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 41144
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 41123 . . 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 41142 . 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 477 . . . . . . . . . 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 41124 . . . . . . . . . . 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 472 . . . . . . . . . 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 2846 . . . . . . . . 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 706 . . . . . . . 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 2009 . . . . . . . . . . . . . 14 𝑘(𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁})
16 nfre1 3151 . . . . . . . . . . . . . . 15 𝑘𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘)
1716nfn 1953 . . . . . . . . . . . . . 14 𝑘 ¬ ∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘)
1815, 17nfan 1998 . . . . . . . . . . . . 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 40191 . . . . . . . . . . . . . . . . 17 (0...𝑁) ⊆ ℝ
20 rabid 3263 . . . . . . . . . . . . . . . . . . . . 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 491 . . . . . . . . . . . . . . . . . . . 20 (𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁} → 𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)))
22 elmapi 8086 . . . . . . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) → 𝑐:(0...𝑀)⟶(0...𝑁))
2524ffvelrnda 6553 . . . . . . . . . . . . . . . . 17 (((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀)) → (𝑐𝑘) ∈ (0...𝑁))
2619, 25sseldi 3761 . . . . . . . . . . . . . . . 16 (((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀)) → (𝑐𝑘) ∈ ℝ)
2726adantlr 706 . . . . . . . . . . . . . . 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 11585 . . . . . . . . . . . . . . . . . . 19 (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ0)
293, 28syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑃 − 1) ∈ ℕ0)
3029nn0red 11603 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑃 − 1) ∈ ℝ)
313nnred 11295 . . . . . . . . . . . . . . . . 17 (𝜑𝑃 ∈ ℝ)
3230, 31ifcld 4290 . . . . . . . . . . . . . . . 16 (𝜑 → if(𝑘 = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
3332ad3antrrr 721 . . . . . . . . . . . . . . 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 3139 . . . . . . . . . . . . . . . . . 18 (∀𝑘 ∈ (0...𝑀) ¬ if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘) ↔ ¬ ∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘))
3534biimpri 219 . . . . . . . . . . . . . . . . 17 (¬ ∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘) → ∀𝑘 ∈ (0...𝑀) ¬ if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘))
3635r19.21bi 3079 . . . . . . . . . . . . . . . 16 ((¬ ∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘) ∧ 𝑘 ∈ (0...𝑀)) → ¬ if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘))
3736adantll 705 . . . . . . . . . . . . . . 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 10445 . . . . . . . . . . . . . 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 401 . . . . . . . . . . . . 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 3104 . . . . . . . . . . . 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))
41 fveq2 6379 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑘 → (𝑐𝑗) = (𝑐𝑘))
4241cbvsumv 14725 . . . . . . . . . . . . . . 15 Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = Σ𝑘 ∈ (0...𝑀)(𝑐𝑘)
4320simprbi 490 . . . . . . . . . . . . . . 15 (𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁} → Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁)
