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Theorem etransclem33 46223
Description: 𝐹 is smooth. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
Hypotheses
Ref Expression
etransclem33.s (𝜑𝑆 ∈ {ℝ, ℂ})
etransclem33.x (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
etransclem33.p (𝜑𝑃 ∈ ℕ)
etransclem33.m (𝜑𝑀 ∈ ℕ0)
etransclem33.f 𝐹 = (𝑥𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥𝑗)↑𝑃)))
etransclem33.n (𝜑𝑁 ∈ ℕ0)
Assertion
Ref Expression
etransclem33 (𝜑 → ((𝑆 D𝑛 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.
StepHypRef Expression
1 eqidd 2736 . . . . . . 7 (𝜑 → (𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚}) = (𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚}))
2 oveq2 7439 . . . . . . . . . 10 (𝑚 = 𝑁 → (0...𝑚) = (0...𝑁))
32oveq1d 7446 . . . . . . . . 9 (𝑚 = 𝑁 → ((0...𝑚) ↑m (0...𝑀)) = ((0...𝑁) ↑m (0...𝑀)))
4 eqeq2 2747 . . . . . . . . 9 (𝑚 = 𝑁 → (Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚 ↔ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑁))
53, 4rabeqbidv 3452 . . . . . . . 8 (𝑚 = 𝑁 → {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚} = {𝑑 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑁})
65adantl 481 . . . . . . 7 ((𝜑𝑚 = 𝑁) → {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚} = {𝑑 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑁})
7 etransclem33.n . . . . . . 7 (𝜑𝑁 ∈ ℕ0)
8 ovex 7464 . . . . . . . . 9 ((0...𝑁) ↑m (0...𝑀)) ∈ V
98rabex 5345 . . . . . . . 8 {𝑑 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑁} ∈ V
109a1i 11 . . . . . . 7 (𝜑 → {𝑑 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑁} ∈ V)
111, 6, 7, 10fvmptd 7023 . . . . . 6 (𝜑 → ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁) = {𝑑 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑁})
12 fzfi 14010 . . . . . . . 8 (0...𝑁) ∈ Fin
13 fzfi 14010 . . . . . . . 8 (0...𝑀) ∈ Fin
14 mapfi 9386 . . . . . . . 8 (((0...𝑁) ∈ Fin ∧ (0...𝑀) ∈ Fin) → ((0...𝑁) ↑m (0...𝑀)) ∈ Fin)
1512, 13, 14mp2an 692 . . . . . . 7 ((0...𝑁) ↑m (0...𝑀)) ∈ Fin
16 ssrab2 4090 . . . . . . 7 {𝑑 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑁} ⊆ ((0...𝑁) ↑m (0...𝑀))
17 ssfi 9212 . . . . . . 7 ((((0...𝑁) ↑m (0...𝑀)) ∈ Fin ∧ {𝑑 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑁} ⊆ ((0...𝑁) ↑m (0...𝑀))) → {𝑑 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑁} ∈ Fin)
1815, 16, 17mp2an 692 . . . . . 6 {𝑑 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑁} ∈ Fin
1911, 18eqeltrdi 2847 . . . . 5 (𝜑 → ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁) ∈ Fin)
2019adantr 480 . . . 4 ((𝜑𝑥𝑋) → ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁) ∈ Fin)
217faccld 14320 . . . . . . . 8 (𝜑 → (!‘𝑁) ∈ ℕ)
2221nncnd 12280 . . . . . . 7 (𝜑 → (!‘𝑁) ∈ ℂ)
2322ad2antrr 726 . . . . . 6 (((𝜑𝑥𝑋) ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) → (!‘𝑁) ∈ ℂ)
2413a1i 11 . . . . . . 7 (((𝜑𝑥𝑋) ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) → (0...𝑀) ∈ Fin)
25 simpr 484 . . . . . . . . . . . . . . 15 ((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) → 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁))
2611adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) → ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁) = {𝑑 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑁})
2725, 26eleqtrd 2841 . . . . . . . . . . . . . 14 ((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) → 𝑐 ∈ {𝑑 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑁})
2816, 27sselid 3993 . . . . . . . . . . . . 13 ((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) → 𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)))
29 elmapi 8888 . . . . . . . . . . . . 13 (𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) → 𝑐:(0...𝑀)⟶(0...𝑁))
