Step | Hyp | Ref
| Expression |
1 | | nn0uz 11598 |
. . . . 5
⊢
ℕ0 = (ℤ≥‘0) |
2 | | 0zd 11266 |
. . . . 5
⊢ (⊤
→ 0 ∈ ℤ) |
3 | | eqidd 2611 |
. . . . 5
⊢
((⊤ ∧ 𝑗
∈ ℕ0) → ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) = ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗)) |
4 | | 0cnd 9912 |
. . . . . . . . 9
⊢ ((𝑘 ∈ ℕ0
∧ (𝑘 = 0 ∨ 2 ∥
𝑘)) → 0 ∈
ℂ) |
5 | | ioran 510 |
. . . . . . . . . 10
⊢ (¬
(𝑘 = 0 ∨ 2 ∥ 𝑘) ↔ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) |
6 | | neg1rr 11002 |
. . . . . . . . . . . . 13
⊢ -1 ∈
ℝ |
7 | | leibpilem1 24467 |
. . . . . . . . . . . . . 14
⊢ ((𝑘 ∈ ℕ0
∧ (¬ 𝑘 = 0 ∧
¬ 2 ∥ 𝑘)) →
(𝑘 ∈ ℕ ∧
((𝑘 − 1) / 2) ∈
ℕ0)) |
8 | 7 | simprd 478 |
. . . . . . . . . . . . 13
⊢ ((𝑘 ∈ ℕ0
∧ (¬ 𝑘 = 0 ∧
¬ 2 ∥ 𝑘)) →
((𝑘 − 1) / 2) ∈
ℕ0) |
9 | | reexpcl 12739 |
. . . . . . . . . . . . 13
⊢ ((-1
∈ ℝ ∧ ((𝑘
− 1) / 2) ∈ ℕ0) → (-1↑((𝑘 − 1) / 2)) ∈
ℝ) |
10 | 6, 8, 9 | sylancr 694 |
. . . . . . . . . . . 12
⊢ ((𝑘 ∈ ℕ0
∧ (¬ 𝑘 = 0 ∧
¬ 2 ∥ 𝑘)) →
(-1↑((𝑘 − 1) /
2)) ∈ ℝ) |
11 | 7 | simpld 474 |
. . . . . . . . . . . 12
⊢ ((𝑘 ∈ ℕ0
∧ (¬ 𝑘 = 0 ∧
¬ 2 ∥ 𝑘)) →
𝑘 ∈
ℕ) |
12 | 10, 11 | nndivred 10946 |
. . . . . . . . . . 11
⊢ ((𝑘 ∈ ℕ0
∧ (¬ 𝑘 = 0 ∧
¬ 2 ∥ 𝑘)) →
((-1↑((𝑘 − 1) /
2)) / 𝑘) ∈
ℝ) |
13 | 12 | recnd 9947 |
. . . . . . . . . 10
⊢ ((𝑘 ∈ ℕ0
∧ (¬ 𝑘 = 0 ∧
¬ 2 ∥ 𝑘)) →
((-1↑((𝑘 − 1) /
2)) / 𝑘) ∈
ℂ) |
14 | 5, 13 | sylan2b 491 |
. . . . . . . . 9
⊢ ((𝑘 ∈ ℕ0
∧ ¬ (𝑘 = 0 ∨ 2
∥ 𝑘)) →
((-1↑((𝑘 − 1) /
2)) / 𝑘) ∈
ℂ) |
15 | 4, 14 | ifclda 4070 |
. . . . . . . 8
⊢ (𝑘 ∈ ℕ0
→ if((𝑘 = 0 ∨ 2
∥ 𝑘), 0,
((-1↑((𝑘 − 1) /
2)) / 𝑘)) ∈
ℂ) |
16 | 15 | adantl 481 |
. . . . . . 7
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) ∈ ℂ) |
17 | | eqid 2610 |
. . . . . . 7
⊢ (𝑘 ∈ ℕ0
↦ if((𝑘 = 0 ∨ 2
∥ 𝑘), 0,
((-1↑((𝑘 − 1) /
2)) / 𝑘))) = (𝑘 ∈ ℕ0
↦ if((𝑘 = 0 ∨ 2
∥ 𝑘), 0,
((-1↑((𝑘 − 1) /
2)) / 𝑘))) |
18 | 16, 17 | fmptd 6292 |
. . . . . 6
⊢ (⊤
→ (𝑘 ∈
ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘))):ℕ0⟶ℂ) |
19 | 18 | ffvelrnda 6267 |
. . . . 5
⊢
((⊤ ∧ 𝑗
∈ ℕ0) → ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) ∈ ℂ) |
20 | | 2nn0 11186 |
. . . . . . . . . . . . . 14
⊢ 2 ∈
ℕ0 |
21 | 20 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (⊤
→ 2 ∈ ℕ0) |
22 | | nn0mulcl 11206 |
. . . . . . . . . . . . 13
⊢ ((2
∈ ℕ0 ∧ 𝑛 ∈ ℕ0) → (2
· 𝑛) ∈
ℕ0) |
23 | 21, 22 | sylan 487 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑛
∈ ℕ0) → (2 · 𝑛) ∈
ℕ0) |
24 | | nn0p1nn 11209 |
. . . . . . . . . . . 12
⊢ ((2
· 𝑛) ∈
ℕ0 → ((2 · 𝑛) + 1) ∈ ℕ) |
25 | 23, 24 | syl 17 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑛
∈ ℕ0) → ((2 · 𝑛) + 1) ∈ ℕ) |
26 | 25 | nnrecred 10943 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑛
∈ ℕ0) → (1 / ((2 · 𝑛) + 1)) ∈ ℝ) |
27 | | eqid 2610 |
. . . . . . . . . 10
⊢ (𝑛 ∈ ℕ0
↦ (1 / ((2 · 𝑛) + 1))) = (𝑛 ∈ ℕ0 ↦ (1 / ((2
· 𝑛) +
1))) |
28 | 26, 27 | fmptd 6292 |
. . . . . . . . 9
⊢ (⊤
→ (𝑛 ∈
ℕ0 ↦ (1 / ((2 · 𝑛) +
1))):ℕ0⟶ℝ) |
29 | | nn0mulcl 11206 |
. . . . . . . . . . . . . 14
⊢ ((2
∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (2
· 𝑘) ∈
ℕ0) |
30 | 21, 29 | sylan 487 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → (2 · 𝑘) ∈
ℕ0) |
31 | 30 | nn0red 11229 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → (2 · 𝑘) ∈ ℝ) |
32 | | peano2nn0 11210 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ ℕ0
→ (𝑘 + 1) ∈
ℕ0) |
33 | 32 | adantl 481 |
. . . . . . . . . . . . . 14
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → (𝑘 + 1) ∈
ℕ0) |
34 | | nn0mulcl 11206 |
. . . . . . . . . . . . . 14
⊢ ((2
∈ ℕ0 ∧ (𝑘 + 1) ∈ ℕ0) → (2
· (𝑘 + 1)) ∈
ℕ0) |
35 | 20, 33, 34 | sylancr 694 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → (2 · (𝑘 + 1)) ∈
ℕ0) |
36 | 35 | nn0red 11229 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → (2 · (𝑘 + 1)) ∈ ℝ) |
37 | | 1red 9934 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → 1 ∈ ℝ) |
38 | | nn0re 11178 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ ℕ0
→ 𝑘 ∈
ℝ) |
39 | 38 | adantl 481 |
. . . . . . . . . . . . . 14
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → 𝑘 ∈ ℝ) |
40 | 39 | lep1d 10834 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → 𝑘 ≤ (𝑘 + 1)) |
41 | | peano2re 10088 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ ℝ → (𝑘 + 1) ∈
ℝ) |
42 | 39, 41 | syl 17 |
. . . . . . . . . . . . . 14
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → (𝑘 + 1) ∈ ℝ) |
43 | | 2re 10967 |
. . . . . . . . . . . . . . 15
⊢ 2 ∈
ℝ |
44 | 43 | a1i 11 |
. . . . . . . . . . . . . 14
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → 2 ∈ ℝ) |
45 | | 2pos 10989 |
. . . . . . . . . . . . . . 15
⊢ 0 <
2 |
46 | 45 | a1i 11 |
. . . . . . . . . . . . . 14
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → 0 < 2) |
47 | | lemul2 10755 |
. . . . . . . . . . . . . 14
⊢ ((𝑘 ∈ ℝ ∧ (𝑘 + 1) ∈ ℝ ∧ (2
∈ ℝ ∧ 0 < 2)) → (𝑘 ≤ (𝑘 + 1) ↔ (2 · 𝑘) ≤ (2 · (𝑘 + 1)))) |
48 | 39, 42, 44, 46, 47 | syl112anc 1322 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → (𝑘 ≤ (𝑘 + 1) ↔ (2 · 𝑘) ≤ (2 · (𝑘 + 1)))) |
49 | 40, 48 | mpbid 221 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → (2 · 𝑘) ≤ (2 · (𝑘 + 1))) |
50 | 31, 36, 37, 49 | leadd1dd 10520 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → ((2 · 𝑘) + 1) ≤ ((2 · (𝑘 + 1)) + 1)) |
51 | | nn0p1nn 11209 |
. . . . . . . . . . . . . 14
⊢ ((2
· 𝑘) ∈
ℕ0 → ((2 · 𝑘) + 1) ∈ ℕ) |
52 | 30, 51 | syl 17 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → ((2 · 𝑘) + 1) ∈ ℕ) |
53 | 52 | nnred 10912 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → ((2 · 𝑘) + 1) ∈ ℝ) |
54 | 52 | nngt0d 10941 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → 0 < ((2 · 𝑘) + 1)) |
55 | | nn0p1nn 11209 |
. . . . . . . . . . . . . 14
⊢ ((2
· (𝑘 + 1)) ∈
ℕ0 → ((2 · (𝑘 + 1)) + 1) ∈ ℕ) |
56 | 35, 55 | syl 17 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → ((2 · (𝑘 + 1)) + 1) ∈ ℕ) |
57 | 56 | nnred 10912 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → ((2 · (𝑘 + 1)) + 1) ∈ ℝ) |
58 | 56 | nngt0d 10941 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → 0 < ((2 · (𝑘 + 1)) + 1)) |
59 | | lerec 10785 |
. . . . . . . . . . . 12
⊢ (((((2
· 𝑘) + 1) ∈
ℝ ∧ 0 < ((2 · 𝑘) + 1)) ∧ (((2 · (𝑘 + 1)) + 1) ∈ ℝ ∧
0 < ((2 · (𝑘 +
1)) + 1))) → (((2 · 𝑘) + 1) ≤ ((2 · (𝑘 + 1)) + 1) ↔ (1 / ((2 · (𝑘 + 1)) + 1)) ≤ (1 / ((2
· 𝑘) +
1)))) |
60 | 53, 54, 57, 58, 59 | syl22anc 1319 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → (((2 · 𝑘) + 1) ≤ ((2 · (𝑘 + 1)) + 1) ↔ (1 / ((2 · (𝑘 + 1)) + 1)) ≤ (1 / ((2
· 𝑘) +
1)))) |
61 | 50, 60 | mpbid 221 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → (1 / ((2 · (𝑘 + 1)) + 1)) ≤ (1 / ((2 · 𝑘) + 1))) |
62 | | oveq2 6557 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = (𝑘 + 1) → (2 · 𝑛) = (2 · (𝑘 + 1))) |