4442, 43syl5reqr 2814 . . . . . . . . . . . . . 14 (𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁} → 𝑁 = Σ𝑘 ∈ (0...𝑀)(𝑐𝑘))
4544ad2antlr 718 . . . . . . . . . . . . 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 6379 . . . . . . . . . . . . . . 15 (𝑘 = → (𝑐𝑘) = (𝑐))
4746cbvsumv 14725 . . . . . . . . . . . . . 14 Σ𝑘 ∈ (0...𝑀)(𝑐𝑘) = Σ ∈ (0...𝑀)(𝑐)
48 fzfid 12985 . . . . . . . . . . . . . . . 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)
4924ffvelrnda 6553 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ ∈ (0...𝑀)) → (𝑐) ∈ (0...𝑁))
5019, 49sseldi 3761 . . . . . . . . . . . . . . . . 17 (((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ ∈ (0...𝑀)) → (𝑐) ∈ ℝ)
5150adantlr 706 . . . . . . . . . . . . . . . 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 4290 . . . . . . . . . . . . . . . . 17 (𝜑 → if( = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
5352ad3antrrr 721 . . . . . . . . . . . . . . . 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 2769 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = → (𝑘 = 0 ↔ = 0))
5554ifbid 4267 . . . . . . . . . . . . . . . . . . 19 (𝑘 = → if(𝑘 = 0, (𝑃 − 1), 𝑃) = if( = 0, (𝑃 − 1), 𝑃))
5646, 55breq12d 4824 . . . . . . . . . . . . . . . . . 18 (𝑘 = → ((𝑐𝑘) ≤ if(𝑘 = 0, (𝑃 − 1), 𝑃) ↔ (𝑐) ≤ if( = 0, (𝑃 − 1), 𝑃)))
5756rspccva 3461 . . . . . . . . . . . . . . . . 17 ((∀𝑘 ∈ (0...𝑀)(𝑐𝑘) ≤ if(𝑘 = 0, (𝑃 − 1), 𝑃) ∧ ∈ (0...𝑀)) → (𝑐) ≤ if( = 0, (𝑃 − 1), 𝑃))
5857adantll 705 . . . . . . . . . . . . . . . 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 14829 . . . . . . . . . . . . . . 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 11927 . . . . . . . . . . . . . . . . . . 19 0 = (ℤ‘0)
614, 60syl6eleq 2854 . . . . . . . . . . . . . . . . . 18 (𝜑𝑀 ∈ (ℤ‘0))
623nnnn0d 11602 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑃 ∈ ℕ0)
6329, 62ifcld 4290 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → if( = 0, (𝑃 − 1), 𝑃) ∈ ℕ0)
6463adantr 472 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∈ (0...𝑀)) → if( = 0, (𝑃 − 1), 𝑃) ∈ ℕ0)
6564nn0cnd 11604 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∈ (0...𝑀)) → if( = 0, (𝑃 − 1), 𝑃) ∈ ℂ)
66 iftrue 4251 . . . . . . . . . . . . . . . . . 18 ( = 0 → if( = 0, (𝑃 − 1), 𝑃) = (𝑃 − 1))
6761, 65, 66fsum1p 14781 . . . . . . . . . . . . . . . . 17 (𝜑 → Σ ∈ (0...𝑀)if( = 0, (𝑃 − 1), 𝑃) = ((𝑃 − 1) + Σ ∈ ((0 + 1)...𝑀)if( = 0, (𝑃 − 1), 𝑃)))
68 0p1e1 11405 . . . . . . . . . . . . . . . . . . . . . 22 (0 + 1) = 1
6968oveq1i 6856 . . . . . . . . . . . . . . . . . . . . 21 ((0 + 1)...𝑀) = (1...𝑀)
7069a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((0 + 1)...𝑀) = (1...𝑀))
7170sumeq1d 14730 . . . . . . . . . . . . . . . . . . 19 (𝜑 → Σ ∈ ((0 + 1)...𝑀)if( = 0, (𝑃 − 1), 𝑃) = Σ ∈ (1...𝑀)if( = 0, (𝑃 − 1), 𝑃))
72 0red 10301 . . . . . . . . . . . . . . . . . . . . . . . . 25 ( ∈ (1...𝑀) → 0 ∈ ℝ)
73 1red 10298 . . . . . . . . . . . . . . . . . . . . . . . . 25 ( ∈ (1...𝑀) → 1 ∈ ℝ)
74 elfzelz 12554 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ( ∈ (1...𝑀) → ∈ ℤ)
7574zred 11734 . . . . . . . . . . . . . . . . . . . . . . . . 25 ( ∈ (1...𝑀) → ∈ ℝ)
76 0lt1 10808 . . . . . . . . . . . . . . . . . . . . . . . . . 26 0 < 1
7776a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25 ( ∈ (1...𝑀) → 0 < 1)
78 elfzle1 12556 . . . . . . . . . . . . . . . . . . . . . . . . 25 ( ∈ (1...𝑀) → 1 ≤ )
7972, 73, 75, 77, 78ltletrd 10455 . . . . . . . . . . . . . . . . . . . . . . . 24 ( ∈ (1...𝑀) → 0 < )
8079gt0ne0d 10850 . . . . . . . . . . . . . . . . . . . . . . 23 ( ∈ (1...𝑀) → ≠ 0)
8180neneqd 2942 . . . . . . . . . . . . . . . . . . . . . 22 ( ∈ (1...𝑀) → ¬ = 0)
8281iffalsed 4256 . . . . . . . . . . . . . . . . . . . . 21 ( ∈ (1...𝑀) → if( = 0, (𝑃 − 1), 𝑃) = 𝑃)
8382adantl 473 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∈ (1...𝑀)) → if( = 0, (𝑃 − 1), 𝑃) = 𝑃)
8483sumeq2dv 14732 . . . . . . . . . . . . . . . . . . 19 (𝜑 → Σ ∈ (1...𝑀)if( = 0, (𝑃 − 1), 𝑃) = Σ ∈ (1...𝑀)𝑃)
85 fzfid 12985 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (1...𝑀) ∈ Fin)
863nncnd 11296 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑃 ∈ ℂ)
87 fsumconst 14820 . . . . . . . . . . . . . . . . . . . . 21 (((1...𝑀) ∈ Fin ∧ 𝑃 ∈ ℂ) → Σ ∈ (1...𝑀)𝑃 = ((♯‘(1...𝑀)) · 𝑃))
8885, 86, 87syl2anc 579 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → Σ ∈ (1...𝑀)𝑃 = ((♯‘(1...𝑀)) · 𝑃))
89 hashfz1 13343 . . . . . . . . . . . . . . . . . . . . . 22 (𝑀 ∈ ℕ0 → (♯‘(1...𝑀)) = 𝑀)
904, 89syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (♯‘(1...𝑀)) = 𝑀)
9190oveq1d 6861 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((♯‘(1...𝑀)) · 𝑃) = (𝑀 · 𝑃))
9288, 91eqtrd 2799 . . . . . . . . . . . . . . . . . . 19 (𝜑 → Σ ∈ (1...𝑀)𝑃 = (𝑀 · 𝑃))
9371, 84, 923eqtrd 2803 . . . . . . . . . . . . . . . . . 18 (𝜑 → Σ ∈ ((0 + 1)...𝑀)if( = 0, (𝑃 − 1), 𝑃) = (𝑀 · 𝑃))
9493oveq2d 6862 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑃 − 1) + Σ ∈ ((0 + 1)...𝑀)if( = 0, (𝑃 − 1), 𝑃)) = ((𝑃 − 1) + (𝑀 · 𝑃)))
9529nn0cnd 11604 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑃 − 1) ∈ ℂ)
964, 62nn0mulcld 11607 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑀 · 𝑃) ∈ ℕ0)
9796nn0cnd 11604 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑀 · 𝑃) ∈ ℂ)
9895, 97addcomd 10496 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑃 − 1) + (𝑀 · 𝑃)) = ((𝑀 · 𝑃) + (𝑃 − 1)))
9967, 94, 983eqtrd 2803 . . . . . . . . . . . . . . . 16 (𝜑 → Σ ∈ (0...𝑀)if( = 0, (𝑃 − 1), 𝑃) = ((𝑀 · 𝑃) + (𝑃 − 1)))
10099ad2antrr 717 . . . . . . . . . . . . . . 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 4837 . . . . . . . . . . . . . 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, 101syl5eqbr 4846 . . . . . . . . . . . . 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 4833 . . . . . . . . . . . 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 585 . . . . . . . . . . 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 11606 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑀 · 𝑃) + (𝑃 − 1)) ∈ ℕ0)
107106nn0red 11603 . . . . . . . . . . . . . 14 (𝜑 → ((𝑀 · 𝑃) + (𝑃 − 1)) ∈ ℝ)
1086nn0red 11603 . . . . . . . . . . . . . 14 (𝜑𝑁 ∈ ℝ)
109107, 108ltnled 10442 . . . . . . . . . . . . 13 (𝜑 → (((𝑀 · 𝑃) + (𝑃 − 1)) < 𝑁 ↔ ¬ 𝑁 ≤ ((𝑀 · 𝑃) + (𝑃 − 1))))
110105, 109mpbid 223 . . . . . . . . . . . 12 (𝜑 → ¬ 𝑁 ≤ ((𝑀 · 𝑃) + (𝑃 − 1)))
111110ad2antrr 717 . . . . . . . . . . 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 852 . . . . . . . . . 10 ((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) → ∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘))
113112adantlr 706 . . . . . . . . 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 2009 . . . . . . . . . . . . 13 𝑗(𝜑𝑥𝑋)
115 nfcv 2907 . . . . . . . . . . . . . . . . 17 𝑗(0...𝑀)
116115nfsum1 14719 . . . . . . . . . . . . . . . 16 𝑗Σ𝑗 ∈ (0...𝑀)(𝑐𝑗)
117116nfeq1 2921 . . . . . . . . . . . . . . 15 𝑗Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁
118 nfcv 2907 . . . . . . . . . . . . . . 15 𝑗((0...𝑁) ↑𝑚 (0...𝑀))
119117, 118nfrab 3271 . . . . . . . . . . . . . 14 𝑗{𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}
120119nfcri 2901 . . . . . . . . . . . . 13 𝑗 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}
121114, 120nfan 1998 . . . . . . . . . . . 12 𝑗((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁})
122 nfv 2009 . . . . . . . . . . . 12 𝑗 𝑘 ∈ (0...𝑀)
123 nfv 2009 . . . . . . . . . . . 12 𝑗if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘)
124121, 122, 123nf3an 2000 . . . . . . . . . . 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 2907 . . . . . . . . . . 11 𝑗(((𝑆 D𝑛 (𝐻𝑘))‘(𝑐𝑘))‘𝑥)
126 fzfid 12985 . . . . . . . . . . 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 721 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → 𝑆 ∈ {ℝ, ℂ})
1282ad3antrrr 721 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
1293ad3antrrr 721 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → 𝑃 ∈ ℕ)
130 etransclem5 41117 . . . . . . . . . . . . . . 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 2787 . . . . . . . . . . . . . 14 𝐻 = (𝑘 ∈ (0...𝑀) ↦ (𝑦𝑋 ↦ ((𝑦𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))
132 simpr 477 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ (0...𝑀))
13323ad2antlr 718 . . . . . . . . . . . . . . . . 17 (((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → 𝑐:(0...𝑀)⟶(0...𝑁))
134 simpr 477 . . . . . . . . . . . . . . . . 17 (((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ (0...𝑀))
135133, 134ffvelrnd 6554 . . . . . . . . . . . . . . . 16 (((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → (𝑐𝑗) ∈ (0...𝑁))
136135adantllr 710 . . . . . . . . . . . . . . 15 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → (𝑐𝑗) ∈ (0...𝑁))
137 elfznn0 12645 . . . . . . . . . . . . . . 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 41132 . . . . . . . . . . . . 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 793 . . . . . . . . . . . . 13 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → 𝑥𝑋)
141139, 140ffvelrnd 6554 . . . . . . . . . . . 12 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑆 D𝑛 (𝐻𝑗))‘(𝑐𝑗))‘𝑥) ∈ ℂ)
1421413ad2antl1 1236 . . . . . . . . . . 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 6379 . . . . . . . . . . . . . 14 (𝑗 = 𝑘 → (𝐻𝑗) = (𝐻𝑘))
144143oveq2d 6862 . . . . . . . . . . . . 13 (𝑗 = 𝑘 → (𝑆 D𝑛 (𝐻𝑗)) = (𝑆 D𝑛 (𝐻𝑘)))
145144, 41fveq12d 6386 . . . . . . . . . . . 12 (𝑗 = 𝑘 → ((𝑆 D𝑛 (𝐻𝑗))‘(𝑐𝑗)) = ((𝑆 D𝑛 (𝐻𝑘))‘(𝑐𝑘)))
146145fveq1d 6381 . . . . . . . . . . 11 (𝑗 = 𝑘 → (((𝑆 D𝑛 (𝐻𝑗))‘(𝑐𝑗))‘𝑥) = (((𝑆 D𝑛 (𝐻𝑘))‘(𝑐𝑘))‘𝑥))
147 simp2 1167 . . . . . . . . . . 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 717 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) → 𝑆 ∈ {ℝ, ℂ})
1491483ad2ant1 1163 . . . . . . . . . . . . 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 717 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
1511503ad2ant1 1163 . . . . . . . . . . . . 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 717 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) → 𝑃 ∈ ℕ)
1531523ad2ant1 1163 . . . . . . . . . . . . 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 41117 . . . . . . . . . . . . . 14 (𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))) = ( ∈ (0...𝑀) ↦ (𝑦𝑋 ↦ ((𝑦)↑if( = 0, (𝑃 − 1), 𝑃))))
1557, 154eqtri 2787 . . . . . . . . . . . . 13 𝐻 = ( ∈ (0...𝑀) ↦ (𝑦𝑋 ↦ ((𝑦)↑if( = 0, (𝑃 − 1), 𝑃))))
156 fzssz 12555 . . . . . . . . . . . . . . . 16 (0...𝑁) ⊆ ℤ
157156, 25sseldi 3761 . . . . . . . . . . . . . . 15 (((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀)) → (𝑐𝑘) ∈ ℤ)
158157adantllr 710 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀)) → (𝑐𝑘) ∈ ℤ)
1591583adant3 1162 . . . . . . . . . . . . 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)
160 simp3 1168 . . . . . . . . . . . . 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))
161149, 151, 153, 155, 147, 159, 160etransclem19 41131 . . . . . . . . . . . 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))
162 eqidd 2766 . . . . . . . . . . . 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)
163 simp1lr 1318 . . . . . . . . . . . 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)
164 0red 10301 . . . . . . . . . . . 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)
165161, 162, 163, 164fvmptd 6481 . . . . . . . . . . 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)
166124, 125, 126, 142, 146, 147, 165fprod0 40490 . . . . . . . . . 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)
167166rexlimdv3a 3180 . . . . . . . . 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))
168113, 167mpd 15 . . . . . . . 8 (((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) → ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘(𝑐𝑗))‘𝑥) = 0)
16914, 168syldan 585 . . . . . . 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)
170169oveq2d 6862 . . . . . 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))
1716faccld 13280 . . . . . . . . . . 11 (𝜑 → (!‘𝑁) ∈ ℕ)
172171nncnd 11296 . . . . . . . . . 10 (𝜑 → (!‘𝑁) ∈ ℂ)
173172adantr 472 . . . . . . . . 9 ((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) → (!‘𝑁) ∈ ℂ)
174 fzfid 12985 . . . . . . . . . 10 ((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) → (0...𝑀) ∈ Fin)
175 simpll 783 . . . . . . . . . . . . . 14 (((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → 𝜑)
17613adantr 472 . . . . . . . . . . . . . 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})
177 simpr 477 . . . . . . . . . . . . . 14 (((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ (0...𝑀))
178175, 176, 177, 135syl21anc 866 . . . . . . . . . . . . 13 (((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (𝑐𝑗) ∈ (0...𝑁))
179178, 137syl 17 . . . . . . . . . . . 12 (((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (𝑐𝑗) ∈ ℕ0)
180179faccld 13280 . . . . . . . . . . 11 (((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝑐𝑗)) ∈ ℕ)
181180nncnd 11296 . . . . . . . . . 10 (((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝑐𝑗)) ∈ ℂ)
182174, 181fprodcl 14979 . . . . . . . . 9 ((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) → ∏𝑗 ∈ (0...𝑀)(!‘(𝑐𝑗)) ∈ ℂ)
183180nnne0d 11326 . . . . . . . . . 10 (((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝑐𝑗)) ≠ 0)
184174, 181, 183fprodn0 15006 . . . . . . . . 9 ((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) → ∏𝑗 ∈ (0...𝑀)(!‘(𝑐𝑗)) ≠ 0)
185173, 182, 184divcld 11059 . . . . . . . 8 ((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐𝑗))) ∈ ℂ)
186185mul01d 10493 . . . . . . 7 ((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) → (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐𝑗))) · 0) = 0)
187186adantlr 706 . . . . . 6 (((𝜑𝑥𝑋) ∧ 𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) → (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐𝑗))) · 0) = 0)
188170, 187eqtrd 2799 . . . . 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)
189188sumeq2dv 14732 . . . 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)
190 eqid 2765 . . . . . . . 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})
191190, 6etransclem16 41128 . . . . . . 7 (𝜑 → ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁) ∈ Fin)
192191olcd 900 . . . . . 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))
193192adantr 472 . . . . 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))
194 sumz 14752 . . . . 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)
195193, 194syl 17 . . . 4 ((𝜑𝑥𝑋) → Σ𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)0 = 0)
196189, 195eqtrd 2799 . . 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)
197196mpteq2dva 4905 . 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))
1989, 197eqtrd 2799 1 (𝜑 → ((𝑆 D𝑛 𝐹)‘𝑁) = (𝑥𝑋 ↦ 0))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wo 873  w3a 1107   = wceq 1652  wcel 2155  wral 3055  wrex 3056  {crab 3059  wss 3734  ifcif 4245  {cpr 4338   class class class wbr 4811  cmpt 4890  wf 6066  cfv 6070  (class class class)co 6846  𝑚 cmap 8064  Fincfn 8164  cc 10191  cr 10192  0cc0 10193  1c1 10194   + caddc 10196   · cmul 10198   < clt 10332  cle 10333  cmin 10524   / cdiv 10942  cn 11278  0cn0 11542  cz 11628  cuz 11891  ...cfz 12538  cexp 13072  !cfa 13269  chash 13326  Σcsu 14715  cprod 14932  t crest 16361  TopOpenctopn 16362  fldccnfld 20033   D𝑛 cdvn 23933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151  ax-inf2 8757  ax-cnex 10249  ax-resscn 10250  ax-1cn 10251  ax-icn 10252  ax-addcl 10253  ax-addrcl 10254  ax-mulcl 10255  ax-mulrcl 10256  ax-mulcom 10257  ax-addass 10258  ax-mulass 10259  ax-distr 10260  ax-i2m1 10261  ax-1ne0 10262  ax-1rid 10263  ax-rnegex 10264  ax-rrecex 10265  ax-cnre 10266  ax-pre-lttri 10267  ax-pre-lttrn 10268  ax-pre-ltadd 10269  ax-pre-mulgt0 10270  ax-pre-sup 10271  ax-addf 10272  ax-mulf 10273
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-iin 4681  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-se 5239  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-isom 6079  df-riota 6807  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-of 7099  df-om 7268  df-1st 7370  df-2nd 7371  df-supp 7502  df-wrecs 7614  df-recs 7676  df-rdg 7714  df-1o 7768  df-2o 7769  df-oadd 7772  df-er 7951  df-map 8066  df-pm 8067  df-ixp 8118  df-en 8165  df-dom 8166  df-sdom 8167  df-fin 8168  df-fsupp 8487  df-fi 8528  df-sup 8559  df-inf 8560  df-oi 8626  df-card 9020  df-cda 9247  df-pnf 10334  df-mnf 10335  df-xr 10336  df-ltxr 10337  df-le 10338  df-sub 10526  df-neg 10527  df-div 10943  df-nn 11279  df-2 11339  df-3 11340  df-4 11341  df-5 11342  df-6 11343  df-7 11344  df-8 11345  df-9 11346  df-n0 11543  df-z 11629  df-dec 11746  df-uz 11892  df-q 11995  df-rp 12034  df-xneg 12151  df-xadd 12152  df-xmul 12153  df-ico 12388  df-icc 12389  df-fz 12539  df-fzo 12679  df-seq 13014  df-exp 13073  df-fac 13270  df-bc 13299  df-hash 13327  df-cj 14138  df-re 14139  df-im 14140  df-sqrt 14274  df-abs 14275  df-clim 14518  df-sum 14716  df-prod 14933  df-struct 16146  df-ndx 16147  df-slot 16148  df-base 16150  df-sets 16151  df-ress 16152  df-plusg 16241  df-mulr 16242  df-starv 16243  df-sca 16244  df-vsca 16245  df-ip 16246  df-tset 16247  df-ple 16248  df-ds 16250  df-unif 16251  df-hom 16252  df-cco 16253  df-rest 16363  df-topn 16364  df-0g 16382  df-gsum 16383  df-topgen 16384  df-pt 16385  df-prds 16388  df-xrs 16442  df-qtop 16447  df-imas 16448  df-xps 16450  df-mre 16526  df-mrc 16527  df-acs 16529  df-mgm 17522  df-sgrp 17564  df-mnd 17575  df-submnd 17616  df-mulg 17822  df-cntz 18027  df-cmn 18475  df-psmet 20025  df-xmet 20026  df-met 20027  df-bl 20028  df-mopn 20029  df-fbas 20030  df-fg 20031  df-cnfld 20034  df-top 20992  df-topon 21009  df-topsp 21031  df-bases 21044  df-cld 21117  df-ntr 21118  df-cls 21119  df-nei 21196  df-lp 21234  df-perf 21235  df-cn 21325  df-cnp 21326  df-haus 21413  df-tx 21659  df-hmeo 21852  df-fil 21943  df-fm 22035  df-flim 22036  df-flf 22037  df-xms 22418  df-ms 22419  df-tms 22420  df-cncf 22974  df-limc 23935  df-dv 23936  df-dvn 23937
This theorem is referenced by:  etransclem46  41158
  Copyright terms: Public domain W3C validator