Theorem etransclem32 47319

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 47298	. . 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 47317	. 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 490	. . . . . . . . . 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 47299	. . . . . . . . . . 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 486	. . . . . . . . . 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 2862	. . . . . . . . 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 728	. . . . . . . 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 1946	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁})
16		nfre1 3287	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑘)
17	16	nfn 1889	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘 ¬ ∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑘)
18	15, 17	nfan 1931	. . . . . . . . . . . . 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 13659	. . . . . . . . . . . . . . . . 17 ⊢ (0...𝑁) ⊆ ℝ
20		rabid 3432	. . . . . . . . . . . . . . . . . . . . 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 502	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁} → 𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)))
22		elmapi 8869	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) → 𝑐:(0...𝑀)⟶(0...𝑁))
23	21, 22	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁} → 𝑐:(0...𝑀)⟶(0...𝑁))
24	23	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) → 𝑐:(0...𝑀)⟶(0...𝑁))
25	24	ffvelcdmda 7084	. . . . . . . . . . . . . . . . 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 3906	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀)) → (𝑐‘𝑘) ∈ ℝ)
27	26	adantlr 728	. . . . . . . . . . . . . . 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 12647	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ0)
29	3, 28	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ0)
30	29	nn0red 12668	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑃 − 1) ∈ ℝ)
31	3	nnred 12350	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑃 ∈ ℝ)
32	30, 31	ifcld 4528	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → if(𝑘 = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
33	32	ad3antrrr 743	. . . . . . . . . . . . . . 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 3088	. . . . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . . . . 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 3254	. . . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . 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 11460	. . . . . . . . . . . . . 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 418	. . . . . . . . . . . . 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 3260	. . . . . . . . . . . 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 503	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁} → Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁)
42		fveq2 6885	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑘 → (𝑐‘𝑗) = (𝑐‘𝑘))
43	42	cbvsumv 15863	. . . . . . . . . . . . . . 15 ⊢ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = Σ𝑘 ∈ (0...𝑀)(𝑐‘𝑘)
44	41, 43	eqtr3di 2810	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁} → 𝑁 = Σ𝑘 ∈ (0...𝑀)(𝑐‘𝑘))
45	44	ad2antlr 740	. . . . . . . . . . . . 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 6885	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = ℎ → (𝑐‘𝑘) = (𝑐‘ℎ))
47	46	cbvsumv 15863	. . . . . . . . . . . . . 14 ⊢ Σ𝑘 ∈ (0...𝑀)(𝑐‘𝑘) = Σℎ ∈ (0...𝑀)(𝑐‘ℎ)
48		fzfid 14116	. . . . . . . . . . . . . . . 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 7084	. . . . . . . . . . . . . . . . . 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 3906	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ ℎ ∈ (0...𝑀)) → (𝑐‘ℎ) ∈ ℝ)
51	50	adantlr 728	. . . . . . . . . . . . . . . 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 4528	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → if(ℎ = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
53	52	ad3antrrr 743	. . . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = ℎ → (𝑘 = 0 ↔ ℎ = 0))
55	54	ifbid 4505	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = ℎ → if(𝑘 = 0, (𝑃 − 1), 𝑃) = if(ℎ = 0, (𝑃 − 1), 𝑃))
56	46, 55	breq12d 5115	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = ℎ → ((𝑐‘𝑘) ≤ if(𝑘 = 0, (𝑃 − 1), 𝑃) ↔ (𝑐‘ℎ) ≤ if(ℎ = 0, (𝑃 − 1), 𝑃)))
57	56	rspccva 3575	. . . . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . . 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 15966	. . . . . . . . . . . . . . 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 13003	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ0 = (ℤ≥‘0)
61	4, 60	eleqtrdi 2870	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘0))
62	3	nnnn0d 12667	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑃 ∈ ℕ0)
