| Step | Hyp | Ref
| Expression |
| 1 | | nn0uz 12920 |
. . . . 5
⊢
ℕ0 = (ℤ≥‘0) |
| 2 | | 0zd 12625 |
. . . . 5
⊢ (⊤
→ 0 ∈ ℤ) |
| 3 | | eqidd 2738 |
. . . . 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 11254 |
. . . . . . . . 9
⊢ ((𝑘 ∈ ℕ0
∧ (𝑘 = 0 ∨ 2 ∥
𝑘)) → 0 ∈
ℂ) |
| 5 | | ioran 986 |
. . . . . . . . . 10
⊢ (¬
(𝑘 = 0 ∨ 2 ∥ 𝑘) ↔ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) |
| 6 | | neg1rr 12381 |
. . . . . . . . . . . . 13
⊢ -1 ∈
ℝ |
| 7 | | leibpilem1 26983 |
. . . . . . . . . . . . . 14
⊢ ((𝑘 ∈ ℕ0
∧ (¬ 𝑘 = 0 ∧
¬ 2 ∥ 𝑘)) →
(𝑘 ∈ ℕ ∧
((𝑘 − 1) / 2) ∈
ℕ0)) |
| 8 | 7 | simprd 495 |
. . . . . . . . . . . . 13
⊢ ((𝑘 ∈ ℕ0
∧ (¬ 𝑘 = 0 ∧
¬ 2 ∥ 𝑘)) →
((𝑘 − 1) / 2) ∈
ℕ0) |
| 9 | | reexpcl 14119 |
. . . . . . . . . . . . 13
⊢ ((-1
∈ ℝ ∧ ((𝑘
− 1) / 2) ∈ ℕ0) → (-1↑((𝑘 − 1) / 2)) ∈
ℝ) |
| 10 | 6, 8, 9 | sylancr 587 |
. . . . . . . . . . . 12
⊢ ((𝑘 ∈ ℕ0
∧ (¬ 𝑘 = 0 ∧
¬ 2 ∥ 𝑘)) →
(-1↑((𝑘 − 1) /
2)) ∈ ℝ) |
| 11 | 7 | simpld 494 |
. . . . . . . . . . . 12
⊢ ((𝑘 ∈ ℕ0
∧ (¬ 𝑘 = 0 ∧
¬ 2 ∥ 𝑘)) →
𝑘 ∈
ℕ) |
| 12 | 10, 11 | nndivred 12320 |
. . . . . . . . . . 11
⊢ ((𝑘 ∈ ℕ0
∧ (¬ 𝑘 = 0 ∧
¬ 2 ∥ 𝑘)) →
((-1↑((𝑘 − 1) /
2)) / 𝑘) ∈
ℝ) |
| 13 | 12 | recnd 11289 |
. . . . . . . . . 10
⊢ ((𝑘 ∈ ℕ0
∧ (¬ 𝑘 = 0 ∧
¬ 2 ∥ 𝑘)) →
((-1↑((𝑘 − 1) /
2)) / 𝑘) ∈
ℂ) |
| 14 | 5, 13 | sylan2b 594 |
. . . . . . . . 9
⊢ ((𝑘 ∈ ℕ0
∧ ¬ (𝑘 = 0 ∨ 2
∥ 𝑘)) →
((-1↑((𝑘 − 1) /
2)) / 𝑘) ∈
ℂ) |
| 15 | 4, 14 | ifclda 4561 |
. . . . . . . 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 | 16 | fmpttd 7135 |
. . . . . 6
⊢ (⊤
→ (𝑘 ∈
ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘))):ℕ0⟶ℂ) |
| 18 | 17 | ffvelcdmda 7104 |
. . . . 5
⊢
((⊤ ∧ 𝑗
∈ ℕ0) → ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) ∈ ℂ) |
| 19 | | 2nn0 12543 |
. . . . . . . . . . . . . 14
⊢ 2 ∈
ℕ0 |
| 20 | 19 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (⊤
→ 2 ∈ ℕ0) |
| 21 | | nn0mulcl 12562 |
. . . . . . . . . . . . 13
⊢ ((2
∈ ℕ0 ∧ 𝑛 ∈ ℕ0) → (2
· 𝑛) ∈
ℕ0) |
| 22 | 20, 21 | sylan 580 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑛
∈ ℕ0) → (2 · 𝑛) ∈
ℕ0) |
| 23 | | nn0p1nn 12565 |
. . . . . . . . . . . 12
⊢ ((2
· 𝑛) ∈
ℕ0 → ((2 · 𝑛) + 1) ∈ ℕ) |
| 24 | 22, 23 | syl 17 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑛
∈ ℕ0) → ((2 · 𝑛) + 1) ∈ ℕ) |
| 25 | 24 | nnrecred 12317 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑛
∈ ℕ0) → (1 / ((2 · 𝑛) + 1)) ∈ ℝ) |
| 26 | 25 | fmpttd 7135 |
. . . . . . . . 9
⊢ (⊤
→ (𝑛 ∈
ℕ0 ↦ (1 / ((2 · 𝑛) +
1))):ℕ0⟶ℝ) |
| 27 | | nn0mulcl 12562 |
. . . . . . . . . . . . . 14
⊢ ((2
∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (2
· 𝑘) ∈
ℕ0) |
| 28 | 20, 27 | sylan 580 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → (2 · 𝑘) ∈
ℕ0) |
| 29 | 28 | nn0red 12588 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → (2 · 𝑘) ∈ ℝ) |
| 30 | | peano2nn0 12566 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ ℕ0
→ (𝑘 + 1) ∈
ℕ0) |
| 31 | 30 | adantl 481 |
. . . . . . . . . . . . . 14
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → (𝑘 + 1) ∈
ℕ0) |
| 32 | | nn0mulcl 12562 |
