dvnprodlem3 - Mathbox for Glauco Siliprandi

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Theorem dvnprodlem3 43118

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 15452	. . . . . . . . 9 ⊢ (𝑠 = ∅ → ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥) = ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥))
2	1	mpteq2dv 5140	. . . . . . . 8 ⊢ (𝑠 = ∅ → (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥)) = (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥)))
3	2	oveq2d 7218	. . . . . . 7 ⊢ (𝑠 = ∅ → (𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥))) = (𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥))))
4	3	fveq1d 6708	. . . . . 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 6706	. . . . . . . . . 10 ⊢ (𝑠 = ∅ → (𝐷‘𝑠) = (𝐷‘∅))
6	5	fveq1d 6708	. . . . . . . . 9 ⊢ (𝑠 = ∅ → ((𝐷‘𝑠)‘𝑘) = ((𝐷‘∅)‘𝑘))
7	6	sumeq1d 15248	. . . . . . . 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 15452	. . . . . . . . . . 11 ⊢ (𝑠 = ∅ → ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡)) = ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡)))
9	8	oveq2d 7218	. . . . . . . . . 10 ⊢ (𝑠 = ∅ → ((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) = ((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))))
10		prodeq1 15452	. . . . . . . . . 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 7220	. . . . . . . . 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 15251	. . . . . . . 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 2774	. . . . . . 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 5140	. . . . . 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 2750	. . . . 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 3111	. . . 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 15452	. . . . . . . . 9 ⊢ (𝑠 = 𝑟 → ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥) = ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥))
18	17	mpteq2dv 5140	. . . . . . . 8 ⊢ (𝑠 = 𝑟 → (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥)) = (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)))
19	18	oveq2d 7218	. . . . . . 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 6708	. . . . . 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 6706	. . . . . . . . . 10 ⊢ (𝑠 = 𝑟 → (𝐷‘𝑠) = (𝐷‘𝑟))
22	21	fveq1d 6708	. . . . . . . . 9 ⊢ (𝑠 = 𝑟 → ((𝐷‘𝑠)‘𝑘) = ((𝐷‘𝑟)‘𝑘))
23	22	sumeq1d 15248	. . . . . . . 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 15452	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑟 → ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡)) = ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡)))
25	24	oveq2d 7218	. . . . . . . . . 10 ⊢ (𝑠 = 𝑟 → ((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) = ((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))))
26		prodeq1 15452	. . . . . . . . . 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 7220	. . . . . . . . 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 15251	. . . . . . . 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 2774	. . . . . . 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 5140	. . . . . 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 2750	. . . . 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 3111	. . . 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 15452	. . . . . . . . 9 ⊢ (𝑠 = (𝑟 ∪ {𝑧}) → ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥) = ∏𝑡 ∈ (𝑟 ∪ {𝑧})((𝐻‘𝑡)‘𝑥))
34	33	mpteq2dv 5140	. . . . . . . 8 ⊢ (𝑠 = (𝑟 ∪ {𝑧}) → (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥)) = (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ (𝑟 ∪ {𝑧})((𝐻‘𝑡)‘𝑥)))
35	34	oveq2d 7218	. . . . . . 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 6708	. . . . . 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 6706	. . . . . . . . . 10 ⊢ (𝑠 = (𝑟 ∪ {𝑧}) → (𝐷‘𝑠) = (𝐷‘(𝑟 ∪ {𝑧})))
38	37	fveq1d 6708	. . . . . . . . 9 ⊢ (𝑠 = (𝑟 ∪ {𝑧}) → ((𝐷‘𝑠)‘𝑘) = ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑘))
39	38	sumeq1d 15248	. . . . . . . 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 15452	. . . . . . . . . . 11 ⊢ (𝑠 = (𝑟 ∪ {𝑧}) → ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡)) = ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡)))
41	40	oveq2d 7218	. . . . . . . . . 10 ⊢ (𝑠 = (𝑟 ∪ {𝑧}) → ((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) = ((!‘𝑘) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))))
42		prodeq1 15452	. . . . . . . . . 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 7220	. . . . . . . . 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 15251	. . . . . . . 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 2774	. . . . . . 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 5140	. . . . . 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 2750	. . . . 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 3111	. . . 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 15452	. . . . . . . . . 10 ⊢ (𝑠 = 𝑇 → ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥) = ∏𝑡 ∈ 𝑇 ((𝐻‘𝑡)‘𝑥))
