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Theorem etransclem32 40801
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 40780 . . 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 40799 . 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 476 . . . . . . . . . 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 40781 . . . . . . . . . . 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 2732 . . . . . . . . 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 751 . . . . . . . 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 1883 . . . . . . . . . . . . . 14 𝑘(𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁})
16 nfre1 3034 . . . . . . . . . . . . . . 15 𝑘𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘)
1716nfn 1824 . . . . . . . . . . . . . 14 𝑘 ¬ ∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘)
1815, 17nfan 1868 . . . . . . . . . . . . 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 39842 . . . . . . . . . . . . . . . . 17 (0...𝑁) ⊆ ℝ
20 rabid 3145 . . . . . . . . . . . . . . . . . . . . 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 475 . . . . . . . . . . . . . . . . . . . 20 (𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁} → 𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)))
22 elmapi 7921 . . . . . . . . . . . . . . . . . . . 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))
2524ffvelrnda 6399 . . . . . . . . . . . . . . . . 17 (((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀)) → (𝑐𝑘) ∈ (0...𝑁))
2619, 25sseldi 3634 . . . . . . . . . . . . . . . 16 (((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀)) → (𝑐𝑘) ∈ ℝ)
2726adantlr 751 . . . . . . . . . . . . . . 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 11372 . . . . . . . . . . . . . . . . . . 19 (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ0)
293, 28syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑃 − 1) ∈ ℕ0)
3029nn0red 11390 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑃 − 1) ∈ ℝ)
313nnred 11073 . . . . . . . . . . . . . . . . 17 (𝜑𝑃 ∈ ℝ)
3230, 31ifcld 4164 . . . . . . . . . . . . . . . 16 (𝜑 → if(𝑘 = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
3332ad3antrrr 766 . . . . . . . . . . . . . . 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 3021 . . . . . . . . . . . . . . . . . 18 (∀𝑘 ∈ (0...𝑀) ¬ if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘) ↔ ¬ ∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘))
3534biimpri 218 . . . . . . . . . . . . . . . . 17 (¬ ∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘) → ∀𝑘 ∈ (0...𝑀) ¬ if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘))
3635r19.21bi 2961 . . . . . . . . . . . . . . . 16 ((¬ ∃𝑘 ∈ (0...𝑀)if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘) ∧ 𝑘 ∈ (0...𝑀)) → ¬ if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘))
3736adantll 750 . . . . . . . . . . . . . . 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 10225 . . . . . . . . . . . . . 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 449 . . . . . . . . . . . . 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 2986 . . . . . . . . . . . 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 6229 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑘 → (𝑐𝑗) = (𝑐𝑘))
4241cbvsumv 14470 . . . . . . . . . . . . . . 15 Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = Σ𝑘 ∈ (0...𝑀)(𝑐𝑘)
4320simprbi 479 . . . . . . . . . . . . . . 15 (𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁} → Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁)
4442, 43syl5reqr 2700 . . . . . . . . . . . . . 14 (𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁} → 𝑁 = Σ𝑘 ∈ (0...𝑀)(𝑐𝑘))
4544ad2antlr 763 . . . . . . . . . . . . 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 6229 . . . . . . . . . . . . . . 15 (𝑘 = → (𝑐𝑘) = (𝑐))