63 | 62 | oveq1d 6564 |
. . . . . . . . . . . . 13
⊢ (𝑛 = (𝑘 + 1) → ((2 · 𝑛) + 1) = ((2 · (𝑘 + 1)) + 1)) |
64 | 63 | oveq2d 6565 |
. . . . . . . . . . . 12
⊢ (𝑛 = (𝑘 + 1) → (1 / ((2 · 𝑛) + 1)) = (1 / ((2 ·
(𝑘 + 1)) +
1))) |
65 | | ovex 6577 |
. . . . . . . . . . . 12
⊢ (1 / ((2
· (𝑘 + 1)) + 1))
∈ V |
66 | 64, 27, 65 | fvmpt 6191 |
. . . . . . . . . . 11
⊢ ((𝑘 + 1) ∈ ℕ0
→ ((𝑛 ∈
ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘(𝑘 + 1)) = (1 / ((2 · (𝑘 + 1)) + 1))) |
67 | 33, 66 | syl 17 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ (1 / ((2
· 𝑛) +
1)))‘(𝑘 + 1)) = (1 /
((2 · (𝑘 + 1)) +
1))) |
68 | | oveq2 6557 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = 𝑘 → (2 · 𝑛) = (2 · 𝑘)) |
69 | 68 | oveq1d 6564 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝑘 → ((2 · 𝑛) + 1) = ((2 · 𝑘) + 1)) |
70 | 69 | oveq2d 6565 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑘 → (1 / ((2 · 𝑛) + 1)) = (1 / ((2 · 𝑘) + 1))) |
71 | | ovex 6577 |
. . . . . . . . . . . 12
⊢ (1 / ((2
· 𝑘) + 1)) ∈
V |
72 | 70, 27, 71 | fvmpt 6191 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ ℕ0
→ ((𝑛 ∈
ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘) = (1 / ((2 · 𝑘) + 1))) |
73 | 72 | adantl 481 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ (1 / ((2
· 𝑛) +
1)))‘𝑘) = (1 / ((2
· 𝑘) +
1))) |
74 | 61, 67, 73 | 3brtr4d 4615 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ (1 / ((2
· 𝑛) +
1)))‘(𝑘 + 1)) ≤
((𝑛 ∈
ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘)) |
75 | | nnuz 11599 |
. . . . . . . . . 10
⊢ ℕ =
(ℤ≥‘1) |
76 | | 1zzd 11285 |
. . . . . . . . . 10
⊢ (⊤
→ 1 ∈ ℤ) |
77 | | ax-1cn 9873 |
. . . . . . . . . . 11
⊢ 1 ∈
ℂ |
78 | | divcnv 14424 |
. . . . . . . . . . 11
⊢ (1 ∈
ℂ → (𝑛 ∈
ℕ ↦ (1 / 𝑛))
⇝ 0) |
79 | 77, 78 | mp1i 13 |
. . . . . . . . . 10
⊢ (⊤
→ (𝑛 ∈ ℕ
↦ (1 / 𝑛)) ⇝
0) |
80 | | nn0ex 11175 |
. . . . . . . . . . . 12
⊢
ℕ0 ∈ V |
81 | 80 | mptex 6390 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ ℕ0
↦ (1 / ((2 · 𝑛) + 1))) ∈ V |
82 | 81 | a1i 11 |
. . . . . . . . . 10
⊢ (⊤
→ (𝑛 ∈
ℕ0 ↦ (1 / ((2 · 𝑛) + 1))) ∈ V) |
83 | | oveq2 6557 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝑘 → (1 / 𝑛) = (1 / 𝑘)) |
84 | | eqid 2610 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ ℕ ↦ (1 /
𝑛)) = (𝑛 ∈ ℕ ↦ (1 / 𝑛)) |
85 | | ovex 6577 |
. . . . . . . . . . . . 13
⊢ (1 /
𝑘) ∈
V |
86 | 83, 84, 85 | fvmpt 6191 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (1 /
𝑛))‘𝑘) = (1 / 𝑘)) |
87 | 86 | adantl 481 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((𝑛
∈ ℕ ↦ (1 / 𝑛))‘𝑘) = (1 / 𝑘)) |
88 | | nnrecre 10934 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ ℕ → (1 /
𝑘) ∈
ℝ) |
89 | 88 | adantl 481 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (1 / 𝑘) ∈ ℝ) |
90 | 87, 89 | eqeltrd 2688 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((𝑛
∈ ℕ ↦ (1 / 𝑛))‘𝑘) ∈ ℝ) |
91 | | nnnn0 11176 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ ℕ → 𝑘 ∈
ℕ0) |
92 | 91 | adantl 481 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑘
∈ ℕ) → 𝑘
∈ ℕ0) |
93 | 92, 72 | syl 17 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((𝑛
∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘) = (1 / ((2 · 𝑘) + 1))) |
94 | 91, 52 | sylan2 490 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((2 · 𝑘) + 1) ∈ ℕ) |
95 | 94 | nnrecred 10943 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (1 / ((2 · 𝑘) + 1)) ∈ ℝ) |
96 | 93, 95 | eqeltrd 2688 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((𝑛
∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘) ∈ ℝ) |
97 | | nnre 10904 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ ℕ → 𝑘 ∈
ℝ) |
98 | 97 | adantl 481 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑘
∈ ℕ) → 𝑘
∈ ℝ) |
99 | 20, 92, 29 | sylancr 694 |
. . . . . . . . . . . . . 14
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (2 · 𝑘) ∈
ℕ0) |
100 | 99 | nn0red 11229 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (2 · 𝑘) ∈ ℝ) |
101 | | peano2re 10088 |
. . . . . . . . . . . . . 14
⊢ ((2
· 𝑘) ∈ ℝ
→ ((2 · 𝑘) + 1)
∈ ℝ) |
102 | 100, 101 | syl 17 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((2 · 𝑘) + 1) ∈ ℝ) |
103 | | nn0addge1 11216 |
. . . . . . . . . . . . . . 15
⊢ ((𝑘 ∈ ℝ ∧ 𝑘 ∈ ℕ0)
→ 𝑘 ≤ (𝑘 + 𝑘)) |
104 | 98, 92, 103 | syl2anc 691 |
. . . . . . . . . . . . . 14
⊢
((⊤ ∧ 𝑘
∈ ℕ) → 𝑘
≤ (𝑘 + 𝑘)) |
105 | 98 | recnd 9947 |
. . . . . . . . . . . . . . 15
⊢
((⊤ ∧ 𝑘
∈ ℕ) → 𝑘
∈ ℂ) |
106 | 105 | 2timesd 11152 |
. . . . . . . . . . . . . 14
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (2 · 𝑘) = (𝑘 + 𝑘)) |
107 | 104, 106 | breqtrrd 4611 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑘
∈ ℕ) → 𝑘
≤ (2 · 𝑘)) |
108 | 100 | lep1d 10834 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (2 · 𝑘) ≤ ((2 · 𝑘) + 1)) |
109 | 98, 100, 102, 107, 108 | letrd 10073 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑘
∈ ℕ) → 𝑘
≤ ((2 · 𝑘) +
1)) |
110 | | nngt0 10926 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ ℕ → 0 <
𝑘) |
111 | 110 | adantl 481 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑘
∈ ℕ) → 0 < 𝑘) |
112 | 94 | nnred 10912 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((2 · 𝑘) + 1) ∈ ℝ) |
113 | 94 | nngt0d 10941 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑘
∈ ℕ) → 0 < ((2 · 𝑘) + 1)) |
114 | | lerec 10785 |
. . . . . . . . . . . . 13
⊢ (((𝑘 ∈ ℝ ∧ 0 <
𝑘) ∧ (((2 ·
𝑘) + 1) ∈ ℝ
∧ 0 < ((2 · 𝑘) + 1))) → (𝑘 ≤ ((2 · 𝑘) + 1) ↔ (1 / ((2 · 𝑘) + 1)) ≤ (1 / 𝑘))) |
115 | 98, 111, 112, 113, 114 | syl22anc 1319 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝑘
≤ ((2 · 𝑘) + 1)
↔ (1 / ((2 · 𝑘)
+ 1)) ≤ (1 / 𝑘))) |
116 | 109, 115 | mpbid 221 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (1 / ((2 · 𝑘) + 1)) ≤ (1 / 𝑘)) |
117 | 116, 93, 87 | 3brtr4d 4615 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((𝑛
∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘) ≤ ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘)) |
118 | 94 | nnrpd 11746 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((2 · 𝑘) + 1) ∈
ℝ+) |
119 | 118 | rpreccld 11758 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (1 / ((2 · 𝑘) + 1)) ∈
ℝ+) |
120 | 119 | rpge0d 11752 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑘
∈ ℕ) → 0 ≤ (1 / ((2 · 𝑘) + 1))) |
121 | 120, 93 | breqtrrd 4611 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑘
∈ ℕ) → 0 ≤ ((𝑛 ∈ ℕ0 ↦ (1 / ((2
· 𝑛) +
1)))‘𝑘)) |
122 | 75, 76, 79, 82, 90, 96, 117, 121 | climsqz2 14220 |
. . . . . . . . 9
⊢ (⊤
→ (𝑛 ∈
ℕ0 ↦ (1 / ((2 · 𝑛) + 1))) ⇝ 0) |
123 | | neg1cn 11001 |
. . . . . . . . . . . . 13
⊢ -1 ∈
ℂ |
124 | 123 | a1i 11 |
. . . . . . . . . . . 12
⊢ (⊤
→ -1 ∈ ℂ) |
125 | | expcl 12740 |
. . . . . . . . . . . 12
⊢ ((-1
∈ ℂ ∧ 𝑘
∈ ℕ0) → (-1↑𝑘) ∈ ℂ) |
126 | 124, 125 | sylan 487 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → (-1↑𝑘) ∈ ℂ) |
127 | 52 | nncnd 10913 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → ((2 · 𝑘) + 1) ∈ ℂ) |
128 | 52 | nnne0d 10942 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → ((2 · 𝑘) + 1) ≠ 0) |
129 | 126, 127,
128 | divrecd 10683 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → ((-1↑𝑘) / ((2 · 𝑘) + 1)) = ((-1↑𝑘) · (1 / ((2 · 𝑘) + 1)))) |
130 | | oveq2 6557 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝑘 → (-1↑𝑛) = (-1↑𝑘)) |
131 | 130, 69 | oveq12d 6567 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑘 → ((-1↑𝑛) / ((2 · 𝑛) + 1)) = ((-1↑𝑘) / ((2 · 𝑘) + 1))) |
132 | | eqid 2610 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ ℕ0
↦ ((-1↑𝑛) / ((2
· 𝑛) + 1))) = (𝑛 ∈ ℕ0
↦ ((-1↑𝑛) / ((2
· 𝑛) +
1))) |
133 | | ovex 6577 |
. . . . . . . . . . . 12
⊢
((-1↑𝑘) / ((2
· 𝑘) + 1)) ∈
V |
134 | 131, 132,
133 | fvmpt 6191 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ ℕ0
→ ((𝑛 ∈
ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))‘𝑘) = ((-1↑𝑘) / ((2 · 𝑘) + 1))) |
135 | 134 | adantl 481 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦
((-1↑𝑛) / ((2 ·
𝑛) + 1)))‘𝑘) = ((-1↑𝑘) / ((2 · 𝑘) + 1))) |
136 | 73 | oveq2d 6565 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → ((-1↑𝑘) · ((𝑛 ∈ ℕ0 ↦ (1 / ((2
· 𝑛) +
1)))‘𝑘)) =
((-1↑𝑘) · (1 /
((2 · 𝑘) +
1)))) |
137 | 129, 135,
136 | 3eqtr4d 2654 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦
((-1↑𝑛) / ((2 ·
𝑛) + 1)))‘𝑘) = ((-1↑𝑘) · ((𝑛 ∈ ℕ0 ↦ (1 / ((2
· 𝑛) +
1)))‘𝑘))) |
138 | 1, 2, 28, 74, 122, 137 | iseralt 14263 |
. . . . . . . 8
⊢ (⊤
→ seq0( + , (𝑛 ∈
ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))) ∈ dom ⇝
) |
139 | | climdm 14133 |
. . . . . . . 8
⊢ (seq0( +
, (𝑛 ∈
ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))) ∈ dom ⇝ ↔ seq0( + ,
(𝑛 ∈
ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))) ⇝ ( ⇝ ‘seq0( + ,
(𝑛 ∈
ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))))) |
140 | 138, 139 | sylib 207 |
. . . . . . 7
⊢ (⊤
→ seq0( + , (𝑛 ∈
ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))) ⇝ ( ⇝ ‘seq0( + ,
(𝑛 ∈
ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))))) |
141 | | fvex 6113 |
. . . . . . . 8
⊢ ( ⇝
‘seq0( + , (𝑛 ∈
ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1))))) ∈ V |
142 | 132, 17, 141 | leibpilem2 24468 |
. . . . . . 7
⊢ (seq0( +
, (𝑛 ∈
ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))) ⇝ ( ⇝ ‘seq0( + ,
(𝑛 ∈
ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1))))) ↔ seq0( + , (𝑘 ∈ ℕ0
↦ if((𝑘 = 0 ∨ 2
∥ 𝑘), 0,
((-1↑((𝑘 − 1) /
2)) / 𝑘)))) ⇝ (
⇝ ‘seq0( + , (𝑛
∈ ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))))) |
143 | 140, 142 | sylib 207 |
. . . . . 6
⊢ (⊤
→ seq0( + , (𝑘 ∈
ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ⇝ ( ⇝ ‘seq0( + ,
(𝑛 ∈
ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))))) |
144 | | seqex 12665 |
. . . . . . 7
⊢ seq0( + ,
(𝑘 ∈
ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ∈ V |
145 | 144, 141 | breldm 5251 |
. . . . . 6
⊢ (seq0( +
, (𝑘 ∈
ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ⇝ ( ⇝ ‘seq0( + ,
(𝑛 ∈
ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1))))) → seq0( + , (𝑘 ∈ ℕ0
↦ if((𝑘 = 0 ∨ 2
∥ 𝑘), 0,
((-1↑((𝑘 − 1) /
2)) / 𝑘)))) ∈ dom
⇝ ) |
146 | 143, 145 | syl 17 |
. . . . 5
⊢ (⊤
→ seq0( + , (𝑘 ∈
ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ∈ dom ⇝ ) |
147 | 1, 2, 3, 19, 146 | isumclim2 14331 |
. . . 4
⊢ (⊤
→ seq0( + , (𝑘 ∈
ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ⇝ Σ𝑗 ∈ ℕ0 ((𝑘 ∈ ℕ0
↦ if((𝑘 = 0 ∨ 2
∥ 𝑘), 0,
((-1↑((𝑘 − 1) /
2)) / 𝑘)))‘𝑗)) |
148 | | eqid 2610 |
. . . . . . . 8
⊢ (𝑥 ∈ (0[,]1) ↦
Σ𝑗 ∈
ℕ0 (((𝑘
∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · (𝑥↑𝑗))) = (𝑥 ∈ (0[,]1) ↦ Σ𝑗 ∈ ℕ0
(((𝑘 ∈
ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · (𝑥↑𝑗))) |
149 | 18, 146, 148 | abelth2 24000 |
. . . . . . 7
⊢ (⊤
→ (𝑥 ∈ (0[,]1)
↦ Σ𝑗 ∈
ℕ0 (((𝑘
∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · (𝑥↑𝑗))) ∈ ((0[,]1)–cn→ℂ)) |
150 | | nnrp 11718 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℝ+) |
151 | 150 | adantl 481 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑛
∈ ℕ) → 𝑛
∈ ℝ+) |
152 | 151 | rpreccld 11758 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑛
∈ ℕ) → (1 / 𝑛) ∈
ℝ+) |
153 | 152 | rpred 11748 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑛
∈ ℕ) → (1 / 𝑛) ∈ ℝ) |
154 | 152 | rpge0d 11752 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑛
∈ ℕ) → 0 ≤ (1 / 𝑛)) |
155 | | nnge1 10923 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ ℕ → 1 ≤
𝑛) |
156 | 155 | adantl 481 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑛
∈ ℕ) → 1 ≤ 𝑛) |
157 | | nnre 10904 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℝ) |
158 | 157 | adantl 481 |
. . . . . . . . . . . . . 14
⊢
((⊤ ∧ 𝑛
∈ ℕ) → 𝑛
∈ ℝ) |
159 | 158 | recnd 9947 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑛
∈ ℕ) → 𝑛
∈ ℂ) |
160 | 159 | mulid1d 9936 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑛
∈ ℕ) → (𝑛
· 1) = 𝑛) |
161 | 156, 160 | breqtrrd 4611 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑛
∈ ℕ) → 1 ≤ (𝑛 · 1)) |
162 | | 1red 9934 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑛
∈ ℕ) → 1 ∈ ℝ) |
163 | | nngt0 10926 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ ℕ → 0 <
𝑛) |
164 | 163 | adantl 481 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑛
∈ ℕ) → 0 < 𝑛) |
165 | | ledivmul 10778 |
. . . . . . . . . . . 12
⊢ ((1
∈ ℝ ∧ 1 ∈ ℝ ∧ (𝑛 ∈ ℝ ∧ 0 < 𝑛)) → ((1 / 𝑛) ≤ 1 ↔ 1 ≤ (𝑛 · 1))) |
166 | 162, 162,
158, 164, 165 | syl112anc 1322 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑛
∈ ℕ) → ((1 / 𝑛) ≤ 1 ↔ 1 ≤ (𝑛 · 1))) |
167 | 161, 166 | mpbird 246 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑛
∈ ℕ) → (1 / 𝑛) ≤ 1) |
168 | | 0re 9919 |
. . . . . . . . . . 11
⊢ 0 ∈
ℝ |
169 | | 1re 9918 |
. . . . . . . . . . 11
⊢ 1 ∈
ℝ |
170 | 168, 169 | elicc2i 12110 |
. . . . . . . . . 10
⊢ ((1 /
𝑛) ∈ (0[,]1) ↔
((1 / 𝑛) ∈ ℝ
∧ 0 ≤ (1 / 𝑛) ∧
(1 / 𝑛) ≤
1)) |
171 | 153, 154,
167, 170 | syl3anbrc 1239 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑛
∈ ℕ) → (1 / 𝑛) ∈ (0[,]1)) |
172 | | iirev 22536 |
. . . . . . . . 9
⊢ ((1 /
𝑛) ∈ (0[,]1) → (1
− (1 / 𝑛)) ∈
(0[,]1)) |
173 | 171, 172 | syl 17 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑛
∈ ℕ) → (1 − (1 / 𝑛)) ∈ (0[,]1)) |
174 | | eqid 2610 |
. . . . . . . 8
⊢ (𝑛 ∈ ℕ ↦ (1
− (1 / 𝑛))) = (𝑛 ∈ ℕ ↦ (1
− (1 / 𝑛))) |
175 | 173, 174 | fmptd 6292 |
. . . . . . 7
⊢ (⊤
→ (𝑛 ∈ ℕ
↦ (1 − (1 / 𝑛))):ℕ⟶(0[,]1)) |
176 | | 1cnd 9935 |
. . . . . . . . 9
⊢ (⊤
→ 1 ∈ ℂ) |
177 | | nnex 10903 |
. . . . . . . . . . 11
⊢ ℕ
∈ V |
178 | 177 | mptex 6390 |
. . . . . . . . . 10
⊢ (𝑛 ∈ ℕ ↦ (1
− (1 / 𝑛))) ∈
V |
179 | 178 | a1i 11 |
. . . . . . . . 9
⊢ (⊤
→ (𝑛 ∈ ℕ
↦ (1 − (1 / 𝑛))) ∈ V) |
180 | 90 | recnd 9947 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((𝑛
∈ ℕ ↦ (1 / 𝑛))‘𝑘) ∈ ℂ) |
181 | 83 | oveq2d 6565 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑘 → (1 − (1 / 𝑛)) = (1 − (1 / 𝑘))) |
182 | | ovex 6577 |
. . . . . . . . . . . 12
⊢ (1
− (1 / 𝑘)) ∈
V |
183 | 181, 174,
182 | fvmpt 6191 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (1
− (1 / 𝑛)))‘𝑘) = (1 − (1 / 𝑘))) |
184 | 86 | oveq2d 6565 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ ℕ → (1
− ((𝑛 ∈ ℕ
↦ (1 / 𝑛))‘𝑘)) = (1 − (1 / 𝑘))) |
185 | 183, 184 | eqtr4d 2647 |
. . . . . . . . . 10
⊢ (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (1
− (1 / 𝑛)))‘𝑘) = (1 − ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘))) |
186 | 185 | adantl 481 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((𝑛
∈ ℕ ↦ (1 − (1 / 𝑛)))‘𝑘) = (1 − ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘))) |
187 | 75, 76, 79, 176, 179, 180, 186 | climsubc2 14217 |
. . . . . . . 8
⊢ (⊤
→ (𝑛 ∈ ℕ
↦ (1 − (1 / 𝑛))) ⇝ (1 − 0)) |
188 | | 1m0e1 11008 |
. . . . . . . 8
⊢ (1
− 0) = 1 |
189 | 187, 188 | syl6breq 4624 |
. . . . . . 7
⊢ (⊤
→ (𝑛 ∈ ℕ
↦ (1 − (1 / 𝑛))) ⇝ 1) |
190 | | 1elunit 12162 |
. . . . . . . 8
⊢ 1 ∈
(0[,]1) |
191 | 190 | a1i 11 |
. . . . . . 7
⊢ (⊤
→ 1 ∈ (0[,]1)) |
192 | 75, 76, 149, 175, 189, 191 | climcncf 22511 |
. . . . . 6
⊢ (⊤
→ ((𝑥 ∈ (0[,]1)
↦ Σ𝑗 ∈
ℕ0 (((𝑘
∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · (𝑥↑𝑗))) ∘ (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))) ⇝ ((𝑥 ∈ (0[,]1) ↦
Σ𝑗 ∈
ℕ0 (((𝑘
∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · (𝑥↑𝑗)))‘1)) |
193 | | eqidd 2611 |
. . . . . . . 8
⊢ (⊤
→ (𝑛 ∈ ℕ
↦ (1 − (1 / 𝑛))) = (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))) |
194 | | eqidd 2611 |
. . . . . . . 8
⊢ (⊤
→ (𝑥 ∈ (0[,]1)
↦ Σ𝑗 ∈
ℕ0 (((𝑘
∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · (𝑥↑𝑗))) = (𝑥 ∈ (0[,]1) ↦ Σ𝑗 ∈ ℕ0
(((𝑘 ∈
ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · (𝑥↑𝑗)))) |
195 | | oveq1 6556 |
. . . . . . . . . 10
⊢ (𝑥 = (1 − (1 / 𝑛)) → (𝑥↑𝑗) = ((1 − (1 / 𝑛))↑𝑗)) |
196 | 195 | oveq2d 6565 |
. . . . . . . . 9
⊢ (𝑥 = (1 − (1 / 𝑛)) → (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · (𝑥↑𝑗)) = (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · ((1 − (1 / 𝑛))↑𝑗))) |
197 | 196 | sumeq2sdv 14282 |
. . . . . . . 8
⊢ (𝑥 = (1 − (1 / 𝑛)) → Σ𝑗 ∈ ℕ0
(((𝑘 ∈
ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · (𝑥↑𝑗)) = Σ𝑗 ∈ ℕ0 (((𝑘 ∈ ℕ0
↦ if((𝑘 = 0 ∨ 2
∥ 𝑘), 0,
((-1↑((𝑘 − 1) /
2)) / 𝑘)))‘𝑗) · ((1 − (1 /
𝑛))↑𝑗))) |
198 | 173, 193,
194, 197 | fmptco 6303 |
. . . . . . 7
⊢ (⊤
→ ((𝑥 ∈ (0[,]1)
↦ Σ𝑗 ∈
ℕ0 (((𝑘
∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · (𝑥↑𝑗))) ∘ (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))) = (𝑛 ∈ ℕ ↦ Σ𝑗 ∈ ℕ0
(((𝑘 ∈
ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · ((1 − (1 / 𝑛))↑𝑗)))) |
199 | | 0zd 11266 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑛
∈ ℕ) → 0 ∈ ℤ) |
200 | 8 | adantll 746 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((⊤ ∧ 𝑘
∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((𝑘 − 1) / 2) ∈
ℕ0) |
201 | 6, 200, 9 | sylancr 694 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((⊤ ∧ 𝑘
∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (-1↑((𝑘 − 1) / 2)) ∈
ℝ) |
202 | 201 | recnd 9947 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((⊤ ∧ 𝑘
∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (-1↑((𝑘 − 1) / 2)) ∈
ℂ) |
203 | 202 | adantllr 751 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (-1↑((𝑘 − 1) / 2)) ∈
ℂ) |
204 | | resubcl 10224 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((1
∈ ℝ ∧ (1 / 𝑛) ∈ ℝ) → (1 − (1 /
𝑛)) ∈
ℝ) |
205 | 169, 153,
204 | sylancr 694 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((⊤ ∧ 𝑛
∈ ℕ) → (1 − (1 / 𝑛)) ∈ ℝ) |
206 | 205 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (1 − (1 / 𝑛)) ∈ ℝ) |
207 | | simplr 788 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → 𝑘 ∈ ℕ0) |
208 | 206, 207 | reexpcld 12887 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((1 − (1 / 𝑛))↑𝑘) ∈ ℝ) |
209 | 208 | recnd 9947 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((1 − (1 / 𝑛))↑𝑘) ∈ ℂ) |
210 | | nn0cn 11179 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 ∈ ℕ0
→ 𝑘 ∈
ℂ) |
211 | 210 | ad2antlr 759 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → 𝑘 ∈ ℂ) |
212 | 11 | adantll 746 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → 𝑘 ∈ ℕ) |
213 | 212 | nnne0d 10942 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → 𝑘 ≠ 0) |
214 | 203, 209,
211, 213 | div12d 10716 |
. . . . . . . . . . . . . . . . . 18
⊢
((((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 /
𝑛))↑𝑘) / 𝑘)) = (((1 − (1 / 𝑛))↑𝑘) · ((-1↑((𝑘 − 1) / 2)) / 𝑘))) |
215 | 13 | adantll 746 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ ℂ) |
216 | 209, 215 | mulcomd 9940 |
. . . . . . . . . . . . . . . . . 18
⊢
((((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (((1 − (1 / 𝑛))↑𝑘) · ((-1↑((𝑘 − 1) / 2)) / 𝑘)) = (((-1↑((𝑘 − 1) / 2)) / 𝑘) · ((1 − (1 / 𝑛))↑𝑘))) |
217 | 214, 216 | eqtrd 2644 |
. . . . . . . . . . . . . . . . 17
⊢
((((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 /
𝑛))↑𝑘) / 𝑘)) = (((-1↑((𝑘 − 1) / 2)) / 𝑘) · ((1 − (1 / 𝑛))↑𝑘))) |
218 | 5, 217 | sylan2b 491 |
. . . . . . . . . . . . . . . 16
⊢
((((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) ∧ ¬ (𝑘 = 0 ∨ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 /
𝑛))↑𝑘) / 𝑘)) = (((-1↑((𝑘 − 1) / 2)) / 𝑘) · ((1 − (1 / 𝑛))↑𝑘))) |
219 | 218 | ifeq2da 4067 |
. . . . . . . . . . . . . . 15
⊢
(((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 /
𝑛))↑𝑘) / 𝑘))) = if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, (((-1↑((𝑘 − 1) / 2)) / 𝑘) · ((1 − (1 / 𝑛))↑𝑘)))) |
220 | 205 | recnd 9947 |
. . . . . . . . . . . . . . . . . 18
⊢
((⊤ ∧ 𝑛
∈ ℕ) → (1 − (1 / 𝑛)) ∈ ℂ) |
221 | | expcl 12740 |
. . . . . . . . . . . . . . . . . 18
⊢ (((1
− (1 / 𝑛)) ∈
ℂ ∧ 𝑘 ∈
ℕ0) → ((1 − (1 / 𝑛))↑𝑘) ∈ ℂ) |
222 | 220, 221 | sylan 487 |
. . . . . . . . . . . . . . . . 17
⊢
(((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) → ((1 − (1 / 𝑛))↑𝑘) ∈ ℂ) |
223 | 222 | mul02d 10113 |
. . . . . . . . . . . . . . . 16
⊢
(((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) → (0 · ((1 − (1 / 𝑛))↑𝑘)) = 0) |
224 | 223 | ifeq1d 4054 |
. . . . . . . . . . . . . . 15
⊢
(((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) → if((𝑘 = 0 ∨ 2 ∥ 𝑘), (0 · ((1 − (1 / 𝑛))↑𝑘)), (((-1↑((𝑘 − 1) / 2)) / 𝑘) · ((1 − (1 / 𝑛))↑𝑘))) = if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, (((-1↑((𝑘 − 1) / 2)) / 𝑘) · ((1 − (1 / 𝑛))↑𝑘)))) |