. . . . . . . . . . . . . 14
⊢ ((2
∈ ℕ0 ∧ (𝑘 + 1) ∈ ℕ0) → (2
· (𝑘 + 1)) ∈
ℕ0) |
| 33 | 19, 31, 32 | sylancr 587 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → (2 · (𝑘 + 1)) ∈
ℕ0) |
| 34 | 33 | nn0red 12588 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → (2 · (𝑘 + 1)) ∈ ℝ) |
| 35 | | 1red 11262 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → 1 ∈ ℝ) |
| 36 | | nn0re 12535 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ ℕ0
→ 𝑘 ∈
ℝ) |
| 37 | 36 | adantl 481 |
. . . . . . . . . . . . . 14
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → 𝑘 ∈ ℝ) |
| 38 | 37 | lep1d 12199 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → 𝑘 ≤ (𝑘 + 1)) |
| 39 | | peano2re 11434 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ ℝ → (𝑘 + 1) ∈
ℝ) |
| 40 | 37, 39 | syl 17 |
. . . . . . . . . . . . . 14
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → (𝑘 + 1) ∈ ℝ) |
| 41 | | 2re 12340 |
. . . . . . . . . . . . . . 15
⊢ 2 ∈
ℝ |
| 42 | 41 | a1i 11 |
. . . . . . . . . . . . . 14
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → 2 ∈ ℝ) |
| 43 | | 2pos 12369 |
. . . . . . . . . . . . . . 15
⊢ 0 <
2 |
| 44 | 43 | a1i 11 |
. . . . . . . . . . . . . 14
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → 0 < 2) |
| 45 | | lemul2 12120 |
. . . . . . . . . . . . . 14
⊢ ((𝑘 ∈ ℝ ∧ (𝑘 + 1) ∈ ℝ ∧ (2
∈ ℝ ∧ 0 < 2)) → (𝑘 ≤ (𝑘 + 1) ↔ (2 · 𝑘) ≤ (2 · (𝑘 + 1)))) |
| 46 | 37, 40, 42, 44, 45 | syl112anc 1376 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → (𝑘 ≤ (𝑘 + 1) ↔ (2 · 𝑘) ≤ (2 · (𝑘 + 1)))) |
| 47 | 38, 46 | mpbid 232 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → (2 · 𝑘) ≤ (2 · (𝑘 + 1))) |
| 48 | 29, 34, 35, 47 | leadd1dd 11877 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → ((2 · 𝑘) + 1) ≤ ((2 · (𝑘 + 1)) + 1)) |
| 49 | | nn0p1nn 12565 |
. . . . . . . . . . . . . 14
⊢ ((2
· 𝑘) ∈
ℕ0 → ((2 · 𝑘) + 1) ∈ ℕ) |
| 50 | 28, 49 | syl 17 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → ((2 · 𝑘) + 1) ∈ ℕ) |
| 51 | 50 | nnred 12281 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → ((2 · 𝑘) + 1) ∈ ℝ) |
| 52 | 50 | nngt0d 12315 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → 0 < ((2 · 𝑘) + 1)) |
| 53 | | nn0p1nn 12565 |
. . . . . . . . . . . . . 14
⊢ ((2
· (𝑘 + 1)) ∈
ℕ0 → ((2 · (𝑘 + 1)) + 1) ∈ ℕ) |
| 54 | 33, 53 | syl 17 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → ((2 · (𝑘 + 1)) + 1) ∈ ℕ) |
| 55 | 54 | nnred 12281 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → ((2 · (𝑘 + 1)) + 1) ∈ ℝ) |
| 56 | 54 | nngt0d 12315 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → 0 < ((2 · (𝑘 + 1)) + 1)) |
| 57 | | lerec 12151 |
. . . . . . . . . . . 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)))) |
| 58 | 51, 52, 55, 56, 57 | syl22anc 839 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → (((2 · 𝑘) + 1) ≤ ((2 · (𝑘 + 1)) + 1) ↔ (1 / ((2 · (𝑘 + 1)) + 1)) ≤ (1 / ((2
· 𝑘) +
1)))) |
| 59 | 48, 58 | mpbid 232 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → (1 / ((2 · (𝑘 + 1)) + 1)) ≤ (1 / ((2 · 𝑘) + 1))) |
