Theorem etransclem32 46295

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 𝐴 𝑐 𝑘 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 46274	. . 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 46293	. 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 484	. . . . . . . . . 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 46275	. . . . . . . . . . 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 480	. . . . . . . . . 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 2836	. . . . . . . . 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 715	. . . . . . . 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 1914	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁})
16		nfre1 3267	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑘)
17	16	nfn 1857	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘 ¬ ∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑘)
18	15, 17	nfan 1899	. . . . . . . . . . . . 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 45343	. . . . . . . . . . . . . . . . 17 ⊢ (0...𝑁) ⊆ ℝ
20		rabid 3437	. . . . . . . . . . . . . . . . . . . . 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 497	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁} → 𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)))
22		elmapi 8863	. . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) → 𝑐:(0...𝑀)⟶(0...𝑁))
25	24	ffvelcdmda 7074	. . . . . . . . . . . . . . . . 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	sselid 3956	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀)) → (𝑐‘𝑘) ∈ ℝ)
27	26	adantlr 715	. . . . . . . . . . . . . . 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 12542	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ0)
29	3, 28	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ0)
30	29	nn0red 12563	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑃 − 1) ∈ ℝ)
31	3	nnred 12255	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑃 ∈ ℝ)
32	30, 31	ifcld 4547	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → if(𝑘 = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
33	32	ad3antrrr 730	. . . . . . . . . . . . . . 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 3062	. . . . . . . . . . . . . . . . . 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 228	. . . . . . . . . . . . . . . . 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 3234	. . . . . . . . . . . . . . . 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 714	. . . . . . . . . . . . . . 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 11385	. . . . . . . . . . . . . 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 412	. . . . . . . . . . . . 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 3240	. . . . . . . . . . . 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	20	simprbi 496	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁} → Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁)
42		fveq2 6876	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑘 → (𝑐‘𝑗) = (𝑐‘𝑘))
43	42	cbvsumv 15712	. . . . . . . . . . . . . . 15 ⊢ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = Σ𝑘 ∈ (0...𝑀)(𝑐‘𝑘)
44	41, 43	eqtr3di 2785	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁} → 𝑁 = Σ𝑘 ∈ (0...𝑀)(𝑐‘𝑘))
45	44	ad2antlr 727	. . . . . . . . . . . . 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 6876	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = ℎ → (𝑐‘𝑘) = (𝑐‘ℎ))
47	46	cbvsumv 15712	. . . . . . . . . . . . . 14 ⊢ Σ𝑘 ∈ (0...𝑀)(𝑐‘𝑘) = Σℎ ∈ (0...𝑀)(𝑐‘ℎ)
48		fzfid 13991	. . . . . . . . . . . . . . . 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	ffvelcdmda 7074	. . . . . . . . . . . . . . . . . 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	sselid 3956	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ ℎ ∈ (0...𝑀)) → (𝑐‘ℎ) ∈ ℝ)
51	50	adantlr 715	. . . . . . . . . . . . . . . 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 4547	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → if(ℎ = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
53	52	ad3antrrr 730	. . . . . . . . . . . . . . . 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 2739	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = ℎ → (𝑘 = 0 ↔ ℎ = 0))
55	54	ifbid 4524	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = ℎ → if(𝑘 = 0, (𝑃 − 1), 𝑃) = if(ℎ = 0, (𝑃 − 1), 𝑃))
56	46, 55	breq12d 5132	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = ℎ → ((𝑐‘𝑘) ≤ if(𝑘 = 0, (𝑃 − 1), 𝑃) ↔ (𝑐‘ℎ) ≤ if(ℎ = 0, (𝑃 − 1), 𝑃)))
57	56	rspccva 3600	. . . . . . . . . . . . . . . . 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 714	. . . . . . . . . . . . . . . 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 15815	. . . . . . . . . . . . . . 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 12894	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ0 = (ℤ≥‘0)
61	4, 60	eleqtrdi 2844	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘0))
62	3	nnnn0d 12562	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑃 ∈ ℕ0)
63	29, 62	ifcld 4547	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → if(ℎ = 0, (𝑃 − 1), 𝑃) ∈ ℕ0)
64	63	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ℎ ∈ (0...𝑀)) → if(ℎ = 0, (𝑃 − 1), 𝑃) ∈ ℕ0)
65	64	nn0cnd 12564	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ℎ ∈ (0...𝑀)) → if(ℎ = 0, (𝑃 − 1), 𝑃) ∈ ℂ)
66		iftrue 4506	. . . . . . . . . . . . . . . . . 18 ⊢ (ℎ = 0 → if(ℎ = 0, (𝑃 − 1), 𝑃) = (𝑃 − 1))
67	61, 65, 66	fsum1p 15769	. . . . . . . . . . . . . . . . 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 12362	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0 + 1) = 1
69	68	oveq1i 7415	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((0 + 1)...𝑀) = (1...𝑀)
70	69	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((0 + 1)...𝑀) = (1...𝑀))
71	70	sumeq1d 15716	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → Σℎ ∈ ((0 + 1)...𝑀)if(ℎ = 0, (𝑃 − 1), 𝑃) = Σℎ ∈ (1...𝑀)if(ℎ = 0, (𝑃 − 1), 𝑃))
72		0red 11238	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (ℎ ∈ (1...𝑀) → 0 ∈ ℝ)
73		1red 11236	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (ℎ ∈ (1...𝑀) → 1 ∈ ℝ)
74		elfzelz 13541	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (ℎ ∈ (1...𝑀) → ℎ ∈ ℤ)
75	74	zred 12697	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (ℎ ∈ (1...𝑀) → ℎ ∈ ℝ)
76		0lt1 11759	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 0 < 1
77	76	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (ℎ ∈ (1...𝑀) → 0 < 1)
78		elfzle1 13544	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (ℎ ∈ (1...𝑀) → 1 ≤ ℎ)
79	72, 73, 75, 77, 78	ltletrd 11395	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (ℎ ∈ (1...𝑀) → 0 < ℎ)
80	79	gt0ne0d 11801	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℎ ∈ (1...𝑀) → ℎ ≠ 0)
81	80	neneqd 2937	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ ∈ (1...𝑀) → ¬ ℎ = 0)
82	81	iffalsed 4511	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℎ ∈ (1...𝑀) → if(ℎ = 0, (𝑃 − 1), 𝑃) = 𝑃)
83	82	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ ℎ ∈ (1...𝑀)) → if(ℎ = 0, (𝑃 − 1), 𝑃) = 𝑃)
84	83	sumeq2dv 15718	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → Σℎ ∈ (1...𝑀)if(ℎ = 0, (𝑃 − 1), 𝑃) = Σℎ ∈ (1...𝑀)𝑃)
85		fzfid 13991	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (1...𝑀) ∈ Fin)
86	3	nncnd 12256	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑃 ∈ ℂ)
87		fsumconst 15806	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((1...𝑀) ∈ Fin ∧ 𝑃 ∈ ℂ) → Σℎ ∈ (1...𝑀)𝑃 = ((♯‘(1...𝑀)) · 𝑃))
88	85, 86, 87	syl2anc 584	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → Σℎ ∈ (1...𝑀)𝑃 = ((♯‘(1...𝑀)) · 𝑃))
89		hashfz1 14364	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑀 ∈ ℕ0 → (♯‘(1...𝑀)) = 𝑀)
90	4, 89	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (♯‘(1...𝑀)) = 𝑀)
91	90	oveq1d 7420	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((♯‘(1...𝑀)) · 𝑃) = (𝑀 · 𝑃))
92	88, 91	eqtrd 2770	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → Σℎ ∈ (1...𝑀)𝑃 = (𝑀 · 𝑃))
93	71, 84, 92	3eqtrd 2774	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → Σℎ ∈ ((0 + 1)...𝑀)if(ℎ = 0, (𝑃 − 1), 𝑃) = (𝑀 · 𝑃))
94	93	oveq2d 7421	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑃 − 1) + Σℎ ∈ ((0 + 1)...𝑀)if(ℎ = 0, (𝑃 − 1), 𝑃)) = ((𝑃 − 1) + (𝑀 · 𝑃)))
95	29	nn0cnd 12564	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑃 − 1) ∈ ℂ)
96	4, 62	nn0mulcld 12567	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑀 · 𝑃) ∈ ℕ0)
97	96	nn0cnd 12564	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑀 · 𝑃) ∈ ℂ)
98	95, 97	addcomd 11437	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑃 − 1) + (𝑀 · 𝑃)) = ((𝑀 · 𝑃) + (𝑃 − 1)))
99	67, 94, 98	3eqtrd 2774	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → Σℎ ∈ (0...𝑀)if(ℎ = 0, (𝑃 − 1), 𝑃) = ((𝑀 · 𝑃) + (𝑃 − 1)))
100	99	ad2antrr 726	. . . . . . . . . . . . . . 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 5145	. . . . . . . . . . . . . 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	eqbrtrid 5154	. . . . . . . . . . . . 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 5141	. . . . . . . . . . . 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 591	. . . . . . . . . . 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 12566	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑀 · 𝑃) + (𝑃 − 1)) ∈ ℕ0)
107	106	nn0red 12563	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑀 · 𝑃) + (𝑃 − 1)) ∈ ℝ)
108	6	nn0red 12563	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℝ)
109	107, 108	ltnled 11382	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((𝑀 · 𝑃) + (𝑃 − 1)) < 𝑁 ↔ ¬ 𝑁 ≤ ((𝑀 · 𝑃) + (𝑃 − 1))))
110	105, 109	mpbid 232	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 𝑁 ≤ ((𝑀 · 𝑃) + (𝑃 − 1)))
111	110	ad2antrr 726	. . . . . . . . . . 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 817	. . . . . . . . . 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 715	. . . . . . . . 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 1914	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑥 ∈ 𝑋)
