dvnprodlem3 - Mathbox for Glauco Siliprandi

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Theorem dvnprodlem3 41698

Description: The multinomial formula for the 𝑘-th derivative of a finite product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

**Hypotheses**
Ref	Expression
dvnprodlem3.s	⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
dvnprodlem3.x	⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
dvnprodlem3.t	⊢ (𝜑 → 𝑇 ∈ Fin)
dvnprodlem3.h	⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡):𝑋⟶ℂ)
dvnprodlem3.n	⊢ (𝜑 → 𝑁 ∈ ℕ₀)
dvnprodlem3.dvnh	⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑗):𝑋⟶ℂ)
dvnprodlem3.f	⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑇 ((𝐻‘𝑡)‘𝑥))
dvnprodlem3.d	⊢ 𝐷 = (𝑠 ∈ 𝒫 𝑇 ↦ (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}))
dvnprodlem3.c	⊢ 𝐶 = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑇) ∣ Σ𝑡 ∈ 𝑇 (𝑐‘𝑡) = 𝑛})

**Assertion**
Ref	Expression
dvnprodlem3	⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ (𝐶‘𝑁)(((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))

Distinct variable groups: 𝐶,𝑐 𝐷,𝑐,𝑗,𝑡,𝑛,𝑠,𝑥 𝐹,𝑠 𝐻,𝑐,𝑗,𝑡,𝑛,𝑠,𝑥 𝑁,𝑐,𝑗,𝑡,𝑛,𝑠,𝑥 𝑆,𝑐,𝑗,𝑡,𝑛,𝑠,𝑥 𝑇,𝑐,𝑗,𝑡,𝑛,𝑠,𝑥 𝑋,𝑐,𝑗,𝑡,𝑛,𝑠,𝑥 𝜑,𝑐,𝑗,𝑡,𝑛,𝑠,𝑥 Allowed substitution hints: 𝐶(𝑥,𝑡,𝑗,𝑛,𝑠) 𝐹(𝑥,𝑡,𝑗,𝑛,𝑐)

Proof of Theorem dvnprodlem3 Dummy variables 𝑑 ℎ 𝑘 𝑙 𝑟 𝑧 𝑦 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prodeq1 15129	. . . . . . . . 9 ⊢ (𝑠 = ∅ → ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥) = ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥))
2	1	mpteq2dv 5028	. . . . . . . 8 ⊢ (𝑠 = ∅ → (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥)) = (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥)))
3	2	oveq2d 6998	. . . . . . 7 ⊢ (𝑠 = ∅ → (𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥))) = (𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥))))
4	3	fveq1d 6506	. . . . . 6 ⊢ (𝑠 = ∅ → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥)))‘𝑘))
5		fveq2 6504	. . . . . . . . . 10 ⊢ (𝑠 = ∅ → (𝐷‘𝑠) = (𝐷‘∅))
6	5	fveq1d 6506	. . . . . . . . 9 ⊢ (𝑠 = ∅ → ((𝐷‘𝑠)‘𝑘) = ((𝐷‘∅)‘𝑘))
7	6	sumeq1d 14924	. . . . . . . 8 ⊢ (𝑠 = ∅ → Σ𝑐 ∈ ((𝐷‘𝑠)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ ((𝐷‘∅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
8		prodeq1 15129	. . . . . . . . . . 11 ⊢ (𝑠 = ∅ → ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡)) = ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡)))
9	8	oveq2d 6998	. . . . . . . . . 10 ⊢ (𝑠 = ∅ → ((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) = ((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))))
10		prodeq1 15129	. . . . . . . . . 10 ⊢ (𝑠 = ∅ → ∏𝑡 ∈ 𝑠 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) = ∏𝑡 ∈ ∅ (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))
11	9, 10	oveq12d 7000	. . . . . . . . 9 ⊢ (𝑠 = ∅ → (((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = (((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
12	11	sumeq2sdv 14927	. . . . . . . 8 ⊢ (𝑠 = ∅ → Σ𝑐 ∈ ((𝐷‘∅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ ((𝐷‘∅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
13	7, 12	eqtrd 2816	. . . . . . 7 ⊢ (𝑠 = ∅ → Σ𝑐 ∈ ((𝐷‘𝑠)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ ((𝐷‘∅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
14	13	mpteq2dv 5028	. . . . . 6 ⊢ (𝑠 = ∅ → (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑠)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘∅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))
15	4, 14	eqeq12d 2795	. . . . 5 ⊢ (𝑠 = ∅ → (((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑠)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) ↔ ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘∅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))))
16	15	ralbidv 3149	. . . 4 ⊢ (𝑠 = ∅ → (∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑠)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) ↔ ∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘∅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))))
17		prodeq1 15129	. . . . . . . . 9 ⊢ (𝑠 = 𝑟 → ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥) = ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥))
18	17	mpteq2dv 5028	. . . . . . . 8 ⊢ (𝑠 = 𝑟 → (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥)) = (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)))
19	18	oveq2d 6998	. . . . . . 7 ⊢ (𝑠 = 𝑟 → (𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥))) = (𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥))))
20	19	fveq1d 6506	. . . . . 6 ⊢ (𝑠 = 𝑟 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)))‘𝑘))
21		fveq2 6504	. . . . . . . . . 10 ⊢ (𝑠 = 𝑟 → (𝐷‘𝑠) = (𝐷‘𝑟))
22	21	fveq1d 6506	. . . . . . . . 9 ⊢ (𝑠 = 𝑟 → ((𝐷‘𝑠)‘𝑘) = ((𝐷‘𝑟)‘𝑘))
23	22	sumeq1d 14924	. . . . . . . 8 ⊢ (𝑠 = 𝑟 → Σ𝑐 ∈ ((𝐷‘𝑠)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
24		prodeq1 15129	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑟 → ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡)) = ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡)))
25	24	oveq2d 6998	. . . . . . . . . 10 ⊢ (𝑠 = 𝑟 → ((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) = ((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))))
26		prodeq1 15129	. . . . . . . . . 10 ⊢ (𝑠 = 𝑟 → ∏𝑡 ∈ 𝑠 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) = ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))
27	25, 26	oveq12d 7000	. . . . . . . . 9 ⊢ (𝑠 = 𝑟 → (((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = (((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
28	27	sumeq2sdv 14927	. . . . . . . 8 ⊢ (𝑠 = 𝑟 → Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
29	23, 28	eqtrd 2816	. . . . . . 7 ⊢ (𝑠 = 𝑟 → Σ𝑐 ∈ ((𝐷‘𝑠)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
30	29	mpteq2dv 5028	. . . . . 6 ⊢ (𝑠 = 𝑟 → (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑠)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))
31	20, 30	eqeq12d 2795	. . . . 5 ⊢ (𝑠 = 𝑟 → (((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑠)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) ↔ ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))))
32	31	ralbidv 3149	. . . 4 ⊢ (𝑠 = 𝑟 → (∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑠)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) ↔ ∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))))
33		prodeq1 15129	. . . . . . . . 9 ⊢ (𝑠 = (𝑟 ∪ {𝑧}) → ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥) = ∏𝑡 ∈ (𝑟 ∪ {𝑧})((𝐻‘𝑡)‘𝑥))
34	33	mpteq2dv 5028	. . . . . . . 8 ⊢ (𝑠 = (𝑟 ∪ {𝑧}) → (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥)) = (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ (𝑟 ∪ {𝑧})((𝐻‘𝑡)‘𝑥)))
35	34	oveq2d 6998	. . . . . . 7 ⊢ (𝑠 = (𝑟 ∪ {𝑧}) → (𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥))) = (𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ (𝑟 ∪ {𝑧})((𝐻‘𝑡)‘𝑥))))
36	35	fveq1d 6506	. . . . . 6 ⊢ (𝑠 = (𝑟 ∪ {𝑧}) → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ (𝑟 ∪ {𝑧})((𝐻‘𝑡)‘𝑥)))‘𝑘))
37		fveq2 6504	. . . . . . . . . 10 ⊢ (𝑠 = (𝑟 ∪ {𝑧}) → (𝐷‘𝑠) = (𝐷‘(𝑟 ∪ {𝑧})))
38	37	fveq1d 6506	. . . . . . . . 9 ⊢ (𝑠 = (𝑟 ∪ {𝑧}) → ((𝐷‘𝑠)‘𝑘) = ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑘))
39	38	sumeq1d 14924	. . . . . . . 8 ⊢ (𝑠 = (𝑟 ∪ {𝑧}) → Σ𝑐 ∈ ((𝐷‘𝑠)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
40		prodeq1 15129	. . . . . . . . . . 11 ⊢ (𝑠 = (𝑟 ∪ {𝑧}) → ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡)) = ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡)))
41	40	oveq2d 6998	. . . . . . . . . 10 ⊢ (𝑠 = (𝑟 ∪ {𝑧}) → ((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) = ((!‘𝑘) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))))