| 60 | | oveq2 7439 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = (𝑘 + 1) → (2 · 𝑛) = (2 · (𝑘 + 1))) |
| 61 | 60 | oveq1d 7446 |
. . . . . . . . . . . . 13
⊢ (𝑛 = (𝑘 + 1) → ((2 · 𝑛) + 1) = ((2 · (𝑘 + 1)) + 1)) |
| 62 | 61 | oveq2d 7447 |
. . . . . . . . . . . 12
⊢ (𝑛 = (𝑘 + 1) → (1 / ((2 · 𝑛) + 1)) = (1 / ((2 ·
(𝑘 + 1)) +
1))) |
| 63 | | eqid 2737 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ ℕ0
↦ (1 / ((2 · 𝑛) + 1))) = (𝑛 ∈ ℕ0 ↦ (1 / ((2
· 𝑛) +
1))) |
| 64 | | ovex 7464 |
. . . . . . . . . . . 12
⊢ (1 / ((2
· (𝑘 + 1)) + 1))
∈ V |
| 65 | 62, 63, 64 | fvmpt 7016 |
. . . . . . . . . . 11
⊢ ((𝑘 + 1) ∈ ℕ0
→ ((𝑛 ∈
ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘(𝑘 + 1)) = (1 / ((2 · (𝑘 + 1)) + 1))) |
| 66 | 31, 65 | syl 17 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ (1 / ((2
· 𝑛) +
1)))‘(𝑘 + 1)) = (1 /
((2 · (𝑘 + 1)) +
1))) |
| 67 | | oveq2 7439 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = 𝑘 → (2 · 𝑛) = (2 · 𝑘)) |
| 68 | 67 | oveq1d 7446 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝑘 → ((2 · 𝑛) + 1) = ((2 · 𝑘) + 1)) |
| 69 | 68 | oveq2d 7447 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑘 → (1 / ((2 · 𝑛) + 1)) = (1 / ((2 · 𝑘) + 1))) |
| 70 | | ovex 7464 |
. . . . . . . . . . . 12
⊢ (1 / ((2
· 𝑘) + 1)) ∈
V |
| 71 | 69, 63, 70 | fvmpt 7016 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ ℕ0
→ ((𝑛 ∈
ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘) = (1 / ((2 · 𝑘) + 1))) |
| 72 | 71 | adantl 481 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ (1 / ((2
· 𝑛) +
1)))‘𝑘) = (1 / ((2
· 𝑘) +
1))) |
| 73 | 59, 66, 72 | 3brtr4d 5175 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ (1 / ((2
· 𝑛) +
1)))‘(𝑘 + 1)) ≤
((𝑛 ∈
ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘)) |
| 74 | | nnuz 12921 |
. . . . . . . . . 10
⊢ ℕ =
(ℤ≥‘1) |
| 75 | | 1zzd 12648 |
. . . . . . . . . 10
⊢ (⊤
→ 1 ∈ ℤ) |
| 76 | | ax-1cn 11213 |
. . . . . . . . . . 11
⊢ 1 ∈
ℂ |
| 77 | | divcnv 15889 |
. . . . . . . . . . 11
⊢ (1 ∈
ℂ → (𝑛 ∈
ℕ ↦ (1 / 𝑛))
⇝ 0) |
| 78 | 76, 77 | mp1i 13 |
. . . . . . . . . 10
⊢ (⊤
→ (𝑛 ∈ ℕ
↦ (1 / 𝑛)) ⇝
0) |
| 79 | | nn0ex 12532 |
. . . . . . . . . . . 12
⊢
ℕ0 ∈ V |
| 80 | 79 | mptex 7243 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ ℕ0
↦ (1 / ((2 · 𝑛) + 1))) ∈ V |
| 81 | 80 | a1i 11 |
. . . . . . . . . 10
⊢ (⊤
→ (𝑛 ∈
ℕ0 ↦ (1 / ((2 · 𝑛) + 1))) ∈ V) |
| 82 | | oveq2 7439 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝑘 → (1 / 𝑛) = (1 / 𝑘)) |
| 83 | | eqid 2737 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ ℕ ↦ (1 /
𝑛)) = (𝑛 ∈ ℕ ↦ (1 / 𝑛)) |
| 84 | | ovex 7464 |
. . . . . . . . . . . . 13
⊢ (1 /
𝑘) ∈
V |
| 85 | 82, 83, 84 | fvmpt 7016 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (1 /
𝑛))‘𝑘) = (1 / 𝑘)) |
| 86 | 85 | adantl 481 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((𝑛
∈ ℕ ↦ (1 / 𝑛))‘𝑘) = (1 / 𝑘)) |
| 87 | | nnrecre 12308 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ ℕ → (1 /
𝑘) ∈
ℝ) |
