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Theorem etransclem27 40990
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 . . . 4 𝐺 = (𝑥𝑋 ↦ Σ𝑙 ∈ dom 𝐶𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝑥))
21a1i 11 . . 3 (𝜑𝐺 = (𝑥𝑋 ↦ Σ𝑙 ∈ dom 𝐶𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝑥)))
3 fveq2 6333 . . . . . 6 (𝑥 = 𝐽 → (((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝑥) = (((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽))
43prodeq2ad 40337 . . . . 5 (𝑥 = 𝐽 → ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝑥) = ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽))
54sumeq2ad 14642 . . . 4 (𝑥 = 𝐽 → Σ𝑙 ∈ dom 𝐶𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝑥) = Σ𝑙 ∈ dom 𝐶𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽))
65adantl 467 . . 3 ((𝜑𝑥 = 𝐽) → Σ𝑙 ∈ dom 𝐶𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝑥) = Σ𝑙 ∈ dom 𝐶𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽))
7 etransclem27.jx . . 3 (𝜑𝐽𝑋)
8 etransclem27.cfi . . . . 5 (𝜑𝐶 ∈ Fin)
9 dmfi 8404 . . . . 5 (𝐶 ∈ Fin → dom 𝐶 ∈ Fin)
108, 9syl 17 . . . 4 (𝜑 → dom 𝐶 ∈ Fin)
11 fzfid 12980 . . . . 5 ((𝜑𝑙 ∈ dom 𝐶) → (0...𝑀) ∈ Fin)
12 etransclem27.s . . . . . . . 8 (𝜑𝑆 ∈ {ℝ, ℂ})
1312ad2antrr 705 . . . . . . 7 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 𝑆 ∈ {ℝ, ℂ})
14 etransclem27.x . . . . . . . 8 (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
1514ad2antrr 705 . . . . . . 7 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
16 etransclem27.p . . . . . . . 8 (𝜑𝑃 ∈ ℕ)
1716ad2antrr 705 . . . . . . 7 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 𝑃 ∈ ℕ)
18 etransclem27.h . . . . . . . 8 𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))
19 etransclem5 40968 . . . . . . . 8 (𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))) = (𝑧 ∈ (0...𝑀) ↦ (𝑦𝑋 ↦ ((𝑦𝑧)↑if(𝑧 = 0, (𝑃 − 1), 𝑃))))
2018, 19eqtri 2793 . . . . . . 7 𝐻 = (𝑧 ∈ (0...𝑀) ↦ (𝑦𝑋 ↦ ((𝑦𝑧)↑if(𝑧 = 0, (𝑃 − 1), 𝑃))))
21 simpr 471 . . . . . . 7 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ (0...𝑀))
22 etransclem27.cf . . . . . . . . . 10 (𝜑𝐶:dom 𝐶⟶(ℕ0𝑚 (0...𝑀)))
2322ffvelrnda 6504 . . . . . . . . 9 ((𝜑𝑙 ∈ dom 𝐶) → (𝐶𝑙) ∈ (ℕ0𝑚 (0...𝑀)))
24 elmapi 8035 . . . . . . . . 9 ((𝐶𝑙) ∈ (ℕ0𝑚 (0...𝑀)) → (𝐶𝑙):(0...𝑀)⟶ℕ0)
2523, 24syl 17 . . . . . . . 8 ((𝜑𝑙 ∈ dom 𝐶) → (𝐶𝑙):(0...𝑀)⟶ℕ0)
2625ffvelrnda 6504 . . . . . . 7 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → ((𝐶𝑙)‘𝑗) ∈ ℕ0)
2713, 15, 17, 20, 21, 26etransclem20 40983 . . . . . 6 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗)):𝑋⟶ℂ)
287ad2antrr 705 . . . . . 6 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 𝐽𝑋)
2927, 28ffvelrnd 6505 . . . . 5 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽) ∈ ℂ)
3011, 29fprodcl 14889 . . . 4 ((𝜑𝑙 ∈ dom 𝐶) → ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽) ∈ ℂ)
3110, 30fsumcl 14672 . . 3 (𝜑 → Σ𝑙 ∈ dom 𝐶𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽) ∈ ℂ)
322, 6, 7, 31fvmptd 6432 . 2 (𝜑 → (𝐺𝐽) = Σ𝑙 ∈ dom 𝐶𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽))
3313, 15, 17, 20, 21, 26, 28etransclem21 40984 . . . . 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))))))
34 iftrue 4232 . . . . . . . 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)
35 0zd 11596 . . . . . . . 8 (if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗) → 0 ∈ ℤ)
3634, 35eqeltrd 2850 . . . . . . 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)
3736adantl 467 . . . . . 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)
38 0zd 11596 . . . . . . 7 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → 0 ∈ ℤ)
39 nnm1nn0 11541 . . . . . . . . . . . . . . . . 17 (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ0)
4016, 39syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑃 − 1) ∈ ℕ0)
4116nnnn0d 11558 . . . . . . . . . . . . . . . 16 (𝜑𝑃 ∈ ℕ0)
4240, 41ifcld 4271 . . . . . . . . . . . . . . 15 (𝜑 → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℕ0)