115		nfcv 2898	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑗(0...𝑀)
116	115	nfsum1 15706	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑗Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗)
117	116	nfeq1 2914	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁
118		nfcv 2898	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗((0...𝑁) ↑m (0...𝑀))
119	117, 118	nfrabw 3454	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗{𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}
120	119	nfcri 2890	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}
121	114, 120	nfan 1899	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁})
122		nfv 1914	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 𝑘 ∈ (0...𝑀)
123		nfv 1914	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑘)
124	121, 122, 123	nf3an 1901	. . . . . . . . . . 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 2898	. . . . . . . . . . 11 ⊢ Ⅎ𝑗(((𝑆 D𝑛 (𝐻‘𝑘))‘(𝑐‘𝑘))‘𝑥)
126		fzfid 13991	. . . . . . . . . . 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 730	. . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → 𝑃 ∈ ℕ)
130		etransclem5 46268	. . . . . . . . . . . . . . 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 2758	. . . . . . . . . . . . . 14 ⊢ 𝐻 = (𝑘 ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))
132		simpr 484	. . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ (0...𝑀))
135	133, 134	ffvelcdmd 7075	. . . . . . . . . . . . . . . 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 719	. . . . . . . . . . . . . . 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 13637	. . . . . . . . . . . . . . 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 46283	. . . . . . . . . . . . 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 775	. . . . . . . . . . . . 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	ffvelcdmd 7075	. . . . . . . . . . . 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 1186	. . . . . . . . . . 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 6876	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → (𝐻‘𝑗) = (𝐻‘𝑘))
144	143	oveq2d 7421	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑘 → (𝑆 D𝑛 (𝐻‘𝑗)) = (𝑆 D𝑛 (𝐻‘𝑘)))
145	144, 42	fveq12d 6883	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → ((𝑆 D𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗)) = ((𝑆 D𝑛 (𝐻‘𝑘))‘(𝑐‘𝑘)))
146	145	fveq1d 6878	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → (((𝑆 D𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑥) = (((𝑆 D𝑛 (𝐻‘𝑘))‘(𝑐‘𝑘))‘𝑥))
147		simp2 1137	. . . . . . . . . . 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 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) → 𝑆 ∈ {ℝ, ℂ})
149	148	3ad2ant1 1133	. . . . . . . . . . . . 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 726	. . . . . . . . . . . . . 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 1133	. . . . . . . . . . . . 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 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) → 𝑃 ∈ ℕ)
153	152	3ad2ant1 1133	. . . . . . . . . . . . 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 46268	. . . . . . . . . . . . . 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 2758	. . . . . . . . . . . . 13 ⊢ 𝐻 = (ℎ ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 − ℎ)↑if(ℎ = 0, (𝑃 − 1), 𝑃))))
156	25	elfzelzd 13542	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀)) → (𝑐‘𝑘) ∈ ℤ)
157	156	adantllr 719	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀)) → (𝑐‘𝑘) ∈ ℤ)
158	157	3adant3 1132	. . . . . . . . . . . . 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)
159		simp3 1138	. . . . . . . . . . . . 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))
160	149, 151, 153, 155, 147, 158, 159	etransclem19 46282	. . . . . . . . . . . 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))
161		eqidd 2736	. . . . . . . . . . . 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)
162		simp1lr 1238	. . . . . . . . . . . 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)
163		0red 11238	. . . . . . . . . . . 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)
164	160, 161, 162, 163	fvmptd 6993	. . . . . . . . . . 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)
165	124, 125, 126, 142, 146, 147, 164	fprod0 45625	. . . . . . . . . 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)
166	165	rexlimdv3a 3145	. . . . . . . . 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))
167	113, 166	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)
168	14, 167	syldan 591	. . . . . . 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)
169	168	oveq2d 7421	. . . . . 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))
170	6	faccld 14302	. . . . . . . . . . 11 ⊢ (𝜑 → (!‘𝑁) ∈ ℕ)
171	170	nncnd 12256	. . . . . . . . . 10 ⊢ (𝜑 → (!‘𝑁) ∈ ℂ)
172	171	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) → (!‘𝑁) ∈ ℂ)
173		fzfid 13991	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) → (0...𝑀) ∈ Fin)
