Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  etransclem27 Structured version   Visualization version   GIF version

Theorem etransclem27 42890
 Description: The 𝑁-th derivative of 𝐹 applied to 𝐽 is an integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
Hypotheses
Ref Expression
etransclem27.s (𝜑𝑆 ∈ {ℝ, ℂ})
etransclem27.x (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
etransclem27.p (𝜑𝑃 ∈ ℕ)
etransclem27.h 𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))
etransclem27.cfi (𝜑𝐶 ∈ Fin)
etransclem27.cf (𝜑𝐶:dom 𝐶⟶(ℕ0m (0...𝑀)))
etransclem27.g 𝐺 = (𝑥𝑋 ↦ Σ𝑙 ∈ dom 𝐶𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝑥))
etransclem27.jx (𝜑𝐽𝑋)
etransclem27.jz (𝜑𝐽 ∈ ℤ)
Assertion
Ref Expression
etransclem27 (𝜑 → (𝐺𝐽) ∈ ℤ)
Distinct variable groups:   𝐶,𝑗,𝑙,𝑥   𝑥,𝐻   𝑗,𝐽,𝑙,𝑥   𝑗,𝑀,𝑥   𝑃,𝑗,𝑥   𝑥,𝑆   𝑗,𝑋,𝑥   𝜑,𝑗,𝑙,𝑥
Allowed substitution hints:   𝑃(𝑙)   𝑆(𝑗,𝑙)   𝐺(𝑥,𝑗,𝑙)   𝐻(𝑗,𝑙)   𝑀(𝑙)   𝑋(𝑙)

Proof of Theorem etransclem27
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 etransclem27.g . . 3 𝐺 = (𝑥𝑋 ↦ Σ𝑙 ∈ dom 𝐶𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝑥))
2 fveq2 6649 . . . . 5 (𝑥 = 𝐽 → (((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝑥) = (((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽))
32prodeq2ad 42221 . . . 4 (𝑥 = 𝐽 → ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝑥) = ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽))
43sumeq2sdv 15056 . . 3 (𝑥 = 𝐽 → Σ𝑙 ∈ dom 𝐶𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝑥) = Σ𝑙 ∈ dom 𝐶𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽))
5 etransclem27.jx . . 3 (𝜑𝐽𝑋)
6 etransclem27.cfi . . . . 5 (𝜑𝐶 ∈ Fin)
7 dmfi 8790 . . . . 5 (𝐶 ∈ Fin → dom 𝐶 ∈ Fin)
86, 7syl 17 . . . 4 (𝜑 → dom 𝐶 ∈ Fin)
9 fzfid 13340 . . . . 5 ((𝜑𝑙 ∈ dom 𝐶) → (0...𝑀) ∈ Fin)
10 etransclem27.s . . . . . . . 8 (𝜑𝑆 ∈ {ℝ, ℂ})
1110ad2antrr 725 . . . . . . 7 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 𝑆 ∈ {ℝ, ℂ})
12 etransclem27.x . . . . . . . 8 (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
1312ad2antrr 725 . . . . . . 7 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
14 etransclem27.p . . . . . . . 8 (𝜑𝑃 ∈ ℕ)
1514ad2antrr 725 . . . . . . 7 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 𝑃 ∈ ℕ)
16 etransclem27.h . . . . . . . 8 𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))
17 etransclem5 42868 . . . . . . . 8 (𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))) = (𝑧 ∈ (0...𝑀) ↦ (𝑦𝑋 ↦ ((𝑦𝑧)↑if(𝑧 = 0, (𝑃 − 1), 𝑃))))
1816, 17eqtri 2824 . . . . . . 7 𝐻 = (𝑧 ∈ (0...𝑀) ↦ (𝑦𝑋 ↦ ((𝑦𝑧)↑if(𝑧 = 0, (𝑃 − 1), 𝑃))))
19 simpr 488 . . . . . . 7 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ (0...𝑀))
20 etransclem27.cf . . . . . . . . . 10 (𝜑𝐶:dom 𝐶⟶(ℕ0m (0...𝑀)))
2120ffvelrnda 6832 . . . . . . . . 9 ((𝜑𝑙 ∈ dom 𝐶) → (𝐶𝑙) ∈ (ℕ0m (0...𝑀)))
22 elmapi 8415 . . . . . . . . 9 ((𝐶𝑙) ∈ (ℕ0m (0...𝑀)) → (𝐶𝑙):(0...𝑀)⟶ℕ0)
2321, 22syl 17 . . . . . . . 8 ((𝜑𝑙 ∈ dom 𝐶) → (𝐶𝑙):(0...𝑀)⟶ℕ0)
2423ffvelrnda 6832 . . . . . . 7 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → ((𝐶𝑙)‘𝑗) ∈ ℕ0)
2511, 13, 15, 18, 19, 24etransclem20 42883 . . . . . 6 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗)):𝑋⟶ℂ)
265ad2antrr 725 . . . . . 6 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 𝐽𝑋)
2725, 26ffvelrnd 6833 . . . . 5 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽) ∈ ℂ)
289, 27fprodcl 15301 . . . 4 ((𝜑𝑙 ∈ dom 𝐶) → ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽) ∈ ℂ)
298, 28fsumcl 15085 . . 3 (𝜑 → Σ𝑙 ∈ dom 𝐶𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽) ∈ ℂ)