42		prodeq1 15129	. . . . . . . . . 10 ⊢ (𝑠 = (𝑟 ∪ {𝑧}) → ∏𝑡 ∈ 𝑠 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) = ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))
43	41, 42	oveq12d 7000	. . . . . . . . 9 ⊢ (𝑠 = (𝑟 ∪ {𝑧}) → (((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = (((!‘𝑘) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
44	43	sumeq2sdv 14927	. . . . . . . 8 ⊢ (𝑠 = (𝑟 ∪ {𝑧}) → Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
45	39, 44	eqtrd 2816	. . . . . . 7 ⊢ (𝑠 = (𝑟 ∪ {𝑧}) → Σ𝑐 ∈ ((𝐷‘𝑠)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
46	45	mpteq2dv 5028	. . . . . 6 ⊢ (𝑠 = (𝑟 ∪ {𝑧}) → (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑠)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))
47	36, 46	eqeq12d 2795	. . . . 5 ⊢ (𝑠 = (𝑟 ∪ {𝑧}) → (((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑠)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) ↔ ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ (𝑟 ∪ {𝑧})((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))))
48	47	ralbidv 3149	. . . 4 ⊢ (𝑠 = (𝑟 ∪ {𝑧}) → (∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑠)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) ↔ ∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ (𝑟 ∪ {𝑧})((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))))
49		prodeq1 15129	. . . . . . . . . 10 ⊢ (𝑠 = 𝑇 → ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥) = ∏𝑡 ∈ 𝑇 ((𝐻‘𝑡)‘𝑥))
50	49	mpteq2dv 5028	. . . . . . . . 9 ⊢ (𝑠 = 𝑇 → (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥)) = (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑇 ((𝐻‘𝑡)‘𝑥)))
51		dvnprodlem3.f	. . . . . . . . . . 11 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑇 ((𝐻‘𝑡)‘𝑥))
52	51	a1i 11	. . . . . . . . . 10 ⊢ (𝑠 = 𝑇 → 𝐹 = (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑇 ((𝐻‘𝑡)‘𝑥)))
53	52	eqcomd 2786	. . . . . . . . 9 ⊢ (𝑠 = 𝑇 → (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑇 ((𝐻‘𝑡)‘𝑥)) = 𝐹)
54	50, 53	eqtrd 2816	. . . . . . . 8 ⊢ (𝑠 = 𝑇 → (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥)) = 𝐹)
55	54	oveq2d 6998	. . . . . . 7 ⊢ (𝑠 = 𝑇 → (𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥))) = (𝑆 D^𝑛 𝐹))
56	55	fveq1d 6506	. . . . . 6 ⊢ (𝑠 = 𝑇 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = ((𝑆 D^𝑛 𝐹)‘𝑘))
57		fveq2 6504	. . . . . . . . . 10 ⊢ (𝑠 = 𝑇 → (𝐷‘𝑠) = (𝐷‘𝑇))
58	57	fveq1d 6506	. . . . . . . . 9 ⊢ (𝑠 = 𝑇 → ((𝐷‘𝑠)‘𝑘) = ((𝐷‘𝑇)‘𝑘))
59	58	sumeq1d 14924	. . . . . . . 8 ⊢ (𝑠 = 𝑇 → Σ𝑐 ∈ ((𝐷‘𝑠)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ ((𝐷‘𝑇)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
60		prodeq1 15129	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑇 → ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡)) = ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡)))
61	60	oveq2d 6998	. . . . . . . . . 10 ⊢ (𝑠 = 𝑇 → ((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) = ((!‘𝑘) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))))
62		prodeq1 15129	. . . . . . . . . 10 ⊢ (𝑠 = 𝑇 → ∏𝑡 ∈ 𝑠 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) = ∏𝑡 ∈ 𝑇 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))
63	61, 62	oveq12d 7000	. . . . . . . . 9 ⊢ (𝑠 = 𝑇 → (((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = (((!‘𝑘) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
64	63	sumeq2sdv 14927	. . . . . . . 8 ⊢ (𝑠 = 𝑇 → Σ𝑐 ∈ ((𝐷‘𝑇)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ ((𝐷‘𝑇)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
65	59, 64	eqtrd 2816	. . . . . . 7 ⊢ (𝑠 = 𝑇 → Σ𝑐 ∈ ((𝐷‘𝑠)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ ((𝐷‘𝑇)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
66	65	mpteq2dv 5028	. . . . . 6 ⊢ (𝑠 = 𝑇 → (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑠)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑇)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))
67	56, 66	eqeq12d 2795	. . . . 5 ⊢ (𝑠 = 𝑇 → (((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑠)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) ↔ ((𝑆 D^𝑛 𝐹)‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑇)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))))
68	67	ralbidv 3149	. . . 4 ⊢ (𝑠 = 𝑇 → (∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑠)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) ↔ ∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 𝐹)‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑇)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))))
69		prod0 15163	. . . . . . . . . . . . 13 ⊢ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥) = 1
70	69	mpteq2i 5024	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥)) = (𝑥 ∈ 𝑋 ↦ 1)
71	70	oveq2i 6993	. . . . . . . . . . 11 ⊢ (𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥))) = (𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 1))
72	71	a1i 11	. . . . . . . . . 10 ⊢ (𝑘 = 0 → (𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥))) = (𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 1)))
73		id 22	. . . . . . . . . 10 ⊢ (𝑘 = 0 → 𝑘 = 0)
74	72, 73	fveq12d 6511	. . . . . . . . 9 ⊢ (𝑘 = 0 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥)))‘𝑘) = ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 1))‘0))
75	74	adantl 474	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 = 0) → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥)))‘𝑘) = ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 1))‘0))
76		dvnprodlem3.s	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
77		recnprss 24220	. . . . . . . . . . 11 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
78	76, 77	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
79		1cnd 10440	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 1 ∈ ℂ)
80	79	fmpttd 6708	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 1):𝑋⟶ℂ)
81		1re 10445	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ
82	81	rgenw 3102	. . . . . . . . . . . . . . . 16 ⊢ ∀𝑥 ∈ 𝑋 1 ∈ ℝ
83		dmmptg 5940	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑥 ∈ 𝑋 1 ∈ ℝ → dom (𝑥 ∈ 𝑋 ↦ 1) = 𝑋)
84	82, 83	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ dom (𝑥 ∈ 𝑋 ↦ 1) = 𝑋
85	84	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom (𝑥 ∈ 𝑋 ↦ 1) = 𝑋)
86	85	feq2d 6335	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 1):dom (𝑥 ∈ 𝑋 ↦ 1)⟶ℂ ↔ (𝑥 ∈ 𝑋 ↦ 1):𝑋⟶ℂ))
87	80, 86	mpbird 249	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 1):dom (𝑥 ∈ 𝑋 ↦ 1)⟶ℂ)
88		restsspw 16567	. . . . . . . . . . . . . . 15 ⊢ ((TopOpen‘ℂ_fld) ↾_t 𝑆) ⊆ 𝒫 𝑆
89		dvnprodlem3.x	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
90	88, 89	sseldi 3858	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 ∈ 𝒫 𝑆)
91		elpwi 4435	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ 𝒫 𝑆 → 𝑋 ⊆ 𝑆)
92	90, 91	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
93	85, 92	eqsstrd 3897	. . . . . . . . . . . 12 ⊢ (𝜑 → dom (𝑥 ∈ 𝑋 ↦ 1) ⊆ 𝑆)
94	87, 93	jca 504	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 1):dom (𝑥 ∈ 𝑋 ↦ 1)⟶ℂ ∧ dom (𝑥 ∈ 𝑋 ↦ 1) ⊆ 𝑆))
95		cnex 10422	. . . . . . . . . . . . 13 ⊢ ℂ ∈ V
96	95	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ℂ ∈ V)
97		elpm2g 8229	. . . . . . . . . . . 12 ⊢ ((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) → ((𝑥 ∈ 𝑋 ↦ 1) ∈ (ℂ ↑_pm 𝑆) ↔ ((𝑥 ∈ 𝑋 ↦ 1):dom (𝑥 ∈ 𝑋 ↦ 1)⟶ℂ ∧ dom (𝑥 ∈ 𝑋 ↦ 1) ⊆ 𝑆)))
98	96, 76, 97	syl2anc 576	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 1) ∈ (ℂ ↑_pm 𝑆) ↔ ((𝑥 ∈ 𝑋 ↦ 1):dom (𝑥 ∈ 𝑋 ↦ 1)⟶ℂ ∧ dom (𝑥 ∈ 𝑋 ↦ 1) ⊆ 𝑆)))
99	94, 98	mpbird 249	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 1) ∈ (ℂ ↑_pm 𝑆))
