Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  etransclem32 Structured version   Visualization version   GIF version

Theorem etransclem32 46221
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 46200 . . 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 46219 . 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 484 . . . . . . . . . 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 46201 . . . . . . . . . . 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 480 . . . . . . . . . 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 2840 . . . . . . . . 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 715 . . . . . . . 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 1911 . . . . . . . . . . . . . 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 1854 . . . . . . . . . . . . . 14 𝑘 ¬ ∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘)
1815, 17nfan 1896 . . . . . . . . . . . . 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 45264 . . . . . . . . . . . . . . . . 17 (0...𝑁) ⊆ ℝ
20 rabid 3454 . . . . . . . . . . . . . . . . . . . . 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 497 . . . . . . . . . . . . . . . . . . . 20 (𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁} → 𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)))
22 elmapi 8887 . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) → 𝑐:(0...𝑀)⟶(0...𝑁))
2524ffvelcdmda 7103 . . . . . . . . . . . . . . . . 17 (((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀)) → (𝑐𝑘) ∈ (0...𝑁))
2619, 25sselid 3992 . . . . . . . . . . . . . . . 16 (((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀)) → (𝑐𝑘) ∈ ℝ)
2726adantlr 715 . . . . . . . . . . . . . . 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 12564 . . . . . . . . . . . . . . . . . . 19 (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ0)
293, 28syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑃 − 1) ∈ ℕ0)
3029nn0red 12585 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑃 − 1) ∈ ℝ)
313nnred 12278 . . . . . . . . . . . . . . . . 17 (𝜑𝑃 ∈ ℝ)
3230, 31ifcld 4576 . . . . . . . . . . . . . . . 16 (𝜑 → if(𝑘 = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
3332ad3antrrr 730 . . . . . . . . . . . . . . 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 3069 . . . . . . . . . . . . . . . . . 18 (∀𝑘 ∈ (0...𝑀) ¬ if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘) ↔ ¬ ∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘))
3534biimpri 228 . . . . . . . . . . . . . . . . 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 714 . . . . . . . . . . . . . . 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 11408 . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . 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 496 . . . . . . . . . . . . . . 15 (𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁} → Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁)
42 fveq2 6906 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑘 → (𝑐𝑗) = (𝑐𝑘))
4342cbvsumv 15728 . . . . . . . . . . . . . . 15 Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = Σ𝑘 ∈ (0...𝑀)(𝑐𝑘)
4441, 43eqtr3di 2789 . . . . . . . . . . . . . 14 (𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁} → 𝑁 = Σ𝑘 ∈ (0...𝑀)(𝑐𝑘))
4544ad2antlr 727 . . . . . . . . . . . . 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 6906 . . . . . . . . . . . . . . 15 (𝑘 = → (𝑐𝑘) = (𝑐))
4746cbvsumv 15728 . . . . . . . . . . . . . 14 Σ𝑘 ∈ (0...𝑀)(𝑐𝑘) = Σ ∈ (0...𝑀)(𝑐)
48 fzfid 14010 . . . . . . . . . . . . . . . 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 7103 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ ∈ (0...𝑀)) → (𝑐) ∈ (0...𝑁))
5019, 49sselid 3992 . . . . . . . . . . . . . . . . 17 (((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ ∈ (0...𝑀)) → (𝑐) ∈ ℝ)
