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Theorem etransclem32 44819
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 (𝜑𝑆 ∈ {ℝ, ℂ})
etransclem32.x (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
etransclem32.p (𝜑𝑃 ∈ ℕ)
etransclem32.m (𝜑𝑀 ∈ ℕ0)
etransclem32.f 𝐹 = (𝑥𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥𝑗)↑𝑃)))
etransclem32.n (𝜑𝑁 ∈ ℕ0)
etransclem32.ngt (𝜑 → ((𝑀 · 𝑃) + (𝑃 − 1)) < 𝑁)
etransclem32.h 𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))
Assertion
Ref Expression
etransclem32 (𝜑 → ((𝑆 D𝑛 𝐹)‘𝑁) = (𝑥𝑋 ↦ 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 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 etransclem32.s . . 3 (𝜑𝑆 ∈ {ℝ, ℂ})
2 etransclem32.x . . 3 (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
3 etransclem32.p . . 3 (𝜑𝑃 ∈ ℕ)
4 etransclem32.m . . 3 (𝜑𝑀 ∈ ℕ0)
5 etransclem32.f . . 3 𝐹 = (𝑥𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥𝑗)↑𝑃)))
6 etransclem32.n . . 3 (𝜑𝑁 ∈ ℕ0)
7 etransclem32.h . . 3 𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))
8 etransclem11 44798 . . 3 (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})
91, 2, 3, 4, 5, 6, 7, 8etransclem30 44817 . 2 (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 (𝐻𝑗))‘(𝑐𝑗))‘𝑥))))
10 simpr 485 . . . . . . . . . 10 ((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))
118, 6etransclem12 44799 . . . . . . . . . . 11 (𝜑 → ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁) = {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁})
1211adantr 481 . . . . . . . . . 10 ((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) = {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁})
1310, 12eleqtrd 2835 . . . . . . . . 9 ((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) → 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁})
1413adantlr 713 . . . . . . . 8 (((𝜑𝑥𝑋) ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) → 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁})
15 nfv 1917 . . . . . . . . . . . . . 14 𝑘(𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁})
16 nfre1 3282 . . . . . . . . . . . . . . 15 𝑘𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘)
1716nfn 1860 . . . . . . . . . . . . . 14 𝑘 ¬ ∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘)
1815, 17nfan 1902 . . . . . . . . . . . . 13 𝑘((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ ¬ ∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘))
19 fzssre 43861 . . . . . . . . . . . . . . . . 17 (0...𝑁) ⊆ ℝ
20 rabid 3452 . . . . . . . . . . . . . . . . . . . . 21 (𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁} ↔ (𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∧ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁))
2120simplbi 498 . . . . . . . . . . . . . . . . . . . 20 (𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁} → 𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)))
22 elmapi 8828 . . . . . . . . . . . . . . . . . . . 20 (𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) → 𝑐:(0...𝑀)⟶(0...𝑁))
2321, 22syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁} → 𝑐:(0...𝑀)⟶(0...𝑁))
2423adantl 482 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) → 𝑐:(0...𝑀)⟶(0...𝑁))
2524ffvelcdmda 7072 . . . . . . . . . . . . . . . . 17 (((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀)) → (𝑐𝑘) ∈ (0...𝑁))
2619, 25sselid 3977 . . . . . . . . . . . . . . . 16 (((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀)) → (𝑐𝑘) ∈ ℝ)