100		dvn0 24239	. . . . . . . . . 10 ⊢ ((𝑆 ⊆ ℂ ∧ (𝑥 ∈ 𝑋 ↦ 1) ∈ (ℂ ↑_pm 𝑆)) → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 1))‘0) = (𝑥 ∈ 𝑋 ↦ 1))
101	78, 99, 100	syl2anc 576	. . . . . . . . 9 ⊢ (𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 1))‘0) = (𝑥 ∈ 𝑋 ↦ 1))
102	101	adantr 473	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 = 0) → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 1))‘0) = (𝑥 ∈ 𝑋 ↦ 1))
103		fveq2 6504	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 0 → ((𝐷‘∅)‘𝑘) = ((𝐷‘∅)‘0))
104	103	adantl 474	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 = 0) → ((𝐷‘∅)‘𝑘) = ((𝐷‘∅)‘0))
105		dvnprodlem3.d	. . . . . . . . . . . . . . . . 17 ⊢ 𝐷 = (𝑠 ∈ 𝒫 𝑇 ↦ (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}))
106		oveq2 6990	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑠 = ∅ → ((0...𝑛) ↑_𝑚 𝑠) = ((0...𝑛) ↑_𝑚 ∅))
107		elmapfn 8235	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 ∈ ((0...𝑛) ↑_𝑚 ∅) → 𝑥 Fn ∅)
108		fn0 6314	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 Fn ∅ ↔ 𝑥 = ∅)
109	107, 108	sylib 210	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 ∈ ((0...𝑛) ↑_𝑚 ∅) → 𝑥 = ∅)
110		velsn 4460	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
111	109, 110	sylibr 226	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 ∈ ((0...𝑛) ↑_𝑚 ∅) → 𝑥 ∈ {∅})
112	110	biimpi 208	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 ∈ {∅} → 𝑥 = ∅)
113		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 = ∅ → 𝑥 = ∅)
114		f0 6394	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ∅:∅⟶(0...𝑛)
115		ovex 7014	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (0...𝑛) ∈ V
116		0ex 5072	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ∅ ∈ V
117	115, 116	elmap 8241	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (∅ ∈ ((0...𝑛) ↑_𝑚 ∅) ↔ ∅:∅⟶(0...𝑛))
118	114, 117	mpbir 223	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ∅ ∈ ((0...𝑛) ↑_𝑚 ∅)
119	118	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 = ∅ → ∅ ∈ ((0...𝑛) ↑_𝑚 ∅))
120	113, 119	eqeltrd 2868	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 = ∅ → 𝑥 ∈ ((0...𝑛) ↑_𝑚 ∅))
121	112, 120	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 ∈ {∅} → 𝑥 ∈ ((0...𝑛) ↑_𝑚 ∅))
122	111, 121	impbii 201	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 ∈ ((0...𝑛) ↑_𝑚 ∅) ↔ 𝑥 ∈ {∅})
123	122	ax-gen 1759	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ∀𝑥(𝑥 ∈ ((0...𝑛) ↑_𝑚 ∅) ↔ 𝑥 ∈ {∅})
124		dfcleq 2774	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((0...𝑛) ↑_𝑚 ∅) = {∅} ↔ ∀𝑥(𝑥 ∈ ((0...𝑛) ↑_𝑚 ∅) ↔ 𝑥 ∈ {∅}))
125	123, 124	mpbir 223	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((0...𝑛) ↑_𝑚 ∅) = {∅}
126	125	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑠 = ∅ → ((0...𝑛) ↑_𝑚 ∅) = {∅})
127	106, 126	eqtrd 2816	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 = ∅ → ((0...𝑛) ↑_𝑚 𝑠) = {∅})
128		rabeq 3408	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((0...𝑛) ↑_𝑚 𝑠) = {∅} → {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛})
129	127, 128	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 = ∅ → {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛})
130		sumeq1 14912	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑠 = ∅ → Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = Σ𝑡 ∈ ∅ (𝑐‘𝑡))
131	130	eqeq1d 2782	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 = ∅ → (Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛))
132	131	rabbidv 3405	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 = ∅ → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛})
133	129, 132	eqtrd 2816	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 = ∅ → {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛})
134	133	mpteq2dv 5028	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = ∅ → (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}) = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛}))
135		0elpw 5114	. . . . . . . . . . . . . . . . . 18 ⊢ ∅ ∈ 𝒫 𝑇
136	135	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∅ ∈ 𝒫 𝑇)
137		nn0ex 11720	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ₀ ∈ V
138	137	mptex 6818	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛}) ∈ V
139	138	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛}) ∈ V)
140	105, 134, 136, 139	fvmptd3 6623	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐷‘∅) = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛}))
141		eqeq2 2791	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 0 → (Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0))
142	141	rabbidv 3405	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 0 → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0})
143	142	adantl 474	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 = 0) → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0})
144		0nn0 11730	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℕ₀
145	144	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 ∈ ℕ₀)
146		p0ex 5141	. . . . . . . . . . . . . . . . . 18 ⊢ {∅} ∈ V
147	146	rabex 5095	. . . . . . . . . . . . . . . . 17 ⊢ {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0} ∈ V
148	147	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0} ∈ V)
149	140, 143, 145, 148	fvmptd 6607	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐷‘∅)‘0) = {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0})
150	149	adantr 473	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 = 0) → ((𝐷‘∅)‘0) = {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0})
151		snidg 4476	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∅ ∈ V → ∅ ∈ {∅})
152	116, 151	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∅ ∈ {∅}
153		eqid 2780	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0 = 0
154	152, 153	pm3.2i 463	. . . . . . . . . . . . . . . . . . 19 ⊢ (∅ ∈ {∅} ∧ 0 = 0)
155		sum0 14944	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0
156	155	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 = ∅ → Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0)
157	156	eqeq1d 2782	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 = ∅ → (Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0 ↔ 0 = 0))
158	157	elrab 3597	. . . . . . . . . . . . . . . . . . 19 ⊢ (∅ ∈ {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0} ↔ (∅ ∈ {∅} ∧ 0 = 0))
159	154, 158	mpbir 223	. . . . . . . . . . . . . . . . . 18 ⊢ ∅ ∈ {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0}
160	159	n0ii 4191	. . . . . . . . . . . . . . . . 17 ⊢ ¬ {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0} = ∅
161		eqid 2780	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0} = {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0}
162		rabrsn 4539	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0} = {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0} → ({𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0} = ∅ ∨ {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0} = {∅}))
163	161, 162	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ({𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0} = ∅ ∨ {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0} = {∅})
164	160, 163	mtpor 1734	. . . . . . . . . . . . . . . 16 ⊢ {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0} = {∅}
165	164	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 = 0) → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0} = {∅})
166		iftrue 4359	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 0 → if(𝑘 = 0, {∅}, ∅) = {∅})
167	166	adantl 474	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 = 0) → if(𝑘 = 0, {∅}, ∅) = {∅})
