Theorem etransclem32 40483

Description: This is the proof for the last equation in the proof of the derivative calculated in [Juillerat] p. 12, just after equation *(6) . (Contributed by Glauco Siliprandi, 5-Apr-2020.)

Hypotheses

Ref	Expression
etransclem32.s	⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
etransclem32.x	⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
etransclem32.p	⊢ (𝜑 → 𝑃 ∈ ℕ)
etransclem32.m	⊢ (𝜑 → 𝑀 ∈ ℕ0)
etransclem32.f	⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃)))
etransclem32.n	⊢ (𝜑 → 𝑁 ∈ ℕ0)
etransclem32.ngt	⊢ (𝜑 → ((𝑀 · 𝑃) + (𝑃 − 1)) < 𝑁)
etransclem32.h	⊢ 𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))

Assertion

Ref	Expression
etransclem32	⊢ (𝜑 → ((𝑆 D𝑛 𝐹)‘𝑁) = (𝑥 ∈ 𝑋 ↦ 0))

Distinct variable groups:   𝑗,𝐻,𝑥   𝑗,𝑀,𝑥   𝑗,𝑁,𝑥   𝑃,𝑗,𝑥   𝑆,𝑗,𝑥   𝑗,𝑋,𝑥   𝜑,𝑗,𝑥

Allowed substitution hints:   𝐹(𝑥,𝑗)

Proof of Theorem etransclem32

Dummy variables 𝑘 𝐴 𝑐 𝑛 𝑑 𝑚 ℎ 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		etransclem32.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
2		etransclem32.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
3		etransclem32.p	. . 3 ⊢ (𝜑 → 𝑃 ∈ ℕ)
4		etransclem32.m	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℕ0)
5		etransclem32.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃)))
6		etransclem32.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
7		etransclem32.h	. . 3 ⊢ 𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))
8		etransclem11 40462	. . 3 ⊢ (𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚}) = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛})
9	1, 2, 3, 4, 5, 6, 7, 8	etransclem30 40481	. 2 ⊢ (𝜑 → ((𝑆 D𝑛 𝐹)‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)(((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) · ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑥))))
10		simpr 477	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) → 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁))
11	8, 6	etransclem12 40463	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁) = {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁})
12	11	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) → ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁) = {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁})
13	10, 12	eleqtrd 2703	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) → 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁})
14	13	adantlr 751	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) → 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁})
15		nfv 1843	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁})
16		nfre1 3005	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑘)
17	16	nfn 1784	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘 ¬ ∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑘)
18	15, 17	nfan 1828	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ ¬ ∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑘))
19		fzssre 39529	. . . . . . . . . . . . . . . . 17 ⊢ (0...𝑁) ⊆ ℝ
20		rabid 3116	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁} ↔ (𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∧ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁))
21	20	simplbi 476	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁} → 𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)))
22		elmapi 7879	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) → 𝑐:(0...𝑀)⟶(0...𝑁))
23	21, 22	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁} → 𝑐:(0...𝑀)⟶(0...𝑁))
24	23	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) → 𝑐:(0...𝑀)⟶(0...𝑁))
25	24	ffvelrnda 6359	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀)) → (𝑐‘𝑘) ∈ (0...𝑁))
26	19, 25	sseldi 3601	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀)) → (𝑐‘𝑘) ∈ ℝ)
27	26	adantlr 751	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ ¬ ∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑘)) ∧ 𝑘 ∈ (0...𝑀)) → (𝑐‘𝑘) ∈ ℝ)
28		nnm1nn0 11334	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ0)
29	3, 28	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ0)
30	29	nn0red 11352	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑃 − 1) ∈ ℝ)
31	3	nnred 11035	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑃 ∈ ℝ)