3028, 29syl 17 . . . . . . . . . . . 12 ((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) → 𝑐:(0...𝑀)⟶(0...𝑁))
3130ffvelcdmda 7104 . . . . . . . . . . 11 (((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (𝑐𝑗) ∈ (0...𝑁))
3231adantllr 719 . . . . . . . . . 10 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (𝑐𝑗) ∈ (0...𝑁))
33 elfznn0 13657 . . . . . . . . . 10 ((𝑐𝑗) ∈ (0...𝑁) → (𝑐𝑗) ∈ ℕ0)
3432, 33syl 17 . . . . . . . . 9 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (𝑐𝑗) ∈ ℕ0)
3534faccld 14320 . . . . . . . 8 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝑐𝑗)) ∈ ℕ)
3635nncnd 12280 . . . . . . 7 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝑐𝑗)) ∈ ℂ)
3724, 36fprodcl 15985 . . . . . 6 (((𝜑𝑥𝑋) ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) → ∏𝑗 ∈ (0...𝑀)(!‘(𝑐𝑗)) ∈ ℂ)
3835nnne0d 12314 . . . . . . 7 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝑐𝑗)) ≠ 0)
3924, 36, 38fprodn0 16012 . . . . . 6 (((𝜑𝑥𝑋) ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) → ∏𝑗 ∈ (0...𝑀)(!‘(𝑐𝑗)) ≠ 0)
4023, 37, 39divcld 12041 . . . . 5 (((𝜑𝑥𝑋) ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐𝑗))) ∈ ℂ)
41 etransclem33.s . . . . . . . . 9 (𝜑𝑆 ∈ {ℝ, ℂ})
4241ad3antrrr 730 . . . . . . . 8 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → 𝑆 ∈ {ℝ, ℂ})
43 etransclem33.x . . . . . . . . 9 (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
4443ad3antrrr 730 . . . . . . . 8 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
45 etransclem33.p . . . . . . . . 9 (𝜑𝑃 ∈ ℕ)
4645ad3antrrr 730 . . . . . . . 8 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → 𝑃 ∈ ℕ)
47 etransclem5 46195 . . . . . . . 8 (𝑘 ∈ (0...𝑀) ↦ (𝑦𝑋 ↦ ((𝑦𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃)))) = (𝑤 ∈ (0...𝑀) ↦ (𝑧𝑋 ↦ ((𝑧𝑤)↑if(𝑤 = 0, (𝑃 − 1), 𝑃))))
48 simpr 484 . . . . . . . 8 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ (0...𝑀))
4942, 44, 46, 47, 48, 34etransclem20 46210 . . . . . . 7 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑆 D𝑛 ((𝑘 ∈ (0...𝑀) ↦ (𝑦𝑋 ↦ ((𝑦𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))‘𝑗))‘(𝑐𝑗)):𝑋⟶ℂ)
50 simpllr 776 . . . . . . 7 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → 𝑥𝑋)
5149, 50ffvelcdmd 7105 . . . . . 6 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑆 D𝑛 ((𝑘 ∈ (0...𝑀) ↦ (𝑦𝑋 ↦ ((𝑦𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))‘𝑗))‘(𝑐𝑗))‘𝑥) ∈ ℂ)
5224, 51fprodcl 15985 . . . . 5 (((𝜑𝑥𝑋) ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) → ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 ((𝑘 ∈ (0...𝑀) ↦ (𝑦𝑋 ↦ ((𝑦𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))‘𝑗))‘(𝑐𝑗))‘𝑥) ∈ ℂ)
5340, 52mulcld 11279 . . . 4 (((𝜑𝑥𝑋) ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) → (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐𝑗))) · ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 ((𝑘 ∈ (0...𝑀) ↦ (𝑦𝑋 ↦ ((𝑦𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))‘𝑗))‘(𝑐𝑗))‘𝑥)) ∈ ℂ)
5420, 53fsumcl 15766 . . 3 ((𝜑𝑥𝑋) → Σ𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)(((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐𝑗))) · ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 ((𝑘 ∈ (0...𝑀) ↦ (𝑦𝑋 ↦ ((𝑦𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))‘𝑗))‘(𝑐𝑗))‘𝑥)) ∈ ℂ)
55 eqid 2735 . . 3 (𝑥𝑋 ↦ Σ𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)(((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐𝑗))) · ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 ((𝑘 ∈ (0...𝑀) ↦ (𝑦𝑋 ↦ ((𝑦𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))‘𝑗))‘(𝑐𝑗))‘𝑥))) = (𝑥𝑋 ↦ Σ𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)(((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐𝑗))) · ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 ((𝑘 ∈ (0...𝑀) ↦ (𝑦𝑋 ↦ ((𝑦𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))‘𝑗))‘(𝑐𝑗))‘𝑥)))