2726adantlr 713 . . . . . . . . . . . . . . 15 ((((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ ¬ ∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘)) ∧ 𝑘 ∈ (0...𝑀)) → (𝑐𝑘) ∈ ℝ)
28 nnm1nn0 12497 . . . . . . . . . . . . . . . . . . 19 (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ0)
293, 28syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑃 − 1) ∈ ℕ0)
3029nn0red 12517 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑃 − 1) ∈ ℝ)
313nnred 12211 . . . . . . . . . . . . . . . . 17 (𝜑𝑃 ∈ ℝ)
3230, 31ifcld 4569 . . . . . . . . . . . . . . . 16 (𝜑 → if(𝑘 = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
3332ad3antrrr 728 . . . . . . . . . . . . . . 15 ((((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ ¬ ∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘)) ∧ 𝑘 ∈ (0...𝑀)) → if(𝑘 = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
34 ralnex 3072 . . . . . . . . . . . . . . . . . 18 (∀𝑘 ∈ (0...𝑀) ¬ if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘) ↔ ¬ ∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘))
3534biimpri 227 . . . . . . . . . . . . . . . . 17 (¬ ∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘) → ∀𝑘 ∈ (0...𝑀) ¬ if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘))
3635r19.21bi 3248 . . . . . . . . . . . . . . . 16 ((¬ ∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘) ∧ 𝑘 ∈ (0...𝑀)) → ¬ if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘))
3736adantll 712 . . . . . . . . . . . . . . 15 ((((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ ¬ ∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘)) ∧ 𝑘 ∈ (0...𝑀)) → ¬ if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘))
3827, 33, 37nltled 11348 . . . . . . . . . . . . . 14 ((((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ ¬ ∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘)) ∧ 𝑘 ∈ (0...𝑀)) → (𝑐𝑘) ≤ if(𝑘 = 0, (𝑃 − 1), 𝑃))
3938ex 413 . . . . . . . . . . . . 13 (((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ ¬ ∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘)) → (𝑘 ∈ (0...𝑀) → (𝑐𝑘) ≤ if(𝑘 = 0, (𝑃 − 1), 𝑃)))
4018, 39ralrimi 3254 . . . . . . . . . . . 12 (((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ ¬ ∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘)) → ∀𝑘 ∈ (0...𝑀)(𝑐𝑘) ≤ if(𝑘 = 0, (𝑃 − 1), 𝑃))
4120simprbi 497 . . . . . . . . . . . . . . 15 (𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁} → Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁)
42 fveq2 6879 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑘 → (𝑐𝑗) = (𝑐𝑘))
4342cbvsumv 15626 . . . . . . . . . . . . . . 15 Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = Σ𝑘 ∈ (0...𝑀)(𝑐𝑘)
4441, 43eqtr3di 2787 . . . . . . . . . . . . . 14 (𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁} → 𝑁 = Σ𝑘 ∈ (0...𝑀)(𝑐𝑘))
4544ad2antlr 725 . . . . . . . . . . . . 13 (((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ ∀𝑘 ∈ (0...𝑀)(𝑐𝑘) ≤ if(𝑘 = 0, (𝑃 − 1), 𝑃)) → 𝑁 = Σ𝑘 ∈ (0...𝑀)(𝑐𝑘))
46 fveq2 6879 . . . . . . . . . . . . . . 15 (𝑘 = → (𝑐𝑘) = (𝑐))
4746cbvsumv 15626 . . . . . . . . . . . . . 14 Σ𝑘 ∈ (0...𝑀)(𝑐𝑘) = Σ ∈ (0...𝑀)(𝑐)
48 fzfid 13922 . . . . . . . . . . . . . . . 16 (((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ ∀𝑘 ∈ (0...𝑀)(𝑐𝑘) ≤ if(𝑘 = 0, (𝑃 − 1), 𝑃)) → (0...𝑀) ∈ Fin)
4924ffvelcdmda 7072 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ ∈ (0...𝑀)) → (𝑐) ∈ (0...𝑁))
5019, 49sselid 3977 . . . . . . . . . . . . . . . . 17 (((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ ∈ (0...𝑀)) → (𝑐) ∈ ℝ)