32	30, 31	ifcld 4131	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → if(𝑘 = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
33	32	ad3antrrr 766	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ ¬ ∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑘)) ∧ 𝑘 ∈ (0...𝑀)) → if(𝑘 = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
34		ralnex 2992	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑘 ∈ (0...𝑀) ¬ if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑘) ↔ ¬ ∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑘))
35	34	biimpri 218	. . . . . . . . . . . . . . . . 17 ⊢ (¬ ∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑘) → ∀𝑘 ∈ (0...𝑀) ¬ if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑘))
36	35	r19.21bi 2932	. . . . . . . . . . . . . . . 16 ⊢ ((¬ ∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑘) ∧ 𝑘 ∈ (0...𝑀)) → ¬ if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑘))
37	36	adantll 750	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ ¬ ∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑘)) ∧ 𝑘 ∈ (0...𝑀)) → ¬ if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑘))
38	27, 33, 37	nltled 10187	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ ¬ ∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑘)) ∧ 𝑘 ∈ (0...𝑀)) → (𝑐‘𝑘) ≤ if(𝑘 = 0, (𝑃 − 1), 𝑃))
39	38	ex 450	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ ¬ ∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑘)) → (𝑘 ∈ (0...𝑀) → (𝑐‘𝑘) ≤ if(𝑘 = 0, (𝑃 − 1), 𝑃)))
40	18, 39	ralrimi 2957	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ ¬ ∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑘)) → ∀𝑘 ∈ (0...𝑀)(𝑐‘𝑘) ≤ if(𝑘 = 0, (𝑃 − 1), 𝑃))
41		fveq2 6191	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑘 → (𝑐‘𝑗) = (𝑐‘𝑘))
42	41	cbvsumv 14426	. . . . . . . . . . . . . . 15 ⊢ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = Σ𝑘 ∈ (0...𝑀)(𝑐‘𝑘)
43	20	simprbi 480	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁} → Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁)
44	42, 43	syl5reqr 2671	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁} → 𝑁 = Σ𝑘 ∈ (0...𝑀)(𝑐‘𝑘))
45	44	ad2antlr 763	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ ∀𝑘 ∈ (0...𝑀)(𝑐‘𝑘) ≤ if(𝑘 = 0, (𝑃 − 1), 𝑃)) → 𝑁 = Σ𝑘 ∈ (0...𝑀)(𝑐‘𝑘))
46		fveq2 6191	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = ℎ → (𝑐‘𝑘) = (𝑐‘ℎ))
47	46	cbvsumv 14426	. . . . . . . . . . . . . 14 ⊢ Σ𝑘 ∈ (0...𝑀)(𝑐‘𝑘) = Σℎ ∈ (0...𝑀)(𝑐‘ℎ)
48		fzfid 12772	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ ∀𝑘 ∈ (0...𝑀)(𝑐‘𝑘) ≤ if(𝑘 = 0, (𝑃 − 1), 𝑃)) → (0...𝑀) ∈ Fin)
49	24	ffvelrnda 6359	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ ℎ ∈ (0...𝑀)) → (𝑐‘ℎ) ∈ (0...𝑁))
50	19, 49	sseldi 3601	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ ℎ ∈ (0...𝑀)) → (𝑐‘ℎ) ∈ ℝ)
51	50	adantlr 751	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ ∀𝑘 ∈ (0...𝑀)(𝑐‘𝑘) ≤ if(𝑘 = 0, (𝑃 − 1), 𝑃)) ∧ ℎ ∈ (0...𝑀)) → (𝑐‘ℎ) ∈ ℝ)
52	30, 31	ifcld 4131	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → if(ℎ = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
53	52	ad3antrrr 766	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ ∀𝑘 ∈ (0...𝑀)(𝑐‘𝑘) ≤ if(𝑘 = 0, (𝑃 − 1), 𝑃)) ∧ ℎ ∈ (0...𝑀)) → if(ℎ = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
54		eqeq1 2626	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = ℎ → (𝑘 = 0 ↔ ℎ = 0))
55	54	ifbid 4108	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = ℎ → if(𝑘 = 0, (𝑃 − 1), 𝑃) = if(ℎ = 0, (𝑃 − 1), 𝑃))
56	46, 55	breq12d 4666	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = ℎ → ((𝑐‘𝑘) ≤ if(𝑘 = 0, (𝑃 − 1), 𝑃) ↔ (𝑐‘ℎ) ≤ if(ℎ = 0, (𝑃 − 1), 𝑃)))
57	56	rspccva 3308	. . . . . . . . . . . . . . . . 17 ⊢ ((∀𝑘 ∈ (0...𝑀)(𝑐‘𝑘) ≤ if(𝑘 = 0, (𝑃 − 1), 𝑃) ∧ ℎ ∈ (0...𝑀)) → (𝑐‘ℎ) ≤ if(ℎ = 0, (𝑃 − 1), 𝑃))
58	57	adantll 750	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ ∀𝑘 ∈ (0...𝑀)(𝑐‘𝑘) ≤ if(𝑘 = 0, (𝑃 − 1), 𝑃)) ∧ ℎ ∈ (0...𝑀)) → (𝑐‘ℎ) ≤ if(ℎ = 0, (𝑃 − 1), 𝑃))
59	48, 51, 53, 58	fsumle 14531	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ ∀𝑘 ∈ (0...𝑀)(𝑐‘𝑘) ≤ if(𝑘 = 0, (𝑃 − 1), 𝑃)) → Σℎ ∈ (0...𝑀)(𝑐‘ℎ) ≤ Σℎ ∈ (0...𝑀)if(ℎ = 0, (𝑃 − 1), 𝑃))
60		nn0uz 11722	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ0 = (ℤ≥‘0)
61	4, 60	syl6eleq 2711	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘0))