50	49	mpteq2dv 5140	. . . . . . . . 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 2740	. . . . . . . . 9 ⊢ (𝑠 = 𝑇 → (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑇 ((𝐻‘𝑡)‘𝑥)) = 𝐹)
54	50, 53	eqtrd 2774	. . . . . . . 8 ⊢ (𝑠 = 𝑇 → (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥)) = 𝐹)
55	54	oveq2d 7218	. . . . . . 7 ⊢ (𝑠 = 𝑇 → (𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥))) = (𝑆 D^𝑛 𝐹))
56	55	fveq1d 6708	. . . . . 6 ⊢ (𝑠 = 𝑇 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = ((𝑆 D^𝑛 𝐹)‘𝑘))
57		fveq2 6706	. . . . . . . . . 10 ⊢ (𝑠 = 𝑇 → (𝐷‘𝑠) = (𝐷‘𝑇))
58	57	fveq1d 6708	. . . . . . . . 9 ⊢ (𝑠 = 𝑇 → ((𝐷‘𝑠)‘𝑘) = ((𝐷‘𝑇)‘𝑘))
59	58	sumeq1d 15248	. . . . . . . 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 15452	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑇 → ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡)) = ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡)))
61	60	oveq2d 7218	. . . . . . . . . 10 ⊢ (𝑠 = 𝑇 → ((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) = ((!‘𝑘) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))))
62		prodeq1 15452	. . . . . . . . . 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 7220	. . . . . . . . 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 15251	. . . . . . . 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 2774	. . . . . . 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 5140	. . . . . 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 2750	. . . . 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 3111	. . . 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 15486	. . . . . . . . . . . . 13 ⊢ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥) = 1
70	69	mpteq2i 5136	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥)) = (𝑥 ∈ 𝑋 ↦ 1)
71	70	oveq2i 7213	. . . . . . . . . . 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 6713	. . . . . . . . 9 ⊢ (𝑘 = 0 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥)))‘𝑘) = ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 1))‘0))
75	74	adantl 485	. . . . . . . 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 24773	. . . . . . . . . . 11 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
78	76, 77	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
79		1cnd 10811	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 1 ∈ ℂ)
80	79	fmpttd 6921	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 1):𝑋⟶ℂ)
81		1re 10816	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ
82	81	rgenw 3066	. . . . . . . . . . . . . . . 16 ⊢ ∀𝑥 ∈ 𝑋 1 ∈ ℝ
83		dmmptg 6094	. . . . . . . . . . . . . . . 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 6520	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 1):dom (𝑥 ∈ 𝑋 ↦ 1)⟶ℂ ↔ (𝑥 ∈ 𝑋 ↦ 1):𝑋⟶ℂ))
87	80, 86	mpbird 260	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 1):dom (𝑥 ∈ 𝑋 ↦ 1)⟶ℂ)
88		restsspw 16908	. . . . . . . . . . . . . . 15 ⊢ ((TopOpen‘ℂ_fld) ↾_t 𝑆) ⊆ 𝒫 𝑆
89		dvnprodlem3.x	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
90	88, 89	sseldi 3889	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 ∈ 𝒫 𝑆)
91		elpwi 4512	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ 𝒫 𝑆 → 𝑋 ⊆ 𝑆)
92	90, 91	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
93	85, 92	eqsstrd 3929	. . . . . . . . . . . 12 ⊢ (𝜑 → dom (𝑥 ∈ 𝑋 ↦ 1) ⊆ 𝑆)
94	87, 93	jca 515	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 1):dom (𝑥 ∈ 𝑋 ↦ 1)⟶ℂ ∧ dom (𝑥 ∈ 𝑋 ↦ 1) ⊆ 𝑆))
95		cnex 10793	. . . . . . . . . . . . 13 ⊢ ℂ ∈ V
96	95	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ℂ ∈ V)
97		elpm2g 8514	. . . . . . . . . . . 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 587	. . . . . . . . . . 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 260	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 1) ∈ (ℂ ↑_pm 𝑆))
100		dvn0 24793	. . . . . . . . . 10 ⊢ ((𝑆 ⊆ ℂ ∧ (𝑥 ∈ 𝑋 ↦ 1) ∈ (ℂ ↑_pm 𝑆)) → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 1))‘0) = (𝑥 ∈ 𝑋 ↦ 1))
101	78, 99, 100	syl2anc 587	. . . . . . . . 9 ⊢ (𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 1))‘0) = (𝑥 ∈ 𝑋 ↦ 1))
102	101	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 = 0) → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 1))‘0) = (𝑥 ∈ 𝑋 ↦ 1))
103		fveq2 6706	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 0 → ((𝐷‘∅)‘𝑘) = ((𝐷‘∅)‘0))
104	103	adantl 485	. . . . . . . . . . . . . 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 7210	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑠 = ∅ → ((0...𝑛) ↑_m 𝑠) = ((0...𝑛) ↑_m ∅))