4342nn0zd 11687 . . . . . . . . . . . . . 14 (𝜑 → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℤ)
4443ad3antrrr 709 . . . . . . . . . . . . 13 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℤ)
4526nn0zd 11687 . . . . . . . . . . . . . . 15 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → ((𝐶𝑙)‘𝑗) ∈ ℤ)
4645adantr 466 . . . . . . . . . . . . . 14 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → ((𝐶𝑙)‘𝑗) ∈ ℤ)
4744, 46zsubcld 11694 . . . . . . . . . . . . 13 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ∈ ℤ)
4838, 44, 473jca 1122 . . . . . . . . . . . 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))
4946zred 11689 . . . . . . . . . . . . . 14 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → ((𝐶𝑙)‘𝑗) ∈ ℝ)
5044zred 11689 . . . . . . . . . . . . . 14 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
51 simpr 471 . . . . . . . . . . . . . 14 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗))
5249, 50, 51nltled 10393 . . . . . . . . . . . . 13 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → ((𝐶𝑙)‘𝑗) ≤ if(𝑗 = 0, (𝑃 − 1), 𝑃))
5350, 49subge0d 10823 . . . . . . . . . . . . 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)))
5452, 53mpbird 247 . . . . . . . . . . . 12 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → 0 ≤ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)))
55 0red 10247 . . . . . . . . . . . . . . 15 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 0 ∈ ℝ)
5626nn0red 11559 . . . . . . . . . . . . . . 15 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → ((𝐶𝑙)‘𝑗) ∈ ℝ)
5742nn0red 11559 . . . . . . . . . . . . . . . 16 (𝜑 → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
5857ad2antrr 705 . . . . . . . . . . . . . . 15 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
5926nn0ge0d 11561 . . . . . . . . . . . . . . 15 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 0 ≤ ((𝐶𝑙)‘𝑗))
6055, 56, 58, 59lesub2dd 10850 . . . . . . . . . . . . . 14 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ≤ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − 0))
6158recnd 10274 . . . . . . . . . . . . . . 15 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℂ)
6261subid1d 10587 . . . . . . . . . . . . . 14 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − 0) = if(𝑗 = 0, (𝑃 − 1), 𝑃))
6360, 62breqtrd 4813 . . . . . . . . . . . . 13 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ≤ if(𝑗 = 0, (𝑃 − 1), 𝑃))
6463adantr 466 . . . . . . . . . . . 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))
6548, 54, 64jca32 505 . . . . . . . . . . 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))))
66 elfz2 12540 . . . . . . . . . . 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))))
6765, 66sylibr 224 . . . . . . . . . 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)))
68 permnn 13317 . . . . . . . . . 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)
6967, 68syl 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)
7069nnzd 11688 . . . . . . . 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)
71 etransclem27.jz . . . . . . . . . . 11 (𝜑𝐽 ∈ ℤ)
7271ad3antrrr 709 . . . . . . . . . 10 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → 𝐽 ∈ ℤ)
73 elfzelz 12549 . . . . . . . . . . 11 (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℤ)
7473ad2antlr 706 . . . . . . . . . 10 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → 𝑗 ∈ ℤ)
7572, 74zsubcld 11694 . . . . . . . . 9 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → (𝐽𝑗) ∈ ℤ)
76 elnn0z 11597 . . . . . . . . . 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))))
7747, 54, 76sylanbrc 572 . . . . . . . . 9 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ∈ ℕ0)
78 zexpcl 13082 . . . . . . . . 9 (((𝐽𝑗) ∈ ℤ ∧ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ∈ ℕ0) → ((𝐽𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗))) ∈ ℤ)
