . . . . . . . . . . . . 19 ⊢ (0 − 0) = 0
145	144	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 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 8226	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 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 9298	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 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 9403	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 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 19955	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) → ((ℂ_fld ↾_s ℕ₀) Σ_g (𝑒 ∘_f + (𝑥 ∘_f − 𝑒))) = (((ℂ_fld ↾_s ℕ₀) Σ_g 𝑒) + ((ℂ_fld ↾_s ℕ₀) Σ_g (𝑥 ∘_f − 𝑒))))
150	97	ffvelcdmda 7103	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) ∧ 𝑏 ∈ 𝐼) → (𝑒‘𝑏) ∈ ℕ₀)
151	150	nn0cnd 12586	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) ∧ 𝑏 ∈ 𝐼) → (𝑒‘𝑏) ∈ ℂ)
152	99	ffvelcdmda 7103	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) ∧ 𝑏 ∈ 𝐼) → (𝑥‘𝑏) ∈ ℕ₀)
153	152	nn0cnd 12586	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) ∧ 𝑏 ∈ 𝐼) → (𝑥‘𝑏) ∈ ℂ)
154	151, 153	pncan3d 11620	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 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 5247	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 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 6921	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) ∧ 𝑏 ∈ 𝐼) → (𝑒‘𝑏) ∈ V)
157		ovexd 7465	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 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 6976	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) → 𝑒 = (𝑏 ∈ 𝐼 ↦ (𝑒‘𝑏)))
159	99	feqmptd 6976	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 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 7716	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 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 7716	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 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 2784	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) → (𝑒 ∘_f + (𝑥 ∘_f − 𝑒)) = 𝑥)
163	162	oveq2d 7446	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) → ((ℂ_fld ↾_s ℕ₀) Σ_g (𝑒 ∘_f + (𝑥 ∘_f − 𝑒))) = ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥))
164	149, 163	eqtr3d 2776	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) → (((ℂ_fld ↾_s ℕ₀) Σ_g 𝑒) + ((ℂ_fld ↾_s ℕ₀) Σ_g (𝑥 ∘_f − 𝑒))) = ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥))
165		simplr 769	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) → ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁))
166	164, 165	eqnetrd 3005	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) → (((ℂ_fld ↾_s ℕ₀) Σ_g 𝑒) + ((ℂ_fld ↾_s ℕ₀) Σ_g (𝑥 ∘_f − 𝑒))) ≠ (𝑀 + 𝑁))
167		oveq12 7439	. . . . . . . . . . . . . 14 ⊢ ((((ℂ_fld ↾_s ℕ₀) Σ_g 𝑒) = 𝑀 ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g (𝑥 ∘_f − 𝑒)) = 𝑁) → (((ℂ_fld ↾_s ℕ₀) Σ_g 𝑒) + ((ℂ_fld ↾_s ℕ₀) Σ_g (𝑥 ∘_f − 𝑒))) = (𝑀 + 𝑁))
168	167	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) → ((((ℂ_fld ↾_s ℕ₀) Σ_g 𝑒) = 𝑀 ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g (𝑥 ∘_f − 𝑒)) = 𝑁) → (((ℂ_fld ↾_s ℕ₀) Σ_g 𝑒) + ((ℂ_fld ↾_s ℕ₀) Σ_g (𝑥 ∘_f − 𝑒))) = (𝑀 + 𝑁)))
169	168	necon3ad 2950	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) → ((((ℂ_fld ↾_s ℕ₀) Σ_g 𝑒) + ((ℂ_fld ↾_s ℕ₀) Σ_g (𝑥 ∘_f − 𝑒))) ≠ (𝑀 + 𝑁) → ¬ (((ℂ_fld ↾_s ℕ₀) Σ_g 𝑒) = 𝑀 ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g (𝑥 ∘_f − 𝑒)) = 𝑁)))
170	166, 169	mpd 15	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) → ¬ (((ℂ_fld ↾_s ℕ₀) Σ_g 𝑒) = 𝑀 ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g (𝑥 ∘_f − 𝑒)) = 𝑁))
171		neorian 3034	. . . . . . . . . . 11 ⊢ ((((ℂ_fld ↾_s ℕ₀) Σ_g 𝑒) ≠ 𝑀 ∨ ((ℂ_fld ↾_s ℕ₀) Σ_g (𝑥 ∘_f − 𝑒)) ≠ 𝑁) ↔ ¬ (((ℂ_fld ↾_s ℕ₀) Σ_g 𝑒) = 𝑀 ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g (𝑥 ∘_f − 𝑒)) = 𝑁))
172	170, 171	sylibr 234	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) → (((ℂ_fld ↾_s ℕ₀) Σ_g 𝑒) ≠ 𝑀 ∨ ((ℂ_fld ↾_s ℕ₀) Σ_g (𝑥 ∘_f − 𝑒)) ≠ 𝑁))
173	52, 74, 172	mpjaodan 960	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥}) → ((𝑃‘𝑒)(._r‘𝑅)(𝑄‘(𝑥 ∘_f − 𝑒))) = (0_g‘𝑅))
174	173	mpteq2dva 5247	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 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} ↦ (0_g‘𝑅)))