5150adantlr 715 . . . . . . . . . . . . . . . 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 4576 . . . . . . . . . . . . . . . . 17 (𝜑 → if( = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
5352ad3antrrr 730 . . . . . . . . . . . . . . . 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 2738 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = → (𝑘 = 0 ↔ = 0))
5554ifbid 4553 . . . . . . . . . . . . . . . . . . 19 (𝑘 = → if(𝑘 = 0, (𝑃 − 1), 𝑃) = if( = 0, (𝑃 − 1), 𝑃))
5646, 55breq12d 5160 . . . . . . . . . . . . . . . . . 18 (𝑘 = → ((𝑐𝑘) ≤ if(𝑘 = 0, (𝑃 − 1), 𝑃) ↔ (𝑐) ≤ if( = 0, (𝑃 − 1), 𝑃)))
5756rspccva 3620 . . . . . . . . . . . . . . . . 17 ((∀𝑘 ∈ (0...𝑀)(𝑐𝑘) ≤ if(𝑘 = 0, (𝑃 − 1), 𝑃) ∧ ∈ (0...𝑀)) → (𝑐) ≤ if( = 0, (𝑃 − 1), 𝑃))
5857adantll 714 . . . . . . . . . . . . . . . 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 15831 . . . . . . . . . . . . . . 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 12917 . . . . . . . . . . . . . . . . . . 19 0 = (ℤ‘0)
614, 60eleqtrdi 2848 . . . . . . . . . . . . . . . . . 18 (𝜑𝑀 ∈ (ℤ‘0))
623nnnn0d 12584 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑃 ∈ ℕ0)
6329, 62ifcld 4576 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → if( = 0, (𝑃 − 1), 𝑃) ∈ ℕ0)
6463adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∈ (0...𝑀)) → if( = 0, (𝑃 − 1), 𝑃) ∈ ℕ0)
6564nn0cnd 12586 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∈ (0...𝑀)) → if( = 0, (𝑃 − 1), 𝑃) ∈ ℂ)
66 iftrue 4536 . . . . . . . . . . . . . . . . . 18 ( = 0 → if( = 0, (𝑃 − 1), 𝑃) = (𝑃 − 1))
6761, 65, 66fsum1p 15785 . . . . . . . . . . . . . . . . 17 (𝜑 → Σ ∈ (0...𝑀)if( = 0, (𝑃 − 1), 𝑃) = ((𝑃 − 1) + Σ ∈ ((0 + 1)...𝑀)if( = 0, (𝑃 − 1), 𝑃)))
68 0p1e1 12385 . . . . . . . . . . . . . . . . . . . . . 22 (0 + 1) = 1
6968oveq1i 7440 . . . . . . . . . . . . . . . . . . . . 21 ((0 + 1)...𝑀) = (1...𝑀)
7069a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((0 + 1)...𝑀) = (1...𝑀))
7170sumeq1d 15732 . . . . . . . . . . . . . . . . . . 19 (𝜑 → Σ ∈ ((0 + 1)...𝑀)if( = 0, (𝑃 − 1), 𝑃) = Σ ∈ (1...𝑀)if( = 0, (𝑃 − 1), 𝑃))
72 0red 11261 . . . . . . . . . . . . . . . . . . . . . . . . 25 ( ∈ (1...𝑀) → 0 ∈ ℝ)
73 1red 11259 . . . . . . . . . . . . . . . . . . . . . . . . 25 ( ∈ (1...𝑀) → 1 ∈ ℝ)
74 elfzelz 13560 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ( ∈ (1...𝑀) → ∈ ℤ)
7574zred 12719 . . . . . . . . . . . . . . . . . . . . . . . . 25 ( ∈ (1...𝑀) → ∈ ℝ)
76 0lt1 11782 . . . . . . . . . . . . . . . . . . . . . . . . . 26 0 < 1
7776a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25 ( ∈ (1...𝑀) → 0 < 1)
78 elfzle1 13563 . . . . . . . . . . . . . . . . . . . . . . . . 25 ( ∈ (1...𝑀) → 1 ≤ )
7972, 73, 75, 77, 78ltletrd 11418 . . . . . . . . . . . . . . . . . . . . . . . 24 ( ∈ (1...𝑀) → 0 < )
8079gt0ne0d 11824 . . . . . . . . . . . . . . . . . . . . . . 23 ( ∈ (1...𝑀) → ≠ 0)
8180neneqd 2942 . . . . . . . . . . . . . . . . . . . . . 22 ( ∈ (1...𝑀) → ¬ = 0)
8281iffalsed 4541 . . . . . . . . . . . . . . . . . . . . 21 ( ∈ (1...𝑀) → if( = 0, (𝑃 − 1), 𝑃) = 𝑃)
8382adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∈ (1...𝑀)) → if( = 0, (𝑃 − 1), 𝑃) = 𝑃)
8483sumeq2dv 15734 . . . . . . . . . . . . . . . . . . 19 (𝜑 → Σ ∈ (1...𝑀)if( = 0, (𝑃 − 1), 𝑃) = Σ ∈ (1...𝑀)𝑃)
85 fzfid 14010 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (1...𝑀) ∈ Fin)
863nncnd 12279 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑃 ∈ ℂ)
87 fsumconst 15822 . . . . . . . . . . . . . . . . . . . . 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 14381 . . . . . . . . . . . . . . . . . . . . . 22 (𝑀 ∈ ℕ0 → (♯‘(1...𝑀)) = 𝑀)
904, 89syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (♯‘(1...𝑀)) = 𝑀)
