dvnprodlem3 - Mathbox for Glauco Siliprandi

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Theorem dvnprodlem3 40163

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 𝑘 𝑑 ℎ 𝑙 𝑟 𝑧 𝑦 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prodeq1 14639	. . . . . . . . 9 ⊢ (𝑠 = ∅ → ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥) = ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥))
2	1	mpteq2dv 4745	. . . . . . . 8 ⊢ (𝑠 = ∅ → (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥)) = (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥)))
3	2	oveq2d 6666	. . . . . . 7 ⊢ (𝑠 = ∅ → (𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥))) = (𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥))))
4	3	fveq1d 6193	. . . . . 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 6191	. . . . . . . . . 10 ⊢ (𝑠 = ∅ → (𝐷‘𝑠) = (𝐷‘∅))
6	5	fveq1d 6193	. . . . . . . . 9 ⊢ (𝑠 = ∅ → ((𝐷‘𝑠)‘𝑘) = ((𝐷‘∅)‘𝑘))
7	6	sumeq1d 14431	. . . . . . . 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 14639	. . . . . . . . . . 11 ⊢ (𝑠 = ∅ → ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡)) = ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡)))
9	8	oveq2d 6666	. . . . . . . . . 10 ⊢ (𝑠 = ∅ → ((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) = ((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))))
10		prodeq1 14639	. . . . . . . . . 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 6668	. . . . . . . . 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	sumeq2ad 14434	. . . . . . . 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 2656	. . . . . . 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 4745	. . . . . 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 2637	. . . . 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 2986	. . . 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 14639	. . . . . . . . 9 ⊢ (𝑠 = 𝑟 → ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥) = ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥))
18	17	mpteq2dv 4745	. . . . . . . 8 ⊢ (𝑠 = 𝑟 → (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥)) = (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)))
19	18	oveq2d 6666	. . . . . . 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 6193	. . . . . 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 6191	. . . . . . . . . 10 ⊢ (𝑠 = 𝑟 → (𝐷‘𝑠) = (𝐷‘𝑟))
22	21	fveq1d 6193	. . . . . . . . 9 ⊢ (𝑠 = 𝑟 → ((𝐷‘𝑠)‘𝑘) = ((𝐷‘𝑟)‘𝑘))
23	22	sumeq1d 14431	. . . . . . . 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 14639	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑟 → ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡)) = ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡)))
25	24	oveq2d 6666	. . . . . . . . . 10 ⊢ (𝑠 = 𝑟 → ((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) = ((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))))
26		prodeq1 14639	. . . . . . . . . 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 6668	. . . . . . . . 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	sumeq2ad 14434	. . . . . . . 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 2656	. . . . . . 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 4745	. . . . . 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 2637	. . . . 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 2986	. . . 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 14639	. . . . . . . . 9 ⊢ (𝑠 = (𝑟 ∪ {𝑧}) → ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥) = ∏𝑡 ∈ (𝑟 ∪ {𝑧})((𝐻‘𝑡)‘𝑥))
34	33	mpteq2dv 4745	. . . . . . . 8 ⊢ (𝑠 = (𝑟 ∪ {𝑧}) → (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥)) = (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ (𝑟 ∪ {𝑧})((𝐻‘𝑡)‘𝑥)))
35	34	oveq2d 6666	. . . . . . 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 6193	. . . . . 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 6191	. . . . . . . . . 10 ⊢ (𝑠 = (𝑟 ∪ {𝑧}) → (𝐷‘𝑠) = (𝐷‘(𝑟 ∪ {𝑧})))
38	37	fveq1d 6193	. . . . . . . . 9 ⊢ (𝑠 = (𝑟 ∪ {𝑧}) → ((𝐷‘𝑠)‘𝑘) = ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑘))
39	38	sumeq1d 14431	. . . . . . . 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 14639	. . . . . . . . . . 11 ⊢ (𝑠 = (𝑟 ∪ {𝑧}) → ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡)) = ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡)))
41	40	oveq2d 6666	. . . . . . . . . 10 ⊢ (𝑠 = (𝑟 ∪ {𝑧}) → ((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) = ((!‘𝑘) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))))
42		prodeq1 14639	. . . . . . . . . 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 6668	. . . . . . . . 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	sumeq2ad 14434	. . . . . . . 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 2656	. . . . . . 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 4745	. . . . . 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 2637	. . . . 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 2986	. . . 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 14639	. . . . . . . . . 10 ⊢ (𝑠 = 𝑇 → ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥) = ∏𝑡 ∈ 𝑇 ((𝐻‘𝑡)‘𝑥))
50	49	mpteq2dv 4745	. . . . . . . . 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 2628	. . . . . . . . 9 ⊢ (𝑠 = 𝑇 → (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑇 ((𝐻‘𝑡)‘𝑥)) = 𝐹)
54	50, 53	eqtrd 2656	. . . . . . . 8 ⊢ (𝑠 = 𝑇 → (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥)) = 𝐹)