63	29, 62	ifcld 4528	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → if(ℎ = 0, (𝑃 − 1), 𝑃) ∈ ℕ0)
64	63	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ℎ ∈ (0...𝑀)) → if(ℎ = 0, (𝑃 − 1), 𝑃) ∈ ℕ0)
65	64	nn0cnd 12669	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ℎ ∈ (0...𝑀)) → if(ℎ = 0, (𝑃 − 1), 𝑃) ∈ ℂ)
66		iftrue 4487	. . . . . . . . . . . . . . . . . 18 ⊢ (ℎ = 0 → if(ℎ = 0, (𝑃 − 1), 𝑃) = (𝑃 − 1))
67	61, 65, 66	fsum1p 15919	. . . . . . . . . . . . . . . . 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 12463	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0 + 1) = 1
69	68	oveq1i 7430	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((0 + 1)...𝑀) = (1...𝑀)
70	69	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((0 + 1)...𝑀) = (1...𝑀))
71	70	sumeq1d 15867	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → Σℎ ∈ ((0 + 1)...𝑀)if(ℎ = 0, (𝑃 − 1), 𝑃) = Σℎ ∈ (1...𝑀)if(ℎ = 0, (𝑃 − 1), 𝑃))
72		0red 11311	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (ℎ ∈ (1...𝑀) → 0 ∈ ℝ)
73		1red 11309	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (ℎ ∈ (1...𝑀) → 1 ∈ ℝ)
74		elfzelz 13656	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (ℎ ∈ (1...𝑀) → ℎ ∈ ℤ)
75	74	zred 12803	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (ℎ ∈ (1...𝑀) → ℎ ∈ ℝ)
76		0lt1 11838	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 0 < 1
77	76	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (ℎ ∈ (1...𝑀) → 0 < 1)
78		elfzle1 13660	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (ℎ ∈ (1...𝑀) → 1 ≤ ℎ)
79	72, 73, 75, 77, 78	ltletrd 11470	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (ℎ ∈ (1...𝑀) → 0 < ℎ)
80	79	gt0ne0d 11880	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℎ ∈ (1...𝑀) → ℎ ≠ 0)
81	80	neneqd 2960	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ ∈ (1...𝑀) → ¬ ℎ = 0)
82	81	iffalsed 4492	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℎ ∈ (1...𝑀) → if(ℎ = 0, (𝑃 − 1), 𝑃) = 𝑃)
83	82	adantl 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ ℎ ∈ (1...𝑀)) → if(ℎ = 0, (𝑃 − 1), 𝑃) = 𝑃)
84	83	sumeq2dv 15869	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → Σℎ ∈ (1...𝑀)if(ℎ = 0, (𝑃 − 1), 𝑃) = Σℎ ∈ (1...𝑀)𝑃)
85		fzfid 14116	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (1...𝑀) ∈ Fin)
86	3	nncnd 12351	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑃 ∈ ℂ)
87		fsumconst 15956	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((1...𝑀) ∈ Fin ∧ 𝑃 ∈ ℂ) → Σℎ ∈ (1...𝑀)𝑃 = ((♯‘(1...𝑀)) · 𝑃))
88	85, 86, 87	syl2anc 596	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → Σℎ ∈ (1...𝑀)𝑃 = ((♯‘(1...𝑀)) · 𝑃))
89		hashfz1 14490	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑀 ∈ ℕ0 → (♯‘(1...𝑀)) = 𝑀)
90	4, 89	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (♯‘(1...𝑀)) = 𝑀)
91	90	oveq1d 7435	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((♯‘(1...𝑀)) · 𝑃) = (𝑀 · 𝑃))
92	88, 91	eqtrd 2795	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → Σℎ ∈ (1...𝑀)𝑃 = (𝑀 · 𝑃))
93	71, 84, 92	3eqtrd 2799	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → Σℎ ∈ ((0 + 1)...𝑀)if(ℎ = 0, (𝑃 − 1), 𝑃) = (𝑀 · 𝑃))
94	93	oveq2d 7436	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑃 − 1) + Σℎ ∈ ((0 + 1)...𝑀)if(ℎ = 0, (𝑃 − 1), 𝑃)) = ((𝑃 − 1) + (𝑀 · 𝑃)))
95	29	nn0cnd 12669	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑃 − 1) ∈ ℂ)
96	4, 62	nn0mulcld 12672	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑀 · 𝑃) ∈ ℕ0)
97	96	nn0cnd 12669	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑀 · 𝑃) ∈ ℂ)
98	95, 97	addcomd 11512	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑃 − 1) + (𝑀 · 𝑃)) = ((𝑀 · 𝑃) + (𝑃 − 1)))
99	67, 94, 98	3eqtrd 2799	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → Σℎ ∈ (0...𝑀)if(ℎ = 0, (𝑃 − 1), 𝑃) = ((𝑀 · 𝑃) + (𝑃 − 1)))
100	99	ad2antrr 739	. . . . . . . . . . . . . . 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 5130	. . . . . . . . . . . . . 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 5139	. . . . . . . . . . . . 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 5126	. . . . . . . . . . . 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 603	. . . . . . . . . . 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 12671	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑀 · 𝑃) + (𝑃 − 1)) ∈ ℕ0)
107	106	nn0red 12668	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑀 · 𝑃) + (𝑃 − 1)) ∈ ℝ)
108	6	nn0red 12668	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℝ)
109	107, 108	ltnled 11457	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((𝑀 · 𝑃) + (𝑃 − 1)) < 𝑁 ↔ ¬ 𝑁 ≤ ((𝑀 · 𝑃) + (𝑃 − 1))))
110	105, 109	mpbid 235	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 𝑁 ≤ ((𝑀 · 𝑃) + (𝑃 − 1)))