168	165, 167	eqtr4d 2819	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 = 0) → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0} = if(𝑘 = 0, {∅}, ∅))
169	104, 150, 168	3eqtrd 2820	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 = 0) → ((𝐷‘∅)‘𝑘) = if(𝑘 = 0, {∅}, ∅))
170	169, 167	eqtrd 2816	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 = 0) → ((𝐷‘∅)‘𝑘) = {∅})
171	170	sumeq1d 14924	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 = 0) → Σ𝑐 ∈ ((𝐷‘∅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ {∅} (((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
172		fveq2 6504	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 0 → (!‘𝑘) = (!‘0))
173		fac0 13457	. . . . . . . . . . . . . . . . . . 19 ⊢ (!‘0) = 1
174	173	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 0 → (!‘0) = 1)
175	172, 174	eqtrd 2816	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 0 → (!‘𝑘) = 1)
176	175	oveq1d 6997	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 0 → ((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) = (1 / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))))
177		prod0 15163	. . . . . . . . . . . . . . . . . 18 ⊢ ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡)) = 1
178	177	oveq2i 6993	. . . . . . . . . . . . . . . . 17 ⊢ (1 / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) = (1 / 1)
179		1div1e1 11137	. . . . . . . . . . . . . . . . 17 ⊢ (1 / 1) = 1
180	178, 179	eqtri 2804	. . . . . . . . . . . . . . . 16 ⊢ (1 / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) = 1
181	176, 180	syl6eq 2832	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 0 → ((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) = 1)
182		prod0 15163	. . . . . . . . . . . . . . . 16 ⊢ ∏𝑡 ∈ ∅ (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) = 1
183	182	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 0 → ∏𝑡 ∈ ∅ (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) = 1)
184	181, 183	oveq12d 7000	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 0 → (((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = (1 · 1))
185	184	ad2antlr 715	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 = 0) ∧ 𝑐 ∈ {∅}) → (((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = (1 · 1))
186		1t1e1 11615	. . . . . . . . . . . . . 14 ⊢ (1 · 1) = 1
187	186	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 = 0) ∧ 𝑐 ∈ {∅}) → (1 · 1) = 1)
188	185, 187	eqtrd 2816	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 = 0) ∧ 𝑐 ∈ {∅}) → (((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = 1)
189	188	sumeq2dv 14926	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 = 0) → Σ𝑐 ∈ {∅} (((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ {∅}1)
190		ax-1cn 10399	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
191		eqidd 2781	. . . . . . . . . . . . . 14 ⊢ (𝑐 = ∅ → 1 = 1)
192	191	sumsn 14967	. . . . . . . . . . . . 13 ⊢ ((∅ ∈ V ∧ 1 ∈ ℂ) → Σ𝑐 ∈ {∅}1 = 1)
193	116, 190, 192	mp2an 680	. . . . . . . . . . . 12 ⊢ Σ𝑐 ∈ {∅}1 = 1
194	193	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 = 0) → Σ𝑐 ∈ {∅}1 = 1)
195	171, 189, 194	3eqtrd 2820	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 = 0) → Σ𝑐 ∈ ((𝐷‘∅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = 1)
196	195	mpteq2dv 5028	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 = 0) → (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘∅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) = (𝑥 ∈ 𝑋 ↦ 1))
197	196	eqcomd 2786	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 = 0) → (𝑥 ∈ 𝑋 ↦ 1) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘∅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))
198	75, 102, 197	3eqtrd 2820	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 = 0) → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘∅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))
199	198	a1d 25	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 0) → (𝑘 ∈ (0...𝑁) → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘∅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))))
200	71	fveq1i 6505	. . . . . . . . 9 ⊢ ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥)))‘𝑘) = ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 1))‘𝑘)
201	200	a1i 11	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥)))‘𝑘) = ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 1))‘𝑘))
202	76	adantr 473	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝑘 = 0) → 𝑆 ∈ {ℝ, ℂ})
203	202	adantr 473	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) → 𝑆 ∈ {ℝ, ℂ})
204	89	adantr 473	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝑘 = 0) → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
205	204	adantr 473	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
206	190	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) → 1 ∈ ℂ)
207		elfznn0 12822	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ₀)
208	207	adantl 474	. . . . . . . . . . . 12 ⊢ ((¬ 𝑘 = 0 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ₀)
209		neqne 2977	. . . . . . . . . . . . 13 ⊢ (¬ 𝑘 = 0 → 𝑘 ≠ 0)
210	209	adantr 473	. . . . . . . . . . . 12 ⊢ ((¬ 𝑘 = 0 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ≠ 0)
211	208, 210	jca 504	. . . . . . . . . . 11 ⊢ ((¬ 𝑘 = 0 ∧ 𝑘 ∈ (0...𝑁)) → (𝑘 ∈ ℕ₀ ∧ 𝑘 ≠ 0))
212		elnnne0 11729	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ ↔ (𝑘 ∈ ℕ₀ ∧ 𝑘 ≠ 0))
213	211, 212	sylibr 226	. . . . . . . . . 10 ⊢ ((¬ 𝑘 = 0 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ)
214	213	adantll 702	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ)
215	203, 205, 206, 214	dvnmptconst 41691	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 1))‘𝑘) = (𝑥 ∈ 𝑋 ↦ 0))
216	140	ad2antrr 714	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) → (𝐷‘∅) = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛}))
217		eqeq2 2791	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑘 → (Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘))
218	217	rabbidv 3405	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑘 → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘})
219	218	adantl 474	. . . . . . . . . . . . . . 15 ⊢ ((¬ 𝑘 = 0 ∧ 𝑛 = 𝑘) → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘})
220		eqidd 2781	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘 → 𝑘 = 𝑘)
221		id 22	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘 → Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘)
222	221	eqcomd 2786	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘 → 𝑘 = Σ𝑡 ∈ ∅ (𝑐‘𝑡))
223	155	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘 → Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0)
224	220, 222, 223	3eqtrd 2820	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘 → 𝑘 = 0)
225	224	adantl 474	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑐 ∈ {∅} ∧ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘) → 𝑘 = 0)
226	225	adantll 702	. . . . . . . . . . . . . . . . . . 19 ⊢ (((¬ 𝑘 = 0 ∧ 𝑐 ∈ {∅}) ∧ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘) → 𝑘 = 0)
227		simpll 755	. . . . . . . . . . . . . . . . . . 19 ⊢ (((¬ 𝑘 = 0 ∧ 𝑐 ∈ {∅}) ∧ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘) → ¬ 𝑘 = 0)
228	226, 227	pm2.65da 805	. . . . . . . . . . . . . . . . . 18 ⊢ ((¬ 𝑘 = 0 ∧ 𝑐 ∈ {∅}) → ¬ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘)
229	228	ralrimiva 3134	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑘 = 0 → ∀𝑐 ∈ {∅} ¬ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘)
230		rabeq0 4227	. . . . . . . . . . . . . . . . 17 ⊢ ({𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘} = ∅ ↔ ∀𝑐 ∈ {∅} ¬ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘)