225 | 219, 224 | eqtr4d 2647 |
. . . . . . . . . . . . . 14
⊢
(((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 /
𝑛))↑𝑘) / 𝑘))) = if((𝑘 = 0 ∨ 2 ∥ 𝑘), (0 · ((1 − (1 / 𝑛))↑𝑘)), (((-1↑((𝑘 − 1) / 2)) / 𝑘) · ((1 − (1 / 𝑛))↑𝑘)))) |
226 | | ovif 6635 |
. . . . . . . . . . . . . 14
⊢
(if((𝑘 = 0 ∨ 2
∥ 𝑘), 0,
((-1↑((𝑘 − 1) /
2)) / 𝑘)) · ((1
− (1 / 𝑛))↑𝑘)) = if((𝑘 = 0 ∨ 2 ∥ 𝑘), (0 · ((1 − (1 / 𝑛))↑𝑘)), (((-1↑((𝑘 − 1) / 2)) / 𝑘) · ((1 − (1 / 𝑛))↑𝑘))) |
227 | 225, 226 | syl6eqr 2662 |
. . . . . . . . . . . . 13
⊢
(((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 /
𝑛))↑𝑘) / 𝑘))) = (if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) · ((1 − (1 / 𝑛))↑𝑘))) |
228 | | simpr 476 |
. . . . . . . . . . . . . 14
⊢
(((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) → 𝑘 ∈ ℕ0) |
229 | | c0ex 9913 |
. . . . . . . . . . . . . . 15
⊢ 0 ∈
V |
230 | | ovex 6577 |
. . . . . . . . . . . . . . 15
⊢
((-1↑((𝑘
− 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)) ∈ V |
231 | 229, 230 | ifex 4106 |
. . . . . . . . . . . . . 14
⊢ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))) ∈ V |
232 | | eqid 2610 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ ℕ0
↦ if((𝑘 = 0 ∨ 2
∥ 𝑘), 0,
((-1↑((𝑘 − 1) /
2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)))) = (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘)))) |
233 | 232 | fvmpt2 6200 |
. . . . . . . . . . . . . 14
⊢ ((𝑘 ∈ ℕ0
∧ if((𝑘 = 0 ∨ 2
∥ 𝑘), 0,
((-1↑((𝑘 − 1) /
2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) ∈ V) → ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) = if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 /
𝑛))↑𝑘) / 𝑘)))) |
234 | 228, 231,
233 | sylancl 693 |
. . . . . . . . . . . . 13
⊢
(((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) → ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) = if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 /
𝑛))↑𝑘) / 𝑘)))) |
235 | | ovex 6577 |
. . . . . . . . . . . . . . . 16
⊢
((-1↑((𝑘
− 1) / 2)) / 𝑘)
∈ V |
236 | 229, 235 | ifex 4106 |
. . . . . . . . . . . . . . 15
⊢ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) ∈ V |
237 | 17 | fvmpt2 6200 |
. . . . . . . . . . . . . . 15
⊢ ((𝑘 ∈ ℕ0
∧ if((𝑘 = 0 ∨ 2
∥ 𝑘), 0,
((-1↑((𝑘 − 1) /
2)) / 𝑘)) ∈ V) →
((𝑘 ∈
ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑘) = if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘))) |
238 | 228, 236,
237 | sylancl 693 |
. . . . . . . . . . . . . 14
⊢
(((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) → ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑘) = if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘))) |
239 | 238 | oveq1d 6564 |
. . . . . . . . . . . . 13
⊢
(((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) → (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑘) · ((1 − (1 / 𝑛))↑𝑘)) = (if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) · ((1 − (1 / 𝑛))↑𝑘))) |
240 | 227, 234,
239 | 3eqtr4d 2654 |
. . . . . . . . . . . 12
⊢
(((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) → ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) = (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑘) · ((1 − (1 / 𝑛))↑𝑘))) |
241 | 240 | ralrimiva 2949 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑛
∈ ℕ) → ∀𝑘 ∈ ℕ0 ((𝑘 ∈ ℕ0
↦ if((𝑘 = 0 ∨ 2
∥ 𝑘), 0,
((-1↑((𝑘 − 1) /
2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) = (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑘) · ((1 − (1 / 𝑛))↑𝑘))) |
242 | | nfv 1830 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑗((𝑘 ∈ ℕ0
↦ if((𝑘 = 0 ∨ 2
∥ 𝑘), 0,
((-1↑((𝑘 − 1) /
2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) = (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑘) · ((1 − (1 / 𝑛))↑𝑘)) |
243 | | nffvmpt1 6111 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑘((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗) |
244 | | nffvmpt1 6111 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑘((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) |
245 | | nfcv 2751 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑘
· |
246 | | nfcv 2751 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑘((1
− (1 / 𝑛))↑𝑗) |
247 | 244, 245,
246 | nfov 6575 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑘(((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · ((1 − (1 / 𝑛))↑𝑗)) |
248 | 243, 247 | nfeq 2762 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑘((𝑘 ∈ ℕ0
↦ if((𝑘 = 0 ∨ 2
∥ 𝑘), 0,
((-1↑((𝑘 − 1) /
2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗) = (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · ((1 − (1 / 𝑛))↑𝑗)) |
249 | | fveq2 6103 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝑗 → ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) = ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗)) |
250 | | fveq2 6103 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 𝑗 → ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑘) = ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗)) |
251 | | oveq2 6557 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 𝑗 → ((1 − (1 / 𝑛))↑𝑘) = ((1 − (1 / 𝑛))↑𝑗)) |
252 | 250, 251 | oveq12d 6567 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝑗 → (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑘) · ((1 − (1 / 𝑛))↑𝑘)) = (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · ((1 − (1 / 𝑛))↑𝑗))) |
253 | 249, 252 | eqeq12d 2625 |
. . . . . . . . . . . 12
⊢ (𝑘 = 𝑗 → (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) = (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑘) · ((1 − (1 / 𝑛))↑𝑘)) ↔ ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗) = (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · ((1 − (1 / 𝑛))↑𝑗)))) |
254 | 242, 248,
253 | cbvral 3143 |
. . . . . . . . . . 11
⊢
(∀𝑘 ∈
ℕ0 ((𝑘
∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 /
𝑛))↑𝑘) / 𝑘))))‘𝑘) = (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑘) · ((1 − (1 / 𝑛))↑𝑘)) ↔ ∀𝑗 ∈ ℕ0 ((𝑘 ∈ ℕ0
↦ if((𝑘 = 0 ∨ 2
∥ 𝑘), 0,
((-1↑((𝑘 − 1) /
2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗) = (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · ((1 − (1 / 𝑛))↑𝑗))) |
255 | 241, 254 | sylib 207 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑛
∈ ℕ) → ∀𝑗 ∈ ℕ0 ((𝑘 ∈ ℕ0
↦ if((𝑘 = 0 ∨ 2
∥ 𝑘), 0,
((-1↑((𝑘 − 1) /
2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗) = (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · ((1 − (1 / 𝑛))↑𝑗))) |
256 | 255 | r19.21bi 2916 |
. . . . . . . . 9
⊢
(((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑗
∈ ℕ0) → ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗) = (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · ((1 − (1 / 𝑛))↑𝑗))) |
257 | | 0cnd 9912 |
. . . . . . . . . . . . 13
⊢
((((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) ∧ (𝑘 = 0 ∨ 2 ∥ 𝑘)) → 0 ∈ ℂ) |
258 | 208, 212 | nndivred 10946 |
. . . . . . . . . . . . . . . 16
⊢
((((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (((1 − (1 / 𝑛))↑𝑘) / 𝑘) ∈ ℝ) |
259 | 258 | recnd 9947 |
. . . . . . . . . . . . . . 15
⊢
((((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (((1 − (1 / 𝑛))↑𝑘) / 𝑘) ∈ ℂ) |
260 | 203, 259 | mulcld 9939 |
. . . . . . . . . . . . . 14
⊢
((((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 /
𝑛))↑𝑘) / 𝑘)) ∈ ℂ) |
261 | 5, 260 | sylan2b 491 |
. . . . . . . . . . . . 13
⊢
((((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) ∧ ¬ (𝑘 = 0 ∨ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 /
𝑛))↑𝑘) / 𝑘)) ∈ ℂ) |
262 | 257, 261 | ifclda 4070 |
. . . . . . . . . . . 12
⊢
(((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 /
𝑛))↑𝑘) / 𝑘))) ∈ ℂ) |
263 | 262, 232 | fmptd 6292 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑛
∈ ℕ) → (𝑘
∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 /
𝑛))↑𝑘) / 𝑘)))):ℕ0⟶ℂ) |
264 | 263 | ffvelrnda 6267 |
. . . . . . . . . 10
⊢
(((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑗
∈ ℕ0) → ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗) ∈ ℂ) |
265 | 256, 264 | eqeltrrd 2689 |
. . . . . . . . 9
⊢
(((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑗
∈ ℕ0) → (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · ((1 − (1 / 𝑛))↑𝑗)) ∈ ℂ) |
266 | | 0nn0 11184 |
. . . . . . . . . . . 12
⊢ 0 ∈
ℕ0 |
267 | 266 | a1i 11 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑛
∈ ℕ) → 0 ∈ ℕ0) |
268 | | 0p1e1 11009 |
. . . . . . . . . . . . 13
⊢ (0 + 1) =
1 |
269 | | seqeq1 12666 |
. . . . . . . . . . . . 13
⊢ ((0 + 1)
= 1 → seq(0 + 1)( + , (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))) = seq1( + , (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘)))))) |
270 | 268, 269 | ax-mp 5 |
. . . . . . . . . . . 12
⊢ seq(0 +
1)( + , (𝑘 ∈
ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 /
𝑛))↑𝑘) / 𝑘))))) = seq1( + , (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))) |
271 | | 1zzd 11285 |
. . . . . . . . . . . . . 14
⊢
((⊤ ∧ 𝑛
∈ ℕ) → 1 ∈ ℤ) |
272 | | elnnuz 11600 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 ∈ ℕ ↔ 𝑗 ∈
(ℤ≥‘1)) |
273 | | nnne0 10930 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑘 ∈ ℕ → 𝑘 ≠ 0) |
274 | 273 | neneqd 2787 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑘 ∈ ℕ → ¬
𝑘 = 0) |
275 | | biorf 419 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (¬
𝑘 = 0 → (2 ∥
𝑘 ↔ (𝑘 = 0 ∨ 2 ∥ 𝑘))) |
276 | 274, 275 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 ∈ ℕ → (2
∥ 𝑘 ↔ (𝑘 = 0 ∨ 2 ∥ 𝑘))) |
277 | 276 | bicomd 212 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 ∈ ℕ → ((𝑘 = 0 ∨ 2 ∥ 𝑘) ↔ 2 ∥ 𝑘)) |
278 | 277 | ifbid 4058 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 ∈ ℕ → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))) = if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 /
𝑛))↑𝑘) / 𝑘)))) |
279 | 91, 231, 233 | sylancl 693 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 ∈ ℕ → ((𝑘 ∈ ℕ0
↦ if((𝑘 = 0 ∨ 2
∥ 𝑘), 0,
((-1↑((𝑘 − 1) /
2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) = if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 /
𝑛))↑𝑘) / 𝑘)))) |
280 | 229, 230 | ifex 4106 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ if(2
∥ 𝑘, 0,
((-1↑((𝑘 − 1) /
2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) ∈ V |
281 | | eqid 2610 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 ∈ ℕ ↦ if(2
∥ 𝑘, 0,
((-1↑((𝑘 − 1) /
2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)))) = (𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘)))) |
282 | 281 | fvmpt2 6200 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑘 ∈ ℕ ∧ if(2
∥ 𝑘, 0,
((-1↑((𝑘 − 1) /
2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) ∈ V) → ((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) = if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 /
𝑛))↑𝑘) / 𝑘)))) |
283 | 280, 282 | mpan2 703 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 ∈ ℕ → ((𝑘 ∈ ℕ ↦ if(2
∥ 𝑘, 0,
((-1↑((𝑘 − 1) /
2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) = if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 /
𝑛))↑𝑘) / 𝑘)))) |
284 | 278, 279,
283 | 3eqtr4d 2654 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 ∈ ℕ → ((𝑘 ∈ ℕ0
↦ if((𝑘 = 0 ∨ 2
∥ 𝑘), 0,
((-1↑((𝑘 − 1) /
2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) = ((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘)) |
285 | 284 | rgen 2906 |
. . . . . . . . . . . . . . . . . 18
⊢
∀𝑘 ∈
ℕ ((𝑘 ∈
ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 /
𝑛))↑𝑘) / 𝑘))))‘𝑘) = ((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) |
286 | 285 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢
((⊤ ∧ 𝑛
∈ ℕ) → ∀𝑘 ∈ ℕ ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) = ((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘)) |
287 | | nfv 1830 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑗((𝑘 ∈ ℕ0
↦ if((𝑘 = 0 ∨ 2
∥ 𝑘), 0,
((-1↑((𝑘 − 1) /
2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) = ((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) |
288 | | nffvmpt1 6111 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑘((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗) |
289 | 243, 288 | nfeq 2762 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑘((𝑘 ∈ ℕ0
↦ if((𝑘 = 0 ∨ 2
∥ 𝑘), 0,
((-1↑((𝑘 − 1) /
2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗) = ((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗) |
290 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 𝑗 → ((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) = ((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗)) |
291 | 249, 290 | eqeq12d 2625 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 𝑗 → (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) = ((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) ↔ ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗) = ((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗))) |
292 | 287, 289,
291 | cbvral 3143 |
. . . . . . . . . . . . . . . . 17
⊢
(∀𝑘 ∈
ℕ ((𝑘 ∈
ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 /
𝑛))↑𝑘) / 𝑘))))‘𝑘) = ((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) ↔ ∀𝑗 ∈ ℕ ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗) = ((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗)) |
293 | 286, 292 | sylib 207 |
. . . . . . . . . . . . . . . 16
⊢
((⊤ ∧ 𝑛