9190oveq1d 7445 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((♯‘(1...𝑀)) · 𝑃) = (𝑀 · 𝑃))
9288, 91eqtrd 2774 . . . . . . . . . . . . . . . . . . 19 (𝜑 → Σ ∈ (1...𝑀)𝑃 = (𝑀 · 𝑃))
9371, 84, 923eqtrd 2778 . . . . . . . . . . . . . . . . . 18 (𝜑 → Σ ∈ ((0 + 1)...𝑀)if( = 0, (𝑃 − 1), 𝑃) = (𝑀 · 𝑃))
9493oveq2d 7446 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑃 − 1) + Σ ∈ ((0 + 1)...𝑀)if( = 0, (𝑃 − 1), 𝑃)) = ((𝑃 − 1) + (𝑀 · 𝑃)))
9529nn0cnd 12586 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑃 − 1) ∈ ℂ)
964, 62nn0mulcld 12589 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑀 · 𝑃) ∈ ℕ0)
9796nn0cnd 12586 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑀 · 𝑃) ∈ ℂ)
9895, 97addcomd 11460 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑃 − 1) + (𝑀 · 𝑃)) = ((𝑀 · 𝑃) + (𝑃 − 1)))
9967, 94, 983eqtrd 2778 . . . . . . . . . . . . . . . 16 (𝜑 → Σ ∈ (0...𝑀)if( = 0, (𝑃 − 1), 𝑃) = ((𝑀 · 𝑃) + (𝑃 − 1)))
10099ad2antrr 726 . . . . . . . . . . . . . . 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 5173 . . . . . . . . . . . . . 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 5182 . . . . . . . . . . . . 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 5169 . . . . . . . . . . . 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 12588 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑀 · 𝑃) + (𝑃 − 1)) ∈ ℕ0)
107106nn0red 12585 . . . . . . . . . . . . . 14 (𝜑 → ((𝑀 · 𝑃) + (𝑃 − 1)) ∈ ℝ)
1086nn0red 12585 . . . . . . . . . . . . . 14 (𝜑𝑁 ∈ ℝ)
109107, 108ltnled 11405 . . . . . . . . . . . . 13 (𝜑 → (((𝑀 · 𝑃) + (𝑃 − 1)) < 𝑁 ↔ ¬ 𝑁 ≤ ((𝑀 · 𝑃) + (𝑃 − 1))))
110105, 109mpbid 232 . . . . . . . . . . . 12 (𝜑 → ¬ 𝑁 ≤ ((𝑀 · 𝑃) + (𝑃 − 1)))
111110ad2antrr 726 . . . . . . . . . . 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 818 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) → ∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘))
113112adantlr 715 . . . . . . . . 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 1911 . . . . . . . . . . . . 13 𝑗(𝜑𝑥𝑋)
115 nfcv 2902 . . . . . . . . . . . . . . . . 17 𝑗(0...𝑀)
116115nfsum1 15722 . . . . . . . . . . . . . . . 16 𝑗Σ𝑗 ∈ (0...𝑀)(𝑐𝑗)
117116nfeq1 2918 . . . . . . . . . . . . . . 15 𝑗Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁
118 nfcv 2902 . . . . . . . . . . . . . . 15 𝑗((0...𝑁) ↑m (0...𝑀))
119117, 118nfrabw 3472 . . . . . . . . . . . . . 14 𝑗{𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}
120119nfcri 2894 . . . . . . . . . . . . 13 𝑗 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}
121114, 120nfan 1896 . . . . . . . . . . . 12 𝑗((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁})
122 nfv 1911 . . . . . . . . . . . 12 𝑗 𝑘 ∈ (0...𝑀)
123 nfv 1911 . . . . . . . . . . . 12 𝑗if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘)
124121, 122, 123nf3an 1898 . . . . . . . . . . 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 2902 . . . . . . . . . . 11 𝑗(((𝑆 D𝑛 (𝐻𝑘))‘(𝑐𝑘))‘𝑥)
126 fzfid 14010 . . . . . . . . . . 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 730 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → 𝑆 ∈ {ℝ, ℂ})
1282ad3antrrr 730 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
1293ad3antrrr 730 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → 𝑃 ∈ ℕ)
130 etransclem5 46194 . . . . . . . . . . . . . . 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 2762 . . . . . . . . . . . . . 14 𝐻 = (𝑘 ∈ (0...𝑀) ↦ (𝑦𝑋 ↦ ((𝑦𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))
132 simpr 484 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ (0...𝑀))
13323ad2antlr 727 . . . . . . . . . . . . . . . . 17 (((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → 𝑐:(0...𝑀)⟶(0...𝑁))
134 simpr 484 . . . . . . . . . . . . . . . . 17 (((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ (0...𝑀))