231	229, 230	sylibr 226	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑘 = 0 → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘} = ∅)
232	231	adantr 473	. . . . . . . . . . . . . . 15 ⊢ ((¬ 𝑘 = 0 ∧ 𝑛 = 𝑘) → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘} = ∅)
233	219, 232	eqtrd 2816	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝑘 = 0 ∧ 𝑛 = 𝑘) → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛} = ∅)
234	233	adantll 702	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑛 = 𝑘) → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛} = ∅)
235	234	adantlr 703	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 = 𝑘) → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛} = ∅)
236	207	adantl 474	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ₀)
237	116	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) → ∅ ∈ V)
238	216, 235, 236, 237	fvmptd 6607	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) → ((𝐷‘∅)‘𝑘) = ∅)
239	238	sumeq1d 14924	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) → Σ𝑐 ∈ ((𝐷‘∅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ ∅ (((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
240		sum0 14944	. . . . . . . . . . 11 ⊢ Σ𝑐 ∈ ∅ (((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = 0
241	240	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) → Σ𝑐 ∈ ∅ (((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = 0)
242	239, 241	eqtr2d 2817	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) → 0 = Σ𝑐 ∈ ((𝐷‘∅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
243	242	mpteq2dv 5028	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) → (𝑥 ∈ 𝑋 ↦ 0) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘∅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))
244	201, 215, 243	3eqtrd 2820	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘∅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))
245	244	ex 405	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑘 = 0) → (𝑘 ∈ (0...𝑁) → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘∅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))))
246	199, 245	pm2.61dan 801	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ (0...𝑁) → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘∅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))))
247	246	ralrimiv 3133	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘∅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))
248		simpll 755	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) ∧ ∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) ∧ 𝑗 ∈ (0...𝑁)) → (𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))))
249		fveq2 6504	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → ((𝐻‘𝑡)‘𝑥) = ((𝐻‘𝑡)‘𝑦))
250	249	prodeq2ad 41339	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥) = ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑦))
251		fveq2 6504	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑢 → (𝐻‘𝑡) = (𝐻‘𝑢))
252	251	fveq1d 6506	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑢 → ((𝐻‘𝑡)‘𝑦) = ((𝐻‘𝑢)‘𝑦))
253	252	cbvprodv 15136	. . . . . . . . . . . . . . . . 17 ⊢ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑦) = ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)
254	253	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑦) = ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦))
255	250, 254	eqtrd 2816	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥) = ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦))
256	255	cbvmptv 5033	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)) = (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦))
257	256	oveq2i 6993	. . . . . . . . . . . . 13 ⊢ (𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥))) = (𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))
258	257	fveq1i 6505	. . . . . . . . . . . 12 ⊢ ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘)
259		fveq2 6504	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑢 → (𝑐‘𝑡) = (𝑐‘𝑢))
260	259	fveq2d 6508	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑢 → (!‘(𝑐‘𝑡)) = (!‘(𝑐‘𝑢)))
261	260	cbvprodv 15136	. . . . . . . . . . . . . . . . . 18 ⊢ ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡)) = ∏𝑢 ∈ 𝑟 (!‘(𝑐‘𝑢))
262	261	oveq2i 6993	. . . . . . . . . . . . . . . . 17 ⊢ ((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) = ((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑐‘𝑢)))
263	262	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → ((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) = ((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑐‘𝑢))))
264		fveq2 6504	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) = (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑦))
265	264	prodeq2ad 41339	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) = ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑦))
266	251	oveq2d 6998	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑢 → (𝑆 D^𝑛 (𝐻‘𝑡)) = (𝑆 D^𝑛 (𝐻‘𝑢)))
267	266, 259	fveq12d 6511	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑢 → ((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)) = ((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑐‘𝑢)))
268	267	fveq1d 6506	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑢 → (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑦) = (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑐‘𝑢))‘𝑦))
269	268	cbvprodv 15136	. . . . . . . . . . . . . . . . . 18 ⊢ ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑦) = ∏𝑢 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑐‘𝑢))‘𝑦)
270	269	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑦) = ∏𝑢 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑐‘𝑢))‘𝑦))
271	265, 270	eqtrd 2816	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) = ∏𝑢 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑐‘𝑢))‘𝑦))
272	263, 271	oveq12d 7000	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = (((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑐‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑐‘𝑢))‘𝑦)))
273	272	sumeq2sdv 14927	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑐‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑐‘𝑢))‘𝑦)))
274		fveq1 6503	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 = 𝑑 → (𝑐‘𝑢) = (𝑑‘𝑢))
275	274	fveq2d 6508	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 = 𝑑 → (!‘(𝑐‘𝑢)) = (!‘(𝑑‘𝑢)))
276	275	prodeq2ad 41339	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = 𝑑 → ∏𝑢 ∈ 𝑟 (!‘(𝑐‘𝑢)) = ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢)))
277	276	oveq2d 6998	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = 𝑑 → ((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑐‘𝑢))) = ((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))))
278	274	fveq2d 6508	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 = 𝑑 → ((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑐‘𝑢)) = ((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢)))
279	278	fveq1d 6506	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = 𝑑 → (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑐‘𝑢))‘𝑦) = (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦))
280	279	prodeq2ad 41339	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = 𝑑 → ∏𝑢 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑐‘𝑢))‘𝑦) = ∏𝑢 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦))
