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Theorem etransclem27 40981
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 6352 . . . . . 6 (𝑥 = 𝐽 → (((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝑥) = (((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽))
43prodeq2ad 40327 . . . . 5 (𝑥 = 𝐽 → ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝑥) = ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽))
54sumeq2ad 14633 . . . 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 473 . . 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 8409 . . . . 5 (𝐶 ∈ Fin → dom 𝐶 ∈ Fin)
108, 9syl 17 . . . 4 (𝜑 → dom 𝐶 ∈ Fin)
11 fzfid 12966 . . . . 5 ((𝜑𝑙 ∈ dom 𝐶) → (0...𝑀) ∈ Fin)
12 etransclem27.s . . . . . . . 8 (𝜑𝑆 ∈ {ℝ, ℂ})
1312ad2antrr 764 . . . . . . 7 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 𝑆 ∈ {ℝ, ℂ})
14 etransclem27.x . . . . . . . 8 (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
1514ad2antrr 764 . . . . . . 7 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
16 etransclem27.p . . . . . . . 8 (𝜑𝑃 ∈ ℕ)
1716ad2antrr 764 . . . . . . 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 40959 . . . . . . . 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 2782 . . . . . . 7 𝐻 = (𝑧 ∈ (0...𝑀) ↦ (𝑦𝑋 ↦ ((𝑦𝑧)↑if(𝑧 = 0, (𝑃 − 1), 𝑃))))
21 simpr 479 . . . . . . 7 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ (0...𝑀))
22 etransclem27.cf . . . . . . . . . 10 (𝜑𝐶:dom 𝐶⟶(ℕ0𝑚 (0...𝑀)))
2322ffvelrnda 6522 . . . . . . . . 9 ((𝜑𝑙 ∈ dom 𝐶) → (𝐶𝑙) ∈ (ℕ0𝑚 (0...𝑀)))
24 elmapi 8045 . . . . . . . . 9 ((𝐶𝑙) ∈ (ℕ0𝑚 (0...𝑀)) → (𝐶𝑙):(0...𝑀)⟶ℕ0)
2523, 24syl 17 . . . . . . . 8 ((𝜑𝑙 ∈ dom 𝐶) → (𝐶𝑙):(0...𝑀)⟶ℕ0)
2625ffvelrnda 6522 . . . . . . 7 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → ((𝐶𝑙)‘𝑗) ∈ ℕ0)
2713, 15, 17, 20, 21, 26etransclem20 40974 . . . . . 6 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗)):𝑋⟶ℂ)
287ad2antrr 764 . . . . . 6 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 𝐽𝑋)
2927, 28ffvelrnd 6523 . . . . 5 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽) ∈ ℂ)
3011, 29fprodcl 14881 . . . 4 ((𝜑𝑙 ∈ dom 𝐶) → ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽) ∈ ℂ)
3110, 30fsumcl 14663 . . 3 (𝜑 → Σ𝑙 ∈ dom 𝐶𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽) ∈ ℂ)
322, 6, 7, 31fvmptd 6450 . 2 (𝜑 → (𝐺𝐽) = Σ𝑙 ∈ dom 𝐶𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽))
3313, 15, 17, 20, 21, 26, 28etransclem21 40975 . . . . 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 4236 . . . . . . . 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 11581 . . . . . . . 8 (if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗) → 0 ∈ ℤ)
3634, 35eqeltrd 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)
3736adantl 473 . . . . . 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 11581 . . . . . . 7 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → 0 ∈ ℤ)
39 nnm1nn0 11526 . . . . . . . . . . . . . . . . 17 (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ0)
4016, 39syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑃 − 1) ∈ ℕ0)
4116nnnn0d 11543 . . . . . . . . . . . . . . . 16 (𝜑𝑃 ∈ ℕ0)
4240, 41ifcld 4275 . . . . . . . . . . . . . . 15 (𝜑 → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℕ0)
4342nn0zd 11672 . . . . . . . . . . . . . 14 (𝜑 → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℤ)
4443ad3antrrr 768 . . . . . . . . . . . . 13 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℤ)
4526nn0zd 11672 . . . . . . . . . . . . . . 15 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → ((𝐶𝑙)‘𝑗) ∈ ℤ)
4645adantr 472 . . . . . . . . . . . . . 14 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → ((𝐶𝑙)‘𝑗) ∈ ℤ)
4744, 46zsubcld 11679 . . . . . . . . . . . . 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 1123 . . . . . . . . . . . 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 11674 . . . . . . . . . . . . . 14 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → ((𝐶𝑙)‘𝑗) ∈ ℝ)
5044zred 11674 . . . . . . . . . . . . . 14 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
51 simpr 479 . . . . . . . . . . . . . 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 10379 . . . . . . . . . . . . 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 10809 . . . . . . . . . . . . 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 10233 . . . . . . . . . . . . . . 15 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 0 ∈ ℝ)
5626nn0red 11544 . . . . . . . . . . . . . . 15 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → ((𝐶𝑙)‘𝑗) ∈ ℝ)
5742nn0red 11544 . . . . . . . . . . . . . . . 16 (𝜑 → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
5857ad2antrr 764 . . . . . . . . . . . . . . 15 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
5926nn0ge0d 11546 . . . . . . . . . . . . . . 15 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 0 ≤ ((𝐶𝑙)‘𝑗))
6055, 56, 58, 59lesub2dd 10836 . . . . . . . . . . . . . 14 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ≤ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − 0))
6158recnd 10260 . . . . . . . . . . . . . . 15 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℂ)
6261subid1d 10573 . . . . . . . . . . . . . 14 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − 0) = if(𝑗 = 0, (𝑃 − 1), 𝑃))
6360, 62breqtrd 4830 . . . . . . . . . . . . 13 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ≤ if(𝑗 = 0, (𝑃 − 1), 𝑃))
6463adantr 472 . . . . . . . . . . . 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 559 . . . . . . . . . . 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 12526 . . . . . . . . . . 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 13307 . . . . . . . . . 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 11673 . . . . . . . 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 768 . . . . . . . . . 10 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → 𝐽 ∈ ℤ)
73 elfzelz 12535 . . . . . . . . . . 11 (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℤ)
7473ad2antlr 765 . . . . . . . . . 10 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → 𝑗 ∈ ℤ)
7572, 74zsubcld 11679 . . . . . . . . 9 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → (𝐽𝑗) ∈ ℤ)
76 elnn0z 11582 . . . . . . . . . 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 701 . . . . . . . . 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 13069 . . . . . . . . 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 696 . . . . . . . 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 11680 . . . . . . 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 4275 . . . . . 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 867 . . . . 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 2839 . . . 4 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽) ∈ ℤ)
8411, 83fprodzcl 14883 . . 3 ((𝜑𝑙 ∈ dom 𝐶) → ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽) ∈ ℤ)
8510, 84fsumzcl 14665 . 2 (𝜑 → Σ𝑙 ∈ dom 𝐶𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽) ∈ ℤ)
8632, 85eqeltrd 2839 1 (𝜑 → (𝐺𝐽) ∈ ℤ)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  w3a 1072   = wceq 1632  wcel 2139  ifcif 4230  {cpr 4323   class class class wbr 4804  cmpt 4881  dom cdm 5266  wf 6045  cfv 6049  (class class class)co 6813  𝑚 cmap 8023  Fincfn 8121  cc 10126  cr 10127  0cc0 10128  1c1 10129   · cmul 10133   < clt 10266  cle 10267  cmin 10458   / cdiv 10876  cn 11212  0cn0 11484  cz 11569  ...cfz 12519  cexp 13054  !cfa 13254  Σcsu 14615  cprod 14834  t crest 16283  TopOpenctopn 16284  fldccnfld 19948   D𝑛 cdvn 23827
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205  ax-pre-sup 10206  ax-addf 10207  ax-mulf 10208
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-of 7062  df-om 7231  df-1st 7333  df-2nd 7334  df-supp 7464  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-2o 7730  df-oadd 7733  df-er 7911  df-map 8025  df-pm 8026  df-ixp 8075  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-fsupp 8441  df-fi 8482  df-sup 8513  df-inf 8514  df-oi 8580  df-card 8955  df-cda 9182  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-div 10877  df-nn 11213  df-2 11271  df-3 11272  df-4 11273  df-5 11274  df-6 11275  df-7 11276  df-8 11277  df-9 11278  df-n0 11485  df-z 11570  df-dec 11686  df-uz 11880  df-q 11982  df-rp 12026  df-xneg 12139  df-xadd 12140  df-xmul 12141  df-icc 12375  df-fz 12520  df-fzo 12660  df-seq 12996  df-exp 13055  df-fac 13255  df-bc 13284  df-hash 13312  df-cj 14038  df-re 14039  df-im 14040  df-sqrt 14174  df-abs 14175  df-clim 14418  df-sum 14616  df-prod 14835  df-struct 16061  df-ndx 16062  df-slot 16063  df-base 16065  df-sets 16066  df-ress 16067  df-plusg 16156  df-mulr 16157  df-starv 16158  df-sca 16159  df-vsca 16160  df-ip 16161  df-tset 16162  df-ple 16163  df-ds 16166  df-unif 16167  df-hom 16168  df-cco 16169  df-rest 16285  df-topn 16286  df-0g 16304  df-gsum 16305  df-topgen 16306  df-pt 16307  df-prds 16310  df-xrs 16364  df-qtop 16369  df-imas 16370  df-xps 16372  df-mre 16448  df-mrc 16449  df-acs 16451  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-submnd 17537  df-mulg 17742  df-cntz 17950  df-cmn 18395  df-psmet 19940  df-xmet 19941  df-met 19942  df-bl 19943  df-mopn 19944  df-fbas 19945  df-fg 19946  df-cnfld 19949  df-top 20901  df-topon 20918  df-topsp 20939  df-bases 20952  df-cld 21025  df-ntr 21026  df-cls 21027  df-nei 21104  df-lp 21142  df-perf 21143  df-cn 21233  df-cnp 21234  df-haus 21321  df-tx 21567  df-hmeo 21760  df-fil 21851  df-fm 21943  df-flim 21944  df-flf 21945  df-xms 22326  df-ms 22327  df-tms 22328  df-cncf 22882  df-limc 23829  df-dv 23830  df-dvn 23831
This theorem is referenced by: (None)
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