111	110	ad2antrr 739	. . . . . . . . . . 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 830	. . . . . . . . . 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 728	. . . . . . . . 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 1946	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑥 ∈ 𝑋)
115		nfcv 2922	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑗(0...𝑀)
116	115	nfsum1 15857	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑗Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗)
117	116	nfeq1 2937	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁
118		nfcv 2922	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗((0...𝑁) ↑m (0...𝑀))
119	117, 118	nfrabw 3447	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗{𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}
120	119	nfcri 2914	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}
121	114, 120	nfan 1931	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁})
122		nfv 1946	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 𝑘 ∈ (0...𝑀)
123		nfv 1946	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑘)
124	121, 122, 123	nf3an 1933	. . . . . . . . . . 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 2922	. . . . . . . . . . 11 ⊢ Ⅎ𝑗(((𝑆 D𝑛 (𝐻‘𝑘))‘(𝑐‘𝑘))‘𝑥)
126		fzfid 14116	. . . . . . . . . . 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 743	. . . . . . . . . . . . . 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 743	. . . . . . . . . . . . . 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 743	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → 𝑃 ∈ ℕ)
130		etransclem5 47292	. . . . . . . . . . . . . . 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 2783	. . . . . . . . . . . . . 14 ⊢ 𝐻 = (𝑘 ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))
132		simpr 490	. . . . . . . . . . . . . 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 740	. . . . . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . . . 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 7085	. . . . . . . . . . . . . . . 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 732	. . . . . . . . . . . . . . 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 13754	. . . . . . . . . . . . . . 15 ⊢ ((𝑐‘𝑗) ∈ (0...𝑁) → (𝑐‘𝑗) ∈ ℕ0)
138	136, 137	syl 18	. . . . . . . . . . . . . 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 47307	. . . . . . . . . . . . 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 788	. . . . . . . . . . . . 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 7085	. . . . . . . . . . . 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 1204	. . . . . . . . . . 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 6885	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → (𝐻‘𝑗) = (𝐻‘𝑘))
144	143	oveq2d 7436	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑘 → (𝑆 D𝑛 (𝐻‘𝑗)) = (𝑆 D𝑛 (𝐻‘𝑘)))
145	144, 42	fveq12d 6892	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → ((𝑆 D𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗)) = ((𝑆 D𝑛 (𝐻‘𝑘))‘(𝑐‘𝑘)))
146	145	fveq1d 6887	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → (((𝑆 D𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑥) = (((𝑆 D𝑛 (𝐻‘𝑘))‘(𝑐‘𝑘))‘𝑥))
147		simp2 1155	. . . . . . . . . . 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 739	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) → 𝑆 ∈ {ℝ, ℂ})
149	148	3ad2ant1 1151	. . . . . . . . . . . . 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 739	. . . . . . . . . . . . . 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 1151	. . . . . . . . . . . . 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 739	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) → 𝑃 ∈ ℕ)
153	152	3ad2ant1 1151	. . . . . . . . . . . . 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 47292	. . . . . . . . . . . . . 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 2783	. . . . . . . . . . . . 13 ⊢ 𝐻 = (ℎ ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 − ℎ)↑if(ℎ = 0, (𝑃 − 1), 𝑃))))
156	25	elfzelzd 13657	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀)) → (𝑐‘𝑘) ∈ ℤ)
157	156	adantllr 732	. . . . . . . . . . . . . 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 1150	. . . . . . . . . . . . 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 1156	. . . . . . . . . . . . 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 47306	. . . . . . . . . . . 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 2761	. . . . . . . . . . . 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 1256	. . . . . . . . . . . 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 11311	. . . . . . . . . . . 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 7001	. . . . . . . . . . 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 46651	. . . . . . . . . 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 3167	. . . . . . . . 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 16	. . . . . . . 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 603	. . . . . . 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 7436	. . . . . 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 14428	. . . . . . . . . . 11 ⊢ (𝜑 → (!‘𝑁) ∈ ℕ)