281	277, 280	oveq12d 7000	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = 𝑑 → (((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑐‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑐‘𝑢))‘𝑦)) = (((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦)))
282	281	cbvsumv 14919	. . . . . . . . . . . . . . 15 ⊢ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑐‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑐‘𝑢))‘𝑦)) = Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦))
283	282	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑐‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑐‘𝑢))‘𝑦)) = Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦)))
284	273, 283	eqtrd 2816	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦)))
285	284	cbvmptv 5033	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦)))
286	258, 285	eqeq12i 2794	. . . . . . . . . . 11 ⊢ (((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) ↔ ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦))))
287	286	ralbii 3117	. . . . . . . . . 10 ⊢ (∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) ↔ ∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦))))
288	287	biimpi 208	. . . . . . . . 9 ⊢ (∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) → ∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦))))
289	288	ad2antlr 715	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) ∧ ∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) ∧ 𝑗 ∈ (0...𝑁)) → ∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦))))
290		simpr 477	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) ∧ ∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) ∧ 𝑗 ∈ (0...𝑁)) → 𝑗 ∈ (0...𝑁))
291	76	ad3antrrr 718	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) ∧ ∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦)))) ∧ 𝑗 ∈ (0...𝑁)) → 𝑆 ∈ {ℝ, ℂ})
292	89	ad3antrrr 718	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) ∧ ∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦)))) ∧ 𝑗 ∈ (0...𝑁)) → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
293		dvnprodlem3.t	. . . . . . . . . 10 ⊢ (𝜑 → 𝑇 ∈ Fin)
294	293	ad3antrrr 718	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) ∧ ∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦)))) ∧ 𝑗 ∈ (0...𝑁)) → 𝑇 ∈ Fin)
295		simp-4l 771	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) ∧ ∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦)))) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → 𝜑)
296		simpr 477	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) ∧ ∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦)))) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
297		dvnprodlem3.h	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡):𝑋⟶ℂ)
298	295, 296, 297	syl2anc 576	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) ∧ ∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦)))) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡):𝑋⟶ℂ)
299		dvnprodlem3.n	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
300	299	ad3antrrr 718	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) ∧ ∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦)))) ∧ 𝑗 ∈ (0...𝑁)) → 𝑁 ∈ ℕ₀)
301		simplll 763	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) ∧ ∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦)))) ∧ 𝑗 ∈ (0...𝑁)) → 𝜑)
302	301	3ad2ant1 1114	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) ∧ ∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦)))) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇 ∧ ℎ ∈ (0...𝑁)) → 𝜑)
303		simp2 1118	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) ∧ ∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦)))) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇 ∧ ℎ ∈ (0...𝑁)) → 𝑡 ∈ 𝑇)
304		simp3 1119	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) ∧ ∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦)))) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇 ∧ ℎ ∈ (0...𝑁)) → ℎ ∈ (0...𝑁))
305		eleq1w 2850	. . . . . . . . . . . . 13 ⊢ (𝑗 = ℎ → (𝑗 ∈ (0...𝑁) ↔ ℎ ∈ (0...𝑁)))
306	305	3anbi3d 1422	. . . . . . . . . . . 12 ⊢ (𝑗 = ℎ → ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑡 ∈ 𝑇 ∧ ℎ ∈ (0...𝑁))))
307		fveq2 6504	. . . . . . . . . . . . 13 ⊢ (𝑗 = ℎ → ((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑗) = ((𝑆 D^𝑛 (𝐻‘𝑡))‘ℎ))
308	307	feq1d 6334	. . . . . . . . . . . 12 ⊢ (𝑗 = ℎ → (((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑗):𝑋⟶ℂ ↔ ((𝑆 D^𝑛 (𝐻‘𝑡))‘ℎ):𝑋⟶ℂ))
309	306, 308	imbi12d 337	. . . . . . . . . . 11 ⊢ (𝑗 = ℎ → (((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑗):𝑋⟶ℂ) ↔ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ ℎ ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘ℎ):𝑋⟶ℂ)))
310		dvnprodlem3.dvnh	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑗):𝑋⟶ℂ)
311	309, 310	chvarv 2328	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ ℎ ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘ℎ):𝑋⟶ℂ)
312	302, 303, 304, 311	syl3anc 1352	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) ∧ ∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦)))) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇 ∧ ℎ ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘ℎ):𝑋⟶ℂ)
313		simprl 759	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) → 𝑟 ⊆ 𝑇)
314	313	ad2antrr 714	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) ∧ ∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦)))) ∧ 𝑗 ∈ (0...𝑁)) → 𝑟 ⊆ 𝑇)
315		simprr 761	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) → 𝑧 ∈ (𝑇 ∖ 𝑟))
316	315	ad2antrr 714	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) ∧ ∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦)))) ∧ 𝑗 ∈ (0...𝑁)) → 𝑧 ∈ (𝑇 ∖ 𝑟))
317	257	eqcomi 2789	. . . . . . . . . . . . . . 15 ⊢ (𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦))) = (𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)))
318	317	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑙 → (𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦))) = (𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥))))
319		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑙 → 𝑘 = 𝑙)
320	318, 319	fveq12d 6511	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑙 → ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)))‘𝑙))
321	285	eqcomi 2789	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦))) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
322	321	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑙 → (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦))) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))
323		fveq2 6504	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑙 → (!‘𝑘) = (!‘𝑙))
324	323	oveq1d 6997	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑙 → ((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) = ((!‘𝑙) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))))
325	324	oveq1d 6997	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑙 → (((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = (((!‘𝑙) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
326	325	sumeq2sdv 14927	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑙 → Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑙) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
327		fveq2 6504	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑙 → ((𝐷‘𝑟)‘𝑘) = ((𝐷‘𝑟)‘𝑙))
328	327	sumeq1d 14924	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑙 → Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑙) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑙)(((!‘𝑙) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