4746cbvsumv 14470 . . . . . . . . . . . . . 14 Σ𝑘 ∈ (0...𝑀)(𝑐𝑘) = Σ ∈ (0...𝑀)(𝑐)
48 fzfid 12812 . . . . . . . . . . . . . . . 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 6399 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ ∈ (0...𝑀)) → (𝑐) ∈ (0...𝑁))
5019, 49sseldi 3634 . . . . . . . . . . . . . . . . 17 (((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ ∈ (0...𝑀)) → (𝑐) ∈ ℝ)
5150adantlr 751 . . . . . . . . . . . . . . . 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 4164 . . . . . . . . . . . . . . . . 17 (𝜑 → if( = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
5352ad3antrrr 766 . . . . . . . . . . . . . . . 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 2655 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = → (𝑘 = 0 ↔ = 0))
5554ifbid 4141 . . . . . . . . . . . . . . . . . . 19 (𝑘 = → if(𝑘 = 0, (𝑃 − 1), 𝑃) = if( = 0, (𝑃 − 1), 𝑃))
5646, 55breq12d 4698 . . . . . . . . . . . . . . . . . 18 (𝑘 = → ((𝑐𝑘) ≤ if(𝑘 = 0, (𝑃 − 1), 𝑃) ↔ (𝑐) ≤ if( = 0, (𝑃 − 1), 𝑃)))
5756rspccva 3339 . . . . . . . . . . . . . . . . 17 ((∀𝑘 ∈ (0...𝑀)(𝑐𝑘) ≤ if(𝑘 = 0, (𝑃 − 1), 𝑃) ∧ ∈ (0...𝑀)) → (𝑐) ≤ if( = 0, (𝑃 − 1), 𝑃))
5857adantll 750 . . . . . . . . . . . . . . . 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 14575 . . . . . . . . . . . . . . 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 11760 . . . . . . . . . . . . . . . . . . 19 0 = (ℤ‘0)
614, 60syl6eleq 2740 . . . . . . . . . . . . . . . . . 18 (𝜑𝑀 ∈ (ℤ‘0))
623nnnn0d 11389 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑃 ∈ ℕ0)
6329, 62ifcld 4164 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → if( = 0, (𝑃 − 1), 𝑃) ∈ ℕ0)
6463adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∈ (0...𝑀)) → if( = 0, (𝑃 − 1), 𝑃) ∈ ℕ0)
6564nn0cnd 11391 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∈ (0...𝑀)) → if( = 0, (𝑃 − 1), 𝑃) ∈ ℂ)
66 iftrue 4125 . . . . . . . . . . . . . . . . . 18 ( = 0 → if( = 0, (𝑃 − 1), 𝑃) = (𝑃 − 1))
6761, 65, 66fsum1p 14526 . . . . . . . . . . . . . . . . 17 (𝜑 → Σ ∈ (0...𝑀)if( = 0, (𝑃 − 1), 𝑃) = ((𝑃 − 1) + Σ ∈ ((0 + 1)...𝑀)if( = 0, (𝑃 − 1), 𝑃)))
68 0p1e1 11170 . . . . . . . . . . . . . . . . . . . . . 22 (0 + 1) = 1
6968oveq1i 6700 . . . . . . . . . . . . . . . . . . . . 21 ((0 + 1)...𝑀) = (1...𝑀)
7069a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((0 + 1)...𝑀) = (1...𝑀))
7170sumeq1d 14475 . . . . . . . . . . . . . . . . . . 19 (𝜑 → Σ ∈ ((0 + 1)...𝑀)if( = 0, (𝑃 − 1), 𝑃) = Σ ∈ (1...𝑀)if( = 0, (𝑃 − 1), 𝑃))
72 0red 10079 . . . . . . . . . . . . . . . . . . . . . . . . 25 ( ∈ (1...𝑀) → 0 ∈ ℝ)
73 1red 10093 . . . . . . . . . . . . . . . . . . . . . . . . 25 ( ∈ (1...𝑀) → 1 ∈ ℝ)
74 elfzelz 12380 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ( ∈ (1...𝑀) → ∈ ℤ)
7574zred 11520 . . . . . . . . . . . . . . . . . . . . . . . . 25 ( ∈ (1...𝑀) → ∈ ℝ)
76 0lt1 10588 . . . . . . . . . . . . . . . . . . . . . . . . . 26 0 < 1
7776a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25 ( ∈ (1...𝑀) → 0 < 1)
78 elfzle1 12382 . . . . . . . . . . . . . . . . . . . . . . . . 25 ( ∈ (1...𝑀) → 1 ≤ )
7972, 73, 75, 77, 78ltletrd 10235 . . . . . . . . . . . . . . . . . . . . . . . 24 ( ∈ (1...𝑀) → 0 < )
8079gt0ne0d 10630 . . . . . . . . . . . . . . . . . . . . . . 23 ( ∈ (1...𝑀) → ≠ 0)
8180neneqd 2828 . . . . . . . . . . . . . . . . . . . . . 22 ( ∈ (1...𝑀) → ¬ = 0)
8281iffalsed 4130 . . . . . . . . . . . . . . . . . . . . 21 ( ∈ (1...𝑀) → if( = 0, (𝑃 − 1), 𝑃) = 𝑃)
8382adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∈ (1...𝑀)) → if( = 0, (𝑃 − 1), 𝑃) = 𝑃)
8483sumeq2dv 14477 . . . . . . . . . . . . . . . . . . 19 (𝜑 → Σ ∈ (1...𝑀)if( = 0, (𝑃 − 1), 𝑃) = Σ ∈ (1...𝑀)𝑃)
85 fzfid 12812 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (1...𝑀) ∈ Fin)
863nncnd 11074 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑃 ∈ ℂ)
