Step | Hyp | Ref
| Expression |
1 | | etransclem22.s |
. . 3
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
2 | | etransclem22.x |
. . 3
⊢ (𝜑 → 𝑋 ∈
((TopOpen‘ℂfld) ↾t 𝑆)) |
3 | | etransclem22.p |
. . 3
⊢ (𝜑 → 𝑃 ∈ ℕ) |
4 | | etransclem22.h |
. . 3
⊢ 𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))) |
5 | | etransclem22.J |
. . 3
⊢ (𝜑 → 𝐽 ∈ (0...𝑀)) |
6 | | etransclem22.n |
. . 3
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
7 | 1, 2, 3, 4, 5, 6 | etransclem17 43496 |
. 2
⊢ (𝜑 → ((𝑆 D𝑛 (𝐻‘𝐽))‘𝑁) = (𝑥 ∈ 𝑋 ↦ if(if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁, 0, (((!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))) · ((𝑥 − 𝐽)↑(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁)))))) |
8 | | simpr 488 |
. . . . . 6
⊢ ((𝜑 ∧ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) |
9 | 8 | iftrued 4462 |
. . . . 5
⊢ ((𝜑 ∧ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → if(if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁, 0, (((!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))) · ((𝑥 − 𝐽)↑(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁)))) = 0) |
10 | 9 | mpteq2dv 5166 |
. . . 4
⊢ ((𝜑 ∧ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (𝑥 ∈ 𝑋 ↦ if(if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁, 0, (((!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))) · ((𝑥 − 𝐽)↑(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))))) = (𝑥 ∈ 𝑋 ↦ 0)) |
11 | 1, 2 | dvdmsscn 43181 |
. . . . . 6
⊢ (𝜑 → 𝑋 ⊆ ℂ) |
12 | | 0cnd 10851 |
. . . . . 6
⊢ (𝜑 → 0 ∈
ℂ) |
13 | | ssid 3938 |
. . . . . . 7
⊢ ℂ
⊆ ℂ |
14 | 13 | a1i 11 |
. . . . . 6
⊢ (𝜑 → ℂ ⊆
ℂ) |
15 | 11, 12, 14 | constcncfg 43117 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 0) ∈ (𝑋–cn→ℂ)) |
16 | 15 | adantr 484 |
. . . 4
⊢ ((𝜑 ∧ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (𝑥 ∈ 𝑋 ↦ 0) ∈ (𝑋–cn→ℂ)) |
17 | 10, 16 | eqeltrd 2839 |
. . 3
⊢ ((𝜑 ∧ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (𝑥 ∈ 𝑋 ↦ if(if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁, 0, (((!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))) · ((𝑥 − 𝐽)↑(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))))) ∈ (𝑋–cn→ℂ)) |
18 | | simpr 488 |
. . . . . 6
⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) |
19 | 18 | iffalsed 4465 |
. . . . 5
⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → if(if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁, 0, (((!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))) · ((𝑥 − 𝐽)↑(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁)))) = (((!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))) · ((𝑥 − 𝐽)↑(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁)))) |
20 | 19 | mpteq2dv 5166 |
. . . 4
⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (𝑥 ∈ 𝑋 ↦ if(if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁, 0, (((!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))) · ((𝑥 − 𝐽)↑(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))))) = (𝑥 ∈ 𝑋 ↦ (((!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))) · ((𝑥 − 𝐽)↑(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))))) |
21 | | nfv 1922 |
. . . . 5
⊢
Ⅎ𝑥(𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) |
22 | 11, 14 | idcncfg 43118 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝑥) ∈ (𝑋–cn→ℂ)) |
23 | 5 | elfzelzd 13138 |
. . . . . . . . 9
⊢ (𝜑 → 𝐽 ∈ ℤ) |
24 | 23 | zcnd 12308 |
. . . . . . . 8
⊢ (𝜑 → 𝐽 ∈ ℂ) |
25 | 11, 24, 14 | constcncfg 43117 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐽) ∈ (𝑋–cn→ℂ)) |
26 | 22, 25 | subcncf 24366 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑥 − 𝐽)) ∈ (𝑋–cn→ℂ)) |
27 | 26 | adantr 484 |
. . . . 5
⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (𝑥 ∈ 𝑋 ↦ (𝑥 − 𝐽)) ∈ (𝑋–cn→ℂ)) |
28 | 13 | a1i 11 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → ℂ ⊆
ℂ) |
29 | | nnm1nn0 12156 |
. . . . . . . . . . . . 13
⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈
ℕ0) |
30 | 3, 29 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑃 − 1) ∈
ℕ0) |
31 | 3 | nnnn0d 12175 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑃 ∈
ℕ0) |
32 | 30, 31 | ifcld 4500 |
. . . . . . . . . . 11
⊢ (𝜑 → if(𝐽 = 0, (𝑃 − 1), 𝑃) ∈
ℕ0) |
33 | 32 | faccld 13875 |
. . . . . . . . . 10
⊢ (𝜑 → (!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) ∈ ℕ) |