301, 4, 5, 29fvmptd3 6772 . 2 (𝜑 → (𝐺𝐽) = Σ𝑙 ∈ dom 𝐶𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽))
3111, 13, 15, 18, 19, 24, 26etransclem21 42884 . . . . 5 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽) = if(if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗), 0, (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)))) · ((𝐽𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗))))))
32 iftrue 4434 . . . . . . . 8 (if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗) → if(if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗), 0, (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)))) · ((𝐽𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗))))) = 0)
33 0zd 11985 . . . . . . . 8 (if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗) → 0 ∈ ℤ)
3432, 33eqeltrd 2893 . . . . . . 7 (if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗) → if(if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗), 0, (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)))) · ((𝐽𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗))))) ∈ ℤ)
3534adantl 485 . . . . . 6 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → if(if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗), 0, (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)))) · ((𝐽𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗))))) ∈ ℤ)
36 0zd 11985 . . . . . . 7 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → 0 ∈ ℤ)
37 nnm1nn0 11930 . . . . . . . . . . . . . . . . 17 (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ0)
3814, 37syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑃 − 1) ∈ ℕ0)
3914nnnn0d 11947 . . . . . . . . . . . . . . . 16 (𝜑𝑃 ∈ ℕ0)
4038, 39ifcld 4473 . . . . . . . . . . . . . . 15 (𝜑 → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℕ0)
4140nn0zd 12077 . . . . . . . . . . . . . 14 (𝜑 → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℤ)
4241ad3antrrr 729 . . . . . . . . . . . . 13 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℤ)
4324nn0zd 12077 . . . . . . . . . . . . . . 15 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → ((𝐶𝑙)‘𝑗) ∈ ℤ)
4443adantr 484 . . . . . . . . . . . . . 14 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → ((𝐶𝑙)‘𝑗) ∈ ℤ)
4542, 44zsubcld 12084 . . . . . . . . . . . . 13 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ∈ ℤ)
4636, 42, 453jca 1125 . . . . . . . . . . . 12 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → (0 ∈ ℤ ∧ if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℤ ∧ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ∈ ℤ))
4744zred 12079 . . . . . . . . . . . . . 14 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → ((𝐶𝑙)‘𝑗) ∈ ℝ)
4842zred 12079 . . . . . . . . . . . . . 14 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
49 simpr 488 . . . . . . . . . . . . . 14 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗))
5047, 48, 49nltled 10783 . . . . . . . . . . . . 13 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → ((𝐶𝑙)‘𝑗) ≤ if(𝑗 = 0, (𝑃 − 1), 𝑃))
5148, 47subge0d 11223 . . . . . . . . . . . . 13 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → (0 ≤ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ↔ ((𝐶𝑙)‘𝑗) ≤ if(𝑗 = 0, (𝑃 − 1), 𝑃)))
5250, 51mpbird 260 . . . . . . . . . . . 12 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → 0 ≤ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)))
53 0red 10637 . . . . . . . . . . . . . . 15 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 0 ∈ ℝ)