62	3	nnnn0d 11351	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑃 ∈ ℕ0)
63	29, 62	ifcld 4131	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → if(ℎ = 0, (𝑃 − 1), 𝑃) ∈ ℕ0)
64	63	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ℎ ∈ (0...𝑀)) → if(ℎ = 0, (𝑃 − 1), 𝑃) ∈ ℕ0)
65	64	nn0cnd 11353	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ℎ ∈ (0...𝑀)) → if(ℎ = 0, (𝑃 − 1), 𝑃) ∈ ℂ)
66		iftrue 4092	. . . . . . . . . . . . . . . . . 18 ⊢ (ℎ = 0 → if(ℎ = 0, (𝑃 − 1), 𝑃) = (𝑃 − 1))
67	61, 65, 66	fsum1p 14482	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → Σℎ ∈ (0...𝑀)if(ℎ = 0, (𝑃 − 1), 𝑃) = ((𝑃 − 1) + Σℎ ∈ ((0 + 1)...𝑀)if(ℎ = 0, (𝑃 − 1), 𝑃)))
68		0p1e1 11132	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0 + 1) = 1
69	68	oveq1i 6660	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((0 + 1)...𝑀) = (1...𝑀)
70	69	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((0 + 1)...𝑀) = (1...𝑀))
71	70	sumeq1d 14431	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → Σℎ ∈ ((0 + 1)...𝑀)if(ℎ = 0, (𝑃 − 1), 𝑃) = Σℎ ∈ (1...𝑀)if(ℎ = 0, (𝑃 − 1), 𝑃))
72		0red 10041	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (ℎ ∈ (1...𝑀) → 0 ∈ ℝ)
73		1red 10055	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (ℎ ∈ (1...𝑀) → 1 ∈ ℝ)
74		elfzelz 12342	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (ℎ ∈ (1...𝑀) → ℎ ∈ ℤ)
75	74	zred 11482	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (ℎ ∈ (1...𝑀) → ℎ ∈ ℝ)
76		0lt1 10550	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 0 < 1
77	76	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (ℎ ∈ (1...𝑀) → 0 < 1)
78		elfzle1 12344	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (ℎ ∈ (1...𝑀) → 1 ≤ ℎ)
79	72, 73, 75, 77, 78	ltletrd 10197	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (ℎ ∈ (1...𝑀) → 0 < ℎ)
80	79	gt0ne0d 10592	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℎ ∈ (1...𝑀) → ℎ ≠ 0)
81	80	neneqd 2799	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ ∈ (1...𝑀) → ¬ ℎ = 0)
82	81	iffalsed 4097	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℎ ∈ (1...𝑀) → if(ℎ = 0, (𝑃 − 1), 𝑃) = 𝑃)
83	82	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ ℎ ∈ (1...𝑀)) → if(ℎ = 0, (𝑃 − 1), 𝑃) = 𝑃)
84	83	sumeq2dv 14433	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → Σℎ ∈ (1...𝑀)if(ℎ = 0, (𝑃 − 1), 𝑃) = Σℎ ∈ (1...𝑀)𝑃)
85		fzfid 12772	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (1...𝑀) ∈ Fin)
86	3	nncnd 11036	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑃 ∈ ℂ)
87		fsumconst 14522	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((1...𝑀) ∈ Fin ∧ 𝑃 ∈ ℂ) → Σℎ ∈ (1...𝑀)𝑃 = ((#‘(1...𝑀)) · 𝑃))
88	85, 86, 87	syl2anc 693	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → Σℎ ∈ (1...𝑀)𝑃 = ((#‘(1...𝑀)) · 𝑃))
89		hashfz1 13134	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑀 ∈ ℕ0 → (#‘(1...𝑀)) = 𝑀)
90	4, 89	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (#‘(1...𝑀)) = 𝑀)
91	90	oveq1d 6665	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((#‘(1...𝑀)) · 𝑃) = (𝑀 · 𝑃))
92	88, 91	eqtrd 2656	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → Σℎ ∈ (1...𝑀)𝑃 = (𝑀 · 𝑃))
93	71, 84, 92	3eqtrd 2660	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → Σℎ ∈ ((0 + 1)...𝑀)if(ℎ = 0, (𝑃 − 1), 𝑃) = (𝑀 · 𝑃))
94	93	oveq2d 6666	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑃 − 1) + Σℎ ∈ ((0 + 1)...𝑀)if(ℎ = 0, (𝑃 − 1), 𝑃)) = ((𝑃 − 1) + (𝑀 · 𝑃)))
95	29	nn0cnd 11353	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑃 − 1) ∈ ℂ)
96	4, 62	nn0mulcld 11356	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑀 · 𝑃) ∈ ℕ0)
97	96	nn0cnd 11353	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑀 · 𝑃) ∈ ℂ)
98	95, 97	addcomd 10238	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑃 − 1) + (𝑀 · 𝑃)) = ((𝑀 · 𝑃) + (𝑃 − 1)))
99	67, 94, 98	3eqtrd 2660	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → Σℎ ∈ (0...𝑀)if(ℎ = 0, (𝑃 − 1), 𝑃) = ((𝑀 · 𝑃) + (𝑃 − 1)))
100	99	ad2antrr 762	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ ∀𝑘 ∈ (0...𝑀)(𝑐‘𝑘) ≤ if(𝑘 = 0, (𝑃 − 1), 𝑃)) → Σℎ ∈ (0...𝑀)if(ℎ = 0, (𝑃 − 1), 𝑃) = ((𝑀 · 𝑃) + (𝑃 − 1)))
101	59, 100	breqtrd 4679	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ ∀𝑘 ∈ (0...𝑀)(𝑐‘𝑘) ≤ if(𝑘 = 0, (𝑃 − 1), 𝑃)) → Σℎ ∈ (0...𝑀)(𝑐‘ℎ) ≤ ((𝑀 · 𝑃) + (𝑃 − 1)))