∈ ℕ) → ∀𝑗 ∈ ℕ ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗) = ((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗)) |
294 | 293 | r19.21bi 2916 |
. . . . . . . . . . . . . . 15
⊢
(((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑗
∈ ℕ) → ((𝑘
∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 /
𝑛))↑𝑘) / 𝑘))))‘𝑗) = ((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗)) |
295 | 272, 294 | sylan2br 492 |
. . . . . . . . . . . . . 14
⊢
(((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑗
∈ (ℤ≥‘1)) → ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗) = ((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗)) |
296 | 271, 295 | seqfeq 12688 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑛
∈ ℕ) → seq1( + , (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))) = seq1( + , (𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘)))))) |
297 | 153, 162,
167 | abssubge0d 14018 |
. . . . . . . . . . . . . . 15
⊢
((⊤ ∧ 𝑛
∈ ℕ) → (abs‘(1 − (1 / 𝑛))) = (1 − (1 / 𝑛))) |
298 | | ltsubrp 11742 |
. . . . . . . . . . . . . . . 16
⊢ ((1
∈ ℝ ∧ (1 / 𝑛) ∈ ℝ+) → (1
− (1 / 𝑛)) <
1) |
299 | 169, 152,
298 | sylancr 694 |
. . . . . . . . . . . . . . 15
⊢
((⊤ ∧ 𝑛
∈ ℕ) → (1 − (1 / 𝑛)) < 1) |
300 | 297, 299 | eqbrtrd 4605 |
. . . . . . . . . . . . . 14
⊢
((⊤ ∧ 𝑛
∈ ℕ) → (abs‘(1 − (1 / 𝑛))) < 1) |
301 | 281 | atantayl2 24465 |
. . . . . . . . . . . . . 14
⊢ (((1
− (1 / 𝑛)) ∈
ℂ ∧ (abs‘(1 − (1 / 𝑛))) < 1) → seq1( + , (𝑘 ∈ ℕ ↦ if(2
∥ 𝑘, 0,
((-1↑((𝑘 − 1) /
2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))) ⇝ (arctan‘(1 − (1 /
𝑛)))) |
302 | 220, 300,
301 | syl2anc 691 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑛
∈ ℕ) → seq1( + , (𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))) ⇝ (arctan‘(1 − (1 /
𝑛)))) |
303 | 296, 302 | eqbrtrd 4605 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑛
∈ ℕ) → seq1( + , (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))) ⇝ (arctan‘(1 − (1 /
𝑛)))) |
304 | 270, 303 | syl5eqbr 4618 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑛
∈ ℕ) → seq(0 + 1)( + , (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))) ⇝ (arctan‘(1 − (1 /
𝑛)))) |
305 | 1, 267, 264, 304 | clim2ser2 14234 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑛
∈ ℕ) → seq0( + , (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))) ⇝ ((arctan‘(1 − (1 /
𝑛))) + (seq0( + , (𝑘 ∈ ℕ0
↦ if((𝑘 = 0 ∨ 2
∥ 𝑘), 0,
((-1↑((𝑘 − 1) /
2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)))))‘0))) |
306 | | 0z 11265 |
. . . . . . . . . . . . . 14
⊢ 0 ∈
ℤ |
307 | | seq1 12676 |
. . . . . . . . . . . . . 14
⊢ (0 ∈
ℤ → (seq0( + , (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘)))))‘0) = ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))‘0)) |
308 | 306, 307 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢ (seq0( +
, (𝑘 ∈
ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 /
𝑛))↑𝑘) / 𝑘)))))‘0) = ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))‘0) |
309 | | iftrue 4042 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑘 = 0 ∨ 2 ∥ 𝑘) → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 /
𝑛))↑𝑘) / 𝑘))) = 0) |
310 | 309 | orcs 408 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 0 → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))) = 0) |
311 | 310, 232,
229 | fvmpt 6191 |
. . . . . . . . . . . . . 14
⊢ (0 ∈
ℕ0 → ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))‘0) = 0) |
312 | 266, 311 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢ ((𝑘 ∈ ℕ0
↦ if((𝑘 = 0 ∨ 2
∥ 𝑘), 0,
((-1↑((𝑘 − 1) /
2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘0) = 0 |
313 | 308, 312 | eqtri 2632 |
. . . . . . . . . . . 12
⊢ (seq0( +
, (𝑘 ∈
ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 /
𝑛))↑𝑘) / 𝑘)))))‘0) = 0 |
314 | 313 | oveq2i 6560 |
. . . . . . . . . . 11
⊢
((arctan‘(1 − (1 / 𝑛))) + (seq0( + , (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘)))))‘0)) = ((arctan‘(1 −
(1 / 𝑛))) +
0) |
315 | | atanrecl 24438 |
. . . . . . . . . . . . . 14
⊢ ((1
− (1 / 𝑛)) ∈
ℝ → (arctan‘(1 − (1 / 𝑛))) ∈ ℝ) |
316 | 205, 315 | syl 17 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑛
∈ ℕ) → (arctan‘(1 − (1 / 𝑛))) ∈ ℝ) |
317 | 316 | recnd 9947 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑛
∈ ℕ) → (arctan‘(1 − (1 / 𝑛))) ∈ ℂ) |
318 | 317 | addid1d 10115 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑛
∈ ℕ) → ((arctan‘(1 − (1 / 𝑛))) + 0) = (arctan‘(1 − (1 /
𝑛)))) |
319 | 314, 318 | syl5eq 2656 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑛
∈ ℕ) → ((arctan‘(1 − (1 / 𝑛))) + (seq0( + , (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘)))))‘0)) = (arctan‘(1 − (1
/ 𝑛)))) |
320 | 305, 319 | breqtrd 4609 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑛
∈ ℕ) → seq0( + , (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))) ⇝ (arctan‘(1 − (1 /
𝑛)))) |
321 | 1, 199, 256, 265, 320 | isumclim 14330 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑛
∈ ℕ) → Σ𝑗 ∈ ℕ0 (((𝑘 ∈ ℕ0
↦ if((𝑘 = 0 ∨ 2
∥ 𝑘), 0,
((-1↑((𝑘 − 1) /
2)) / 𝑘)))‘𝑗) · ((1 − (1 /
𝑛))↑𝑗)) = (arctan‘(1 − (1 / 𝑛)))) |
322 | 321 | mpteq2dva 4672 |
. . . . . . 7
⊢ (⊤
→ (𝑛 ∈ ℕ
↦ Σ𝑗 ∈
ℕ0 (((𝑘
∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · ((1 − (1 / 𝑛))↑𝑗))) = (𝑛 ∈ ℕ ↦ (arctan‘(1
− (1 / 𝑛))))) |
323 | 198, 322 | eqtrd 2644 |
. . . . . 6
⊢ (⊤
→ ((𝑥 ∈ (0[,]1)
↦ Σ𝑗 ∈
ℕ0 (((𝑘
∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · (𝑥↑𝑗))) ∘ (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))) = (𝑛 ∈ ℕ ↦ (arctan‘(1
− (1 / 𝑛))))) |
324 | | oveq1 6556 |
. . . . . . . . . . . 12
⊢ (𝑥 = 1 → (𝑥↑𝑗) = (1↑𝑗)) |
325 | | nn0z 11277 |
. . . . . . . . . . . . 13
⊢ (𝑗 ∈ ℕ0
→ 𝑗 ∈
ℤ) |
326 | | 1exp 12751 |
. . . . . . . . . . . . 13
⊢ (𝑗 ∈ ℤ →
(1↑𝑗) =
1) |
327 | 325, 326 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝑗 ∈ ℕ0
→ (1↑𝑗) =
1) |
328 | 324, 327 | sylan9eq 2664 |
. . . . . . . . . . 11
⊢ ((𝑥 = 1 ∧ 𝑗 ∈ ℕ0) → (𝑥↑𝑗) = 1) |
329 | 328 | oveq2d 6565 |
. . . . . . . . . 10
⊢ ((𝑥 = 1 ∧ 𝑗 ∈ ℕ0) → (((𝑘 ∈ ℕ0
↦ if((𝑘 = 0 ∨ 2
∥ 𝑘), 0,
((-1↑((𝑘 − 1) /
2)) / 𝑘)))‘𝑗) · (𝑥↑𝑗)) = (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · 1)) |
330 | 18 | trud 1484 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ ℕ0
↦ if((𝑘 = 0 ∨ 2
∥ 𝑘), 0,
((-1↑((𝑘 − 1) /
2)) / 𝑘))):ℕ0⟶ℂ |
331 | 330 | ffvelrni 6266 |
. . . . . . . . . . . 12
⊢ (𝑗 ∈ ℕ0
→ ((𝑘 ∈
ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) ∈ ℂ) |
332 | 331 | mulid1d 9936 |
. . . . . . . . . . 11
⊢ (𝑗 ∈ ℕ0
→ (((𝑘 ∈
ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · 1) = ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗)) |
333 | 332 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝑥 = 1 ∧ 𝑗 ∈ ℕ0) → (((𝑘 ∈ ℕ0