107		elmapfn 8535	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 ∈ ((0...𝑛) ↑_m ∅) → 𝑥 Fn ∅)
108		fn0 6498	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 Fn ∅ ↔ 𝑥 = ∅)
109	107, 108	sylib 221	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 ∈ ((0...𝑛) ↑_m ∅) → 𝑥 = ∅)
110		velsn 4547	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
111	109, 110	sylibr 237	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 ∈ ((0...𝑛) ↑_m ∅) → 𝑥 ∈ {∅})
112	110	biimpi 219	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 ∈ {∅} → 𝑥 = ∅)
113		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 = ∅ → 𝑥 = ∅)
114		f0 6589	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ∅:∅⟶(0...𝑛)
115		ovex 7235	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (0...𝑛) ∈ V
116		0ex 5189	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ∅ ∈ V
117	115, 116	elmap 8541	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (∅ ∈ ((0...𝑛) ↑_m ∅) ↔ ∅:∅⟶(0...𝑛))
118	114, 117	mpbir 234	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ∅ ∈ ((0...𝑛) ↑_m ∅)
119	118	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 = ∅ → ∅ ∈ ((0...𝑛) ↑_m ∅))
120	113, 119	eqeltrd 2834	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 = ∅ → 𝑥 ∈ ((0...𝑛) ↑_m ∅))
121	112, 120	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 ∈ {∅} → 𝑥 ∈ ((0...𝑛) ↑_m ∅))
122	111, 121	impbii 212	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 ∈ ((0...𝑛) ↑_m ∅) ↔ 𝑥 ∈ {∅})
123	122	ax-gen 1803	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ∀𝑥(𝑥 ∈ ((0...𝑛) ↑_m ∅) ↔ 𝑥 ∈ {∅})
124		dfcleq 2727	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((0...𝑛) ↑_m ∅) = {∅} ↔ ∀𝑥(𝑥 ∈ ((0...𝑛) ↑_m ∅) ↔ 𝑥 ∈ {∅}))
125	123, 124	mpbir 234	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((0...𝑛) ↑_m ∅) = {∅}
126	125	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑠 = ∅ → ((0...𝑛) ↑_m ∅) = {∅})
127	106, 126	eqtrd 2774	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 = ∅ → ((0...𝑛) ↑_m 𝑠) = {∅})
128		rabeq 3387	. . . . . . . . . . . . . . . . . . . 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 15235	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑠 = ∅ → Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = Σ𝑡 ∈ ∅ (𝑐‘𝑡))
131	130	eqeq1d 2736	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 = ∅ → (Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛))
132	131	rabbidv 3383	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 = ∅ → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛})
133	129, 132	eqtrd 2774	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 = ∅ → {𝑐 ∈ ((0...𝑛) ↑_m 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛})
134	133	mpteq2dv 5140	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = ∅ → (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_m 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}) = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛}))
135		0elpw 5236	. . . . . . . . . . . . . . . . . 18 ⊢ ∅ ∈ 𝒫 𝑇
136	135	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∅ ∈ 𝒫 𝑇)
137		nn0ex 12079	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ₀ ∈ V
138	137	mptex 7028	. . . . . . . . . . . . . . . . . 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 6830	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐷‘∅) = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛}))
141		eqeq2 2746	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 0 → (Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0))
142	141	rabbidv 3383	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 0 → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0})
143	142	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 = 0) → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0})
144		0nn0 12088	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℕ₀
145	144	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 ∈ ℕ₀)
146		p0ex 5266	. . . . . . . . . . . . . . . . . 18 ⊢ {∅} ∈ V
147	146	rabex 5214	. . . . . . . . . . . . . . . . 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 6814	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐷‘∅)‘0) = {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0})
150	149	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 = 0) → ((𝐷‘∅)‘0) = {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0})
151		snidg 4565	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∅ ∈ V → ∅ ∈ {∅})
152	116, 151	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∅ ∈ {∅}
153		eqid 2734	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0 = 0
154	152, 153	pm3.2i 474	. . . . . . . . . . . . . . . . . . 19 ⊢ (∅ ∈ {∅} ∧ 0 = 0)
155		sum0 15268	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0