135133, 134ffvelcdmd 7104 . . . . . . . . . . . . . . . 16 (((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → (𝑐𝑗) ∈ (0...𝑁))
136135adantllr 719 . . . . . . . . . . . . . . 15 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → (𝑐𝑗) ∈ (0...𝑁))
137 elfznn0 13656 . . . . . . . . . . . . . . 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 46209 . . . . . . . . . . . . 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 776 . . . . . . . . . . . . 13 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → 𝑥𝑋)
141139, 140ffvelcdmd 7104 . . . . . . . . . . . 12 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑆 D𝑛 (𝐻𝑗))‘(𝑐𝑗))‘𝑥) ∈ ℂ)
1421413ad2antl1 1184 . . . . . . . . . . 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 6906 . . . . . . . . . . . . . 14 (𝑗 = 𝑘 → (𝐻𝑗) = (𝐻𝑘))
144143oveq2d 7446 . . . . . . . . . . . . 13 (𝑗 = 𝑘 → (𝑆 D𝑛 (𝐻𝑗)) = (𝑆 D𝑛 (𝐻𝑘)))
145144, 42fveq12d 6913 . . . . . . . . . . . 12 (𝑗 = 𝑘 → ((𝑆 D𝑛 (𝐻𝑗))‘(𝑐𝑗)) = ((𝑆 D𝑛 (𝐻𝑘))‘(𝑐𝑘)))
146145fveq1d 6908 . . . . . . . . . . 11 (𝑗 = 𝑘 → (((𝑆 D𝑛 (𝐻𝑗))‘(𝑐𝑗))‘𝑥) = (((𝑆 D𝑛 (𝐻𝑘))‘(𝑐𝑘))‘𝑥))
147 simp2 1136 . . . . . . . . . . 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 726 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) → 𝑆 ∈ {ℝ, ℂ})
1491483ad2ant1 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)) → 𝑆 ∈ {ℝ, ℂ})
1502ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
1511503ad2ant1 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)) → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
1523ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) → 𝑃 ∈ ℕ)
1531523ad2ant1 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)) → 𝑃 ∈ ℕ)
154 etransclem5 46194 . . . . . . . . . . . . . 14 (𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))) = ( ∈ (0...𝑀) ↦ (𝑦𝑋 ↦ ((𝑦)↑if( = 0, (𝑃 − 1), 𝑃))))
1557, 154eqtri 2762 . . . . . . . . . . . . 13 𝐻 = ( ∈ (0...𝑀) ↦ (𝑦𝑋 ↦ ((𝑦)↑if( = 0, (𝑃 − 1), 𝑃))))
15625elfzelzd 13561 . . . . . . . . . . . . . . 15 (((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀)) → (𝑐𝑘) ∈ ℤ)
157156adantllr 719 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀)) → (𝑐𝑘) ∈ ℤ)
1581573adant3 1131 . . . . . . . . . . . . 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 1137 . . . . . . . . . . . . 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 46208 . . . . . . . . . . . 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 2735 . . . . . . . . . . . 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 1236 . . . . . . . . . . . 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 11261 . . . . . . . . . . . 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 7022 . . . . . . . . . . 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 45551 . . . . . . . . . 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 3156 . . . . . . . . 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 7446 . . . . . 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 14319 . . . . . . . . . . 11 (𝜑 → (!‘𝑁) ∈ ℕ)
171170nncnd 12279 . . . . . . . . . 10 (𝜑 → (!‘𝑁) ∈ ℂ)
172171adantr 480 . . . . . . . . 9 ((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) → (!‘𝑁) ∈ ℂ)
173 fzfid 14010 . . . . . . . . . 10 ((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) → (0...𝑀) ∈ Fin)
174 simpll 767 . . . . . . . . . . . . . 14 (((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → 𝜑)
17513adantr 480 . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . 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 838 . . . . . . . . . . . . 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 14319 . . . . . . . . . . 11 (((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝑐𝑗)) ∈ ℕ)
180179nncnd 12279 . . . . . . . . . 10 (((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝑐𝑗)) ∈ ℂ)