5654, 55fmptd 7134 . 2 (𝜑 → (𝑥𝑋 ↦ Σ𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)(((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐𝑗))) · ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 ((𝑘 ∈ (0...𝑀) ↦ (𝑦𝑋 ↦ ((𝑦𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))‘𝑗))‘(𝑐𝑗))‘𝑥))):𝑋⟶ℂ)
57 etransclem33.m . . . 4 (𝜑𝑀 ∈ ℕ0)
58 etransclem33.f . . . 4 𝐹 = (𝑥𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥𝑗)↑𝑃)))
59 etransclem5 46195 . . . 4 (𝑘 ∈ (0...𝑀) ↦ (𝑦𝑋 ↦ ((𝑦𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃)))) = (𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))
60 etransclem11 46201 . . . 4 (𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚}) = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑛})
6141, 43, 45, 57, 58, 7, 59, 60etransclem30 46220 . . 3 (𝜑 → ((𝑆 D𝑛 𝐹)‘𝑁) = (𝑥𝑋 ↦ Σ𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)(((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐𝑗))) · ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 ((𝑘 ∈ (0...𝑀) ↦ (𝑦𝑋 ↦ ((𝑦𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))‘𝑗))‘(𝑐𝑗))‘𝑥))))
6261feq1d 6721 . 2 (𝜑 → (((𝑆 D𝑛 𝐹)‘𝑁):𝑋⟶ℂ ↔ (𝑥𝑋 ↦ Σ𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)(((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐𝑗))) · ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 ((𝑘 ∈ (0...𝑀) ↦ (𝑦𝑋 ↦ ((𝑦𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))‘𝑗))‘(𝑐𝑗))‘𝑥))):𝑋⟶ℂ))
6356, 62mpbird 257 1 (𝜑 → ((𝑆 D𝑛 𝐹)‘𝑁):𝑋⟶ℂ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  {crab 3433  Vcvv 3478  wss 3963  ifcif 4531  {cpr 4633  cmpt 5231  wf 6559  cfv 6563  (class class class)co 7431  m cmap 8865  Fincfn 8984  cc 11151  cr 11152  0cc0 11153  1c1 11154   · cmul 11158  cmin 11490   / cdiv 11918  cn 12264  0cn0 12524  ...cfz 13544  cexp 14099  !cfa 14309  Σcsu 15719  cprod 15936  t crest 17467  TopOpenctopn 17468  fldccnfld 21382   D𝑛 cdvn 25914
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231  ax-addf 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-pm 8868  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-fi 9449  df-sup 9480  df-inf 9481  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-q 12989  df-rp 13033  df-xneg 13152  df-xadd 13153  df-xmul 13154  df-ico 13390  df-icc 13391  df-fz 13545  df-fzo 13692  df-seq 14040  df-exp 14100  df-fac 14310  df-bc 14339  df-hash 14367  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-clim 15521  df-sum 15720  df-prod 15937  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-starv 17313  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-unif 17321  df-hom 17322  df-cco 17323  df-rest 17469  df-topn 17470  df-0g 17488  df-gsum 17489  df-topgen 17490  df-pt 17491  df-prds 17494  df-xrs 17549  df-qtop 17554  df-imas 17555  df-xps 17557  df-mre 17631  df-mrc 17632  df-acs 17634  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-submnd 18810  df-mulg 19099  df-cntz 19348  df-cmn 19815  df-psmet 21374  df-xmet 21375  df-met 21376  df-bl 21377  df-mopn 21378  df-fbas 21379  df-fg 21380  df-cnfld 21383  df-top 22916  df-topon 22933  df-topsp 22955  df-bases 22969  df-cld 23043  df-ntr 23044  df-cls 23045  df-nei 23122  df-lp 23160  df-perf 23161  df-cn 23251  df-cnp 23252  df-haus 23339  df-tx 23586  df-hmeo 23779  df-fil 23870  df-fm 23962  df-flim 23963  df-flf 23964  df-xms 24346  df-ms 24347  df-tms 24348  df-cncf 24918  df-limc 25916  df-dv 25917  df-dvn 25918
This theorem is referenced by:  etransclem39  46229  etransclem43  46233  etransclem45  46235  etransclem46  46236  etransclem47  46237
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