87 fsumconst 14566 . . . . . . . . . . . . . . . . . . . . 21 (((1...𝑀) ∈ Fin ∧ 𝑃 ∈ ℂ) → Σ ∈ (1...𝑀)𝑃 = ((#‘(1...𝑀)) · 𝑃))
8885, 86, 87syl2anc 694 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → Σ ∈ (1...𝑀)𝑃 = ((#‘(1...𝑀)) · 𝑃))
89 hashfz1 13174 . . . . . . . . . . . . . . . . . . . . . 22 (𝑀 ∈ ℕ0 → (#‘(1...𝑀)) = 𝑀)
904, 89syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (#‘(1...𝑀)) = 𝑀)
9190oveq1d 6705 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((#‘(1...𝑀)) · 𝑃) = (𝑀 · 𝑃))
9288, 91eqtrd 2685 . . . . . . . . . . . . . . . . . . 19 (𝜑 → Σ ∈ (1...𝑀)𝑃 = (𝑀 · 𝑃))
9371, 84, 923eqtrd 2689 . . . . . . . . . . . . . . . . . 18 (𝜑 → Σ ∈ ((0 + 1)...𝑀)if( = 0, (𝑃 − 1), 𝑃) = (𝑀 · 𝑃))
9493oveq2d 6706 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑃 − 1) + Σ ∈ ((0 + 1)...𝑀)if( = 0, (𝑃 − 1), 𝑃)) = ((𝑃 − 1) + (𝑀 · 𝑃)))
9529nn0cnd 11391 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑃 − 1) ∈ ℂ)
964, 62nn0mulcld 11394 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑀 · 𝑃) ∈ ℕ0)
9796nn0cnd 11391 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑀 · 𝑃) ∈ ℂ)
9895, 97addcomd 10276 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑃 − 1) + (𝑀 · 𝑃)) = ((𝑀 · 𝑃) + (𝑃 − 1)))
9967, 94, 983eqtrd 2689 . . . . . . . . . . . . . . . 16 (𝜑 → Σ ∈ (0...𝑀)if( = 0, (𝑃 − 1), 𝑃) = ((𝑀 · 𝑃) + (𝑃 − 1)))
10099ad2antrr 762 . . . . . . . . . . . . . . 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 4711 . . . . . . . . . . . . . 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 4720 . . . . . . . . . . . . 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 4707 . . . . . . . . . . . 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 486 . . . . . . . . . . 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 11393 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑀 · 𝑃) + (𝑃 − 1)) ∈ ℕ0)
107106nn0red 11390 . . . . . . . . . . . . . 14 (𝜑 → ((𝑀 · 𝑃) + (𝑃 − 1)) ∈ ℝ)
1086nn0red 11390 . . . . . . . . . . . . . 14 (𝜑𝑁 ∈ ℝ)
109107, 108ltnled 10222 . . . . . . . . . . . . 13 (𝜑 → (((𝑀 · 𝑃) + (𝑃 − 1)) < 𝑁 ↔ ¬ 𝑁 ≤ ((𝑀 · 𝑃) + (𝑃 − 1))))
110105, 109mpbid 222 . . . . . . . . . . . 12 (𝜑 → ¬ 𝑁 ≤ ((𝑀 · 𝑃) + (𝑃 − 1)))
111110ad2antrr 762 . . . . . . . . . . 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 751 . . . . . . . . 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 1883 . . . . . . . . . . . . 13 𝑗(𝜑𝑥𝑋)
115 nfcv 2793 . . . . . . . . . . . . . . . . 17 𝑗(0...𝑀)
116115nfsum1 14464 . . . . . . . . . . . . . . . 16 𝑗Σ𝑗 ∈ (0...𝑀)(𝑐𝑗)
117116nfeq1 2807 . . . . . . . . . . . . . . 15 𝑗Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁
118 nfcv 2793 . . . . . . . . . . . . . . 15 𝑗((0...𝑁) ↑𝑚 (0...𝑀))
119117, 118nfrab 3153 . . . . . . . . . . . . . 14 𝑗{𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}
120119nfcri 2787 . . . . . . . . . . . . 13 𝑗 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}
121114, 120nfan 1868 . . . . . . . . . . . 12 𝑗((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁})
122 nfv 1883 . . . . . . . . . . . 12 𝑗 𝑘 ∈ (0...𝑀)
123 nfv 1883 . . . . . . . . . . . 12 𝑗if(𝑘 = 0, (𝑃 − 1), 𝑃) < (𝑐𝑘)
124121, 122, 123nf3an 1871 . . . . . . . . . . 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 2793 . . . . . . . . . . 11 𝑗(((𝑆 D𝑛 (𝐻𝑘))‘(𝑐𝑘))‘𝑥)
126 fzfid 12812 . . . . . . . . . . 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 766 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → 𝑆 ∈ {ℝ, ℂ})
1282ad3antrrr 766 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
1293ad3antrrr 766 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → 𝑃 ∈ ℕ)
130 etransclem5 40774 . . . . . . . . . . . . . . 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 2673 . . . . . . . . . . . . . 14 𝐻 = (𝑘 ∈ (0...𝑀) ↦ (𝑦𝑋 ↦ ((𝑦𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))