5424nn0red 11948 . . . . . . . . . . . . . . 15 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → ((𝐶𝑙)‘𝑗) ∈ ℝ)
5540nn0red 11948 . . . . . . . . . . . . . . . 16 (𝜑 → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
5655ad2antrr 725 . . . . . . . . . . . . . . 15 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
5724nn0ge0d 11950 . . . . . . . . . . . . . . 15 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 0 ≤ ((𝐶𝑙)‘𝑗))
5853, 54, 56, 57lesub2dd 11250 . . . . . . . . . . . . . 14 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ≤ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − 0))
5956recnd 10662 . . . . . . . . . . . . . . 15 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℂ)
6059subid1d 10979 . . . . . . . . . . . . . 14 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − 0) = if(𝑗 = 0, (𝑃 − 1), 𝑃))
6158, 60breqtrd 5059 . . . . . . . . . . . . 13 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ≤ if(𝑗 = 0, (𝑃 − 1), 𝑃))
6261adantr 484 . . . . . . . . . . . 12 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ≤ if(𝑗 = 0, (𝑃 − 1), 𝑃))
6346, 52, 62jca32 519 . . . . . . . . . . 11 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → ((0 ∈ ℤ ∧ if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℤ ∧ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ∈ ℤ) ∧ (0 ≤ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ∧ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ≤ if(𝑗 = 0, (𝑃 − 1), 𝑃))))
64 elfz2 12896 . . . . . . . . . . 11 ((if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ∈ (0...if(𝑗 = 0, (𝑃 − 1), 𝑃)) ↔ ((0 ∈ ℤ ∧ if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℤ ∧ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ∈ ℤ) ∧ (0 ≤ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ∧ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ≤ if(𝑗 = 0, (𝑃 − 1), 𝑃))))
6563, 64sylibr 237 . . . . . . . . . 10 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ∈ (0...if(𝑗 = 0, (𝑃 − 1), 𝑃)))
66 permnn 13686 . . . . . . . . . 10 ((if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ∈ (0...if(𝑗 = 0, (𝑃 − 1), 𝑃)) → ((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)))) ∈ ℕ)
6765, 66syl 17 . . . . . . . . 9 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → ((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)))) ∈ ℕ)
6867nnzd 12078 . . . . . . . 8 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → ((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)))) ∈ ℤ)
69 etransclem27.jz . . . . . . . . . . 11 (𝜑𝐽 ∈ ℤ)
7069ad3antrrr 729 . . . . . . . . . 10 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → 𝐽 ∈ ℤ)
71 elfzelz 12906 . . . . . . . . . . 11 (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℤ)
7271ad2antlr 726 . . . . . . . . . 10 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → 𝑗 ∈ ℤ)
7370, 72zsubcld 12084 . . . . . . . . 9 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → (𝐽𝑗) ∈ ℤ)
74 elnn0z 11986 . . . . . . . . . 10 ((if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ∈ ℕ0 ↔ ((if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ∈ ℤ ∧ 0 ≤ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗))))
7545, 52, 74sylanbrc 586 . . . . . . . . 9 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ∈ ℕ0)
76 zexpcl 13444 . . . . . . . . 9 (((𝐽𝑗) ∈ ℤ ∧ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ∈ ℕ0) → ((𝐽𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗))) ∈ ℤ)
7773, 75, 76syl2anc 587 . . . . . . . 8 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → ((𝐽𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗))) ∈ ℤ)