5150adantlr 713 . . . . . . . . . . . . . . . 16 ((((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ ∀𝑘 ∈ (0...𝑀)(𝑐𝑘) ≤ if(𝑘 = 0, (𝑃 − 1), 𝑃)) ∧ ∈ (0...𝑀)) → (𝑐) ∈ ℝ)
5230, 31ifcld 4569 . . . . . . . . . . . . . . . . 17 (𝜑 → if( = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
5352ad3antrrr 728 . . . . . . . . . . . . . . . 16 ((((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ ∀𝑘 ∈ (0...𝑀)(𝑐𝑘) ≤ if(𝑘 = 0, (𝑃 − 1), 𝑃)) ∧ ∈ (0...𝑀)) → if( = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
54 eqeq1 2736 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = → (𝑘 = 0 ↔ = 0))
5554ifbid 4546 . . . . . . . . . . . . . . . . . . 19 (𝑘 = → if(𝑘 = 0, (𝑃 − 1), 𝑃) = if( = 0, (𝑃 − 1), 𝑃))
5646, 55breq12d 5155 . . . . . . . . . . . . . . . . . 18 (𝑘 = → ((𝑐𝑘) ≤ if(𝑘 = 0, (𝑃 − 1), 𝑃) ↔ (𝑐) ≤ if( = 0, (𝑃 − 1), 𝑃)))
5756rspccva 3609 . . . . . . . . . . . . . . . . 17 ((∀𝑘 ∈ (0...𝑀)(𝑐𝑘) ≤ if(𝑘 = 0, (𝑃 − 1), 𝑃) ∧ ∈ (0...𝑀)) → (𝑐) ≤ if( = 0, (𝑃 − 1), 𝑃))
5857adantll 712 . . . . . . . . . . . . . . . 16 ((((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ ∀𝑘 ∈ (0...𝑀)(𝑐𝑘) ≤ if(𝑘 = 0, (𝑃 − 1), 𝑃)) ∧ ∈ (0...𝑀)) → (𝑐) ≤ if( = 0, (𝑃 − 1), 𝑃))
5948, 51, 53, 58fsumle 15729 . . . . . . . . . . . . . . 15 (((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ ∀𝑘 ∈ (0...𝑀)(𝑐𝑘) ≤ if(𝑘 = 0, (𝑃 − 1), 𝑃)) → Σ ∈ (0...𝑀)(𝑐) ≤ Σ ∈ (0...𝑀)if( = 0, (𝑃 − 1), 𝑃))
60 nn0uz 12848 . . . . . . . . . . . . . . . . . . 19 0 = (ℤ‘0)
614, 60eleqtrdi 2843 . . . . . . . . . . . . . . . . . 18 (𝜑𝑀 ∈ (ℤ‘0))
623nnnn0d 12516 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑃 ∈ ℕ0)
6329, 62ifcld 4569 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → if( = 0, (𝑃 − 1), 𝑃) ∈ ℕ0)
6463adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∈ (0...𝑀)) → if( = 0, (𝑃 − 1), 𝑃) ∈ ℕ0)
6564nn0cnd 12518 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∈ (0...𝑀)) → if( = 0, (𝑃 − 1), 𝑃) ∈ ℂ)
66 iftrue 4529 . . . . . . . . . . . . . . . . . 18 ( = 0 → if( = 0, (𝑃 − 1), 𝑃) = (𝑃 − 1))
6761, 65, 66fsum1p 15683 . . . . . . . . . . . . . . . . 17 (𝜑 → Σ ∈ (0...𝑀)if( = 0, (𝑃 − 1), 𝑃) = ((𝑃 − 1) + Σ ∈ ((0 + 1)...𝑀)if( = 0, (𝑃 − 1), 𝑃)))
68 0p1e1 12318 . . . . . . . . . . . . . . . . . . . . . 22 (0 + 1) = 1
6968oveq1i 7404 . . . . . . . . . . . . . . . . . . . . 21 ((0 + 1)...𝑀) = (1...𝑀)
7069a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((0 + 1)...𝑀) = (1...𝑀))
7170sumeq1d 15631 . . . . . . . . . . . . . . . . . . 19 (𝜑 → Σ ∈ ((0 + 1)...𝑀)if( = 0, (𝑃 − 1), 𝑃) = Σ ∈ (1...𝑀)if( = 0, (𝑃 − 1), 𝑃))
72 0red 11201 . . . . . . . . . . . . . . . . . . . . . . . . 25 ( ∈ (1...𝑀) → 0 ∈ ℝ)
73 1red 11199 . . . . . . . . . . . . . . . . . . . . . . . . 25 ( ∈ (1...𝑀) → 1 ∈ ℝ)
74 elfzelz 13485 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ( ∈ (1...𝑀) → ∈ ℤ)
7574zred 12650 . . . . . . . . . . . . . . . . . . . . . . . . 25 ( ∈ (1...𝑀) → ∈ ℝ)
76 0lt1 11720 . . . . . . . . . . . . . . . . . . . . . . . . . 26 0 < 1
7776a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25 ( ∈ (1...𝑀) → 0 < 1)
78 elfzle1 13488 . . . . . . . . . . . . . . . . . . . . . . . . 25 ( ∈ (1...𝑀) → 1 ≤ )
7972, 73, 75, 77, 78ltletrd 11358 . . . . . . . . . . . . . . . . . . . . . . . 24 ( ∈ (1...𝑀) → 0 < )
8079gt0ne0d 11762 . . . . . . . . . . . . . . . . . . . . . . 23 ( ∈ (1...𝑀) → ≠ 0)
8180neneqd 2945 . . . . . . . . . . . . . . . . . . . . . 22 ( ∈ (1...𝑀) → ¬ = 0)
8281iffalsed 4534 . . . . . . . . . . . . . . . . . . . . 21 ( ∈ (1...𝑀) → if( = 0, (𝑃 − 1), 𝑃) = 𝑃)
8382adantl 482 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∈ (1...𝑀)) → if( = 0, (𝑃 − 1), 𝑃) = 𝑃)