102	47, 101	syl5eqbr 4688	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ ∀𝑘 ∈ (0...𝑀)(𝑐‘𝑘) ≤ if(𝑘 = 0, (𝑃 − 1), 𝑃)) → Σ𝑘 ∈ (0...𝑀)(𝑐‘𝑘) ≤ ((𝑀 · 𝑃) + (𝑃 − 1)))
103	45, 102	eqbrtrd 4675	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ ∀𝑘 ∈ (0...𝑀)(𝑐‘𝑘) ≤ if(𝑘 = 0, (𝑃 − 1), 𝑃)) → 𝑁 ≤ ((𝑀 · 𝑃) + (𝑃 − 1)))
104	40, 103	syldan 487	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ ¬ ∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑘)) → 𝑁 ≤ ((𝑀 · 𝑃) + (𝑃 − 1)))
105		etransclem32.ngt	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑀 · 𝑃) + (𝑃 − 1)) < 𝑁)
106	96, 29	nn0addcld 11355	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑀 · 𝑃) + (𝑃 − 1)) ∈ ℕ0)
107	106	nn0red 11352	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑀 · 𝑃) + (𝑃 − 1)) ∈ ℝ)
108	6	nn0red 11352	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℝ)
109	107, 108	ltnled 10184	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((𝑀 · 𝑃) + (𝑃 − 1)) < 𝑁 ↔ ¬ 𝑁 ≤ ((𝑀 · 𝑃) + (𝑃 − 1))))
110	105, 109	mpbid 222	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 𝑁 ≤ ((𝑀 · 𝑃) + (𝑃 − 1)))
111	110	ad2antrr 762	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ ¬ ∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑘)) → ¬ 𝑁 ≤ ((𝑀 · 𝑃) + (𝑃 − 1)))
112	104, 111	condan 835	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) → ∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑘))
113	112	adantlr 751	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) → ∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑘))
114		nfv 1843	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑥 ∈ 𝑋)
115		nfcv 2764	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑗(0...𝑀)
116	115	nfsum1 14420	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑗Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗)
117	116	nfeq1 2778	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁
118		nfcv 2764	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗((0...𝑁) ↑𝑚 (0...𝑀))
119	117, 118	nfrab 3123	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗{𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}
120	119	nfcri 2758	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}
121	114, 120	nfan 1828	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁})
122		nfv 1843	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 𝑘 ∈ (0...𝑀)
123		nfv 1843	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑘)
124	121, 122, 123	nf3an 1831	. . . . . . . . . . 11 ⊢ Ⅎ𝑗(((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀) ∧ if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑘))
125		nfcv 2764	. . . . . . . . . . 11 ⊢ Ⅎ𝑗(((𝑆 D𝑛 (𝐻‘𝑘))‘(𝑐‘𝑘))‘𝑥)
126		fzfid 12772	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀) ∧ if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑘)) → (0...𝑀) ∈ Fin)
127	1	ad3antrrr 766	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → 𝑆 ∈ {ℝ, ℂ})
128	2	ad3antrrr 766	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
129	3	ad3antrrr 766	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → 𝑃 ∈ ℕ)
130		etransclem5 40456	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (0...𝑀) ↦ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))) = (𝑘 ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))
131	7, 130	eqtri 2644	. . . . . . . . . . . . . 14 ⊢ 𝐻 = (𝑘 ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))
132		simpr 477	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ (0...𝑀))
133	23	ad2antlr 763	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → 𝑐:(0...𝑀)⟶(0...𝑁))
134		simpr 477	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ (0...𝑀))
135	133, 134	ffvelrnd 6360	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → (𝑐‘𝑗) ∈ (0...𝑁))
136	135	adantllr 755	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → (𝑐‘𝑗) ∈ (0...𝑁))
137		elfznn0 12433	. . . . . . . . . . . . . . 15 ⊢ ((𝑐‘𝑗) ∈ (0...𝑁) → (𝑐‘𝑗) ∈ ℕ0)
138	136, 137	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → (𝑐‘𝑗) ∈ ℕ0)
139	127, 128, 129, 131, 132, 138	etransclem20 40471	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗)):𝑋⟶ℂ)
140		simpllr 799	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → 𝑥 ∈ 𝑋)
141	139, 140	ffvelrnd 6360	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑆 D𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑥) ∈ ℂ)