55	54	oveq2d 6666	. . . . . . 7 ⊢ (𝑠 = 𝑇 → (𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥))) = (𝑆 D^𝑛 𝐹))
56	55	fveq1d 6193	. . . . . 6 ⊢ (𝑠 = 𝑇 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = ((𝑆 D^𝑛 𝐹)‘𝑘))
57		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑠 = 𝑇 → (𝐷‘𝑠) = (𝐷‘𝑇))
58	57	fveq1d 6193	. . . . . . . . 9 ⊢ (𝑠 = 𝑇 → ((𝐷‘𝑠)‘𝑘) = ((𝐷‘𝑇)‘𝑘))
59	58	sumeq1d 14431	. . . . . . . 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 14639	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑇 → ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡)) = ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡)))
61	60	oveq2d 6666	. . . . . . . . . 10 ⊢ (𝑠 = 𝑇 → ((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) = ((!‘𝑘) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))))
62		prodeq1 14639	. . . . . . . . . 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 6668	. . . . . . . . 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	sumeq2ad 14434	. . . . . . . 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 2656	. . . . . . 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 4745	. . . . . 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 2637	. . . . 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 2986	. . . 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 14673	. . . . . . . . . . . . 13 ⊢ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥) = 1
70	69	mpteq2i 4741	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥)) = (𝑥 ∈ 𝑋 ↦ 1)
71	70	oveq2i 6661	. . . . . . . . . . 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 6197	. . . . . . . . 9 ⊢ (𝑘 = 0 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥)))‘𝑘) = ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 1))‘0))
75	74	adantl 482	. . . . . . . 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 23668	. . . . . . . . . . 11 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
78	76, 77	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
79		1cnd 10056	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 1 ∈ ℂ)
80		eqid 2622	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑋 ↦ 1) = (𝑥 ∈ 𝑋 ↦ 1)
81	79, 80	fmptd 6385	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 1):𝑋⟶ℂ)
82		1re 10039	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ
83	82	rgenw 2924	. . . . . . . . . . . . . . . 16 ⊢ ∀𝑥 ∈ 𝑋 1 ∈ ℝ
84		dmmptg 5632	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑥 ∈ 𝑋 1 ∈ ℝ → dom (𝑥 ∈ 𝑋 ↦ 1) = 𝑋)
85	83, 84	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ dom (𝑥 ∈ 𝑋 ↦ 1) = 𝑋
86	85	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom (𝑥 ∈ 𝑋 ↦ 1) = 𝑋)
87	86	feq2d 6031	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 1):dom (𝑥 ∈ 𝑋 ↦ 1)⟶ℂ ↔ (𝑥 ∈ 𝑋 ↦ 1):𝑋⟶ℂ))
88	81, 87	mpbird 247	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 1):dom (𝑥 ∈ 𝑋 ↦ 1)⟶ℂ)
89		restsspw 16092	. . . . . . . . . . . . . . 15 ⊢ ((TopOpen‘ℂ_fld) ↾_t 𝑆) ⊆ 𝒫 𝑆
90		dvnprodlem3.x	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
91	89, 90	sseldi 3601	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 ∈ 𝒫 𝑆)
92		elpwi 4168	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ 𝒫 𝑆 → 𝑋 ⊆ 𝑆)
93	91, 92	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
94	86, 93	eqsstrd 3639	. . . . . . . . . . . 12 ⊢ (𝜑 → dom (𝑥 ∈ 𝑋 ↦ 1) ⊆ 𝑆)
95	88, 94	jca 554	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 1):dom (𝑥 ∈ 𝑋 ↦ 1)⟶ℂ ∧ dom (𝑥 ∈ 𝑋 ↦ 1) ⊆ 𝑆))
96		cnex 10017	. . . . . . . . . . . . 13 ⊢ ℂ ∈ V
97	96	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ℂ ∈ V)
98		elpm2g 7874	. . . . . . . . . . . 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)))
99	97, 76, 98	syl2anc 693	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 1) ∈ (ℂ ↑_pm 𝑆) ↔ ((𝑥 ∈ 𝑋 ↦ 1):dom (𝑥 ∈ 𝑋 ↦ 1)⟶ℂ ∧ dom (𝑥 ∈ 𝑋 ↦ 1) ⊆ 𝑆)))
100	95, 99	mpbird 247	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 1) ∈ (ℂ ↑_pm 𝑆))
101		dvn0 23687	. . . . . . . . . 10 ⊢ ((𝑆 ⊆ ℂ ∧ (𝑥 ∈ 𝑋 ↦ 1) ∈ (ℂ ↑_pm 𝑆)) → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 1))‘0) = (𝑥 ∈ 𝑋 ↦ 1))
102	78, 100, 101	syl2anc 693	. . . . . . . . 9 ⊢ (𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 1))‘0) = (𝑥 ∈ 𝑋 ↦ 1))
103	102	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 = 0) → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 1))‘0) = (𝑥 ∈ 𝑋 ↦ 1))
104		fveq2 6191	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 0 → ((𝐷‘∅)‘𝑘) = ((𝐷‘∅)‘0))
105	104	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 = 0) → ((𝐷‘∅)‘𝑘) = ((𝐷‘∅)‘0))
106		dvnprodlem3.d	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐷 = (𝑠 ∈ 𝒫 𝑇 ↦ (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}))
107	106	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐷 = (𝑠 ∈ 𝒫 𝑇 ↦ (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛})))
108		oveq2 6658	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑠 = ∅ → ((0...𝑛) ↑_𝑚 𝑠) = ((0...𝑛) ↑_𝑚 ∅))
109		elmapfn 7880	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑥 ∈ ((0...𝑛) ↑_𝑚 ∅) → 𝑥 Fn ∅)
110		fn0 6011	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑥 Fn ∅ ↔ 𝑥 = ∅)
111	109, 110	sylib 208	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 ∈ ((0...𝑛) ↑_𝑚 ∅) → 𝑥 = ∅)
112		velsn 4193	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
113	111, 112	sylibr 224	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 ∈ ((0...𝑛) ↑_𝑚 ∅) → 𝑥 ∈ {∅})
114	112	biimpi 206	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 ∈ {∅} → 𝑥 = ∅)
115		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑥 = ∅ → 𝑥 = ∅)
116		f0 6086	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ∅:∅⟶(0...𝑛)
117		ovex 6678	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (0...𝑛) ∈ V
118		0ex 4790	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ∅ ∈ V
119	117, 118	elmap 7886	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (∅ ∈ ((0...𝑛) ↑_𝑚 ∅) ↔ ∅:∅⟶(0...𝑛))
120	116, 119	mpbir 221	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ∅ ∈ ((0...𝑛) ↑_𝑚 ∅)
121	120	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑥 = ∅ → ∅ ∈ ((0...𝑛) ↑_𝑚 ∅))
122	115, 121	eqeltrd 2701	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 = ∅ → 𝑥 ∈ ((0...𝑛) ↑_𝑚 ∅))
123	114, 122	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 ∈ {∅} → 𝑥 ∈ ((0...𝑛) ↑_𝑚 ∅))
124	113, 123	impbii 199	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 ∈ ((0...𝑛) ↑_𝑚 ∅) ↔ 𝑥 ∈ {∅})
125	124	ax-gen 1722	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ∀𝑥(𝑥 ∈ ((0...𝑛) ↑_𝑚 ∅) ↔ 𝑥 ∈ {∅})
126		dfcleq 2616	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((0...𝑛) ↑_𝑚 ∅) = {∅} ↔ ∀𝑥(𝑥 ∈ ((0...𝑛) ↑_𝑚 ∅) ↔ 𝑥 ∈ {∅}))
127	125, 126	mpbir 221	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((0...𝑛) ↑_𝑚 ∅) = {∅}
128	127	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑠 = ∅ → ((0...𝑛) ↑_𝑚 ∅) = {∅})
129	108, 128	eqtrd 2656	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑠 = ∅ → ((0...𝑛) ↑_𝑚 𝑠) = {∅})
130		rabeq 3192	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((0...𝑛) ↑_𝑚 𝑠) = {∅} → {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛})
131	129, 130	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 = ∅ → {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛})
132		sumeq1 14419	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑠 = ∅ → Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = Σ𝑡 ∈ ∅ (𝑐‘𝑡))
133	132	eqeq1d 2624	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑠 = ∅ → (Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛))
134	133	rabbidv 3189	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 = ∅ → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛})
135	131, 134	eqtrd 2656	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 = ∅ → {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛})
136	135	mpteq2dv 4745	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 = ∅ → (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}) = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛}))
137	136	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 = ∅) → (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}) = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛}))
138		0elpw 4834	. . . . . . . . . . . . . . . . . 18 ⊢ ∅ ∈ 𝒫 𝑇
139	138	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∅ ∈ 𝒫 𝑇)
140		nn0ex 11298	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ₀ ∈ V
141	140	mptex 6486	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛}) ∈ V
142	141	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛}) ∈ V)
143	107, 137, 139, 142	fvmptd 6288	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐷‘∅) = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛}))
144		eqeq2 2633	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 0 → (Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0))
145	144	rabbidv 3189	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 0 → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0})
146	145	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 = 0) → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0})
147		0nn0 11307	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℕ₀
148	147	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 ∈ ℕ₀)
149		p0ex 4853	. . . . . . . . . . . . . . . . . 18 ⊢ {∅} ∈ V
150	149	rabex 4813	. . . . . . . . . . . . . . . . 17 ⊢ {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0} ∈ V
151	150	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0} ∈ V)
152	143, 146, 148, 151	fvmptd 6288	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐷‘∅)‘0) = {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0})
153	152	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 = 0) → ((𝐷‘∅)‘0) = {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0})
154		snidg 4206	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∅ ∈ V → ∅ ∈ {∅})
155	118, 154	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∅ ∈ {∅}
156		eqid 2622	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0 = 0
157	155, 156	pm3.2i 471	. . . . . . . . . . . . . . . . . . 19 ⊢ (∅ ∈ {∅} ∧ 0 = 0)
158		sum0 14452	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0
159	158	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 = ∅ → Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0)
160	159	eqeq1d 2624	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 = ∅ → (Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0 ↔ 0 = 0))
161	160	elrab 3363	. . . . . . . . . . . . . . . . . . 19 ⊢ (∅ ∈ {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0} ↔ (∅ ∈ {∅} ∧ 0 = 0))
162	157, 161	mpbir 221	. . . . . . . . . . . . . . . . . 18 ⊢ ∅ ∈ {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0}
163	162	n0ii 3922	. . . . . . . . . . . . . . . . 17 ⊢ ¬ {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0} = ∅