156	155	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 = ∅ → Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0)
157	156	eqeq1d 2736	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 = ∅ → (Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0 ↔ 0 = 0))
158	157	elrab 3595	. . . . . . . . . . . . . . . . . . 19 ⊢ (∅ ∈ {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0} ↔ (∅ ∈ {∅} ∧ 0 = 0))
159	154, 158	mpbir 234	. . . . . . . . . . . . . . . . . 18 ⊢ ∅ ∈ {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0}
160	159	n0ii 4241	. . . . . . . . . . . . . . . . 17 ⊢ ¬ {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0} = ∅
161		eqid 2734	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0} = {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0}
162		rabrsn 4630	. . . . . . . . . . . . . . . . . 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 1778	. . . . . . . . . . . . . . . 16 ⊢ {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0} = {∅}
165	164	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 = 0) → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0} = {∅})
166		iftrue 4435	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 0 → if(𝑘 = 0, {∅}, ∅) = {∅})
167	166	adantl 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 = 0) → if(𝑘 = 0, {∅}, ∅) = {∅})
168	165, 167	eqtr4d 2777	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 = 0) → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0} = if(𝑘 = 0, {∅}, ∅))
169	104, 150, 168	3eqtrd 2778	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 = 0) → ((𝐷‘∅)‘𝑘) = if(𝑘 = 0, {∅}, ∅))
170	169, 167	eqtrd 2774	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 = 0) → ((𝐷‘∅)‘𝑘) = {∅})
171	170	sumeq1d 15248	. . . . . . . . . . 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 6706	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 0 → (!‘𝑘) = (!‘0))
173		fac0 13825	. . . . . . . . . . . . . . . . . . 19 ⊢ (!‘0) = 1
174	173	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 0 → (!‘0) = 1)
175	172, 174	eqtrd 2774	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 0 → (!‘𝑘) = 1)
176	175	oveq1d 7217	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 0 → ((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) = (1 / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))))
177		prod0 15486	. . . . . . . . . . . . . . . . . 18 ⊢ ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡)) = 1
178	177	oveq2i 7213	. . . . . . . . . . . . . . . . 17 ⊢ (1 / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) = (1 / 1)
179		1div1e1 11505	. . . . . . . . . . . . . . . . 17 ⊢ (1 / 1) = 1
180	178, 179	eqtri 2762	. . . . . . . . . . . . . . . 16 ⊢ (1 / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) = 1
181	176, 180	eqtrdi 2790	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 0 → ((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) = 1)
182		prod0 15486	. . . . . . . . . . . . . . . 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 7220	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 0 → (((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = (1 · 1))
185	184	ad2antlr 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 = 0) ∧ 𝑐 ∈ {∅}) → (((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = (1 · 1))
186		1t1e1 11975	. . . . . . . . . . . . . 14 ⊢ (1 · 1) = 1
187	186	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 = 0) ∧ 𝑐 ∈ {∅}) → (1 · 1) = 1)
188	185, 187	eqtrd 2774	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 = 0) ∧ 𝑐 ∈ {∅}) → (((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = 1)
189	188	sumeq2dv 15250	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 = 0) → Σ𝑐 ∈ {∅} (((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ {∅}1)
190		ax-1cn 10770	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
191		eqidd 2735	. . . . . . . . . . . . . 14 ⊢ (𝑐 = ∅ → 1 = 1)
192	191	sumsn 15291	. . . . . . . . . . . . 13 ⊢ ((∅ ∈ V ∧ 1 ∈ ℂ) → Σ𝑐 ∈ {∅}1 = 1)
193	116, 190, 192	mp2an 692	. . . . . . . . . . . 12 ⊢ Σ𝑐 ∈ {∅}1 = 1
194	193	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 = 0) → Σ𝑐 ∈ {∅}1 = 1)
195	171, 189, 194	3eqtrd 2778	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 = 0) → Σ𝑐 ∈ ((𝐷‘∅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = 1)
196	195	mpteq2dv 5140	. . . . . . . . 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 2740	. . . . . . . 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 2778	. . . . . . 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 6707	. . . . . . . . 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 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝑘 = 0) → 𝑆 ∈ {ℝ, ℂ})
203	202	adantr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) → 𝑆 ∈ {ℝ, ℂ})