171	170	nncnd 12351	. . . . . . . . . 10 ⊢ (𝜑 → (!‘𝑁) ∈ ℂ)
172	171	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) → (!‘𝑁) ∈ ℂ)
173		fzfid 14116	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) → (0...𝑀) ∈ Fin)
174		simpll 779	. . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . 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 851	. . . . . . . . . . . . 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 18	. . . . . . . . . . . 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 14428	. . . . . . . . . . 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 12351	. . . . . . . . . 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 16119	. . . . . . . . 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 12388	. . . . . . . . . 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 16146	. . . . . . . . 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 12093	. . . . . . . 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 11509	. . . . . . 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 728	. . . . . 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 2795	. . . . 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 15869	. . . 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 2760	. . . . . . . 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 47303	. . . . . . 7 ⊢ (𝜑 → ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁) ∈ Fin)
191	190	olcd 888	. . . . . 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 486	. . . . 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 15888	. . . . 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 18	. . . 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 2795	. . 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 5197	. 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 2795	1 ⊢ (𝜑 → ((𝑆 D𝑛 𝐹)‘𝑁) = (𝑥 ∈ 𝑋 ↦ 0))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  {crab 3412   ⊆ wss 3895  ifcif 4481  {cpr 4585   class class class wbr 5102   ↦ cmpt 5185  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420   ↑_m cmap 8847  Fincfn 8973  ℂcc 11198  ℝcr 11199  0cc0 11200  1c1 11201   + caddc 11203   · cmul 11205   < clt 11343   ≤ cle 11344   − cmin 11541   / cdiv 11973  ℕcn 12335  ℕ₀cn0 12606  ℤcz 12693  ℤ_≥cuz 12965  ...cfz 13639  ↑cexp 14204  !cfa 14417  ♯chash 14474  Σcsu 15853  ∏cprod 16072   ↾_t crest 17591  TopOpenctopn 17592  ℂ_fldccnfld 21678   D^𝑛 cdvn 26184

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-inf2 9642  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277  ax-pre-sup 11278  ax-addf 11279

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-of 7693  df-om 7878  df-1st 8001  df-2nd 8002  df-supp 8178  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-er 8717  df-map 8849  df-pm 8850  df-ixp 8926  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-fsupp 9354  df-fi 9403  df-sup 9434  df-inf 9435  df-oi 9504  df-card 10020  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-div 11974  df-nn 12336  df-2 12405  df-3 12406  df-4 12407  df-5 12408  df-6 12409  df-7 12410  df-8 12411  df-9 12412  df-n0 12607  df-z 12694  df-dec 12815  df-uz 12966  df-q 13076  df-rp 13121  df-xneg 13241  df-xadd 13242  df-xmul 13243  df-ico 13482  df-icc 13483  df-fz 13640  df-fzo 13789  df-seq 14145  df-exp 14205  df-fac 14418  df-bc 14447  df-hash 14475  df-cj 15266  df-re 15267  df-im 15268  df-sqrt 15402  df-abs 15403  df-clim 15655  df-sum 15854  df-prod 16073  df-struct 17325  df-sets 17342  df-slot 17360  df-ndx 17372  df-base 17388  df-ress 17409  df-plusg 17441  df-mulr 17442  df-starv 17443  df-sca 17444  df-vsca 17445  df-ip 17446  df-tset 17447  df-ple 17448  df-ds 17450  df-unif 17451  df-hom 17452  df-cco 17453  df-rest 17593  df-topn 17594  df-0g 17612  df-gsum 17613  df-topgen 17614  df-pt 17615  df-prds 17618  df-xrs 17674  df-qtop 17679  df-imas 17680  df-xps 17682  df-mre 17756  df-mrc 17757  df-acs 17759  df-mgm 18816  df-sgrp 18908  df-mnd 18924  df-submnd 18979  df-mulg 19278  df-cntz 19531  df-cmn 19996  df-psmet 21670  df-xmet 21671  df-met 21672  df-bl 21673  df-mopn 21674  df-fbas 21675  df-fg 21676  df-cnfld 21679  df-top 23212  df-topon 23229  df-topsp 23251  df-bases 23264  df-cld 23337  df-ntr 23338  df-cls 23339  df-nei 23416  df-lp 23454  df-perf 23455  df-cn 23545  df-cnp 23546  df-haus 23633  df-tx 23881  df-hmeo 24074  df-fil 24165  df-fm 24257  df-flim 24258  df-flf 24259  df-xms 24639  df-ms 24640  df-tms 24641  df-cncf 25199  df-limc 26186  df-dv 26187  df-dvn 26188

This theorem is used by:  etransclem46  47333