8483sumeq2dv 15633 . . . . . . . . . . . . . . . . . . 19 (𝜑 → Σ ∈ (1...𝑀)if( = 0, (𝑃 − 1), 𝑃) = Σ ∈ (1...𝑀)𝑃)
85 fzfid 13922 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (1...𝑀) ∈ Fin)
863nncnd 12212 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑃 ∈ ℂ)
87 fsumconst 15720 . . . . . . . . . . . . . . . . . . . . 21 (((1...𝑀) ∈ Fin ∧ 𝑃 ∈ ℂ) → Σ ∈ (1...𝑀)𝑃 = ((♯‘(1...𝑀)) · 𝑃))
8885, 86, 87syl2anc 584 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → Σ ∈ (1...𝑀)𝑃 = ((♯‘(1...𝑀)) · 𝑃))
89 hashfz1 14290 . . . . . . . . . . . . . . . . . . . . . 22 (𝑀 ∈ ℕ0 → (♯‘(1...𝑀)) = 𝑀)
904, 89syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (♯‘(1...𝑀)) = 𝑀)
9190oveq1d 7409 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((♯‘(1...𝑀)) · 𝑃) = (𝑀 · 𝑃))
9288, 91eqtrd 2772 . . . . . . . . . . . . . . . . . . 19 (𝜑 → Σ ∈ (1...𝑀)𝑃 = (𝑀 · 𝑃))
9371, 84, 923eqtrd 2776 . . . . . . . . . . . . . . . . . 18 (𝜑 → Σ ∈ ((0 + 1)...𝑀)if( = 0, (𝑃 − 1), 𝑃) = (𝑀 · 𝑃))
9493oveq2d 7410 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑃 − 1) + Σ ∈ ((0 + 1)...𝑀)if( = 0, (𝑃 − 1), 𝑃)) = ((𝑃 − 1) + (𝑀 · 𝑃)))
9529nn0cnd 12518 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑃 − 1) ∈ ℂ)
964, 62nn0mulcld 12521 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑀 · 𝑃) ∈ ℕ0)
9796nn0cnd 12518 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑀 · 𝑃) ∈ ℂ)
9895, 97addcomd 11400 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑃 − 1) + (𝑀 · 𝑃)) = ((𝑀 · 𝑃) + (𝑃 − 1)))
9967, 94, 983eqtrd 2776 . . . . . . . . . . . . . . . 16 (𝜑 → Σ ∈ (0...𝑀)if( = 0, (𝑃 − 1), 𝑃) = ((𝑀 · 𝑃) + (𝑃 − 1)))
10099ad2antrr 724 . . . . . . . . . . . . . . 15 (((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ ∀𝑘 ∈ (0...𝑀)(𝑐𝑘) ≤ if(𝑘 = 0, (𝑃 − 1), 𝑃)) → Σ ∈ (0...𝑀)if( = 0, (𝑃 − 1), 𝑃) = ((𝑀 · 𝑃) + (𝑃 − 1)))
10159, 100breqtrd 5168 . . . . . . . . . . . . . 14 (((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ ∀𝑘 ∈ (0...𝑀)(𝑐𝑘) ≤ if(𝑘 = 0, (𝑃 − 1), 𝑃)) → Σ ∈ (0...𝑀)(𝑐) ≤ ((𝑀 · 𝑃) + (𝑃 − 1)))
10247, 101eqbrtrid 5177 . . . . . . . . . . . . 13 (((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ ∀𝑘 ∈ (0...𝑀)(𝑐𝑘) ≤ if(𝑘 = 0, (𝑃 − 1), 𝑃)) → Σ𝑘 ∈ (0...𝑀)(𝑐𝑘) ≤ ((𝑀 · 𝑃) + (𝑃 − 1)))
10345, 102eqbrtrd 5164 . . . . . . . . . . . 12 (((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ ∀𝑘 ∈ (0...𝑀)(𝑐𝑘) ≤ if(𝑘 = 0, (𝑃 − 1), 𝑃)) → 𝑁 ≤ ((𝑀 · 𝑃) + (𝑃 − 1)))
10440, 103syldan 591 . . . . . . . . . . 11 (((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ ¬ ∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘)) → 𝑁 ≤ ((𝑀 · 𝑃) + (𝑃 − 1)))
105 etransclem32.ngt . . . . . . . . . . . . 13 (𝜑 → ((𝑀 · 𝑃) + (𝑃 − 1)) < 𝑁)
10696, 29nn0addcld 12520 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑀 · 𝑃) + (𝑃 − 1)) ∈ ℕ0)
107106nn0red 12517 . . . . . . . . . . . . . 14 (𝜑 → ((𝑀 · 𝑃) + (𝑃 − 1)) ∈ ℝ)
1086nn0red 12517 . . . . . . . . . . . . . 14 (𝜑𝑁 ∈ ℝ)
109107, 108ltnled 11345 . . . . . . . . . . . . 13 (𝜑 → (((𝑀 · 𝑃) + (𝑃 − 1)) < 𝑁 ↔ ¬ 𝑁 ≤ ((𝑀 · 𝑃) + (𝑃 − 1))))
110105, 109mpbid 231 . . . . . . . . . . . 12 (𝜑 → ¬ 𝑁 ≤ ((𝑀 · 𝑃) + (𝑃 − 1)))
111110ad2antrr 724 . . . . . . . . . . 11 (((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ ¬ ∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘)) → ¬ 𝑁 ≤ ((𝑀 · 𝑃) + (𝑃 − 1)))
112104, 111condan 816 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) → ∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘))
113112adantlr 713 . . . . . . . . 9 (((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) → ∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘))