132 simpr 476 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ (0...𝑀))
13323ad2antlr 763 . . . . . . . . . . . . . . . . 17 (((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → 𝑐:(0...𝑀)⟶(0...𝑁))
134 simpr 476 . . . . . . . . . . . . . . . . 17 (((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ (0...𝑀))
135133, 134ffvelrnd 6400 . . . . . . . . . . . . . . . 16 (((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → (𝑐𝑗) ∈ (0...𝑁))
136135adantllr 755 . . . . . . . . . . . . . . 15 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → (𝑐𝑗) ∈ (0...𝑁))
137 elfznn0 12471 . . . . . . . . . . . . . . 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 40789 . . . . . . . . . . . . 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 815 . . . . . . . . . . . . 13 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → 𝑥𝑋)
141139, 140ffvelrnd 6400 . . . . . . . . . . . 12 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑆 D𝑛 (𝐻𝑗))‘(𝑐𝑗))‘𝑥) ∈ ℂ)
1421413ad2antl1 1243 . . . . . . . . . . 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 6229 . . . . . . . . . . . . . 14 (𝑗 = 𝑘 → (𝐻𝑗) = (𝐻𝑘))
144143oveq2d 6706 . . . . . . . . . . . . 13 (𝑗 = 𝑘 → (𝑆 D𝑛 (𝐻𝑗)) = (𝑆 D𝑛 (𝐻𝑘)))
145144, 41fveq12d 6235 . . . . . . . . . . . 12 (𝑗 = 𝑘 → ((𝑆 D𝑛 (𝐻𝑗))‘(𝑐𝑗)) = ((𝑆 D𝑛 (𝐻𝑘))‘(𝑐𝑘)))
146145fveq1d 6231 . . . . . . . . . . 11 (𝑗 = 𝑘 → (((𝑆 D𝑛 (𝐻𝑗))‘(𝑐𝑗))‘𝑥) = (((𝑆 D𝑛 (𝐻𝑘))‘(𝑐𝑘))‘𝑥))
147 simp2 1082 . . . . . . . . . . 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 762 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) → 𝑆 ∈ {ℝ, ℂ})
1491483ad2ant1 1102 . . . . . . . . . . . . 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 762 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
1511503ad2ant1 1102 . . . . . . . . . . . . 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 762 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) → 𝑃 ∈ ℕ)
1531523ad2ant1 1102 . . . . . . . . . . . . 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 40774 . . . . . . . . . . . . . 14 (𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))) = ( ∈ (0...𝑀) ↦ (𝑦𝑋 ↦ ((𝑦)↑if( = 0, (𝑃 − 1), 𝑃))))
1557, 154eqtri 2673 . . . . . . . . . . . . 13 𝐻 = ( ∈ (0...𝑀) ↦ (𝑦𝑋 ↦ ((𝑦)↑if( = 0, (𝑃 − 1), 𝑃))))
156 fzssz 12381 . . . . . . . . . . . . . . . 16 (0...𝑁) ⊆ ℤ
157156, 25sseldi 3634 . . . . . . . . . . . . . . 15 (((𝜑𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀)) → (𝑐𝑘) ∈ ℤ)
158157adantllr 755 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁}) ∧ 𝑘 ∈ (0...𝑀)) → (𝑐𝑘) ∈ ℤ)
1591583adant3 1101 . . . . . . . . . . . . 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 1083 . . . . . . . . . . . . 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 40788 . . . . . . . . . . . 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 2652 . . . . . . . . . . . 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 1145 . . . . . . . . . . . 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 10079 . . . . . . . . . . . 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 6327 . . . . . . . . . . 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 40146 . . . . . . . . . 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 3062 . . . . . . . . 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 486 . . . . . . 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 6706 . . . . . 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 13111 . . . . . . . . . . 11 (𝜑 → (!‘𝑁) ∈ ℕ)
172171nncnd 11074 . . . . . . . . . 10 (𝜑 → (!‘𝑁) ∈ ℂ)
173172adantr 480 . . . . . . . . 9 ((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) → (!‘𝑁) ∈ ℂ)
174 fzfid 12812 . . . . . . . . . 10 ((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) → (0...𝑀) ∈ Fin)
175 simpll 805 . . . . . . . . . . . . . 14 (((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → 𝜑)
