| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | etransclem27.g | . . 3
⊢ 𝐺 = (𝑥 ∈ 𝑋 ↦ Σ𝑙 ∈ dom 𝐶∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝑥)) | 
| 2 |  | fveq2 6906 | . . . . 5
⊢ (𝑥 = 𝐽 → (((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝑥) = (((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽)) | 
| 3 | 2 | prodeq2ad 45607 | . . . 4
⊢ (𝑥 = 𝐽 → ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝑥) = ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽)) | 
| 4 | 3 | sumeq2sdv 15739 | . . 3
⊢ (𝑥 = 𝐽 → Σ𝑙 ∈ dom 𝐶∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝑥) = Σ𝑙 ∈ dom 𝐶∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽)) | 
| 5 |  | etransclem27.jx | . . 3
⊢ (𝜑 → 𝐽 ∈ 𝑋) | 
| 6 |  | etransclem27.cfi | . . . . 5
⊢ (𝜑 → 𝐶 ∈ Fin) | 
| 7 |  | dmfi 9375 | . . . . 5
⊢ (𝐶 ∈ Fin → dom 𝐶 ∈ Fin) | 
| 8 | 6, 7 | syl 17 | . . . 4
⊢ (𝜑 → dom 𝐶 ∈ Fin) | 
| 9 |  | fzfid 14014 | . . . . 5
⊢ ((𝜑 ∧ 𝑙 ∈ dom 𝐶) → (0...𝑀) ∈ Fin) | 
| 10 |  | etransclem27.s | . . . . . . . 8
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) | 
| 11 | 10 | ad2antrr 726 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 𝑆 ∈ {ℝ, ℂ}) | 
| 12 |  | etransclem27.x | . . . . . . . 8
⊢ (𝜑 → 𝑋 ∈
((TopOpen‘ℂfld) ↾t 𝑆)) | 
| 13 | 12 | ad2antrr 726 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 𝑋 ∈
((TopOpen‘ℂfld) ↾t 𝑆)) | 
| 14 |  | etransclem27.p | . . . . . . . 8
⊢ (𝜑 → 𝑃 ∈ ℕ) | 
| 15 | 14 | ad2antrr 726 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 𝑃 ∈ ℕ) | 
| 16 |  | etransclem27.h | . . . . . . . 8
⊢ 𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))) | 
| 17 |  | etransclem5 46254 | . . . . . . . 8
⊢ (𝑗 ∈ (0...𝑀) ↦ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))) = (𝑧 ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑧)↑if(𝑧 = 0, (𝑃 − 1), 𝑃)))) | 
| 18 | 16, 17 | eqtri 2765 | . . . . . . 7
⊢ 𝐻 = (𝑧 ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑧)↑if(𝑧 = 0, (𝑃 − 1), 𝑃)))) | 
| 19 |  | simpr 484 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ (0...𝑀)) | 
| 20 |  | etransclem27.cf | . . . . . . . . . 10
⊢ (𝜑 → 𝐶:dom 𝐶⟶(ℕ0
↑m (0...𝑀))) | 
| 21 | 20 | ffvelcdmda 7104 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑙 ∈ dom 𝐶) → (𝐶‘𝑙) ∈ (ℕ0
↑m (0...𝑀))) | 
| 22 |  | elmapi 8889 | . . . . . . . . 9
⊢ ((𝐶‘𝑙) ∈ (ℕ0
↑m (0...𝑀))
→ (𝐶‘𝑙):(0...𝑀)⟶ℕ0) | 
| 23 | 21, 22 | syl 17 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑙 ∈ dom 𝐶) → (𝐶‘𝑙):(0...𝑀)⟶ℕ0) | 
| 24 | 23 | ffvelcdmda 7104 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → ((𝐶‘𝑙)‘𝑗) ∈
ℕ0) | 
| 25 | 11, 13, 15, 18, 19, 24 | etransclem20 46269 | . . . . . 6
⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗)):𝑋⟶ℂ) | 
| 26 | 5 | ad2antrr 726 | . . . . . 6
⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 𝐽 ∈ 𝑋) | 
| 27 | 25, 26 | ffvelcdmd 7105 | . . . . 5
⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽) ∈ ℂ) | 
| 28 | 9, 27 | fprodcl 15988 | . . . 4
⊢ ((𝜑 ∧ 𝑙 ∈ dom 𝐶) → ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽) ∈ ℂ) | 
| 29 | 8, 28 | fsumcl 15769 | . . 3
⊢ (𝜑 → Σ𝑙 ∈ dom 𝐶∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽) ∈ ℂ) | 
| 30 | 1, 4, 5, 29 | fvmptd3 7039 | . 2
⊢ (𝜑 → (𝐺‘𝐽) = Σ𝑙 ∈ dom 𝐶∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽)) | 
| 31 | 11, 13, 15, 18, 19, 24, 26 | etransclem21 46270 | . . . . 5
⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽) = if(if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗), 0, (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)))) · ((𝐽 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)))))) | 