329	326, 328	eqtrd 2816	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑙 → Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑙)(((!‘𝑙) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
330	329	mpteq2dv 5028	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑙 → (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑙)(((!‘𝑙) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))
331	322, 330	eqtrd 2816	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑙 → (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦))) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑙)(((!‘𝑙) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))
332	320, 331	eqeq12d 2795	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑙 → (((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦))) ↔ ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)))‘𝑙) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑙)(((!‘𝑙) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))))
333	332	cbvralv 3385	. . . . . . . . . . 11 ⊢ (∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦))) ↔ ∀𝑙 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)))‘𝑙) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑙)(((!‘𝑙) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))
334	333	biimpi 208	. . . . . . . . . 10 ⊢ (∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦))) → ∀𝑙 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)))‘𝑙) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑙)(((!‘𝑙) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))
335	334	ad2antlr 715	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) ∧ ∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦)))) ∧ 𝑗 ∈ (0...𝑁)) → ∀𝑙 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)))‘𝑙) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑙)(((!‘𝑙) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))
336		simpr 477	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) ∧ ∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦)))) ∧ 𝑗 ∈ (0...𝑁)) → 𝑗 ∈ (0...𝑁))
337		fveq1 6503	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑐 → (𝑑‘𝑧) = (𝑐‘𝑧))
338	337	oveq2d 6998	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑐 → (𝑗 − (𝑑‘𝑧)) = (𝑗 − (𝑐‘𝑧)))
339		reseq1 5694	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑐 → (𝑑 ↾ 𝑟) = (𝑐 ↾ 𝑟))
340	338, 339	opeq12d 4690	. . . . . . . . . 10 ⊢ (𝑑 = 𝑐 → ⟨(𝑗 − (𝑑‘𝑧)), (𝑑 ↾ 𝑟)⟩ = ⟨(𝑗 − (𝑐‘𝑧)), (𝑐 ↾ 𝑟)⟩)
341	340	cbvmptv 5033	. . . . . . . . 9 ⊢ (𝑑 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑗) ↦ ⟨(𝑗 − (𝑑‘𝑧)), (𝑑 ↾ 𝑟)⟩) = (𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑗) ↦ ⟨(𝑗 − (𝑐‘𝑧)), (𝑐 ↾ 𝑟)⟩)
342	291, 292, 294, 298, 300, 312, 105, 314, 316, 335, 336, 341	dvnprodlem2 41697	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) ∧ ∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦)))) ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ (𝑟 ∪ {𝑧})((𝐻‘𝑡)‘𝑥)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑗)(((!‘𝑗) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))
343	248, 289, 290, 342	syl21anc 826	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) ∧ ∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ (𝑟 ∪ {𝑧})((𝐻‘𝑡)‘𝑥)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑗)(((!‘𝑗) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))
344	343	ralrimiva 3134	. . . . . 6 ⊢ (((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) ∧ ∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) → ∀𝑗 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ (𝑟 ∪ {𝑧})((𝐻‘𝑡)‘𝑥)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑗)(((!‘𝑗) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))
345		fveq2 6504	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ (𝑟 ∪ {𝑧})((𝐻‘𝑡)‘𝑥)))‘𝑗) = ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ (𝑟 ∪ {𝑧})((𝐻‘𝑡)‘𝑥)))‘𝑘))
346		fveq2 6504	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑘 → (!‘𝑗) = (!‘𝑘))
347	346	oveq1d 6997	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → ((!‘𝑗) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) = ((!‘𝑘) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))))
348	347	oveq1d 6997	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → (((!‘𝑗) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = (((!‘𝑘) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
349	348	sumeq2sdv 14927	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑗)(((!‘𝑗) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑗)(((!‘𝑘) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
350		fveq2 6504	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑗) = ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑘))
351	350	sumeq1d 14924	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑗)(((!‘𝑘) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
352	349, 351	eqtrd 2816	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑗)(((!‘𝑗) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
353	352	mpteq2dv 5028	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑗)(((!‘𝑗) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))
354	345, 353	eqeq12d 2795	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ (𝑟 ∪ {𝑧})((𝐻‘𝑡)‘𝑥)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑗)(((!‘𝑗) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) ↔ ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ (𝑟 ∪ {𝑧})((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))))
355	354	cbvralv 3385	. . . . . 6 ⊢ (∀𝑗 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ (𝑟 ∪ {𝑧})((𝐻‘𝑡)‘𝑥)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑗)(((!‘𝑗) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) ↔ ∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ (𝑟 ∪ {𝑧})((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))
356	344, 355	sylib 210	. . . . 5 ⊢ (((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) ∧ ∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) → ∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ (𝑟 ∪ {𝑧})((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))
357	356	ex 405	. . . 4 ⊢ ((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) → (∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) → ∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ (𝑟 ∪ {𝑧})((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))))
358	16, 32, 48, 68, 247, 357, 293	findcard2d 8561	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 𝐹)‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑇)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))
359		nn0uz 12100	. . . . 5 ⊢ ℕ₀ = (ℤ_≥‘0)
360	299, 359	syl6eleq 2878	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘0))
361		eluzfz2 12737	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘0) → 𝑁 ∈ (0...𝑁))
362	360, 361	syl 17	. . 3 ⊢ (𝜑 → 𝑁 ∈ (0...𝑁))
363		fveq2 6504	. . . . 5 ⊢ (𝑘 = 𝑁 → ((𝑆 D^𝑛 𝐹)‘𝑘) = ((𝑆 D^𝑛 𝐹)‘𝑁))
364		fveq2 6504	. . . . . . . 8 ⊢ (𝑘 = 𝑁 → ((𝐷‘𝑇)‘𝑘) = ((𝐷‘𝑇)‘𝑁))
365	364	sumeq1d 14924	. . . . . . 7 ⊢ (𝑘 = 𝑁 → Σ𝑐 ∈ ((𝐷‘𝑇)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ ((𝐷‘𝑇)‘𝑁)(((!‘𝑘) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
366		fveq2 6504	. . . . . . . . . 10 ⊢ (𝑘 = 𝑁 → (!‘𝑘) = (!‘𝑁))
367	366	oveq1d 6997	. . . . . . . . 9 ⊢ (𝑘 = 𝑁 → ((!‘𝑘) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) = ((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))))
368	367	oveq1d 6997	. . . . . . . 8 ⊢ (𝑘 = 𝑁 → (((!‘𝑘) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = (((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
369	368	sumeq2sdv 14927	. . . . . . 7 ⊢ (𝑘 = 𝑁 → Σ𝑐 ∈ ((𝐷‘𝑇)‘𝑁)(((!‘𝑘) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ ((𝐷‘𝑇)‘𝑁)(((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