7975, 77, 78syl2anc 573 . . . . . . . 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)
8070, 79zmulcld 11695 . . . . . . 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)
8138, 80ifcld 4271 . . . . . 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)
8237, 81pm2.61dan 814 . . . . 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)
8333, 82eqeltrd 2850 . . . 4 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽) ∈ ℤ)
8411, 83fprodzcl 14891 . . 3 ((𝜑𝑙 ∈ dom 𝐶) → ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽) ∈ ℤ)
8510, 84fsumzcl 14674 . 2 (𝜑 → Σ𝑙 ∈ dom 𝐶𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽) ∈ ℤ)
8632, 85eqeltrd 2850 1 (𝜑 → (𝐺𝐽) ∈ ℤ)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  ifcif 4226  {cpr 4319   class class class wbr 4787  cmpt 4864  dom cdm 5250  wf 6026  cfv 6030  (class class class)co 6796  𝑚 cmap 8013  Fincfn 8113  cc 10140  cr 10141  0cc0 10142  1c1 10143   · cmul 10147   < clt 10280  cle 10281  cmin 10472   / cdiv 10890  cn 11226  0cn0 11499  cz 11584  ...cfz 12533  cexp 13067  !cfa 13264  Σcsu 14624  cprod 14842  t crest 16289  TopOpenctopn 16290  fldccnfld 19961   D𝑛 cdvn 23848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-inf2 8706  ax-cnex 10198  ax-resscn 10199  ax-1cn 10200  ax-icn 10201  ax-addcl 10202  ax-addrcl 10203  ax-mulcl 10204  ax-mulrcl 10205  ax-mulcom 10206  ax-addass 10207  ax-mulass 10208  ax-distr 10209  ax-i2m1 10210  ax-1ne0 10211  ax-1rid 10212  ax-rnegex 10213  ax-rrecex 10214  ax-cnre 10215  ax-pre-lttri 10216  ax-pre-lttrn 10217  ax-pre-ltadd 10218  ax-pre-mulgt0 10219  ax-pre-sup 10220  ax-addf 10221  ax-mulf 10222
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-iin 4658  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-se 5210  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-isom 6039  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-of 7048  df-om 7217  df-1st 7319  df-2nd 7320  df-supp 7451  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-1o 7717  df-2o 7718  df-oadd 7721  df-er 7900  df-map 8015  df-pm 8016  df-ixp 8067  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-fsupp 8436  df-fi 8477  df-sup 8508  df-inf 8509  df-oi 8575  df-card 8969  df-cda 9196  df-pnf 10282  df-mnf 10283  df-xr 10284  df-ltxr 10285  df-le 10286  df-sub 10474  df-neg 10475  df-div 10891  df-nn 11227  df-2 11285  df-3 11286  df-4 11287  df-5 11288  df-6 11289  df-7 11290  df-8 11291  df-9 11292  df-n0 11500  df-z 11585  df-dec 11701  df-uz 11894  df-q 11997  df-rp 12036  df-xneg 12151  df-xadd 12152  df-xmul 12153  df-icc 12387  df-fz 12534  df-fzo 12674  df-seq 13009  df-exp 13068  df-fac 13265  df-bc 13294  df-hash 13322  df-cj 14047  df-re 14048  df-im 14049  df-sqrt 14183  df-abs 14184  df-clim 14427  df-sum 14625  df-prod 14843  df-struct 16066  df-ndx 16067  df-slot 16068  df-base 16070  df-sets 16071  df-ress 16072  df-plusg 16162  df-mulr 16163  df-starv 16164  df-sca 16165  df-vsca 16166  df-ip 16167  df-tset 16168  df-ple 16169  df-ds 16172  df-unif 16173  df-hom 16174  df-cco 16175  df-rest 16291  df-topn 16292  df-0g 16310  df-gsum 16311  df-topgen 16312  df-pt 16313  df-prds 16316  df-xrs 16370  df-qtop 16375  df-imas 16376  df-xps 16378  df-mre 16454  df-mrc 16455  df-acs 16457  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-submnd 17544  df-mulg 17749  df-cntz 17957  df-cmn 18402  df-psmet 19953  df-xmet 19954  df-met 19955  df-bl 19956  df-mopn 19957  df-fbas 19958  df-fg 19959  df-cnfld 19962  df-top 20919  df-topon 20936  df-topsp 20958  df-bases 20971  df-cld 21044  df-ntr 21045  df-cls 21046  df-nei 21123  df-lp 21161  df-perf 21162  df-cn 21252  df-cnp 21253  df-haus 21340  df-tx 21586  df-hmeo 21779  df-fil 21870  df-fm 21962  df-flim 21963  df-flf 21964  df-xms 22345  df-ms 22346  df-tms 22347  df-cncf 22901  df-limc 23850  df-dv 23851  df-dvn 23852
This theorem is referenced by: (None)
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