| 32 |  | iftrue 4531 | . . . . . . . 8
⊢ (if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗) → if(if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗), 0, (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)))) · ((𝐽 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗))))) = 0) | 
| 33 |  | 0zd 12625 | . . . . . . . 8
⊢ (if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗) → 0 ∈ ℤ) | 
| 34 | 32, 33 | eqeltrd 2841 | . . . . . . 7
⊢ (if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗) → if(if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗), 0, (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)))) · ((𝐽 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗))))) ∈ ℤ) | 
| 35 | 34 | adantl 481 | . . . . . 6
⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → if(if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗), 0, (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)))) · ((𝐽 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗))))) ∈ ℤ) | 
| 36 |  | 0zd 12625 | . . . . . . 7
⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → 0 ∈ ℤ) | 
| 37 |  | nnm1nn0 12567 | . . . . . . . . . . . . . . 15
⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈
ℕ0) | 
| 38 | 14, 37 | syl 17 | . . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑃 − 1) ∈
ℕ0) | 
| 39 | 14 | nnnn0d 12587 | . . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑃 ∈
ℕ0) | 
| 40 | 38, 39 | ifcld 4572 | . . . . . . . . . . . . 13
⊢ (𝜑 → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈
ℕ0) | 
| 41 | 40 | nn0zd 12639 | . . . . . . . . . . . 12
⊢ (𝜑 → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℤ) | 
| 42 | 41 | ad3antrrr 730 | . . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℤ) | 
| 43 | 24 | nn0zd 12639 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → ((𝐶‘𝑙)‘𝑗) ∈ ℤ) | 
| 44 | 43 | adantr 480 | . . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → ((𝐶‘𝑙)‘𝑗) ∈ ℤ) | 
| 45 | 42, 44 | zsubcld 12727 | . . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)) ∈ ℤ) | 
| 46 | 44 | zred 12722 | . . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → ((𝐶‘𝑙)‘𝑗) ∈ ℝ) | 
| 47 | 42 | zred 12722 | . . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℝ) | 
| 48 |  | simpr 484 | . . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) | 
| 49 | 46, 47, 48 | nltled 11411 | . . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → ((𝐶‘𝑙)‘𝑗) ≤ if(𝑗 = 0, (𝑃 − 1), 𝑃)) | 
| 50 | 47, 46 | subge0d 11853 | . . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → (0 ≤ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)) ↔ ((𝐶‘𝑙)‘𝑗) ≤ if(𝑗 = 0, (𝑃 − 1), 𝑃))) | 
| 51 | 49, 50 | mpbird 257 | . . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → 0 ≤ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗))) | 
| 52 |  | 0red 11264 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 0 ∈ ℝ) | 
| 53 | 24 | nn0red 12588 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → ((𝐶‘𝑙)‘𝑗) ∈ ℝ) | 
| 54 | 40 | nn0red 12588 | . . . . . . . . . . . . . . 15
⊢ (𝜑 → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℝ) | 
| 55 | 54 | ad2antrr 726 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℝ) | 
| 56 | 24 | nn0ge0d 12590 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 0 ≤ ((𝐶‘𝑙)‘𝑗)) | 
| 57 | 52, 53, 55, 56 | lesub2dd 11880 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)) ≤ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − 0)) | 