↦ if((𝑘 = 0 ∨ 2
∥ 𝑘), 0,
((-1↑((𝑘 − 1) /
2)) / 𝑘)))‘𝑗) · 1) = ((𝑘 ∈ ℕ0
↦ if((𝑘 = 0 ∨ 2
∥ 𝑘), 0,
((-1↑((𝑘 − 1) /
2)) / 𝑘)))‘𝑗)) |
334 | 329, 333 | eqtrd 2644 |
. . . . . . . . 9
⊢ ((𝑥 = 1 ∧ 𝑗 ∈ ℕ0) → (((𝑘 ∈ ℕ0
↦ if((𝑘 = 0 ∨ 2
∥ 𝑘), 0,
((-1↑((𝑘 − 1) /
2)) / 𝑘)))‘𝑗) · (𝑥↑𝑗)) = ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗)) |
335 | 334 | sumeq2dv 14281 |
. . . . . . . 8
⊢ (𝑥 = 1 → Σ𝑗 ∈ ℕ0
(((𝑘 ∈
ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · (𝑥↑𝑗)) = Σ𝑗 ∈ ℕ0 ((𝑘 ∈ ℕ0
↦ if((𝑘 = 0 ∨ 2
∥ 𝑘), 0,
((-1↑((𝑘 − 1) /
2)) / 𝑘)))‘𝑗)) |
336 | | sumex 14266 |
. . . . . . . 8
⊢
Σ𝑗 ∈
ℕ0 ((𝑘
∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) ∈ V |
337 | 335, 148,
336 | fvmpt 6191 |
. . . . . . 7
⊢ (1 ∈
(0[,]1) → ((𝑥 ∈
(0[,]1) ↦ Σ𝑗
∈ ℕ0 (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · (𝑥↑𝑗)))‘1) = Σ𝑗 ∈ ℕ0 ((𝑘 ∈ ℕ0
↦ if((𝑘 = 0 ∨ 2
∥ 𝑘), 0,
((-1↑((𝑘 − 1) /
2)) / 𝑘)))‘𝑗)) |
338 | 190, 337 | mp1i 13 |
. . . . . 6
⊢ (⊤
→ ((𝑥 ∈ (0[,]1)
↦ Σ𝑗 ∈
ℕ0 (((𝑘
∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · (𝑥↑𝑗)))‘1) = Σ𝑗 ∈ ℕ0 ((𝑘 ∈ ℕ0
↦ if((𝑘 = 0 ∨ 2
∥ 𝑘), 0,
((-1↑((𝑘 − 1) /
2)) / 𝑘)))‘𝑗)) |
339 | 192, 323,
338 | 3brtr3d 4614 |
. . . . 5
⊢ (⊤
→ (𝑛 ∈ ℕ
↦ (arctan‘(1 − (1 / 𝑛)))) ⇝ Σ𝑗 ∈ ℕ0 ((𝑘 ∈ ℕ0
↦ if((𝑘 = 0 ∨ 2
∥ 𝑘), 0,
((-1↑((𝑘 − 1) /
2)) / 𝑘)))‘𝑗)) |
340 | | eqid 2610 |
. . . . . . . . 9
⊢ (ℂ
∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0)) |
341 | | eqid 2610 |
. . . . . . . . 9
⊢ {𝑥 ∈ ℂ ∣ (1 +
(𝑥↑2)) ∈ (ℂ
∖ (-∞(,]0))} = {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ
∖ (-∞(,]0))} |
342 | 340, 341 | atancn 24463 |
. . . . . . . 8
⊢ (arctan
↾ {𝑥 ∈ ℂ
∣ (1 + (𝑥↑2))
∈ (ℂ ∖ (-∞(,]0))}) ∈ ({𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ
∖ (-∞(,]0))}–cn→ℂ) |
343 | 342 | a1i 11 |
. . . . . . 7
⊢ (⊤
→ (arctan ↾ {𝑥
∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖
(-∞(,]0))}) ∈ ({𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ
∖ (-∞(,]0))}–cn→ℂ)) |
344 | | unitssre 12190 |
. . . . . . . . 9
⊢ (0[,]1)
⊆ ℝ |
345 | 340, 341 | ressatans 24461 |
. . . . . . . . 9
⊢ ℝ
⊆ {𝑥 ∈ ℂ
∣ (1 + (𝑥↑2))
∈ (ℂ ∖ (-∞(,]0))} |
346 | 344, 345 | sstri 3577 |
. . . . . . . 8
⊢ (0[,]1)
⊆ {𝑥 ∈ ℂ
∣ (1 + (𝑥↑2))
∈ (ℂ ∖ (-∞(,]0))} |
347 | | fss 5969 |
. . . . . . . 8
⊢ (((𝑛 ∈ ℕ ↦ (1
− (1 / 𝑛))):ℕ⟶(0[,]1) ∧ (0[,]1)
⊆ {𝑥 ∈ ℂ
∣ (1 + (𝑥↑2))
∈ (ℂ ∖ (-∞(,]0))}) → (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))):ℕ⟶{𝑥 ∈ ℂ ∣ (1 +
(𝑥↑2)) ∈ (ℂ
∖ (-∞(,]0))}) |
348 | 175, 346,
347 | sylancl 693 |
. . . . . . 7
⊢ (⊤
→ (𝑛 ∈ ℕ
↦ (1 − (1 / 𝑛))):ℕ⟶{𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ
∖ (-∞(,]0))}) |
349 | 345, 169 | sselii 3565 |
. . . . . . . 8
⊢ 1 ∈
{𝑥 ∈ ℂ ∣
(1 + (𝑥↑2)) ∈
(ℂ ∖ (-∞(,]0))} |
350 | 349 | a1i 11 |
. . . . . . 7
⊢ (⊤
→ 1 ∈ {𝑥 ∈
ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖
(-∞(,]0))}) |
351 | 75, 76, 343, 348, 189, 350 | climcncf 22511 |
. . . . . 6
⊢ (⊤
→ ((arctan ↾ {𝑥
∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖
(-∞(,]0))}) ∘ (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))) ⇝ ((arctan ↾
{𝑥 ∈ ℂ ∣
(1 + (𝑥↑2)) ∈
(ℂ ∖ (-∞(,]0))})‘1)) |
352 | 346, 173 | sseldi 3566 |
. . . . . . 7
⊢
((⊤ ∧ 𝑛
∈ ℕ) → (1 − (1 / 𝑛)) ∈ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ
∖ (-∞(,]0))}) |
353 | | cncff 22504 |
. . . . . . . . . 10
⊢ ((arctan
↾ {𝑥 ∈ ℂ
∣ (1 + (𝑥↑2))
∈ (ℂ ∖ (-∞(,]0))}) ∈ ({𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ
∖ (-∞(,]0))}–cn→ℂ) → (arctan ↾ {𝑥 ∈ ℂ ∣ (1 +
(𝑥↑2)) ∈ (ℂ
∖ (-∞(,]0))}):{𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ
∖ (-∞(,]0))}⟶ℂ) |
354 | 342, 353 | mp1i 13 |
. . . . . . . . 9
⊢ (⊤
→ (arctan ↾ {𝑥
∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖
(-∞(,]0))}):{𝑥 ∈
ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖
(-∞(,]0))}⟶ℂ) |
355 | 354 | feqmptd 6159 |
. . . . . . . 8
⊢ (⊤
→ (arctan ↾ {𝑥
∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖
(-∞(,]0))}) = (𝑘
∈ {𝑥 ∈ ℂ
∣ (1 + (𝑥↑2))
∈ (ℂ ∖ (-∞(,]0))} ↦ ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 +
(𝑥↑2)) ∈ (ℂ
∖ (-∞(,]0))})‘𝑘))) |
356 | | fvres 6117 |
. . . . . . . . 9
⊢ (𝑘 ∈ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ
∖ (-∞(,]0))} → ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ
∖ (-∞(,]0))})‘𝑘) = (arctan‘𝑘)) |
357 | 356 | mpteq2ia 4668 |
. . . . . . . 8
⊢ (𝑘 ∈ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ
∖ (-∞(,]0))} ↦ ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ
∖ (-∞(,]0))})‘𝑘)) = (𝑘 ∈ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ
∖ (-∞(,]0))} ↦ (arctan‘𝑘)) |
358 | 355, 357 | syl6eq 2660 |
. . . . . . 7
⊢ (⊤
→ (arctan ↾ {𝑥
∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖
(-∞(,]0))}) = (𝑘
∈ {𝑥 ∈ ℂ
∣ (1 + (𝑥↑2))
∈ (ℂ ∖ (-∞(,]0))} ↦ (arctan‘𝑘))) |
359 | | fveq2 6103 |
. . . . . . 7
⊢ (𝑘 = (1 − (1 / 𝑛)) → (arctan‘𝑘) = (arctan‘(1 − (1
/ 𝑛)))) |
360 | 352, 193,
358, 359 | fmptco 6303 |
. . . . . 6
⊢ (⊤
→ ((arctan ↾ {𝑥
∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖
(-∞(,]0))}) ∘ (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))) = (𝑛 ∈ ℕ ↦ (arctan‘(1
− (1 / 𝑛))))) |
361 | | fvres 6117 |
. . . . . . . 8
⊢ (1 ∈
{𝑥 ∈ ℂ ∣
(1 + (𝑥↑2)) ∈
(ℂ ∖ (-∞(,]0))} → ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ
∖ (-∞(,]0))})‘1) = (arctan‘1)) |
362 | 349, 361 | mp1i 13 |
. . . . . . 7
⊢ (⊤
→ ((arctan ↾ {𝑥
∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖
(-∞(,]0))})‘1) = (arctan‘1)) |
363 | | atan1 24455 |
. . . . . . 7
⊢
(arctan‘1) = (π / 4) |
364 | 362, 363 | syl6eq 2660 |
. . . . . 6
⊢ (⊤
→ ((arctan ↾ {𝑥
∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖
(-∞(,]0))})‘1) = (π / 4)) |
365 | 351, 360,
364 | 3brtr3d 4614 |
. . . . 5
⊢ (⊤
→ (𝑛 ∈ ℕ
↦ (arctan‘(1 − (1 / 𝑛)))) ⇝ (π / 4)) |
366 | | climuni 14131 |
. . . . 5
⊢ (((𝑛 ∈ ℕ ↦
(arctan‘(1 − (1 / 𝑛)))) ⇝ Σ𝑗 ∈ ℕ0 ((𝑘 ∈ ℕ0
↦ if((𝑘 = 0 ∨ 2
∥ 𝑘), 0,
((-1↑((𝑘 − 1) /
2)) / 𝑘)))‘𝑗) ∧ (𝑛 ∈ ℕ ↦ (arctan‘(1
− (1 / 𝑛)))) ⇝
(π / 4)) → Σ𝑗
∈ ℕ0 ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) = (π / 4)) |
367 | 339, 365,
366 | syl2anc 691 |
. . . 4
⊢ (⊤
→ Σ𝑗 ∈
ℕ0 ((𝑘
∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) = (π / 4)) |
368 | 147, 367 | breqtrd 4609 |
. . 3
⊢ (⊤
→ seq0( + , (𝑘 ∈
ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ⇝ (π / 4)) |
369 | 368 | trud 1484 |
. 2
⊢ seq0( + ,
(𝑘 ∈
ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ⇝ (π / 4) |
370 | | leibpi.1 |
. . 3
⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦
((-1↑𝑛) / ((2 ·
𝑛) + 1))) |
371 | | ovex 6577 |
. . 3
⊢ (π /
4) ∈ V |
372 | 370, 17, 371 | leibpilem2 24468 |
. 2
⊢ (seq0( +
, 𝐹) ⇝ (π / 4)
↔ seq0( + , (𝑘 ∈
ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ⇝ (π / 4)) |
373 | 369, 372 | mpbir 220 |
1
⊢ seq0( + ,
𝐹) ⇝ (π /
4) |