174		simpll 766	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → 𝜑)
175	13	adantr 480	. . . . . . . . . . . . . 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})
176		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ (0...𝑀))
177	174, 175, 176, 135	syl21anc 837	. . . . . . . . . . . . 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))
178	177, 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)
179	178	faccld 14302	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝑐‘𝑗)) ∈ ℕ)
180	179	nncnd 12256	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝑐‘𝑗)) ∈ ℂ)
181	173, 180	fprodcl 15968	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) → ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗)) ∈ ℂ)
182	179	nnne0d 12290	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝑐‘𝑗)) ≠ 0)
183	173, 180, 182	fprodn0 15995	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) → ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗)) ≠ 0)
184	172, 181, 183	divcld 12017	. . . . . . . 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)
185	184	mul01d 11434	. . . . . . 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)
186	185	adantlr 715	. . . . . 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)
187	169, 186	eqtrd 2770	. . . . 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)
188	187	sumeq2dv 15718	. . . 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)
189		eqid 2735	. . . . . . . 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})
190	189, 6	etransclem16 46279	. . . . . . 7 ⊢ (𝜑 → ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁) ∈ Fin)
191	190	olcd 874	. . . . . 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))
192	191	adantr 480	. . . . 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))
193		sumz 15738	. . . . 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)
194	192, 193	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → Σ𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)0 = 0)
195	188, 194	eqtrd 2770	. . 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)
196	195	mpteq2dva 5214	. 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))
197	9, 196	eqtrd 2770	1 ⊢ (𝜑 → ((𝑆 D𝑛 𝐹)‘𝑁) = (𝑥 ∈ 𝑋 ↦ 0))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  ∀wral 3051  ∃wrex 3060  {crab 3415   ⊆ wss 3926  ifcif 4500  {cpr 4603   class class class wbr 5119   ↦ cmpt 5201  ⟶wf 6527  ‘cfv 6531  (class class class)co 7405   ↑_m cmap 8840  Fincfn 8959  ℂcc 11127  ℝcr 11128  0cc0 11129  1c1 11130   + caddc 11132   · cmul 11134   < clt 11269   ≤ cle 11270   − cmin 11466   / cdiv 11894  ℕcn 12240  ℕ₀cn0 12501  ℤcz 12588  ℤ_≥cuz 12852  ...cfz 13524  ↑cexp 14079  !cfa 14291  ♯chash 14348  Σcsu 15702  ∏cprod 15919   ↾_t crest 17434  TopOpenctopn 17435  ℂ_fldccnfld 21315   D^𝑛 cdvn 25817

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-inf2 9655  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206  ax-pre-sup 11207  ax-addf 11208

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-iin 4970  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7671  df-om 7862  df-1st 7988  df-2nd 7989  df-supp 8160  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-er 8719  df-map 8842  df-pm 8843  df-ixp 8912  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-fsupp 9374  df-fi 9423  df-sup 9454  df-inf 9455  df-oi 9524  df-card 9953  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-div 11895  df-nn 12241  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12502  df-z 12589  df-dec 12709  df-uz 12853  df-q 12965  df-rp 13009  df-xneg 13128  df-xadd 13129  df-xmul 13130  df-ico 13368  df-icc 13369  df-fz 13525  df-fzo 13672  df-seq 14020  df-exp 14080  df-fac 14292  df-bc 14321  df-hash 14349  df-cj 15118  df-re 15119  df-im 15120  df-sqrt 15254  df-abs 15255  df-clim 15504  df-sum 15703  df-prod 15920  df-struct 17166  df-sets 17183  df-slot 17201  df-ndx 17213  df-base 17229  df-ress 17252  df-plusg 17284  df-mulr 17285  df-starv 17286  df-sca 17287  df-vsca 17288  df-ip 17289  df-tset 17290  df-ple 17291  df-ds 17293  df-unif 17294  df-hom 17295  df-cco 17296  df-rest 17436  df-topn 17437  df-0g 17455  df-gsum 17456  df-topgen 17457  df-pt 17458  df-prds 17461  df-xrs 17516  df-qtop 17521  df-imas 17522  df-xps 17524  df-mre 17598  df-mrc 17599  df-acs 17601  df-mgm 18618  df-sgrp 18697  df-mnd 18713  df-submnd 18762  df-mulg 19051  df-cntz 19300  df-cmn 19763  df-psmet 21307  df-xmet 21308  df-met 21309  df-bl 21310  df-mopn 21311  df-fbas 21312  df-fg 21313  df-cnfld 21316  df-top 22832  df-topon 22849  df-topsp 22871  df-bases 22884  df-cld 22957  df-ntr 22958  df-cls 22959  df-nei 23036  df-lp 23074  df-perf 23075  df-cn 23165  df-cnp 23166  df-haus 23253  df-tx 23500  df-hmeo 23693  df-fil 23784  df-fm 23876  df-flim 23877  df-flf 23878  df-xms 24259  df-ms 24260  df-tms 24261  df-cncf 24822  df-limc 25819  df-dv 25820  df-dvn 25821

This theorem is referenced by:  etransclem46  46309