| 88 | 87 | adantl 481 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (1 / 𝑘) ∈ ℝ) |
| 89 | 86, 88 | eqeltrd 2841 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((𝑛
∈ ℕ ↦ (1 / 𝑛))‘𝑘) ∈ ℝ) |
| 90 | | nnnn0 12533 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ ℕ → 𝑘 ∈
ℕ0) |
| 91 | 90 | adantl 481 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑘
∈ ℕ) → 𝑘
∈ ℕ0) |
| 92 | 91, 71 | syl 17 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((𝑛
∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘) = (1 / ((2 · 𝑘) + 1))) |
| 93 | 90, 50 | sylan2 593 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((2 · 𝑘) + 1) ∈ ℕ) |
| 94 | 93 | nnrecred 12317 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (1 / ((2 · 𝑘) + 1)) ∈ ℝ) |
| 95 | 92, 94 | eqeltrd 2841 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((𝑛
∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘) ∈ ℝ) |
| 96 | | nnre 12273 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ ℕ → 𝑘 ∈
ℝ) |
| 97 | 96 | adantl 481 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑘
∈ ℕ) → 𝑘
∈ ℝ) |
| 98 | 19, 91, 27 | sylancr 587 |
. . . . . . . . . . . . . 14
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (2 · 𝑘) ∈
ℕ0) |
| 99 | 98 | nn0red 12588 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (2 · 𝑘) ∈ ℝ) |
| 100 | | peano2re 11434 |
. . . . . . . . . . . . . 14
⊢ ((2
· 𝑘) ∈ ℝ
→ ((2 · 𝑘) + 1)
∈ ℝ) |
| 101 | 99, 100 | syl 17 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((2 · 𝑘) + 1) ∈ ℝ) |
| 102 | | nn0addge1 12572 |
. . . . . . . . . . . . . . 15
⊢ ((𝑘 ∈ ℝ ∧ 𝑘 ∈ ℕ0)
→ 𝑘 ≤ (𝑘 + 𝑘)) |
| 103 | 97, 91, 102 | syl2anc 584 |
. . . . . . . . . . . . . 14
⊢
((⊤ ∧ 𝑘
∈ ℕ) → 𝑘
≤ (𝑘 + 𝑘)) |
| 104 | 97 | recnd 11289 |
. . . . . . . . . . . . . . 15
⊢
((⊤ ∧ 𝑘
∈ ℕ) → 𝑘
∈ ℂ) |
| 105 | 104 | 2timesd 12509 |
. . . . . . . . . . . . . 14
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (2 · 𝑘) = (𝑘 + 𝑘)) |
| 106 | 103, 105 | breqtrrd 5171 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑘
∈ ℕ) → 𝑘
≤ (2 · 𝑘)) |
| 107 | 99 | lep1d 12199 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (2 · 𝑘) ≤ ((2 · 𝑘) + 1)) |
| 108 | 97, 99, 101, 106, 107 | letrd 11418 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑘
∈ ℕ) → 𝑘
≤ ((2 · 𝑘) +
1)) |
| 109 | | nngt0 12297 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ ℕ → 0 <
𝑘) |
| 110 | 109 | adantl 481 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑘
∈ ℕ) → 0 < 𝑘) |
| 111 | 93 | nnred 12281 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((2 · 𝑘) + 1) ∈ ℝ) |
| 112 | 93 | nngt0d 12315 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑘
∈ ℕ) → 0 < ((2 · 𝑘) + 1)) |
| 113 | | lerec 12151 |
. . . . . . . . . . . . 13
⊢ (((𝑘 ∈ ℝ ∧ 0 <
𝑘) ∧ (((2 ·
𝑘) + 1) ∈ ℝ
∧ 0 < ((2 · 𝑘) + 1))) → (𝑘 ≤ ((2 · 𝑘) + 1) ↔ (1 / ((2 · 𝑘) + 1)) ≤ (1 / 𝑘))) |
| 114 | 97, 110, 111, 112, 113 | syl22anc 839 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝑘
≤ ((2 · 𝑘) + 1)
↔ (1 / ((2 · 𝑘)
+ 1)) ≤ (1 / 𝑘))) |
| 115 | 108, 114 | mpbid 232 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (1 / ((2 · 𝑘) + 1)) ≤ (1 / 𝑘)) |