114 nfv 1917 . . . . . . . . . . . . 13 𝑗(𝜑𝑥𝑋)
115 nfcv 2903 . . . . . . . . . . . . . . . . 17 𝑗(0...𝑀)
116115nfsum1 15620 . . . . . . . . . . . . . . . 16 𝑗Σ𝑗 ∈ (0...𝑀)(𝑐𝑗)
117116nfeq1 2918 . . . . . . . . . . . . . . 15 𝑗Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁
118 nfcv 2903 . . . . . . . . . . . . . . 15 𝑗((0...𝑁) ↑m (0...𝑀))
119117, 118nfrabw 3468 . . . . . . . . . . . . . 14 𝑗{𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}
120119nfcri 2890 . . . . . . . . . . . . 13 𝑗 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}
121114, 120nfan 1902 . . . . . . . . . . . 12 𝑗((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁})
122 nfv 1917 . . . . . . . . . . . 12 𝑗 𝑘 ∈ (0...𝑀)
123 nfv 1917 . . . . . . . . . . . 12 𝑗if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘)
124121, 122, 123nf3an 1904 . . . . . . . . . . 11 𝑗(((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀) ∧ if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘))
125 nfcv 2903 . . . . . . . . . . 11 𝑗(((𝑆 D𝑛 (𝐻𝑘))‘(𝑐𝑘))‘𝑥)
126 fzfid 13922 . . . . . . . . . . 11 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀) ∧ if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘)) → (0...𝑀) ∈ Fin)
1271ad3antrrr 728 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → 𝑆 ∈ {ℝ, ℂ})
1282ad3antrrr 728 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
1293ad3antrrr 728 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → 𝑃 ∈ ℕ)
130 etransclem5 44792 . . . . . . . . . . . . . . 15 (𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))) = (𝑘 ∈ (0...𝑀) ↦ (𝑦𝑋 ↦ ((𝑦𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))
1317, 130eqtri 2760 . . . . . . . . . . . . . 14 𝐻 = (𝑘 ∈ (0...𝑀) ↦ (𝑦𝑋 ↦ ((𝑦𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))
132 simpr 485 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ (0...𝑀))
13323ad2antlr 725 . . . . . . . . . . . . . . . . 17 (((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → 𝑐:(0...𝑀)⟶(0...𝑁))
134 simpr 485 . . . . . . . . . . . . . . . . 17 (((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ (0...𝑀))
135133, 134ffvelcdmd 7073 . . . . . . . . . . . . . . . 16 (((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → (𝑐𝑗) ∈ (0...𝑁))
136135adantllr 717 . . . . . . . . . . . . . . 15 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → (𝑐𝑗) ∈ (0...𝑁))
137 elfznn0 13578 . . . . . . . . . . . . . . 15 ((𝑐𝑗) ∈ (0...𝑁) → (𝑐𝑗) ∈ ℕ0)
138136, 137syl 17 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → (𝑐𝑗) ∈ ℕ0)
139127, 128, 129, 131, 132, 138etransclem20 44807 . . . . . . . . . . . . 13 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝐻𝑗))‘(𝑐𝑗)):𝑋⟶ℂ)
140 simpllr 774 . . . . . . . . . . . . 13 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → 𝑥𝑋)
141139, 140ffvelcdmd 7073 . . . . . . . . . . . 12 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑆 D𝑛 (𝐻𝑗))‘(𝑐𝑗))‘𝑥) ∈ ℂ)
1421413ad2antl1 1185 . . . . . . . . . . 11 (((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀) ∧ if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘)) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑆 D𝑛 (𝐻𝑗))‘(𝑐𝑗))‘𝑥) ∈ ℂ)
143 fveq2 6879 . . . . . . . . . . . . . 14 (𝑗 = 𝑘 → (𝐻𝑗) = (𝐻𝑘))
144143oveq2d 7410 . . . . . . . . . . . . 13 (𝑗 = 𝑘 → (𝑆 D𝑛 (𝐻𝑗)) = (𝑆 D𝑛 (𝐻𝑘)))
145144, 42fveq12d 6886 . . . . . . . . . . . 12 (𝑗 = 𝑘 → ((𝑆 D𝑛 (𝐻𝑗))‘(𝑐𝑗)) = ((𝑆 D𝑛 (𝐻𝑘))‘(𝑐𝑘)))
146145fveq1d 6881 . . . . . . . . . . 11 (𝑗 = 𝑘 → (((𝑆 D𝑛 (𝐻𝑗))‘(𝑐𝑗))‘𝑥) = (((𝑆 D𝑛 (𝐻𝑘))‘(𝑐𝑘))‘𝑥))