164		eqid 2622	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0} = {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0}
165		rabrsn 4259	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0} = {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0} → ({𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0} = ∅ ∨ {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0} = {∅}))
166	164, 165	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ({𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0} = ∅ ∨ {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0} = {∅})
167	163, 166	mtpor 1695	. . . . . . . . . . . . . . . 16 ⊢ {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0} = {∅}
168	167	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 = 0) → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0} = {∅})
169		iftrue 4092	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 0 → if(𝑘 = 0, {∅}, ∅) = {∅})
170	169	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 = 0) → if(𝑘 = 0, {∅}, ∅) = {∅})
171	168, 170	eqtr4d 2659	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 = 0) → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0} = if(𝑘 = 0, {∅}, ∅))
172	105, 153, 171	3eqtrd 2660	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 = 0) → ((𝐷‘∅)‘𝑘) = if(𝑘 = 0, {∅}, ∅))
173	172, 170	eqtrd 2656	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 = 0) → ((𝐷‘∅)‘𝑘) = {∅})
174	173	sumeq1d 14431	. . . . . . . . . . 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)))
175		fveq2 6191	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 0 → (!‘𝑘) = (!‘0))
176		fac0 13063	. . . . . . . . . . . . . . . . . . 19 ⊢ (!‘0) = 1
177	176	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 0 → (!‘0) = 1)
178	175, 177	eqtrd 2656	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 0 → (!‘𝑘) = 1)
179	178	oveq1d 6665	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 0 → ((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) = (1 / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))))
180		prod0 14673	. . . . . . . . . . . . . . . . . 18 ⊢ ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡)) = 1
181	180	oveq2i 6661	. . . . . . . . . . . . . . . . 17 ⊢ (1 / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) = (1 / 1)
182		1div1e1 10717	. . . . . . . . . . . . . . . . 17 ⊢ (1 / 1) = 1
183	181, 182	eqtri 2644	. . . . . . . . . . . . . . . 16 ⊢ (1 / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) = 1
184	179, 183	syl6eq 2672	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 0 → ((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) = 1)
185		prod0 14673	. . . . . . . . . . . . . . . 16 ⊢ ∏𝑡 ∈ ∅ (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) = 1
186	185	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 0 → ∏𝑡 ∈ ∅ (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) = 1)
187	184, 186	oveq12d 6668	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 0 → (((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = (1 · 1))
188	187	ad2antlr 763	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 = 0) ∧ 𝑐 ∈ {∅}) → (((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = (1 · 1))
189		1t1e1 11175	. . . . . . . . . . . . . 14 ⊢ (1 · 1) = 1
190	189	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 = 0) ∧ 𝑐 ∈ {∅}) → (1 · 1) = 1)
191	188, 190	eqtrd 2656	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 = 0) ∧ 𝑐 ∈ {∅}) → (((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = 1)
192	191	sumeq2dv 14433	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 = 0) → Σ𝑐 ∈ {∅} (((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ {∅}1)
193		ax-1cn 9994	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
194		eqidd 2623	. . . . . . . . . . . . . 14 ⊢ (𝑐 = ∅ → 1 = 1)
195	194	sumsn 14475	. . . . . . . . . . . . 13 ⊢ ((∅ ∈ V ∧ 1 ∈ ℂ) → Σ𝑐 ∈ {∅}1 = 1)
196	118, 193, 195	mp2an 708	. . . . . . . . . . . 12 ⊢ Σ𝑐 ∈ {∅}1 = 1
197	196	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 = 0) → Σ𝑐 ∈ {∅}1 = 1)
198	174, 192, 197	3eqtrd 2660	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 = 0) → Σ𝑐 ∈ ((𝐷‘∅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = 1)
199	198	mpteq2dv 4745	. . . . . . . . 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))
200	199	eqcomd 2628	. . . . . . . 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))))
201	75, 103, 200	3eqtrd 2660	. . . . . . 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))))
202	201	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)))))
203	71	fveq1i 6192	. . . . . . . . 9 ⊢ ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥)))‘𝑘) = ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 1))‘𝑘)
204	203	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))
205	76	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝑘 = 0) → 𝑆 ∈ {ℝ, ℂ})
206	205	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) → 𝑆 ∈ {ℝ, ℂ})
207	90	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝑘 = 0) → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
208	207	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
209	193	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) → 1 ∈ ℂ)
210		elfznn0 12433	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ₀)
211	210	adantl 482	. . . . . . . . . . . 12 ⊢ ((¬ 𝑘 = 0 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ₀)