| 58 | 55 | recnd 11289 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℂ) | 
| 59 | 58 | subid1d 11609 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − 0) = if(𝑗 = 0, (𝑃 − 1), 𝑃)) | 
| 60 | 57, 59 | breqtrd 5169 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)) ≤ if(𝑗 = 0, (𝑃 − 1), 𝑃)) | 
| 61 | 60 | adantr 480 | . . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)) ≤ if(𝑗 = 0, (𝑃 − 1), 𝑃)) | 
| 62 | 36, 42, 45, 51, 61 | elfzd 13555 | . . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)) ∈ (0...if(𝑗 = 0, (𝑃 − 1), 𝑃))) | 
| 63 |  | permnn 14365 | . . . . . . . . . 10
⊢
((if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)) ∈ (0...if(𝑗 = 0, (𝑃 − 1), 𝑃)) → ((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)))) ∈ ℕ) | 
| 64 | 62, 63 | syl 17 | . . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → ((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)))) ∈ ℕ) | 
| 65 | 64 | nnzd 12640 | . . . . . . . 8
⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → ((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)))) ∈ ℤ) | 
| 66 |  | etransclem27.jz | . . . . . . . . . . 11
⊢ (𝜑 → 𝐽 ∈ ℤ) | 
| 67 | 66 | ad3antrrr 730 | . . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → 𝐽 ∈ ℤ) | 
| 68 |  | elfzelz 13564 | . . . . . . . . . . 11
⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℤ) | 
| 69 | 68 | ad2antlr 727 | . . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → 𝑗 ∈ ℤ) | 
| 70 | 67, 69 | zsubcld 12727 | . . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → (𝐽 − 𝑗) ∈ ℤ) | 
| 71 |  | elnn0z 12626 | . . . . . . . . . 10
⊢
((if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)) ∈ ℕ0 ↔
((if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)) ∈ ℤ ∧ 0 ≤ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)))) | 
| 72 | 45, 51, 71 | sylanbrc 583 | . . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)) ∈
ℕ0) | 
| 73 |  | zexpcl 14117 | . . . . . . . . 9
⊢ (((𝐽 − 𝑗) ∈ ℤ ∧ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)) ∈ ℕ0) → ((𝐽 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗))) ∈ ℤ) | 
| 74 | 70, 72, 73 | syl2anc 584 | . . . . . . . 8
⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → ((𝐽 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗))) ∈ ℤ) | 
| 75 | 65, 74 | zmulcld 12728 | . . . . . . 7
⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)))) · ((𝐽 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)))) ∈ ℤ) | 
| 76 | 36, 75 | ifcld 4572 | . . . . . 6
⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → if(if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗), 0, (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)))) · ((𝐽 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗))))) ∈ ℤ) | 
| 77 | 35, 76 | pm2.61dan 813 | . . . . 5
⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → if(if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗), 0, (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)))) · ((𝐽 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗))))) ∈ ℤ) | 
| 78 | 31, 77 | eqeltrd 2841 | . . . 4
⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽) ∈ ℤ) | 
| 79 | 9, 78 | fprodzcl 15990 | . . 3
⊢ ((𝜑 ∧ 𝑙 ∈ dom 𝐶) → ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽) ∈ ℤ) | 
| 80 | 8, 79 | fsumzcl 15771 | . 2
⊢ (𝜑 → Σ𝑙 ∈ dom 𝐶∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽) ∈ ℤ) | 
| 81 | 30, 80 | eqeltrd 2841 | 1
⊢ (𝜑 → (𝐺‘𝐽) ∈ ℤ) |