204	89	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝑘 = 0) → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
205	204	adantr 484	. . . . . . . . 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 13188	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ₀)
208	207	adantl 485	. . . . . . . . . . . 12 ⊢ ((¬ 𝑘 = 0 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ₀)
209		neqne 2943	. . . . . . . . . . . . 13 ⊢ (¬ 𝑘 = 0 → 𝑘 ≠ 0)
210	209	adantr 484	. . . . . . . . . . . 12 ⊢ ((¬ 𝑘 = 0 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ≠ 0)
211	208, 210	jca 515	. . . . . . . . . . 11 ⊢ ((¬ 𝑘 = 0 ∧ 𝑘 ∈ (0...𝑁)) → (𝑘 ∈ ℕ₀ ∧ 𝑘 ≠ 0))
212		elnnne0 12087	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ ↔ (𝑘 ∈ ℕ₀ ∧ 𝑘 ≠ 0))
213	211, 212	sylibr 237	. . . . . . . . . 10 ⊢ ((¬ 𝑘 = 0 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ)
214	213	adantll 714	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ)
215	203, 205, 206, 214	dvnmptconst 43111	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 1))‘𝑘) = (𝑥 ∈ 𝑋 ↦ 0))
216	140	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) → (𝐷‘∅) = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛}))
217		eqeq2 2746	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑘 → (Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘))
218	217	rabbidv 3383	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑘 → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘})
219	218	adantl 485	. . . . . . . . . . . . . . 15 ⊢ ((¬ 𝑘 = 0 ∧ 𝑛 = 𝑘) → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘})
220		eqidd 2735	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘 → 𝑘 = 𝑘)
221		id 22	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘 → Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘)
222	221	eqcomd 2740	. . . . . . . . . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘 → 𝑘 = 0)
225	224	adantl 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑐 ∈ {∅} ∧ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘) → 𝑘 = 0)
226	225	adantll 714	. . . . . . . . . . . . . . . . . . 19 ⊢ (((¬ 𝑘 = 0 ∧ 𝑐 ∈ {∅}) ∧ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘) → 𝑘 = 0)
227		simpll 767	. . . . . . . . . . . . . . . . . . 19 ⊢ (((¬ 𝑘 = 0 ∧ 𝑐 ∈ {∅}) ∧ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘) → ¬ 𝑘 = 0)
228	226, 227	pm2.65da 817	. . . . . . . . . . . . . . . . . 18 ⊢ ((¬ 𝑘 = 0 ∧ 𝑐 ∈ {∅}) → ¬ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘)
229	228	ralrimiva 3098	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑘 = 0 → ∀𝑐 ∈ {∅} ¬ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘)
230		rabeq0 4289	. . . . . . . . . . . . . . . . 17 ⊢ ({𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘} = ∅ ↔ ∀𝑐 ∈ {∅} ¬ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘)
231	229, 230	sylibr 237	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑘 = 0 → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘} = ∅)
232	231	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((¬ 𝑘 = 0 ∧ 𝑛 = 𝑘) → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘} = ∅)
233	219, 232	eqtrd 2774	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝑘 = 0 ∧ 𝑛 = 𝑘) → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛} = ∅)
234	233	adantll 714	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑛 = 𝑘) → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛} = ∅)
235	234	adantlr 715	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 = 𝑘) → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛} = ∅)
236	207	adantl 485	. . . . . . . . . . . 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 6814	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) → ((𝐷‘∅)‘𝑘) = ∅)
239	238	sumeq1d 15248	. . . . . . . . . 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 15268	. . . . . . . . . . 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 2775	. . . . . . . . 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 5140	. . . . . . . 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 2778	. . . . . . 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 416	. . . . . 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 813	. . . . 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 3097	. . . 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 767	. . . . . . . 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 6706	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → ((𝐻‘𝑡)‘𝑥) = ((𝐻‘𝑡)‘𝑦))
250	249	prodeq2ad 42762	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥) = ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑦))
251		fveq2 6706	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑢 → (𝐻‘𝑡) = (𝐻‘𝑢))
252	251	fveq1d 6708	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑢 → ((𝐻‘𝑡)‘𝑦) = ((𝐻‘𝑢)‘𝑦))