147 simp2 1137 . . . . . . . . . . 11 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀) ∧ if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘)) → 𝑘 ∈ (0...𝑀))
1481ad2antrr 724 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) → 𝑆 ∈ {ℝ, ℂ})
1491483ad2ant1 1133 . . . . . . . . . . . . 13 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀) ∧ if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘)) → 𝑆 ∈ {ℝ, ℂ})
1502ad2antrr 724 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
1511503ad2ant1 1133 . . . . . . . . . . . . 13 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀) ∧ if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘)) → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
1523ad2antrr 724 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) → 𝑃 ∈ ℕ)
1531523ad2ant1 1133 . . . . . . . . . . . . 13 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀) ∧ if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘)) → 𝑃 ∈ ℕ)
154 etransclem5 44792 . . . . . . . . . . . . . 14 (𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))) = ( ∈ (0...𝑀) ↦ (𝑦𝑋 ↦ ((𝑦)↑if( = 0, (𝑃 − 1), 𝑃))))
1557, 154eqtri 2760 . . . . . . . . . . . . 13 𝐻 = ( ∈ (0...𝑀) ↦ (𝑦𝑋 ↦ ((𝑦)↑if( = 0, (𝑃 − 1), 𝑃))))
15625elfzelzd 13486 . . . . . . . . . . . . . . 15 (((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀)) → (𝑐𝑘) ∈ ℤ)
157156adantllr 717 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀)) → (𝑐𝑘) ∈ ℤ)
1581573adant3 1132 . . . . . . . . . . . . 13 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀) ∧ if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘)) → (𝑐𝑘) ∈ ℤ)
159 simp3 1138 . . . . . . . . . . . . 13 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀) ∧ if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘)) → if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘))
160149, 151, 153, 155, 147, 158, 159etransclem19 44806 . . . . . . . . . . . 12 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀) ∧ if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘)) → ((𝑆 D𝑛 (𝐻𝑘))‘(𝑐𝑘)) = (𝑦𝑋 ↦ 0))
161 eqidd 2733 . . . . . . . . . . . 12 (((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀) ∧ if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘)) ∧ 𝑦 = 𝑥) → 0 = 0)
162 simp1lr 1237 . . . . . . . . . . . 12 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀) ∧ if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘)) → 𝑥𝑋)
163 0red 11201 . . . . . . . . . . . 12 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀) ∧ if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘)) → 0 ∈ ℝ)
164160, 161, 162, 163fvmptd 6992 . . . . . . . . . . 11 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀) ∧ if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘)) → (((𝑆 D𝑛 (𝐻𝑘))‘(𝑐𝑘))‘𝑥) = 0)
165124, 125, 126, 142, 146, 147, 164fprod0 44149 . . . . . . . . . 10 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀) ∧ if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘)) → ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘(𝑐𝑗))‘𝑥) = 0)
166165rexlimdv3a 3159 . . . . . . . . 9 (((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) → (∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘) → ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘(𝑐𝑗))‘𝑥) = 0))
167113, 166mpd 15 . . . . . . . 8 (((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) → ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘(𝑐𝑗))‘𝑥) = 0)