212		neqne 2802	. . . . . . . . . . . . 13 ⊢ (¬ 𝑘 = 0 → 𝑘 ≠ 0)
213	212	adantr 481	. . . . . . . . . . . 12 ⊢ ((¬ 𝑘 = 0 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ≠ 0)
214	211, 213	jca 554	. . . . . . . . . . 11 ⊢ ((¬ 𝑘 = 0 ∧ 𝑘 ∈ (0...𝑁)) → (𝑘 ∈ ℕ₀ ∧ 𝑘 ≠ 0))
215		elnnne0 11306	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ ↔ (𝑘 ∈ ℕ₀ ∧ 𝑘 ≠ 0))
216	214, 215	sylibr 224	. . . . . . . . . 10 ⊢ ((¬ 𝑘 = 0 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ)
217	216	adantll 750	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ)
218	206, 208, 209, 217	dvnmptconst 40156	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 1))‘𝑘) = (𝑥 ∈ 𝑋 ↦ 0))
219	143	ad2antrr 762	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) → (𝐷‘∅) = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛}))
220		eqeq2 2633	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑘 → (Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘))
221	220	rabbidv 3189	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑘 → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘})
222	221	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((¬ 𝑘 = 0 ∧ 𝑛 = 𝑘) → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘})
223		eqidd 2623	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘 → 𝑘 = 𝑘)
224		id 22	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘 → Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘)
225	224	eqcomd 2628	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘 → 𝑘 = Σ𝑡 ∈ ∅ (𝑐‘𝑡))
226	158	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘 → Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0)
227	223, 225, 226	3eqtrd 2660	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘 → 𝑘 = 0)
228	227	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑐 ∈ {∅} ∧ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘) → 𝑘 = 0)
229	228	adantll 750	. . . . . . . . . . . . . . . . . . 19 ⊢ (((¬ 𝑘 = 0 ∧ 𝑐 ∈ {∅}) ∧ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘) → 𝑘 = 0)
230		simpll 790	. . . . . . . . . . . . . . . . . . 19 ⊢ (((¬ 𝑘 = 0 ∧ 𝑐 ∈ {∅}) ∧ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘) → ¬ 𝑘 = 0)
231	229, 230	pm2.65da 600	. . . . . . . . . . . . . . . . . 18 ⊢ ((¬ 𝑘 = 0 ∧ 𝑐 ∈ {∅}) → ¬ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘)
232	231	ralrimiva 2966	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑘 = 0 → ∀𝑐 ∈ {∅} ¬ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘)
233		rabeq0 3957	. . . . . . . . . . . . . . . . 17 ⊢ ({𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘} = ∅ ↔ ∀𝑐 ∈ {∅} ¬ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘)
234	232, 233	sylibr 224	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑘 = 0 → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘} = ∅)
235	234	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((¬ 𝑘 = 0 ∧ 𝑛 = 𝑘) → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘} = ∅)
236	222, 235	eqtrd 2656	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝑘 = 0 ∧ 𝑛 = 𝑘) → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛} = ∅)
237	236	adantll 750	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑛 = 𝑘) → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛} = ∅)
238	237	adantlr 751	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 = 𝑘) → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛} = ∅)
239	210	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ₀)
240	118	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) → ∅ ∈ V)
241	219, 238, 239, 240	fvmptd 6288	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) → ((𝐷‘∅)‘𝑘) = ∅)
242	241	sumeq1d 14431	. . . . . . . . . 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)))
243		sum0 14452	. . . . . . . . . . 11 ⊢ Σ𝑐 ∈ ∅ (((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = 0
244	243	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) → Σ𝑐 ∈ ∅ (((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = 0)
245	242, 244	eqtr2d 2657	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) → 0 = Σ𝑐 ∈ ((𝐷‘∅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
246	245	mpteq2dv 4745	. . . . . . . 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))))
247	204, 218, 246	3eqtrd 2660	. . . . . . 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))))
248	247	ex 450	. . . . . 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)))))
249	202, 248	pm2.61dan 832	. . . . 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)))))
250	249	ralrimiv 2965	. . . 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))))
251		simpll 790	. . . . . . . 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))))
252		fveq2 6191	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → ((𝐻‘𝑡)‘𝑥) = ((𝐻‘𝑡)‘𝑦))
253	252	prodeq2ad 39824	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥) = ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑦))
254		fveq2 6191	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑢 → (𝐻‘𝑡) = (𝐻‘𝑢))
255	254	fveq1d 6193	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑢 → ((𝐻‘𝑡)‘𝑦) = ((𝐻‘𝑢)‘𝑦))
256	255	cbvprodv 14646	. . . . . . . . . . . . . . . . 17 ⊢ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑦) = ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)
257	256	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑦) = ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦))