175	174	oveq2d 7446	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) → (𝑅 Σ_g (𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥} ↦ ((𝑃‘𝑒)(._r‘𝑅)(𝑄‘(𝑥 ∘_f − 𝑒))))) = (𝑅 Σ_g (𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥} ↦ (0_g‘𝑅))))
176		ringmnd 20260	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
177	41, 176	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ Mnd)
178	177	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) → 𝑅 ∈ Mnd)
179		ovex 7463	. . . . . . . . . 10 ⊢ (ℕ₀ ↑_m 𝐼) ∈ V
180	179	rabex 5344	. . . . . . . . 9 ⊢ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∈ V
181	180	rabex 5344	. . . . . . . 8 ⊢ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥} ∈ V
182	35	gsumz 18861	. . . . . . . 8 ⊢ ((𝑅 ∈ Mnd ∧ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥} ∈ V) → (𝑅 Σ_g (𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥} ↦ (0_g‘𝑅))) = (0_g‘𝑅))
183	178, 181, 182	sylancl 586	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) → (𝑅 Σ_g (𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥} ↦ (0_g‘𝑅))) = (0_g‘𝑅))
184	175, 183	eqtrd 2774	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁)) → (𝑅 Σ_g (𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥} ↦ ((𝑃‘𝑒)(._r‘𝑅)(𝑄‘(𝑥 ∘_f − 𝑒))))) = (0_g‘𝑅))
185	184	ex 412	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → (((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) ≠ (𝑀 + 𝑁) → (𝑅 Σ_g (𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥} ↦ ((𝑃‘𝑒)(._r‘𝑅)(𝑄‘(𝑥 ∘_f − 𝑒))))) = (0_g‘𝑅)))
186	185	necon1d 2959	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → ((𝑅 Σ_g (𝑒 ∈ {𝑐 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ 𝑐 ∘_r ≤ 𝑥} ↦ ((𝑃‘𝑒)(._r‘𝑅)(𝑄‘(𝑥 ∘_f − 𝑒))))) ≠ (0_g‘𝑅) → ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) = (𝑀 + 𝑁)))
187	22, 186	sylbid 240	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → (((𝑃 · 𝑄)‘𝑥) ≠ (0_g‘𝑅) → ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) = (𝑀 + 𝑁)))
188	187	ralrimiva 3143	. 2 ⊢ (𝜑 → ∀𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} (((𝑃 · 𝑄)‘𝑥) ≠ (0_g‘𝑅) → ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) = (𝑀 + 𝑁)))
189	12, 13	mhprcl 22164	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
190	12, 15	mhprcl 22164	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
191	189, 190	nn0addcld 12588	. . 3 ⊢ (𝜑 → (𝑀 + 𝑁) ∈ ℕ₀)
192	7, 93, 41	mplringd 22060	. . . 4 ⊢ (𝜑 → 𝑌 ∈ Ring)
193	8, 10, 192, 14, 16	ringcld 20276	. . 3 ⊢ (𝜑 → (𝑃 · 𝑄) ∈ (Base‘𝑌))
194	12, 7, 8, 35, 11, 191, 193	ismhp3 22163	. 2 ⊢ (𝜑 → ((𝑃 · 𝑄) ∈ (𝐻‘(𝑀 + 𝑁)) ↔ ∀𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} (((𝑃 · 𝑄)‘𝑥) ≠ (0_g‘𝑅) → ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑥) = (𝑀 + 𝑁))))
195	188, 194	mpbird 257	1 ⊢ (𝜑 → (𝑃 · 𝑄) ∈ (𝐻‘(𝑀 + 𝑁)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∀wral 3058 {crab 3432 Vcvv 3477 ∖ cdif 3959 ∪ cun 3960 {csn 4630 class class class wbr 5147 ↦ cmpt 5230 ^◡ccnv 5687 “ cima 5691 Fun wfun 6556 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 ∘_f cof 7694 ∘_r cofr 7695 supp csupp 8183 ↑_m cmap 8864 Fincfn 8983 finSupp cfsupp 9398 0cc0 11152 + caddc 11155 ≤ cle 11293 − cmin 11489 ℕcn 12263 ℕ₀cn0 12523 Basecbs 17244 ↾_s cress 17273 +_gcplusg 17297 ._rcmulr 17298 0_gc0g 17485 Σ_g cgsu 17486 Mndcmnd 18759 SubMndcsubmnd 18807 CMndccmn 19812 Ringcrg 20250 ℂ_fldccnfld 21381 mPoly cmpl 21943 mHomP cmhp 22150

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-addf 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-ofr 7697 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-map 8866 df-pm 8867 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-sup 9479 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-fz 13544 df-fzo 13691 df-seq 14039 df-hash 14366 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-0g 17487 df-gsum 17488 df-prds 17493 df-pws 17495 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-mhm 18808 df-submnd 18809 df-grp 18966 df-minusg 18967 df-mulg 19098 df-subg 19153 df-ghm 19243 df-cntz 19347 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 df-cring 20253 df-subrng 20562 df-subrg 20586 df-cnfld 21382 df-psr 21946 df-mpl 21948 df-mhp 22157

This theorem is referenced by: mhppwdeg 22171

W3C validator