142	141	3ad2antl1 1223	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀) ∧ if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑘)) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑆 D𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑥) ∈ ℂ)
143		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → (𝐻‘𝑗) = (𝐻‘𝑘))
144	143	oveq2d 6666	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑘 → (𝑆 D𝑛 (𝐻‘𝑗)) = (𝑆 D𝑛 (𝐻‘𝑘)))
145	144, 41	fveq12d 6197	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → ((𝑆 D𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗)) = ((𝑆 D𝑛 (𝐻‘𝑘))‘(𝑐‘𝑘)))
146	145	fveq1d 6193	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → (((𝑆 D𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑥) = (((𝑆 D𝑛 (𝐻‘𝑘))‘(𝑐‘𝑘))‘𝑥))
147		simp2 1062	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀) ∧ if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑘)) → 𝑘 ∈ (0...𝑀))
148	1	ad2antrr 762	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) → 𝑆 ∈ {ℝ, ℂ})
149	148	3ad2ant1 1082	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀) ∧ if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑘)) → 𝑆 ∈ {ℝ, ℂ})
150	2	ad2antrr 762	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
151	150	3ad2ant1 1082	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀) ∧ if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑘)) → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
152	3	ad2antrr 762	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) → 𝑃 ∈ ℕ)
153	152	3ad2ant1 1082	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀) ∧ if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑘)) → 𝑃 ∈ ℕ)
154		etransclem5 40456	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (0...𝑀) ↦ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))) = (ℎ ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 − ℎ)↑if(ℎ = 0, (𝑃 − 1), 𝑃))))
155	7, 154	eqtri 2644	. . . . . . . . . . . . 13 ⊢ 𝐻 = (ℎ ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 − ℎ)↑if(ℎ = 0, (𝑃 − 1), 𝑃))))
156		fzssz 12343	. . . . . . . . . . . . . . . 16 ⊢ (0...𝑁) ⊆ ℤ
157	156, 25	sseldi 3601	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀)) → (𝑐‘𝑘) ∈ ℤ)
158	157	adantllr 755	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀)) → (𝑐‘𝑘) ∈ ℤ)
159	158	3adant3 1081	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀) ∧ if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑘)) → (𝑐‘𝑘) ∈ ℤ)
160		simp3 1063	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀) ∧ if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑘)) → if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑘))
161	149, 151, 153, 155, 147, 159, 160	etransclem19 40470	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀) ∧ if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑘)) → ((𝑆 D𝑛 (𝐻‘𝑘))‘(𝑐‘𝑘)) = (𝑦 ∈ 𝑋 ↦ 0))
162		eqidd 2623	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀) ∧ if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑘)) ∧ 𝑦 = 𝑥) → 0 = 0)
163		simp1lr 1125	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀) ∧ if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑘)) → 𝑥 ∈ 𝑋)
164		0red 10041	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀) ∧ if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑘)) → 0 ∈ ℝ)
165	161, 162, 163, 164	fvmptd 6288	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀) ∧ if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑘)) → (((𝑆 D𝑛 (𝐻‘𝑘))‘(𝑐‘𝑘))‘𝑥) = 0)
166	124, 125, 126, 142, 146, 147, 165	fprod0 39828	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀) ∧ if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑘)) → ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑥) = 0)
167	166	rexlimdv3a 3033	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) → (∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑘) → ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑥) = 0))
168	113, 167	mpd 15	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) → ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑥) = 0)
169	14, 168	syldan 487	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) → ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑥) = 0)
170	169	oveq2d 6666	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) → (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) · ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑥)) = (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) · 0))
171	6	faccld 13071	. . . . . . . . . . 11 ⊢ (𝜑 → (!‘𝑁) ∈ ℕ)
172	171	nncnd 11036	. . . . . . . . . 10 ⊢ (𝜑 → (!‘𝑁) ∈ ℂ)
173	172	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) → (!‘𝑁) ∈ ℂ)
174		fzfid 12772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) → (0...𝑀) ∈ Fin)
175		simpll 790	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → 𝜑)
176	13	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁})
177		simpr 477	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ (0...𝑀))
178	175, 176, 177, 135	syl21anc 1325	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (𝑐‘𝑗) ∈ (0...𝑁))
179	178, 137	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (𝑐‘𝑗) ∈ ℕ0)
180	179	faccld 13071	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝑐‘𝑗)) ∈ ℕ)
181	180	nncnd 11036	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝑐‘𝑗)) ∈ ℂ)
182	174, 181	fprodcl 14682	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) → ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗)) ∈ ℂ)
183	180	nnne0d 11065	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝑐‘𝑗)) ≠ 0)
184	174, 181, 183	fprodn0 14709	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) → ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗)) ≠ 0)
185	173, 182, 184	divcld 10801	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) ∈ ℂ)
186	185	mul01d 10235	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) → (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) · 0) = 0)
187	186	adantlr 751	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) → (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) · 0) = 0)
188	170, 187	eqtrd 2656	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) → (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) · ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑥)) = 0)
189	188	sumeq2dv 14433	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → Σ𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)(((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) · ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑥)) = Σ𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)0)
190		eqid 2622	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚}) = (𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})
191	190, 6	etransclem16 40467	. . . . . . 7 ⊢ (𝜑 → ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁) ∈ Fin)
192	191	olcd 408	. . . . . 6 ⊢ (𝜑 → (((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁) ⊆ (ℤ≥‘𝐴) ∨ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁) ∈ Fin))
193	192	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁) ⊆ (ℤ≥‘𝐴) ∨ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁) ∈ Fin))
194		sumz 14453	. . . . 5 ⊢ ((((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁) ⊆ (ℤ≥‘𝐴) ∨ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁) ∈ Fin) → Σ𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)0 = 0)
195	193, 194	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → Σ𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)0 = 0)
196	189, 195	eqtrd 2656	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → Σ𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)(((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) · ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑥)) = 0)
197	196	mpteq2dva 4744	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)(((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) · ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑥))) = (𝑥 ∈ 𝑋 ↦ 0))
198	9, 197	eqtrd 2656	1 ⊢ (𝜑 → ((𝑆 D𝑛 𝐹)‘𝑁) = (𝑥 ∈ 𝑋 ↦ 0))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  {crab 2916   ⊆ wss 3574  ifcif 4086  {cpr 4179   class class class wbr 4653   ↦ cmpt 4729  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↑_𝑚 cmap 7857  Fincfn 7955  ℂcc 9934  ℝcr 9935  0cc0 9936  1c1 9937   + caddc 9939   · cmul 9941   < clt 10074   ≤ cle 10075   − cmin 10266   / cdiv 10684  ℕcn 11020  ℕ₀cn0 11292  ℤcz 11377  ℤ_≥cuz 11687  ...cfz 12326  ↑cexp 12860  !cfa 13060  #chash 13117  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:  etransclem46  40497