253	252	cbvprodv 15459	. . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥) = ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦))
256	255	cbvmptv 5147	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)) = (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦))
257	256	oveq2i 7213	. . . . . . . . . . . . 13 ⊢ (𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥))) = (𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))
258	257	fveq1i 6707	. . . . . . . . . . . 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 6706	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑢 → (𝑐‘𝑡) = (𝑐‘𝑢))
260	259	fveq2d 6710	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑢 → (!‘(𝑐‘𝑡)) = (!‘(𝑐‘𝑢)))
261	260	cbvprodv 15459	. . . . . . . . . . . . . . . . . 18 ⊢ ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡)) = ∏𝑢 ∈ 𝑟 (!‘(𝑐‘𝑢))
262	261	oveq2i 7213	. . . . . . . . . . . . . . . . 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 6706	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) = (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑦))
265	264	prodeq2ad 42762	. . . . . . . . . . . . . . . . 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 7218	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑢 → (𝑆 D^𝑛 (𝐻‘𝑡)) = (𝑆 D^𝑛 (𝐻‘𝑢)))
267	266, 259	fveq12d 6713	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑢 → ((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)) = ((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑐‘𝑢)))
268	267	fveq1d 6708	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑢 → (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑦) = (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑐‘𝑢))‘𝑦))
269	268	cbvprodv 15459	. . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . 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 7220	. . . . . . . . . . . . . . 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 15251	. . . . . . . . . . . . . 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 6705	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 = 𝑑 → (𝑐‘𝑢) = (𝑑‘𝑢))
275	274	fveq2d 6710	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 = 𝑑 → (!‘(𝑐‘𝑢)) = (!‘(𝑑‘𝑢)))
276	275	prodeq2ad 42762	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = 𝑑 → ∏𝑢 ∈ 𝑟 (!‘(𝑐‘𝑢)) = ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢)))
277	276	oveq2d 7218	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = 𝑑 → ((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑐‘𝑢))) = ((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))))
278	274	fveq2d 6710	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 = 𝑑 → ((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑐‘𝑢)) = ((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢)))
279	278	fveq1d 6708	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = 𝑑 → (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑐‘𝑢))‘𝑦) = (((𝑆 D^𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦))
280	279	prodeq2ad 42762	. . . . . . . . . . . . . . . . 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 7220	. . . . . . . . . . . . . . . 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 15243	. . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . 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 5147	. . . . . . . . . . . 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 2752	. . . . . . . . . . 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 3081	. . . . . . . . . 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 219	. . . . . . . . 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 727	. . . . . . . 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 488	. . . . . . . 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 730	. . . . . . . . 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 730	. . . . . . . . 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 730	. . . . . . . . 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 783	. . . . . . . . . 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 488	. . . . . . . . . 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 587	. . . . . . . . 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 730	. . . . . . . . 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 775	. . . . . . . . . . 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 1135	. . . . . . . . . 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 1139	. . . . . . . . . 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 1140	. . . . . . . . . 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 2816	. . . . . . . . . . . . 13 ⊢ (𝑗 = ℎ → (𝑗 ∈ (0...𝑁) ↔ ℎ ∈ (0...𝑁)))
306	305	3anbi3d 1444	. . . . . . . . . . . 12 ⊢ (𝑗 = ℎ → ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑡 ∈ 𝑇 ∧ ℎ ∈ (0...𝑁))))
307		fveq2 6706	. . . . . . . . . . . . 13 ⊢ (𝑗 = ℎ → ((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑗) = ((𝑆 D^𝑛 (𝐻‘𝑡))‘ℎ))
308	307	feq1d 6519	. . . . . . . . . . . 12 ⊢ (𝑗 = ℎ → (((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑗):𝑋⟶ℂ ↔ ((𝑆 D^𝑛 (𝐻‘𝑡))‘ℎ):𝑋⟶ℂ))
309	306, 308	imbi12d 348	. . . . . . . . . . 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	chvarvv 2007	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ ℎ ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘ℎ):𝑋⟶ℂ)
312	302, 303, 304, 311	syl3anc 1373	. . . . . . . . 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 771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) → 𝑟 ⊆ 𝑇)
314	313	ad2antrr 726	. . . . . . . . 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 773	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) → 𝑧 ∈ (𝑇 ∖ 𝑟))
316	315	ad2antrr 726	. . . . . . . . 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 2743	. . . . . . . . . . . . . . 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 6713	. . . . . . . . . . . . 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 2743	. . . . . . . . . . . . . . 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 6706	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑙 → (!‘𝑘) = (!‘𝑙))
324	323	oveq1d 7217	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑙 → ((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) = ((!‘𝑙) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))))
325	324	oveq1d 7217	. . . . . . . . . . . . . . . . 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 15251	. . . . . . . . . . . . . . . 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 6706	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑙 → ((𝐷‘𝑟)‘𝑘) = ((𝐷‘𝑟)‘𝑙))
328	327	sumeq1d 15248	. . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . 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 5140	. . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . 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 2750	. . . . . . . . . . . 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	cbvralvw 3351	. . . . . . . . . . 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 219	. . . . . . . . . 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 727	. . . . . . . . 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 488	. . . . . . . . 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 6705	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑐 → (𝑑‘𝑧) = (𝑐‘𝑧))
338	337	oveq2d 7218	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑐 → (𝑗 − (𝑑‘𝑧)) = (𝑗 − (𝑐‘𝑧)))
339		reseq1 5834	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑐 → (𝑑 ↾ 𝑟) = (𝑐 ↾ 𝑟))
340	338, 339	opeq12d 4782	. . . . . . . . . 10 ⊢ (𝑑 = 𝑐 → ⟨(𝑗 − (𝑑‘𝑧)), (𝑑 ↾ 𝑟)⟩ = ⟨(𝑗 − (𝑐‘𝑧)), (𝑐 ↾ 𝑟)⟩)
341	340	cbvmptv 5147	. . . . . . . . 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 43117	. . . . . . . 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 838	. . . . . . 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 3098	. . . . . 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 6706	. . . . . . . 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 6706	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑘 → (!‘𝑗) = (!‘𝑘))
347	346	oveq1d 7217	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → ((!‘𝑗) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) = ((!‘𝑘) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))))
348	347	oveq1d 7217	. . . . . . . . . . 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 15251	. . . . . . . . . 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 6706	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑗) = ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑘))
351	350	sumeq1d 15248	. . . . . . . . . 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 2774	. . . . . . . . 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 5140	. . . . . . . 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 2750	. . . . . . 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	cbvralvw 3351	. . . . . 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 221	. . . . 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 416	. . . 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 8833	. . 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 12459	. . . . 5 ⊢ ℕ₀ = (ℤ_≥‘0)
360	299, 359	eleqtrdi 2844	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘0))
361		eluzfz2 13103	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘0) → 𝑁 ∈ (0...𝑁))
362	360, 361	syl 17	. . 3 ⊢ (𝜑 → 𝑁 ∈ (0...𝑁))
363		fveq2 6706	. . . . 5 ⊢ (𝑘 = 𝑁 → ((𝑆 D^𝑛 𝐹)‘𝑘) = ((𝑆 D^𝑛 𝐹)‘𝑁))
364		fveq2 6706	. . . . . . . 8 ⊢ (𝑘 = 𝑁 → ((𝐷‘𝑇)‘𝑘) = ((𝐷‘𝑇)‘𝑁))
