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Theorem etransclem27 46217
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 6907 . . . . 5 (𝑥 = 𝐽 → (((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝑥) = (((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽))
32prodeq2ad 45548 . . . 4 (𝑥 = 𝐽 → ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝑥) = ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽))
43sumeq2sdv 15736 . . 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 9373 . . . . 5 (𝐶 ∈ Fin → dom 𝐶 ∈ Fin)
86, 7syl 17 . . . 4 (𝜑 → dom 𝐶 ∈ Fin)
9 fzfid 14011 . . . . 5 ((𝜑𝑙 ∈ dom 𝐶) → (0...𝑀) ∈ Fin)
10 etransclem27.s . . . . . . . 8 (𝜑𝑆 ∈ {ℝ, ℂ})
1110ad2antrr 726 . . . . . . 7 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 𝑆 ∈ {ℝ, ℂ})
12 etransclem27.x . . . . . . . 8 (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
1312ad2antrr 726 . . . . . . 7 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
14 etransclem27.p . . . . . . . 8 (𝜑𝑃 ∈ ℕ)
1514ad2antrr 726 . . . . . . 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 46195 . . . . . . . 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 2763 . . . . . . 7 𝐻 = (𝑧 ∈ (0...𝑀) ↦ (𝑦𝑋 ↦ ((𝑦𝑧)↑if(𝑧 = 0, (𝑃 − 1), 𝑃))))
19 simpr 484 . . . . . . 7 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ (0...𝑀))
20 etransclem27.cf . . . . . . . . . 10 (𝜑𝐶:dom 𝐶⟶(ℕ0m (0...𝑀)))
2120ffvelcdmda 7104 . . . . . . . . 9 ((𝜑𝑙 ∈ dom 𝐶) → (𝐶𝑙) ∈ (ℕ0m (0...𝑀)))
22 elmapi 8888 . . . . . . . . 9 ((𝐶𝑙) ∈ (ℕ0m (0...𝑀)) → (𝐶𝑙):(0...𝑀)⟶ℕ0)
2321, 22syl 17 . . . . . . . 8 ((𝜑𝑙 ∈ dom 𝐶) → (𝐶𝑙):(0...𝑀)⟶ℕ0)
2423ffvelcdmda 7104 . . . . . . 7 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → ((𝐶𝑙)‘𝑗) ∈ ℕ0)
2511, 13, 15, 18, 19, 24etransclem20 46210 . . . . . 6 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗)):𝑋⟶ℂ)
265ad2antrr 726 . . . . . 6 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 𝐽𝑋)
2725, 26ffvelcdmd 7105 . . . . 5 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽) ∈ ℂ)
289, 27fprodcl 15985 . . . 4 ((𝜑𝑙 ∈ dom 𝐶) → ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽) ∈ ℂ)
298, 28fsumcl 15766 . . 3 (𝜑 → Σ𝑙 ∈ dom 𝐶𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽) ∈ ℂ)
301, 4, 5, 29fvmptd3 7039 . 2 (𝜑 → (𝐺𝐽) = Σ𝑙 ∈ dom 𝐶𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽))
3111, 13, 15, 18, 19, 24, 26etransclem21 46211 . . . . 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 4537 . . . . . . . 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 12623 . . . . . . . 8 (if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗) → 0 ∈ ℤ)
3432, 33eqeltrd 2839 . . . . . . 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 481 . . . . . 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 12623 . . . . . . 7 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → 0 ∈ ℤ)
37 nnm1nn0 12565 . . . . . . . . . . . . . . 15 (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ0)
3814, 37syl 17 . . . . . . . . . . . . . 14 (𝜑 → (𝑃 − 1) ∈ ℕ0)
3914nnnn0d 12585 . . . . . . . . . . . . . 14 (𝜑𝑃 ∈ ℕ0)
4038, 39ifcld 4577 . . . . . . . . . . . . 13 (𝜑 → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℕ0)
4140nn0zd 12637 . . . . . . . . . . . 12 (𝜑 → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℤ)
4241ad3antrrr 730 . . . . . . . . . . 11 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℤ)
4324nn0zd 12637 . . . . . . . . . . . . 13 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → ((𝐶𝑙)‘𝑗) ∈ ℤ)
4443adantr 480 . . . . . . . . . . . 12 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → ((𝐶𝑙)‘𝑗) ∈ ℤ)
4542, 44zsubcld 12725 . . . . . . . . . . 11 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ∈ ℤ)
4644zred 12720 . . . . . . . . . . . . 13 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → ((𝐶𝑙)‘𝑗) ∈ ℝ)
4742zred 12720 . . . . . . . . . . . . 13 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
48 simpr 484 . . . . . . . . . . . . 13 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗))
4946, 47, 48nltled 11409 . . . . . . . . . . . 12 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → ((𝐶𝑙)‘𝑗) ≤ if(𝑗 = 0, (𝑃 − 1), 𝑃))
5047, 46subge0d 11851 . . . . . . . . . . . 12 ((((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)))
5149, 50mpbird 257 . . . . . . . . . . 11 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → 0 ≤ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)))
52 0red 11262 . . . . . . . . . . . . . 14 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 0 ∈ ℝ)
5324nn0red 12586 . . . . . . . . . . . . . 14 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → ((𝐶𝑙)‘𝑗) ∈ ℝ)
5440nn0red 12586 . . . . . . . . . . . . . . 15 (𝜑 → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