258	253, 257	eqtrd 2656	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥) = ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦))
259	258	cbvmptv 4750	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)) = (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦))
260	259	oveq2i 6661	. . . . . . . . . . . . 13 ⊢ (𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥))) = (𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))
261	260	fveq1i 6192	. . . . . . . . . . . 12 ⊢ ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘)
262		fveq2 6191	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑢 → (𝑐‘𝑡) = (𝑐‘𝑢))
263	262	fveq2d 6195	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑢 → (!‘(𝑐‘𝑡)) = (!‘(𝑐‘𝑢)))
264	263	cbvprodv 14646	. . . . . . . . . . . . . . . . . 18 ⊢ ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡)) = ∏𝑢 ∈ 𝑟 (!‘(𝑐‘𝑢))
265	264	oveq2i 6661	. . . . . . . . . . . . . . . . 17 ⊢ ((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) = ((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑐‘𝑢)))
266	265	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → ((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) = ((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑐‘𝑢))))
267		fveq2 6191	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) = (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑦))
268	267	prodeq2ad 39824	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) = ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑦))
269	254	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑢 → (𝑆 D^𝑛 (𝐻‘𝑡)) = (𝑆 D^𝑛 (𝐻‘𝑢)))
270	269, 262	fveq12d 6197	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑢 → ((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)) = ((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑐‘𝑢)))
271	270	fveq1d 6193	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑢 → (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑦) = (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑐‘𝑢))‘𝑦))
272	271	cbvprodv 14646	. . . . . . . . . . . . . . . . . 18 ⊢ ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑦) = ∏𝑢 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑐‘𝑢))‘𝑦)
273	272	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))
274	268, 273	eqtrd 2656	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → ∏𝑡 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) = ∏𝑢 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑐‘𝑢))‘𝑦))
275	266, 274	oveq12d 6668	. . . . . . . . . . . . . . 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)))
276	275	sumeq2ad 14434	. . . . . . . . . . . . . 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)))
277		fveq1 6190	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 = 𝑑 → (𝑐‘𝑢) = (𝑑‘𝑢))
278	277	fveq2d 6195	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 = 𝑑 → (!‘(𝑐‘𝑢)) = (!‘(𝑑‘𝑢)))
279	278	prodeq2ad 39824	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = 𝑑 → ∏𝑢 ∈ 𝑟 (!‘(𝑐‘𝑢)) = ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢)))
280	279	oveq2d 6666	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = 𝑑 → ((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑐‘𝑢))) = ((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))))
281	277	fveq2d 6195	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 = 𝑑 → ((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑐‘𝑢)) = ((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢)))
282	281	fveq1d 6193	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = 𝑑 → (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑐‘𝑢))‘𝑦) = (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦))
283	282	prodeq2ad 39824	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = 𝑑 → ∏𝑢 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑐‘𝑢))‘𝑦) = ∏𝑢 ∈ 𝑟 (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦))
284	280, 283	oveq12d 6668	. . . . . . . . . . . . . . . 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)))
285	284	cbvsumv 14426	. . . . . . . . . . . . . . 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))
286	285	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)))
287	276, 286	eqtrd 2656	. . . . . . . . . . . . 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)))
288	287	cbvmptv 4750	. . . . . . . . . . . 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)))
289	261, 288	eqeq12i 2636	. . . . . . . . . . 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))))
290	289	ralbii 2980	. . . . . . . . . 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))))
291	290	biimpi 206	. . . . . . . . 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))))
292	291	ad2antlr 763	. . . . . . . 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))))
293		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))
294	76	ad3antrrr 766	. . . . . . . . 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})
295	90	ad3antrrr 766	. . . . . . . . 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))
296		dvnprodlem3.t	. . . . . . . . . 10 ⊢ (𝜑 → 𝑇 ∈ Fin)
297	296	ad3antrrr 766	. . . . . . . . 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)
298		simp-4l 806	. . . . . . . . . 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)
299		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)
300		dvnprodlem3.h	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡):𝑋⟶ℂ)
301	298, 299, 300	syl2anc 693	. . . . . . . . 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)