181173, 180fprodcl 15984 . . . . . . . . 9 ((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) → ∏𝑗 ∈ (0...𝑀)(!‘(𝑐𝑗)) ∈ ℂ)
182179nnne0d 12313 . . . . . . . . . 10 (((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝑐𝑗)) ≠ 0)
183173, 180, 182fprodn0 16011 . . . . . . . . 9 ((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) → ∏𝑗 ∈ (0...𝑀)(!‘(𝑐𝑗)) ≠ 0)
184172, 181, 183divcld 12040 . . . . . . . 8 ((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐𝑗))) ∈ ℂ)
185184mul01d 11457 . . . . . . 7 ((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) → (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐𝑗))) · 0) = 0)
186185adantlr 715 . . . . . 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 2774 . . . . 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 15734 . . . 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 2734 . . . . . . . 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 46205 . . . . . . 7 (𝜑 → ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁) ∈ Fin)
191190olcd 874 . . . . . 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 480 . . . . 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 15754 . . . . 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 2774 . . 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 5247 . 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 2774 1 (𝜑 → ((𝑆 D𝑛 𝐹)‘𝑁) = (𝑥𝑋 ↦ 0))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 847  w3a 1086   = wceq 1536  wcel 2105  wral 3058  wrex 3067  {crab 3432  wss 3962  ifcif 4530  {cpr 4632   class class class wbr 5147  cmpt 5230  wf 6558  cfv 6562  (class class class)co 7430  m cmap 8864  Fincfn 8983  cc 11150  cr 11151  0cc0 11152  1c1 11153   + caddc 11155   · cmul 11157   < clt 11292  cle 11293  cmin 11489   / cdiv 11917  cn 12263  0cn0 12523  cz 12610  cuz 12875  ...cfz 13543  cexp 14098  !cfa 14308  chash 14365  Σcsu 15718  cprod 15935  t crest 17466  TopOpenctopn 17467  fldccnfld 21381   D𝑛 cdvn 25913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230  ax-addf 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-om 7887  df-1st 8012  df-2nd 8013  df-supp 8184  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8743  df-map 8866  df-pm 8867  df-ixp 8936  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fsupp 9399  df-fi 9448  df-sup 9479  df-inf 9480  df-oi 9547  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-dec 12731  df-uz 12876  df-q 12988  df-rp 13032  df-xneg 13151  df-xadd 13152  df-xmul 13153  df-ico 13389  df-icc 13390  df-fz 13544  df-fzo 13691  df-seq 14039  df-exp 14099  df-fac 14309  df-bc 14338  df-hash 14366  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-clim 15520  df-sum 15719  df-prod 15936  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-mulr 17311  df-starv 17312  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-hom 17321  df-cco 17322  df-rest 17468  df-topn 17469  df-0g 17487  df-gsum 17488  df-topgen 17489  df-pt 17490  df-prds 17493  df-xrs 17548  df-qtop 17553  df-imas 17554  df-xps 17556  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-submnd 18809  df-mulg 19098  df-cntz 19347  df-cmn 19814  df-psmet 21373  df-xmet 21374  df-met 21375  df-bl 21376  df-mopn 21377  df-fbas 21378  df-fg 21379  df-cnfld 21382  df-top 22915  df-topon 22932  df-topsp 22954  df-bases 22968  df-cld 23042  df-ntr 23043  df-cls 23044  df-nei 23121  df-lp 23159  df-perf 23160  df-cn 23250  df-cnp 23251  df-haus 23338  df-tx 23585  df-hmeo 23778  df-fil 23869  df-fm 23961  df-flim 23962  df-flf 23963  df-xms 24345  df-ms 24346  df-tms 24347  df-cncf 24917  df-limc 25915  df-dv 25916  df-dvn 25917
This theorem is referenced by:  etransclem46  46235
  Copyright terms: Public domain W3C validator