34 | 33 | nncnd 11871 |
. . . . . . . . 9
⊢ (𝜑 → (!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) ∈ ℂ) |
35 | 34 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) ∈ ℂ) |
36 | 32 | nn0zd 12305 |
. . . . . . . . . . . . 13
⊢ (𝜑 → if(𝐽 = 0, (𝑃 − 1), 𝑃) ∈ ℤ) |
37 | 6 | nn0zd 12305 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑁 ∈ ℤ) |
38 | 36, 37 | zsubcld 12312 |
. . . . . . . . . . . 12
⊢ (𝜑 → (if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁) ∈ ℤ) |
39 | 38 | adantr 484 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁) ∈ ℤ) |
40 | 6 | nn0red 12176 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑁 ∈ ℝ) |
41 | 40 | adantr 484 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → 𝑁 ∈ ℝ) |
42 | 32 | nn0red 12176 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → if(𝐽 = 0, (𝑃 − 1), 𝑃) ∈ ℝ) |
43 | 42 | adantr 484 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → if(𝐽 = 0, (𝑃 − 1), 𝑃) ∈ ℝ) |
44 | 41, 43, 18 | nltled 11007 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → 𝑁 ≤ if(𝐽 = 0, (𝑃 − 1), 𝑃)) |
45 | 43, 41 | subge0d 11447 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (0 ≤ (if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁) ↔ 𝑁 ≤ if(𝐽 = 0, (𝑃 − 1), 𝑃))) |
46 | 44, 45 | mpbird 260 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → 0 ≤ (if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁)) |
47 | | elnn0z 12214 |
. . . . . . . . . . 11
⊢
((if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁) ∈ ℕ0 ↔
((if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁) ∈ ℤ ∧ 0 ≤ (if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))) |
48 | 39, 46, 47 | sylanbrc 586 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁) ∈
ℕ0) |
49 | 48 | faccld 13875 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁)) ∈ ℕ) |
50 | 49 | nncnd 11871 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁)) ∈ ℂ) |
51 | 49 | nnne0d 11905 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁)) ≠ 0) |
52 | 35, 50, 51 | divcld 11633 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → ((!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))) ∈ ℂ) |
53 | 28, 52, 28 | constcncfg 43117 |
. . . . . 6
⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (𝑦 ∈ ℂ ↦ ((!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁)))) ∈ (ℂ–cn→ℂ)) |
54 | | expcncf 23847 |
. . . . . . 7
⊢
((if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁) ∈ ℕ0 → (𝑦 ∈ ℂ ↦ (𝑦↑(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))) ∈ (ℂ–cn→ℂ)) |
55 | 48, 54 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (𝑦 ∈ ℂ ↦ (𝑦↑(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))) ∈ (ℂ–cn→ℂ)) |
56 | 53, 55 | mulcncf 24367 |
. . . . 5
⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (𝑦 ∈ ℂ ↦ (((!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))) · (𝑦↑(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁)))) ∈ (ℂ–cn→ℂ)) |
57 | | oveq1 7239 |
. . . . . 6
⊢ (𝑦 = (𝑥 − 𝐽) → (𝑦↑(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁)) = ((𝑥 − 𝐽)↑(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))) |
58 | 57 | oveq2d 7248 |
. . . . 5
⊢ (𝑦 = (𝑥 − 𝐽) → (((!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))) · (𝑦↑(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))) = (((!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))) · ((𝑥 − 𝐽)↑(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁)))) |
59 | 21, 27, 56, 28, 58 | cncfcompt2 23829 |
. . . 4
⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (𝑥 ∈ 𝑋 ↦ (((!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))) · ((𝑥 − 𝐽)↑(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁)))) ∈ (𝑋–cn→ℂ)) |
60 | 20, 59 | eqeltrd 2839 |
. . 3
⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (𝑥 ∈ 𝑋 ↦ if(if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁, 0, (((!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))) · ((𝑥 − 𝐽)↑(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))))) ∈ (𝑋–cn→ℂ)) |
61 | 17, 60 | pm2.61dan 813 |
. 2
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ if(if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁, 0, (((!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))) · ((𝑥 − 𝐽)↑(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))))) ∈ (𝑋–cn→ℂ)) |
62 | 7, 61 | eqeltrd 2839 |
1
⊢ (𝜑 → ((𝑆 D𝑛 (𝐻‘𝐽))‘𝑁) ∈ (𝑋–cn→ℂ)) |