302		dvnprodlem3.n	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
303	302	ad3antrrr 766	. . . . . . . . 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)
304		simplll 798	. . . . . . . . . . 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)
305	304	3ad2ant1 1082	. . . . . . . . . 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)
306		simp2 1062	. . . . . . . . . 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)
307		simp3 1063	. . . . . . . . . 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))
308		eleq1 2689	. . . . . . . . . . . . 13 ⊢ (𝑗 = ℎ → (𝑗 ∈ (0...𝑁) ↔ ℎ ∈ (0...𝑁)))
309	308	3anbi3d 1405	. . . . . . . . . . . 12 ⊢ (𝑗 = ℎ → ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑡 ∈ 𝑇 ∧ ℎ ∈ (0...𝑁))))
310		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑗 = ℎ → ((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑗) = ((𝑆 D^𝑛 (𝐻‘𝑡))‘ℎ))
311	310	feq1d 6030	. . . . . . . . . . . 12 ⊢ (𝑗 = ℎ → (((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑗):𝑋⟶ℂ ↔ ((𝑆 D^𝑛 (𝐻‘𝑡))‘ℎ):𝑋⟶ℂ))
312	309, 311	imbi12d 334	. . . . . . . . . . 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)))
313		dvnprodlem3.dvnh	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑗):𝑋⟶ℂ)
314	312, 313	chvarv 2263	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ ℎ ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘ℎ):𝑋⟶ℂ)
315	305, 306, 307, 314	syl3anc 1326	. . . . . . . . 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)
316		simprl 794	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) → 𝑟 ⊆ 𝑇)
317	316	ad2antrr 762	. . . . . . . . 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)
318		simprr 796	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) → 𝑧 ∈ (𝑇 ∖ 𝑟))
319	318	ad2antrr 762	. . . . . . . . 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))
320	260	eqcomi 2631	. . . . . . . . . . . . . . 15 ⊢ (𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦))) = (𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)))
321	320	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))))
322		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑙 → 𝑘 = 𝑙)
323	321, 322	fveq12d 6197	. . . . . . . . . . . . 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))
324	288	eqcomi 2631	. . . . . . . . . . . . . . 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)))
325	324	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))))
326		fveq2 6191	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑙 → (!‘𝑘) = (!‘𝑙))
327	326	oveq1d 6665	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑙 → ((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) = ((!‘𝑙) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))))
328	327	oveq1d 6665	. . . . . . . . . . . . . . . . 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)))
329	328	sumeq2ad 14434	. . . . . . . . . . . . . . . 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)))
330		fveq2 6191	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑙 → ((𝐷‘𝑟)‘𝑘) = ((𝐷‘𝑟)‘𝑙))
331	330	sumeq1d 14431	. . . . . . . . . . . . . . . 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)))
332	329, 331	eqtrd 2656	. . . . . . . . . . . . . . 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)))
333	332	mpteq2dv 4745	. . . . . . . . . . . . . 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))))
334	325, 333	eqtrd 2656	. . . . . . . . . . . . 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))))
335	323, 334	eqeq12d 2637	. . . . . . . . . . . 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)))))
336	335	cbvralv 3171	. . . . . . . . . . 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))))
337	336	biimpi 206	. . . . . . . . . 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))))
338	337	ad2antlr 763	. . . . . . . . 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))))
339		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))
340		fveq1 6190	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑐 → (𝑑‘𝑧) = (𝑐‘𝑧))
341	340	oveq2d 6666	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑐 → (𝑗 − (𝑑‘𝑧)) = (𝑗 − (𝑐‘𝑧)))
342		reseq1 5390	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑐 → (𝑑 ↾ 𝑟) = (𝑐 ↾ 𝑟))
343	341, 342	opeq12d 4410	. . . . . . . . . 10 ⊢ (𝑑 = 𝑐 → ⟨(𝑗 − (𝑑‘𝑧)), (𝑑 ↾ 𝑟)⟩ = ⟨(𝑗 − (𝑐‘𝑧)), (𝑐 ↾ 𝑟)⟩)
344	343	cbvmptv 4750	. . . . . . . . 9 ⊢ (𝑑 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑗) ↦ ⟨(𝑗 − (𝑑‘𝑧)), (𝑑 ↾ 𝑟)⟩) = (𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑗) ↦ ⟨(𝑗 − (𝑐‘𝑧)), (𝑐 ↾ 𝑟)⟩)
345	294, 295, 297, 301, 303, 315, 106, 317, 319, 338, 339, 344	dvnprodlem2 40162	. . . . . . . 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))))
346	251, 292, 293, 345	syl21anc 1325	. . . . . . 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))))
347	346	ralrimiva 2966	. . . . . 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))))
348		fveq2 6191	. . . . . . . 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))
349		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑘 → (!‘𝑗) = (!‘𝑘))
350	349	oveq1d 6665	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → ((!‘𝑗) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) = ((!‘𝑘) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))))
351	350	oveq1d 6665	. . . . . . . . . . 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)))
352	351	sumeq2ad 14434	. . . . . . . . . 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)))
353		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑗) = ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑘))
354	353	sumeq1d 14431	. . . . . . . . . 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)))
355	352, 354	eqtrd 2656	. . . . . . . . 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)))
356	355	mpteq2dv 4745	. . . . . . . 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))))
357	348, 356	eqeq12d 2637	. . . . . . 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)))))
358	357	cbvralv 3171	. . . . . 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))))
359	347, 358	sylib 208	. . . . 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))))
360	359	ex 450	. . . 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)))))
361	16, 32, 48, 68, 250, 360, 296	findcard2d 8202	. . 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))))
362		nn0uz 11722	. . . . 5 ⊢ ℕ₀ = (ℤ_≥‘0)
363	302, 362	syl6eleq 2711	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘0))
364		eluzfz2 12349	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘0) → 𝑁 ∈ (0...𝑁))
365	363, 364	syl 17	. . 3 ⊢ (𝜑 → 𝑁 ∈ (0...𝑁))
366		fveq2 6191	. . . . 5 ⊢ (𝑘 = 𝑁 → ((𝑆 D^𝑛 𝐹)‘𝑘) = ((𝑆 D^𝑛 𝐹)‘𝑁))
367		fveq2 6191	. . . . . . . 8 ⊢ (𝑘 = 𝑁 → ((𝐷‘𝑇)‘𝑘) = ((𝐷‘𝑇)‘𝑁))
368	367	sumeq1d 14431	. . . . . . 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)))
369		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑘 = 𝑁 → (!‘𝑘) = (!‘𝑁))
370	369	oveq1d 6665	. . . . . . . . 9 ⊢ (𝑘 = 𝑁 → ((!‘𝑘) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) = ((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))))
371	370	oveq1d 6665	. . . . . . . 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)))
372	371	sumeq2ad 14434	. . . . . . 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)))
373	368, 372	eqtrd 2656	. . . . . 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)))
374	373	mpteq2dv 4745	. . . . 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))))
375	366, 374	eqeq12d 2637	. . . 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)))))
376	375	rspccva 3308	. . 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))))
377	361, 365, 376	syl2anc 693	. 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))))
378		oveq2 6658	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑇 → ((0...𝑛) ↑_𝑚 𝑠) = ((0...𝑛) ↑_𝑚 𝑇))
379		rabeq 3192	. . . . . . . . . . 11 ⊢ (((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})
380	378, 379	syl 17	. . . . . . . . . 10 ⊢ (𝑠 = 𝑇 → {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑇) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛})
381		sumeq1 14419	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑇 → Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = Σ𝑡 ∈ 𝑇 (𝑐‘𝑡))
382	381	eqeq1d 2624	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑇 → (Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ 𝑇 (𝑐‘𝑡) = 𝑛))
383	382	rabbidv 3189	. . . . . . . . . 10 ⊢ (𝑠 = 𝑇 → {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑇) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑇) ∣ Σ𝑡 ∈ 𝑇 (𝑐‘𝑡) = 𝑛})
384	380, 383	eqtrd 2656	. . . . . . . . 9 ⊢ (𝑠 = 𝑇 → {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑇) ∣ Σ𝑡 ∈ 𝑇 (𝑐‘𝑡) = 𝑛})
385	384	mpteq2dv 4745	. . . . . . . 8 ⊢ (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}))
386	385	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 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}))
387		pwidg 4173	. . . . . . . 8 ⊢ (𝑇 ∈ Fin → 𝑇 ∈ 𝒫 𝑇)
388	296, 387	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ 𝒫 𝑇)
389	140	mptex 6486	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑇) ∣ Σ𝑡 ∈ 𝑇 (𝑐‘𝑡) = 𝑛}) ∈ V
390	389	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑇) ∣ Σ𝑡 ∈ 𝑇 (𝑐‘𝑡) = 𝑛}) ∈ V)
391	107, 386, 388, 390	fvmptd 6288	. . . . . 6 ⊢ (𝜑 → (𝐷‘𝑇) = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑇) ∣ Σ𝑡 ∈ 𝑇 (𝑐‘𝑡) = 𝑛}))
392		dvnprodlem3.c	. . . . . . 7 ⊢ 𝐶 = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑇) ∣ Σ𝑡 ∈ 𝑇 (𝑐‘𝑡) = 𝑛})
393	392	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐶 = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑇) ∣ Σ𝑡 ∈ 𝑇 (𝑐‘𝑡) = 𝑛}))
394	391, 393	eqtr4d 2659	. . . . 5 ⊢ (𝜑 → (𝐷‘𝑇) = 𝐶)
395	394	fveq1d 6193	. . . 4 ⊢ (𝜑 → ((𝐷‘𝑇)‘𝑁) = (𝐶‘𝑁))
396	395	sumeq1d 14431	. . 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)))
397	396	mpteq2dv 4745	. 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))))
398	377, 397	eqtrd 2656	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 196 ∨ wo 383 ∧ wa 384 ∧ w3a 1037 ∀wal 1481 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 {crab 2916 Vcvv 3200 ∖ cdif 3571 ∪ cun 3572 ⊆ wss 3574 ∅c0 3915 ifcif 4086 𝒫 cpw 4158 {csn 4177 {cpr 4179 ⟨cop 4183 ↦ cmpt 4729 dom cdm 5114 ↾ cres 5116 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↑_𝑚 cmap 7857 ↑_pm cpm 7858 Fincfn 7955 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 · cmul 9941 − cmin 10266 / cdiv 10684 ℕcn 11020 ℕ₀cn0 11292 ℤ_≥cuz 11687 ...cfz 12326 !cfa 13060 Σcsu 14416 ∏cprod 14635 ↾_t crest 16081 TopOpenctopn 16082 ℂ_fldccnfld 19746 D^𝑛 cdvn 23628

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-prod 14636 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-limc 23630 df-dv 23631 df-dvn 23632

This theorem is referenced by: dvnprod 40164

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