16814, 167syldan 591 . . . . . . 7 (((𝜑𝑥𝑋) ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) → ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘(𝑐𝑗))‘𝑥) = 0)
169168oveq2d 7410 . . . . . 6 (((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 (𝐻𝑗))‘(𝑐𝑗))‘𝑥)) = (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐𝑗))) · 0))
1706faccld 14228 . . . . . . . . . . 11 (𝜑 → (!‘𝑁) ∈ ℕ)
171170nncnd 12212 . . . . . . . . . 10 (𝜑 → (!‘𝑁) ∈ ℂ)
172171adantr 481 . . . . . . . . 9 ((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) → (!‘𝑁) ∈ ℂ)
173 fzfid 13922 . . . . . . . . . 10 ((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) → (0...𝑀) ∈ Fin)
174 simpll 765 . . . . . . . . . . . . . 14 (((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → 𝜑)
17513adantr 481 . . . . . . . . . . . . . 14 (((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁})
176 simpr 485 . . . . . . . . . . . . . 14 (((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ (0...𝑀))
177174, 175, 176, 135syl21anc 836 . . . . . . . . . . . . 13 (((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (𝑐𝑗) ∈ (0...𝑁))
178177, 137syl 17 . . . . . . . . . . . 12 (((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (𝑐𝑗) ∈ ℕ0)
179178faccld 14228 . . . . . . . . . . 11 (((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝑐𝑗)) ∈ ℕ)
180179nncnd 12212 . . . . . . . . . 10 (((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝑐𝑗)) ∈ ℂ)
181173, 180fprodcl 15880 . . . . . . . . 9 ((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) → ∏𝑗 ∈ (0...𝑀)(!‘(𝑐𝑗)) ∈ ℂ)
182179nnne0d 12246 . . . . . . . . . 10 (((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝑐𝑗)) ≠ 0)
183173, 180, 182fprodn0 15907 . . . . . . . . 9 ((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) → ∏𝑗 ∈ (0...𝑀)(!‘(𝑐𝑗)) ≠ 0)
184172, 181, 183divcld 11974 . . . . . . . 8 ((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐𝑗))) ∈ ℂ)
185184mul01d 11397 . . . . . . 7 ((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) → (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐𝑗))) · 0) = 0)
186185adantlr 713 . . . . . 6 (((𝜑𝑥𝑋) ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) → (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐𝑗))) · 0) = 0)
187169, 186eqtrd 2772 . . . . 5 (((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 (𝐻𝑗))‘(𝑐𝑗))‘𝑥)) = 0)
188187sumeq2dv 15633 . . . 4 ((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 (𝐻𝑗))‘(𝑐𝑗))‘𝑥)) = Σ𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)0)
189 eqid 2732 . . . . . . . 8 (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})
190189, 6etransclem16 44803 . . . . . . 7 (𝜑 → ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁) ∈ Fin)
191190olcd 872 . . . . . 6 (𝜑 → (((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁) ⊆ (ℤ𝐴) ∨ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁) ∈ Fin))
192191adantr 481 . . . . 5 ((𝜑𝑥𝑋) → (((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁) ⊆ (ℤ𝐴) ∨ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁) ∈ Fin))
193 sumz 15652 . . . . 5 ((((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁) ⊆ (ℤ𝐴) ∨ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁) ∈ Fin) → Σ𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)0 = 0)
194192, 193syl 17 . . . 4 ((𝜑𝑥𝑋) → Σ𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)0 = 0)
195188, 194eqtrd 2772 . . 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 (𝐻𝑗))‘(𝑐𝑗))‘𝑥)) = 0)
196195mpteq2dva 5242 . 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 (𝐻𝑗))‘(𝑐𝑗))‘𝑥))) = (𝑥𝑋 ↦ 0))