365	364	sumeq1d 15248	. . . . . . 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 6706	. . . . . . . . . 10 ⊢ (𝑘 = 𝑁 → (!‘𝑘) = (!‘𝑁))
367	366	oveq1d 7217	. . . . . . . . 9 ⊢ (𝑘 = 𝑁 → ((!‘𝑘) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) = ((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))))
368	367	oveq1d 7217	. . . . . . . 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 15251	. . . . . . 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 2774	. . . . . 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 5140	. . . . 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 2750	. . . 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 3529	. . 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 587	. 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 7210	. . . . . . . . . 10 ⊢ (𝑠 = 𝑇 → ((0...𝑛) ↑_m 𝑠) = ((0...𝑛) ↑_m 𝑇))
376		rabeq 3387	. . . . . . . . . 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 15235	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑇 → Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = Σ𝑡 ∈ 𝑇 (𝑐‘𝑡))
379	378	eqeq1d 2736	. . . . . . . . . 10 ⊢ (𝑠 = 𝑇 → (Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ 𝑇 (𝑐‘𝑡) = 𝑛))
380	379	rabbidv 3383	. . . . . . . . 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 2774	. . . . . . . 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 5140	. . . . . . 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 4525	. . . . . . . 8 ⊢ (𝑇 ∈ Fin → 𝑇 ∈ 𝒫 𝑇)
384	293, 383	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ 𝒫 𝑇)
385	137	mptex 7028	. . . . . . . 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 6830	. . . . . 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 2777	. . . . 5 ⊢ (𝜑 → (𝐷‘𝑇) = 𝐶)
391	390	fveq1d 6708	. . . 4 ⊢ (𝜑 → ((𝐷‘𝑇)‘𝑁) = (𝐶‘𝑁))
392	391	sumeq1d 15248	. . 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 5140	. 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 2774	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 209 ∧ wa 399 ∨ wo 847 ∧ w3a 1089 ∀wal 1541 = wceq 1543 ∈ wcel 2110 ≠ wne 2935 ∀wral 3054 {crab 3058 Vcvv 3401 ∖ cdif 3854 ∪ cun 3855 ⊆ wss 3857 ∅c0 4227 ifcif 4429 𝒫 cpw 4503 {csn 4531 {cpr 4533 ⟨cop 4537 ↦ cmpt 5124 dom cdm 5540 ↾ cres 5542 Fn wfn 6364 ⟶wf 6365 ‘cfv 6369 (class class class)co 7202 ↑_m cmap 8497 ↑_pm cpm 8498 Fincfn 8615 ℂcc 10710 ℝcr 10711 0cc0 10712 1c1 10713 · cmul 10717 − cmin 11045 / cdiv 11472 ℕcn 11813 ℕ₀cn0 12073 ℤ_≥cuz 12421 ...cfz 13078 !cfa 13822 Σcsu 15232 ∏cprod 15448 ↾_t crest 16897 TopOpenctopn 16898 ℂ_fldccnfld 20335 D^𝑛 cdvn 24733

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-inf2 9245 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-pre-sup 10790 ax-addf 10791 ax-mulf 10792

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-iin 4897 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-se 5499 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-isom 6378 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-of 7458 df-om 7634 df-1st 7750 df-2nd 7751 df-supp 7893 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-2o 8192 df-er 8380 df-map 8499 df-pm 8500 df-ixp 8568 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-fsupp 8975 df-fi 9016 df-sup 9047 df-inf 9048 df-oi 9115 df-card 9538 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-5 11879 df-6 11880 df-7 11881 df-8 11882 df-9 11883 df-n0 12074 df-z 12160 df-dec 12277 df-uz 12422 df-q 12528 df-rp 12570 df-xneg 12687 df-xadd 12688 df-xmul 12689 df-ico 12924 df-icc 12925 df-fz 13079 df-fzo 13222 df-seq 13558 df-exp 13619 df-fac 13823 df-bc 13852 df-hash 13880 df-cj 14645 df-re 14646 df-im 14647 df-sqrt 14781 df-abs 14782 df-clim 15032 df-sum 15233 df-prod 15449 df-struct 16686 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-ress 16692 df-plusg 16780 df-mulr 16781 df-starv 16782 df-sca 16783 df-vsca 16784 df-ip 16785 df-tset 16786 df-ple 16787 df-ds 16789 df-unif 16790 df-hom 16791 df-cco 16792 df-rest 16899 df-topn 16900 df-0g 16918 df-gsum 16919 df-topgen 16920 df-pt 16921 df-prds 16924 df-xrs 16979 df-qtop 16984 df-imas 16985 df-xps 16987 df-mre 17061 df-mrc 17062 df-acs 17064 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-submnd 18191 df-mulg 18461 df-cntz 18683 df-cmn 19144 df-psmet 20327 df-xmet 20328 df-met 20329 df-bl 20330 df-mopn 20331 df-fbas 20332 df-fg 20333 df-cnfld 20336 df-top 21763 df-topon 21780 df-topsp 21802 df-bases 21815 df-cld 21888 df-ntr 21889 df-cls 21890 df-nei 21967 df-lp 22005 df-perf 22006 df-cn 22096 df-cnp 22097 df-haus 22184 df-tx 22431 df-hmeo 22624 df-fil 22715 df-fm 22807 df-flim 22808 df-flf 22809 df-xms 23190 df-ms 23191 df-tms 23192 df-cncf 23747 df-limc 24735 df-dv 24736 df-dvn 24737

This theorem is referenced by: dvnprod 43119

W3C validator