| 116 | 115, 92, 86 | 3brtr4d 5175 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((𝑛
∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘) ≤ ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘)) |
| 117 | 93 | nnrpd 13075 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((2 · 𝑘) + 1) ∈
ℝ+) |
| 118 | 117 | rpreccld 13087 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (1 / ((2 · 𝑘) + 1)) ∈
ℝ+) |
| 119 | 118 | rpge0d 13081 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑘
∈ ℕ) → 0 ≤ (1 / ((2 · 𝑘) + 1))) |
| 120 | 119, 92 | breqtrrd 5171 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑘
∈ ℕ) → 0 ≤ ((𝑛 ∈ ℕ0 ↦ (1 / ((2
· 𝑛) +
1)))‘𝑘)) |
| 121 | 74, 75, 78, 81, 89, 95, 116, 120 | climsqz2 15678 |
. . . . . . . . 9
⊢ (⊤
→ (𝑛 ∈
ℕ0 ↦ (1 / ((2 · 𝑛) + 1))) ⇝ 0) |
| 122 | | neg1cn 12380 |
. . . . . . . . . . . . 13
⊢ -1 ∈
ℂ |
| 123 | 122 | a1i 11 |
. . . . . . . . . . . 12
⊢ (⊤
→ -1 ∈ ℂ) |
| 124 | | expcl 14120 |
. . . . . . . . . . . 12
⊢ ((-1
∈ ℂ ∧ 𝑘
∈ ℕ0) → (-1↑𝑘) ∈ ℂ) |
| 125 | 123, 124 | sylan 580 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → (-1↑𝑘) ∈ ℂ) |
| 126 | 50 | nncnd 12282 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → ((2 · 𝑘) + 1) ∈ ℂ) |
| 127 | 50 | nnne0d 12316 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → ((2 · 𝑘) + 1) ≠ 0) |
| 128 | 125, 126,
127 | divrecd 12046 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → ((-1↑𝑘) / ((2 · 𝑘) + 1)) = ((-1↑𝑘) · (1 / ((2 · 𝑘) + 1)))) |
| 129 | | oveq2 7439 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝑘 → (-1↑𝑛) = (-1↑𝑘)) |
| 130 | 129, 68 | oveq12d 7449 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑘 → ((-1↑𝑛) / ((2 · 𝑛) + 1)) = ((-1↑𝑘) / ((2 · 𝑘) + 1))) |
| 131 | | eqid 2737 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ ℕ0
↦ ((-1↑𝑛) / ((2
· 𝑛) + 1))) = (𝑛 ∈ ℕ0
↦ ((-1↑𝑛) / ((2
· 𝑛) +
1))) |
| 132 | | ovex 7464 |
. . . . . . . . . . . 12
⊢
((-1↑𝑘) / ((2
· 𝑘) + 1)) ∈
V |
| 133 | 130, 131,
132 | fvmpt 7016 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ ℕ0
→ ((𝑛 ∈
ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))‘𝑘) = ((-1↑𝑘) / ((2 · 𝑘) + 1))) |
| 134 | 133 | adantl 481 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦
((-1↑𝑛) / ((2 ·
𝑛) + 1)))‘𝑘) = ((-1↑𝑘) / ((2 · 𝑘) + 1))) |
| 135 | 72 | oveq2d 7447 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → ((-1↑𝑘) · ((𝑛 ∈ ℕ0 ↦ (1 / ((2
· 𝑛) +
1)))‘𝑘)) =
((-1↑𝑘) · (1 /
((2 · 𝑘) +
1)))) |
| 136 | 128, 134,
135 | 3eqtr4d 2787 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦
((-1↑𝑛) / ((2 ·
𝑛) + 1)))‘𝑘) = ((-1↑𝑘) · ((𝑛 ∈ ℕ0 ↦ (1 / ((2
· 𝑛) +
1)))‘𝑘))) |
| 137 | 1, 2, 26, 73, 121, 136 | iseralt 15721 |
. . . . . . . 8
⊢ (⊤
→ seq0( + , (𝑛 ∈
ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))) ∈ dom ⇝
) |
| 138 | | climdm 15590 |
. . . . . . . 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)))))) |
| 139 | 137, 138 | sylib 218 |
. . . . . . 7
⊢ (⊤
→ seq0( + , (𝑛 ∈
ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))) ⇝ ( ⇝ ‘seq0( + ,