5554ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
5624nn0ge0d 12588 . . . . . . . . . . . . . 14 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 0 ≤ ((𝐶𝑙)‘𝑗))
5752, 53, 55, 56lesub2dd 11878 . . . . . . . . . . . . 13 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ≤ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − 0))
5855recnd 11287 . . . . . . . . . . . . . 14 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℂ)
5958subid1d 11607 . . . . . . . . . . . . 13 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − 0) = if(𝑗 = 0, (𝑃 − 1), 𝑃))
6057, 59breqtrd 5174 . . . . . . . . . . . 12 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ≤ if(𝑗 = 0, (𝑃 − 1), 𝑃))
6160adantr 480 . . . . . . . . . . 11 ((((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))
6236, 42, 45, 51, 61elfzd 13552 . . . . . . . . . 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)))
63 permnn 14362 . . . . . . . . . 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)
6462, 63syl 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)
6564nnzd 12638 . . . . . . . 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)
66 etransclem27.jz . . . . . . . . . . 11 (𝜑𝐽 ∈ ℤ)
6766ad3antrrr 730 . . . . . . . . . 10 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → 𝐽 ∈ ℤ)
68 elfzelz 13561 . . . . . . . . . . 11 (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℤ)
6968ad2antlr 727 . . . . . . . . . 10 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → 𝑗 ∈ ℤ)
7067, 69zsubcld 12725 . . . . . . . . 9 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → (𝐽𝑗) ∈ ℤ)
71 elnn0z 12624 . . . . . . . . . 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))))
7245, 51, 71sylanbrc 583 . . . . . . . . 9 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ∈ ℕ0)
73 zexpcl 14114 . . . . . . . . 9 (((𝐽𝑗) ∈ ℤ ∧ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ∈ ℕ0) → ((𝐽𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗))) ∈ ℤ)
7470, 72, 73syl2anc 584 . . . . . . . 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)
7565, 74zmulcld 12726 . . . . . . 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)
7636, 75ifcld 4577 . . . . . 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)
7735, 76pm2.61dan 813 . . . . 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)
7831, 77eqeltrd 2839 . . . 4 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽) ∈ ℤ)
799, 78fprodzcl 15987 . . 3 ((𝜑𝑙 ∈ dom 𝐶) → ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽) ∈ ℤ)
808, 79fsumzcl 15768 . 2 (𝜑 → Σ𝑙 ∈ dom 𝐶𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽) ∈ ℤ)
8130, 80eqeltrd 2839 1 (𝜑 → (𝐺𝐽) ∈ ℤ)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1537  wcel 2106  ifcif 4531  {cpr 4633   class class class wbr 5148  cmpt 5231  dom cdm 5689  wf 6559  cfv 6563  (class class class)co 7431  m cmap 8865  Fincfn 8984  cc 11151  cr 11152  0cc0 11153  1c1 11154   · cmul 11158   < clt 11293  cle 11294  cmin 11490   / cdiv 11918  cn 12264  0cn0 12524  cz 12611  ...cfz 13544  cexp 14099  !cfa 14309  Σcsu 15719  cprod 15936  t crest 17467  TopOpenctopn 17468  fldccnfld 21382   D𝑛 cdvn 25914
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231  ax-addf 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-pm 8868  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-fi 9449  df-sup 9480  df-inf 9481  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-q 12989  df-rp 13033  df-xneg 13152  df-xadd 13153  df-xmul 13154  df-icc 13391  df-fz 13545  df-fzo 13692  df-seq 14040  df-exp 14100  df-fac 14310  df-bc 14339  df-hash 14367  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-clim 15521  df-sum 15720  df-prod 15937  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-starv 17313  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-unif 17321  df-hom 17322  df-cco 17323  df-rest 17469  df-topn 17470  df-0g 17488  df-gsum 17489  df-topgen 17490  df-pt 17491  df-prds 17494  df-xrs 17549  df-qtop 17554  df-imas 17555  df-xps 17557  df-mre 17631  df-mrc 17632  df-acs 17634  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-submnd 18810  df-mulg 19099  df-cntz 19348  df-cmn 19815  df-psmet 21374  df-xmet 21375  df-met 21376  df-bl 21377  df-mopn 21378  df-fbas 21379  df-fg 21380  df-cnfld 21383  df-top 22916  df-topon 22933  df-topsp 22955  df-bases 22969  df-cld 23043  df-ntr 23044  df-cls 23045  df-nei 23122  df-lp 23160  df-perf 23161  df-cn 23251  df-cnp 23252  df-haus 23339  df-tx 23586  df-hmeo 23779  df-fil 23870  df-fm 23962  df-flim 23963  df-flf 23964  df-xms 24346  df-ms 24347  df-tms 24348  df-cncf 24918  df-limc 25916  df-dv 25917  df-dvn 25918
This theorem is referenced by: (None)
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