17613adantr 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})
177 simpr 476 . . . . . . . . . . . . . 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 1365 . . . . . . . . . . . . 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 13111 . . . . . . . . . . 11 (((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝑐𝑗)) ∈ ℕ)
181180nncnd 11074 . . . . . . . . . 10 (((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝑐𝑗)) ∈ ℂ)
182174, 181fprodcl 14726 . . . . . . . . 9 ((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) → ∏𝑗 ∈ (0...𝑀)(!‘(𝑐𝑗)) ∈ ℂ)
183180nnne0d 11103 . . . . . . . . . 10 (((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝑐𝑗)) ≠ 0)
184174, 181, 183fprodn0 14753 . . . . . . . . 9 ((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) → ∏𝑗 ∈ (0...𝑀)(!‘(𝑐𝑗)) ≠ 0)
185173, 182, 184divcld 10839 . . . . . . . 8 ((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐𝑗))) ∈ ℂ)
186185mul01d 10273 . . . . . . 7 ((𝜑𝑐 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁)) → (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐𝑗))) · 0) = 0)
187186adantlr 751 . . . . . 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 2685 . . . . 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 14477 . . . 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 2651 . . . . . . . 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 40785 . . . . . . 7 (𝜑 → ((𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑𝑘) = 𝑚})‘𝑁) ∈ Fin)
192191olcd 407 . . . . . 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 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))
194 sumz 14497 . . . . 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 2685 . . 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 4777 . 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 2685 1 (𝜑 → ((𝑆 D𝑛 𝐹)‘𝑁) = (𝑥𝑋 ↦ 0))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 382  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wrex 2942  {crab 2945  wss 3607  ifcif 4119  {cpr 4212   class class class wbr 4685  cmpt 4762  wf 5922  cfv 5926  (class class class)co 6690  𝑚 cmap 7899  Fincfn 7997  cc 9972  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979   < clt 10112  cle 10113  cmin 10304   / cdiv 10722  cn 11058  0cn0 11330  cz 11415  cuz 11725  ...cfz 12364  cexp 12900  !cfa 13100  #chash 13157  Σcsu 14460  cprod 14679  t crest 16128  TopOpenctopn 16129  fldccnfld 19794   D𝑛 cdvn 23673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-addf 10053  ax-mulf 10054
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-seq 12842  df-exp 12901  df-fac 13101  df-bc 13130  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-sum 14461  df-prod 14680  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-starv 16003  df-sca 16004  df-vsca 16005  df-ip 16006  df-tset 16007  df-ple 16008  df-ds 16011  df-unif 16012  df-hom 16013  df-cco 16014  df-rest 16130  df-topn 16131  df-0g 16149  df-gsum 16150  df-topgen 16151  df-pt 16152  df-prds 16155  df-xrs 16209  df-qtop 16214  df-imas 16215  df-xps 16217  df-mre 16293  df-mrc 16294  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-submnd 17383  df-mulg 17588  df-cntz 17796  df-cmn 18241  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-fbas 19791  df-fg 19792  df-cnfld 19795  df-top 20747  df-topon 20764  df-topsp 20785  df-bases 20798  df-cld 20871  df-ntr 20872  df-cls 20873  df-nei 20950  df-lp 20988  df-perf 20989  df-cn 21079  df-cnp 21080  df-haus 21167  df-tx 21413  df-hmeo 21606  df-fil 21697  df-fm 21789  df-flim 21790  df-flf 21791  df-xms 22172  df-ms 22173  df-tms 22174  df-cncf 22728  df-limc 23675  df-dv 23676  df-dvn 23677
This theorem is referenced by:  etransclem46  40815
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