1979, 196eqtrd 2772 1 (𝜑 → ((𝑆 D𝑛 𝐹)‘𝑁) = (𝑥𝑋 ↦ 0))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  wral 3061  wrex 3070  {crab 3432  wss 3945  ifcif 4523  {cpr 4625   class class class wbr 5142  cmpt 5225  wf 6529  cfv 6533  (class class class)co 7394  m cmap 8805  Fincfn 8924  cc 11092  cr 11093  0cc0 11094  1c1 11095   + caddc 11097   · cmul 11099   < clt 11232  cle 11233  cmin 11428   / cdiv 11855  cn 12196  0cn0 12456  cz 12542  cuz 12806  ...cfz 13468  cexp 14011  !cfa 14217  chash 14274  Σcsu 15616  cprod 15833  t crest 17350  TopOpenctopn 17351  fldccnfld 20880   D𝑛 cdvn 25312
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 2703  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5357  ax-pr 5421  ax-un 7709  ax-inf2 9620  ax-cnex 11150  ax-resscn 11151  ax-1cn 11152  ax-icn 11153  ax-addcl 11154  ax-addrcl 11155  ax-mulcl 11156  ax-mulrcl 11157  ax-mulcom 11158  ax-addass 11159  ax-mulass 11160  ax-distr 11161  ax-i2m1 11162  ax-1ne0 11163  ax-1rid 11164  ax-rnegex 11165  ax-rrecex 11166  ax-cnre 11167  ax-pre-lttri 11168  ax-pre-lttrn 11169  ax-pre-ltadd 11170  ax-pre-mulgt0 11171  ax-pre-sup 11172  ax-addf 11173  ax-mulf 11174
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-int 4945  df-iun 4993  df-iin 4994  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-se 5626  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7350  df-ov 7397  df-oprab 7398  df-mpo 7399  df-of 7654  df-om 7840  df-1st 7959  df-2nd 7960  df-supp 8131  df-frecs 8250  df-wrecs 8281  df-recs 8355  df-rdg 8394  df-1o 8450  df-2o 8451  df-er 8688  df-map 8807  df-pm 8808  df-ixp 8877  df-en 8925  df-dom 8926  df-sdom 8927  df-fin 8928  df-fsupp 9347  df-fi 9390  df-sup 9421  df-inf 9422  df-oi 9489  df-card 9918  df-pnf 11234  df-mnf 11235  df-xr 11236  df-ltxr 11237  df-le 11238  df-sub 11430  df-neg 11431  df-div 11856  df-nn 12197  df-2 12259  df-3 12260  df-4 12261  df-5 12262  df-6 12263  df-7 12264  df-8 12265  df-9 12266  df-n0 12457  df-z 12543  df-dec 12662  df-uz 12807  df-q 12917  df-rp 12959  df-xneg 13076  df-xadd 13077  df-xmul 13078  df-ico 13314  df-icc 13315  df-fz 13469  df-fzo 13612  df-seq 13951  df-exp 14012  df-fac 14218  df-bc 14247  df-hash 14275  df-cj 15030  df-re 15031  df-im 15032  df-sqrt 15166  df-abs 15167  df-clim 15416  df-sum 15617  df-prod 15834  df-struct 17064  df-sets 17081  df-slot 17099  df-ndx 17111  df-base 17129  df-ress 17158  df-plusg 17194  df-mulr 17195  df-starv 17196  df-sca 17197  df-vsca 17198  df-ip 17199  df-tset 17200  df-ple 17201  df-ds 17203  df-unif 17204  df-hom 17205  df-cco 17206  df-rest 17352  df-topn 17353  df-0g 17371  df-gsum 17372  df-topgen 17373  df-pt 17374  df-prds 17377  df-xrs 17432  df-qtop 17437  df-imas 17438  df-xps 17440  df-mre 17514  df-mrc 17515  df-acs 17517  df-mgm 18545  df-sgrp 18594  df-mnd 18605  df-submnd 18650  df-mulg 18925  df-cntz 19149  df-cmn 19616  df-psmet 20872  df-xmet 20873  df-met 20874  df-bl 20875  df-mopn 20876  df-fbas 20877  df-fg 20878  df-cnfld 20881  df-top 22327  df-topon 22344  df-topsp 22366  df-bases 22380  df-cld 22454  df-ntr 22455  df-cls 22456  df-nei 22533  df-lp 22571  df-perf 22572  df-cn 22662  df-cnp 22663  df-haus 22750  df-tx 22997  df-hmeo 23190  df-fil 23281  df-fm 23373  df-flim 23374  df-flf 23375  df-xms 23757  df-ms 23758  df-tms 23759  df-cncf 24325  df-limc 25314  df-dv 25315  df-dvn 25316
This theorem is referenced by:  etransclem46  44833
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