7868, 77zmulcld 12085 . . . . . . 7 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)))) · ((𝐽𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)))) ∈ ℤ)
7936, 78ifcld 4473 . . . . . 6 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → if(if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗), 0, (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)))) · ((𝐽𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗))))) ∈ ℤ)
8035, 79pm2.61dan 812 . . . . 5 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → if(if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗), 0, (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)))) · ((𝐽𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗))))) ∈ ℤ)
8131, 80eqeltrd 2893 . . . 4 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽) ∈ ℤ)
829, 81fprodzcl 15303 . . 3 ((𝜑𝑙 ∈ dom 𝐶) → ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽) ∈ ℤ)
838, 82fsumzcl 15087 . 2 (𝜑 → Σ𝑙 ∈ dom 𝐶𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽) ∈ ℤ)
8430, 83eqeltrd 2893 1 (𝜑 → (𝐺𝐽) ∈ ℤ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2112  ifcif 4428  {cpr 4530   class class class wbr 5033   ↦ cmpt 5113  dom cdm 5523  ⟶wf 6324  ‘cfv 6328  (class class class)co 7139   ↑m cmap 8393  Fincfn 8496  ℂcc 10528  ℝcr 10529  0cc0 10530  1c1 10531   · cmul 10535   < clt 10668   ≤ cle 10669   − cmin 10863   / cdiv 11290  ℕcn 11629  ℕ0cn0 11889  ℤcz 11973  ...cfz 12889  ↑cexp 13429  !cfa 13633  Σcsu 15037  ∏cprod 15254   ↾t crest 16689  TopOpenctopn 16690  ℂfldccnfld 20094   D𝑛 cdvn 24470 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-inf2 9092  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-pre-sup 10608  ax-addf 10609  ax-mulf 10610 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-of 7393  df-om 7565  df-1st 7675  df-2nd 7676  df-supp 7818  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-er 8276  df-map 8395  df-pm 8396  df-ixp 8449  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fsupp 8822  df-fi 8863  df-sup 8894  df-inf 8895  df-oi 8962  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-nn 11630  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-q 12341  df-rp 12382  df-xneg 12499  df-xadd 12500  df-xmul 12501  df-icc 12737  df-fz 12890  df-fzo 13033  df-seq 13369  df-exp 13430  df-fac 13634  df-bc 13663  df-hash 13691  df-cj 14453  df-re 14454  df-im 14455  df-sqrt 14589  df-abs 14590  df-clim 14840  df-sum 15038  df-prod 15255  df-struct 16480  df-ndx 16481  df-slot 16482  df-base 16484  df-sets 16485  df-ress 16486  df-plusg 16573  df-mulr 16574  df-starv 16575  df-sca 16576  df-vsca 16577  df-ip 16578  df-tset 16579  df-ple 16580  df-ds 16582  df-unif 16583  df-hom 16584  df-cco 16585  df-rest 16691  df-topn 16692  df-0g 16710  df-gsum 16711  df-topgen 16712  df-pt 16713  df-prds 16716  df-xrs 16770  df-qtop 16775  df-imas 16776  df-xps 16778  df-mre 16852  df-mrc 16853  df-acs 16855  df-mgm 17847  df-sgrp 17896  df-mnd 17907  df-submnd 17952  df-mulg 18220  df-cntz 18442  df-cmn 18903  df-psmet 20086  df-xmet 20087  df-met 20088  df-bl 20089  df-mopn 20090  df-fbas 20091  df-fg 20092  df-cnfld 20095  df-top 21502  df-topon 21519  df-topsp 21541  df-bases 21554  df-cld 21627  df-ntr 21628  df-cls 21629  df-nei 21706  df-lp 21744  df-perf 21745  df-cn 21835  df-cnp 21836  df-haus 21923  df-tx 22170  df-hmeo 22363  df-fil 22454  df-fm 22546  df-flim 22547  df-flf 22548  df-xms 22930  df-ms 22931  df-tms 22932  df-cncf 23486  df-limc 24472  df-dv 24473  df-dvn 24474 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator