etransclem33 - Mathbox for Glauco Siliprandi

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Theorem etransclem33 44498

Description: 𝐹 is smooth. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

**Hypotheses**
Ref	Expression
etransclem33.s	⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
etransclem33.x	⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
etransclem33.p	⊢ (𝜑 → 𝑃 ∈ ℕ)
etransclem33.m	⊢ (𝜑 → 𝑀 ∈ ℕ₀)
etransclem33.f	⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃)))
etransclem33.n	⊢ (𝜑 → 𝑁 ∈ ℕ₀)

**Assertion**
Ref	Expression
etransclem33	⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑁):𝑋⟶ℂ)

Distinct variable groups: 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 etransclem33 Dummy variables 𝑐 𝑑 𝑘 𝑚 𝑛 𝑤 𝑧 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2737	. . . . . . 7 ⊢ (𝜑 → (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}))
2		oveq2 7365	. . . . . . . . . 10 ⊢ (𝑚 = 𝑁 → (0...𝑚) = (0...𝑁))
3	2	oveq1d 7372	. . . . . . . . 9 ⊢ (𝑚 = 𝑁 → ((0...𝑚) ↑_m (0...𝑀)) = ((0...𝑁) ↑_m (0...𝑀)))
4		eqeq2 2748	. . . . . . . . 9 ⊢ (𝑚 = 𝑁 → (Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚 ↔ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑁))
5	3, 4	rabeqbidv 3424	. . . . . . . 8 ⊢ (𝑚 = 𝑁 → {𝑑 ∈ ((0...𝑚) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚} = {𝑑 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑁})
6	5	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 = 𝑁) → {𝑑 ∈ ((0...𝑚) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚} = {𝑑 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑁})
7		etransclem33.n	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
8		ovex 7390	. . . . . . . . 9 ⊢ ((0...𝑁) ↑_m (0...𝑀)) ∈ V
9	8	rabex 5289	. . . . . . . 8 ⊢ {𝑑 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑁} ∈ V
10	9	a1i 11	. . . . . . 7 ⊢ (𝜑 → {𝑑 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑁} ∈ V)
11	1, 6, 7, 10	fvmptd 6955	. . . . . 6 ⊢ (𝜑 → ((𝑚 ∈ ℕ₀ ↦ {𝑑 ∈ ((0...𝑚) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁) = {𝑑 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑁})
12		fzfi 13877	. . . . . . . 8 ⊢ (0...𝑁) ∈ Fin
13		fzfi 13877	. . . . . . . 8 ⊢ (0...𝑀) ∈ Fin
14		mapfi 9292	. . . . . . . 8 ⊢ (((0...𝑁) ∈ Fin ∧ (0...𝑀) ∈ Fin) → ((0...𝑁) ↑_m (0...𝑀)) ∈ Fin)
15	12, 13, 14	mp2an 690	. . . . . . 7 ⊢ ((0...𝑁) ↑_m (0...𝑀)) ∈ Fin
16		ssrab2 4037	. . . . . . 7 ⊢ {𝑑 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑁} ⊆ ((0...𝑁) ↑_m (0...𝑀))
17		ssfi 9117	. . . . . . 7 ⊢ ((((0...𝑁) ↑_m (0...𝑀)) ∈ Fin ∧ {𝑑 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑁} ⊆ ((0...𝑁) ↑_m (0...𝑀))) → {𝑑 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑁} ∈ Fin)
18	15, 16, 17	mp2an 690	. . . . . 6 ⊢ {𝑑 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑁} ∈ Fin
19	11, 18	eqeltrdi 2846	. . . . 5 ⊢ (𝜑 → ((𝑚 ∈ ℕ₀ ↦ {𝑑 ∈ ((0...𝑚) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁) ∈ Fin)
20	19	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑚 ∈ ℕ₀ ↦ {𝑑 ∈ ((0...𝑚) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁) ∈ Fin)
21	7	faccld 14184	. . . . . . . 8 ⊢ (𝜑 → (!‘𝑁) ∈ ℕ)
22	21	nncnd 12169	. . . . . . 7 ⊢ (𝜑 → (!‘𝑁) ∈ ℂ)
23	22	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝑚 ∈ ℕ₀ ↦ {𝑑 ∈ ((0...𝑚) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) → (!‘𝑁) ∈ ℂ)
24	13	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝑚 ∈ ℕ₀ ↦ {𝑑 ∈ ((0...𝑚) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) → (0...𝑀) ∈ Fin)
25		simpr 485	. . . . . . . . . . . . . . 15 ⊢ ((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))
26	11	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((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) = {𝑑 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑁})
27	25, 26	eleqtrd 2840	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝑚 ∈ ℕ₀ ↦ {𝑑 ∈ ((0...𝑚) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) → 𝑐 ∈ {𝑑 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑁})
28	16, 27	sselid 3942	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝑚 ∈ ℕ₀ ↦ {𝑑 ∈ ((0...𝑚) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) → 𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)))
29		elmapi 8787	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) → 𝑐:(0...𝑀)⟶(0...𝑁))
30	28, 29	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝑚 ∈ ℕ₀ ↦ {𝑑 ∈ ((0...𝑚) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) → 𝑐:(0...𝑀)⟶(0...𝑁))
31	30	ffvelcdmda 7035	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝑚 ∈ ℕ₀ ↦ {𝑑 ∈ ((0...𝑚) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (𝑐‘𝑗) ∈ (0...𝑁))
32	31	adantllr 717	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝑚 ∈ ℕ₀ ↦ {𝑑 ∈ ((0...𝑚) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (𝑐‘𝑗) ∈ (0...𝑁))
33		elfznn0 13534	. . . . . . . . . 10 ⊢ ((𝑐‘𝑗) ∈ (0...𝑁) → (𝑐‘𝑗) ∈ ℕ₀)
34	32, 33	syl 17	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝑚 ∈ ℕ₀ ↦ {𝑑 ∈ ((0...𝑚) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (𝑐‘𝑗) ∈ ℕ₀)
35	34	faccld 14184	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝑚 ∈ ℕ₀ ↦ {𝑑 ∈ ((0...𝑚) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝑐‘𝑗)) ∈ ℕ)
36	35	nncnd 12169	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝑚 ∈ ℕ₀ ↦ {𝑑 ∈ ((0...𝑚) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝑐‘𝑗)) ∈ ℂ)
37	24, 36	fprodcl 15835	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝑚 ∈ ℕ₀ ↦ {𝑑 ∈ ((0...𝑚) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) → ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗)) ∈ ℂ)
38	35	nnne0d 12203	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝑚 ∈ ℕ₀ ↦ {𝑑 ∈ ((0...𝑚) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝑐‘𝑗)) ≠ 0)
39	24, 36, 38	fprodn0 15862	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝑚 ∈ ℕ₀ ↦ {𝑑 ∈ ((0...𝑚) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) → ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗)) ≠ 0)
40	23, 37, 39	divcld 11931	. . . . 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))) ∈ ℂ)
41		etransclem33.s	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
42	41	ad3antrrr 728	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝑚 ∈ ℕ₀ ↦ {𝑑 ∈ ((0...𝑚) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → 𝑆 ∈ {ℝ, ℂ})
43		etransclem33.x	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
44	43	ad3antrrr 728	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝑚 ∈ ℕ₀ ↦ {𝑑 ∈ ((0...𝑚) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
45		etransclem33.p	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℕ)
46	45	ad3antrrr 728	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝑚 ∈ ℕ₀ ↦ {𝑑 ∈ ((0...𝑚) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → 𝑃 ∈ ℕ)
47		etransclem5 44470	. . . . . . . 8 ⊢ (𝑘 ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃)))) = (𝑤 ∈ (0...𝑀) ↦ (𝑧 ∈ 𝑋 ↦ ((𝑧 − 𝑤)↑if(𝑤 = 0, (𝑃 − 1), 𝑃))))
48		simpr 485	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝑚 ∈ ℕ₀ ↦ {𝑑 ∈ ((0...𝑚) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ (0...𝑀))
49	42, 44, 46, 47, 48, 34	etransclem20 44485	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝑚 ∈ ℕ₀ ↦ {𝑑 ∈ ((0...𝑚) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑆 D^𝑛 ((𝑘 ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))‘𝑗))‘(𝑐‘𝑗)):𝑋⟶ℂ)
50		simpllr 774	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝑚 ∈ ℕ₀ ↦ {𝑑 ∈ ((0...𝑚) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → 𝑥 ∈ 𝑋)
51	49, 50	ffvelcdmd 7036	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝑚 ∈ ℕ₀ ↦ {𝑑 ∈ ((0...𝑚) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑆 D^𝑛 ((𝑘 ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))‘𝑗))‘(𝑐‘𝑗))‘𝑥) ∈ ℂ)
52	24, 51	fprodcl 15835	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝑚 ∈ ℕ₀ ↦ {𝑑 ∈ ((0...𝑚) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})‘𝑁)) → ∏𝑗 ∈ (0...𝑀)(((𝑆 D^𝑛 ((𝑘 ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))‘𝑗))‘(𝑐‘𝑗))‘𝑥) ∈ ℂ)
53	40, 52	mulcld 11175	. . . 4 ⊢ (((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 D^𝑛 ((𝑘 ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))‘𝑗))‘(𝑐‘𝑗))‘𝑥)) ∈ ℂ)
54	20, 53	fsumcl 15618	. . 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 D^𝑛 ((𝑘 ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))‘𝑗))‘(𝑐‘𝑗))‘𝑥)) ∈ ℂ)
55		eqid 2736	. . 3 ⊢ (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 D^𝑛 ((𝑘 ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))‘𝑗))‘(𝑐‘𝑗))‘𝑥))) = (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 D^𝑛 ((𝑘 ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))‘𝑗))‘(𝑐‘𝑗))‘𝑥)))
56	54, 55	fmptd 7062	. 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 D^𝑛 ((𝑘 ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))‘𝑗))‘(𝑐‘𝑗))‘𝑥))):𝑋⟶ℂ)
57		etransclem33.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
58		etransclem33.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃)))
59		etransclem5 44470	. . . 4 ⊢ (𝑘 ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃)))) = (𝑗 ∈ (0...𝑀) ↦ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))
60		etransclem11 44476	. . . 4 ⊢ (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})
61	41, 43, 45, 57, 58, 7, 59, 60	etransclem30 44495	. . 3 ⊢ (𝜑 → ((𝑆 D^𝑛 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 D^𝑛 ((𝑘 ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))‘𝑗))‘(𝑐‘𝑗))‘𝑥))))
62	61	feq1d 6653	. 2 ⊢ (𝜑 → (((𝑆 D^𝑛 𝐹)‘𝑁):𝑋⟶ℂ ↔ (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 D^𝑛 ((𝑘 ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))‘𝑗))‘(𝑐‘𝑗))‘𝑥))):𝑋⟶ℂ))
63	56, 62	mpbird 256	1 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑁):𝑋⟶ℂ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {crab 3407 Vcvv 3445 ⊆ wss 3910 ifcif 4486 {cpr 4588 ↦ cmpt 5188 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 ↑_m cmap 8765 Fincfn 8883 ℂcc 11049 ℝcr 11050 0cc0 11051 1c1 11052 · cmul 11056 − cmin 11385 / cdiv 11812 ℕcn 12153 ℕ₀cn0 12413 ...cfz 13424 ↑cexp 13967 !cfa 14173 Σcsu 15570 ∏cprod 15788 ↾_t crest 17302 TopOpenctopn 17303 ℂ_fldccnfld 20796 D^𝑛 cdvn 25228

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-seq 13907 df-exp 13968 df-fac 14174 df-bc 14203 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-clim 15370 df-sum 15571 df-prod 15789 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-xrs 17384 df-qtop 17389 df-imas 17390 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-mulg 18873 df-cntz 19097 df-cmn 19564 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-fbas 20793 df-fg 20794 df-cnfld 20797 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cld 22370 df-ntr 22371 df-cls 22372 df-nei 22449 df-lp 22487 df-perf 22488 df-cn 22578 df-cnp 22579 df-haus 22666 df-tx 22913 df-hmeo 23106 df-fil 23197 df-fm 23289 df-flim 23290 df-flf 23291 df-xms 23673 df-ms 23674 df-tms 23675 df-cncf 24241 df-limc 25230 df-dv 25231 df-dvn 25232

This theorem is referenced by: etransclem39 44504 etransclem43 44508 etransclem45 44510 etransclem46 44511 etransclem47 44512

W3C validator