370	365, 369	eqtrd 2816	. . . . . 6 ⊢ (𝑘 = 𝑁 → Σ𝑐 ∈ ((𝐷‘𝑇)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ ((𝐷‘𝑇)‘𝑁)(((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
371	370	mpteq2dv 5028	. . . . 5 ⊢ (𝑘 = 𝑁 → (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑇)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑇)‘𝑁)(((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))
372	363, 371	eqeq12d 2795	. . . 4 ⊢ (𝑘 = 𝑁 → (((𝑆 D^𝑛 𝐹)‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑇)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) ↔ ((𝑆 D^𝑛 𝐹)‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑇)‘𝑁)(((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))))
373	372	rspccva 3536	. . 3 ⊢ ((∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 𝐹)‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑇)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) ∧ 𝑁 ∈ (0...𝑁)) → ((𝑆 D^𝑛 𝐹)‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑇)‘𝑁)(((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))
374	358, 362, 373	syl2anc 576	. 2 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑇)‘𝑁)(((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))
375		oveq2 6990	. . . . . . . . . 10 ⊢ (𝑠 = 𝑇 → ((0...𝑛) ↑_𝑚 𝑠) = ((0...𝑛) ↑_𝑚 𝑇))
376		rabeq 3408	. . . . . . . . . 10 ⊢ (((0...𝑛) ↑_𝑚 𝑠) = ((0...𝑛) ↑_𝑚 𝑇) → {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑇) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛})
377	375, 376	syl 17	. . . . . . . . 9 ⊢ (𝑠 = 𝑇 → {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑇) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛})
378		sumeq1 14912	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑇 → Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = Σ𝑡 ∈ 𝑇 (𝑐‘𝑡))
379	378	eqeq1d 2782	. . . . . . . . . 10 ⊢ (𝑠 = 𝑇 → (Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ 𝑇 (𝑐‘𝑡) = 𝑛))
380	379	rabbidv 3405	. . . . . . . . 9 ⊢ (𝑠 = 𝑇 → {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑇) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑇) ∣ Σ𝑡 ∈ 𝑇 (𝑐‘𝑡) = 𝑛})
381	377, 380	eqtrd 2816	. . . . . . . 8 ⊢ (𝑠 = 𝑇 → {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑇) ∣ Σ𝑡 ∈ 𝑇 (𝑐‘𝑡) = 𝑛})
382	381	mpteq2dv 5028	. . . . . . 7 ⊢ (𝑠 = 𝑇 → (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}) = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑇) ∣ Σ𝑡 ∈ 𝑇 (𝑐‘𝑡) = 𝑛}))
383		pwidg 4440	. . . . . . . 8 ⊢ (𝑇 ∈ Fin → 𝑇 ∈ 𝒫 𝑇)
384	293, 383	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ 𝒫 𝑇)
385	137	mptex 6818	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑇) ∣ Σ𝑡 ∈ 𝑇 (𝑐‘𝑡) = 𝑛}) ∈ V
386	385	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑇) ∣ Σ𝑡 ∈ 𝑇 (𝑐‘𝑡) = 𝑛}) ∈ V)
387	105, 382, 384, 386	fvmptd3 6623	. . . . . 6 ⊢ (𝜑 → (𝐷‘𝑇) = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑇) ∣ Σ𝑡 ∈ 𝑇 (𝑐‘𝑡) = 𝑛}))
388		dvnprodlem3.c	. . . . . . 7 ⊢ 𝐶 = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑇) ∣ Σ𝑡 ∈ 𝑇 (𝑐‘𝑡) = 𝑛})
389	388	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐶 = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑇) ∣ Σ𝑡 ∈ 𝑇 (𝑐‘𝑡) = 𝑛}))
390	387, 389	eqtr4d 2819	. . . . 5 ⊢ (𝜑 → (𝐷‘𝑇) = 𝐶)
391	390	fveq1d 6506	. . . 4 ⊢ (𝜑 → ((𝐷‘𝑇)‘𝑁) = (𝐶‘𝑁))
392	391	sumeq1d 14924	. . 3 ⊢ (𝜑 → Σ𝑐 ∈ ((𝐷‘𝑇)‘𝑁)(((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ (𝐶‘𝑁)(((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
393	392	mpteq2dv 5028	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑇)‘𝑁)(((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ (𝐶‘𝑁)(((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))
394	374, 393	eqtrd 2816	1 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ (𝐶‘𝑁)(((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 387 ∨ wo 834 ∧ w3a 1069 ∀wal 1506 = wceq 1508 ∈ wcel 2051 ≠ wne 2969 ∀wral 3090 {crab 3094 Vcvv 3417 ∖ cdif 3828 ∪ cun 3829 ⊆ wss 3831 ∅c0 4181 ifcif 4353 𝒫 cpw 4425 {csn 4444 {cpr 4446 ⟨cop 4450 ↦ cmpt 5013 dom cdm 5411 ↾ cres 5413 Fn wfn 6188 ⟶wf 6189 ‘cfv 6193 (class class class)co 6982 ↑_𝑚 cmap 8212 ↑_pm cpm 8213 Fincfn 8312 ℂcc 10339 ℝcr 10340 0cc0 10341 1c1 10342 · cmul 10346 − cmin 10676 / cdiv 11104 ℕcn 11445 ℕ₀cn0 11713 ℤ_≥cuz 12064 ...cfz 12714 !cfa 13454 Σcsu 14909 ∏cprod 15125 ↾_t crest 16556 TopOpenctopn 16557 ℂ_fldccnfld 20262 D^𝑛 cdvn 24180

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2752 ax-rep 5053 ax-sep 5064 ax-nul 5071 ax-pow 5123 ax-pr 5190 ax-un 7285 ax-inf2 8904 ax-cnex 10397 ax-resscn 10398 ax-1cn 10399 ax-icn 10400 ax-addcl 10401 ax-addrcl 10402 ax-mulcl 10403 ax-mulrcl 10404 ax-mulcom 10405 ax-addass 10406 ax-mulass 10407 ax-distr 10408 ax-i2m1 10409 ax-1ne0 10410 ax-1rid 10411 ax-rnegex 10412 ax-rrecex 10413 ax-cnre 10414 ax-pre-lttri 10415 ax-pre-lttrn 10416 ax-pre-ltadd 10417 ax-pre-mulgt0 10418 ax-pre-sup 10419 ax-addf 10420 ax-mulf 10421

This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-fal 1521 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2551 df-eu 2589 df-clab 2761 df-cleq 2773 df-clel 2848 df-nfc 2920 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3419 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4182 df-if 4354 df-pw 4427 df-sn 4445 df-pr 4447 df-tp 4449 df-op 4451 df-uni 4718 df-int 4755 df-iun 4799 df-iin 4800 df-br 4935 df-opab 4997 df-mpt 5014 df-tr 5036 df-id 5316 df-eprel 5321 df-po 5330 df-so 5331 df-fr 5370 df-se 5371 df-we 5372 df-xp 5417 df-rel 5418 df-cnv 5419 df-co 5420 df-dm 5421 df-rn 5422 df-res 5423 df-ima 5424 df-pred 5991 df-ord 6037 df-on 6038 df-lim 6039 df-suc 6040 df-iota 6157 df-fun 6195 df-fn 6196 df-f 6197 df-f1 6198 df-fo 6199 df-f1o 6200 df-fv 6201 df-isom 6202 df-riota 6943 df-ov 6985 df-oprab 6986 df-mpo 6987 df-of 7233 df-om 7403 df-1st 7507 df-2nd 7508 df-supp 7640 df-wrecs 7756 df-recs 7818 df-rdg 7856 df-1o 7911 df-2o 7912 df-oadd 7915 df-er 8095 df-map 8214 df-pm 8215 df-ixp 8266 df-en 8313 df-dom 8314 df-sdom 8315 df-fin 8316 df-fsupp 8635 df-fi 8676 df-sup 8707 df-inf 8708 df-oi 8775 df-card 9168 df-cda 9394 df-pnf 10482 df-mnf 10483 df-xr 10484 df-ltxr 10485 df-le 10486 df-sub 10678 df-neg 10679 df-div 11105 df-nn 11446 df-2 11509 df-3 11510 df-4 11511 df-5 11512 df-6 11513 df-7 11514 df-8 11515 df-9 11516 df-n0 11714 df-z 11800 df-dec 11918 df-uz 12065 df-q 12169 df-rp 12211 df-xneg 12330 df-xadd 12331 df-xmul 12332 df-ico 12566 df-icc 12567 df-fz 12715 df-fzo 12856 df-seq 13191 df-exp 13251 df-fac 13455 df-bc 13484 df-hash 13512 df-cj 14325 df-re 14326 df-im 14327 df-sqrt 14461 df-abs 14462 df-clim 14712 df-sum 14910 df-prod 15126 df-struct 16347 df-ndx 16348 df-slot 16349 df-base 16351 df-sets 16352 df-ress 16353 df-plusg 16440 df-mulr 16441 df-starv 16442 df-sca 16443 df-vsca 16444 df-ip 16445 df-tset 16446 df-ple 16447 df-ds 16449 df-unif 16450 df-hom 16451 df-cco 16452 df-rest 16558 df-topn 16559 df-0g 16577 df-gsum 16578 df-topgen 16579 df-pt 16580 df-prds 16583 df-xrs 16637 df-qtop 16642 df-imas 16643 df-xps 16645 df-mre 16727 df-mrc 16728 df-acs 16730 df-mgm 17722 df-sgrp 17764 df-mnd 17775 df-submnd 17816 df-mulg 18024 df-cntz 18230 df-cmn 18680 df-psmet 20254 df-xmet 20255 df-met 20256 df-bl 20257 df-mopn 20258 df-fbas 20259 df-fg 20260 df-cnfld 20263 df-top 21221 df-topon 21238 df-topsp 21260 df-bases 21273 df-cld 21346 df-ntr 21347 df-cls 21348 df-nei 21425 df-lp 21463 df-perf 21464 df-cn 21554 df-cnp 21555 df-haus 21642 df-tx 21889 df-hmeo 22082 df-fil 22173 df-fm 22265 df-flim 22266 df-flf 22267 df-xms 22648 df-ms 22649 df-tms 22650 df-cncf 23204 df-limc 24182 df-dv 24183 df-dvn 24184

This theorem is referenced by: dvnprod 41699

W3C validator