(𝑛 ∈
ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))))) |
| 140 | | eqid 2737 |
. . . . . . . 8
⊢ (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))) |
| 141 | | fvex 6919 |
. . . . . . . 8
⊢ ( ⇝
‘seq0( + , (𝑛 ∈
ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1))))) ∈ V |
| 142 | 131, 140,
141 | leibpilem2 26984 |
. . . . . . 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 | 139, 142 | sylib 218 |
. . . . . 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 14044 |
. . . . . . 7
⊢ seq0( + ,
(𝑘 ∈
ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ∈ V |
| 145 | 144, 141 | breldm 5919 |
. . . . . 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, 18, 146 | isumclim2 15794 |
. . . 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 2737 |
. . . . . . . 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 | 17, 146, 148 | abelth2 26486 |
. . . . . . 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 13046 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℝ+) |
| 151 | 150 | adantl 481 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑛
∈ ℕ) → 𝑛
∈ ℝ+) |
| 152 | 151 | rpreccld 13087 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑛
∈ ℕ) → (1 / 𝑛) ∈
ℝ+) |
| 153 | 152 | rpred 13077 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑛
∈ ℕ) → (1 / 𝑛) ∈ ℝ) |
| 154 | 152 | rpge0d 13081 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑛
∈ ℕ) → 0 ≤ (1 / 𝑛)) |
| 155 | | nnge1 12294 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ ℕ → 1 ≤
𝑛) |
| 156 | 155 | adantl 481 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑛
∈ ℕ) → 1 ≤ 𝑛) |
| 157 | | nnre 12273 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℝ) |
| 158 | 157 | adantl 481 |
. . . . . . . . . . . . . 14
⊢
((⊤ ∧ 𝑛
∈ ℕ) → 𝑛
∈ ℝ) |
| 159 | 158 | recnd 11289 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑛
∈ ℕ) → 𝑛
∈ ℂ) |
| 160 | 159 | mulridd 11278 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑛
∈ ℕ) → (𝑛
· 1) = 𝑛) |
| 161 | 156, 160 | breqtrrd 5171 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑛
∈ ℕ) → 1 ≤ (𝑛 · 1)) |
| 162 | | 1red 11262 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑛
∈ ℕ) → 1 ∈ ℝ) |
| 163 | | nngt0 12297 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ ℕ → 0 <
𝑛) |
| 164 | 163 | adantl 481 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑛
∈ ℕ) → 0 < 𝑛) |
| 165 | | ledivmul 12144 |
. . . . . . . . . . . 12
⊢ ((1
∈ ℝ ∧ 1 ∈ ℝ ∧ (𝑛 ∈ ℝ ∧ 0 < 𝑛)) → ((1 / 𝑛) ≤ 1 ↔ 1 ≤ (𝑛 · 1))) |
| 166 | 162, 162,
158, 164, 165 | syl112anc 1376 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑛
∈ ℕ) → ((1 / 𝑛) ≤ 1 ↔ 1 ≤ (𝑛 · 1))) |
| 167 | 161, 166 | mpbird 257 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑛
∈ ℕ) → (1 / 𝑛) ≤ 1) |
| 168 | | elicc01 13506 |
. . . . . . . . . 10
⊢ ((1 /
𝑛) ∈ (0[,]1) ↔
((1 / 𝑛) ∈ ℝ
∧ 0 ≤ (1 / 𝑛) ∧
(1 / 𝑛) ≤
1)) |
| 169 | 153, 154,
167, 168 | syl3anbrc 1344 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑛
∈ ℕ) → (1 / 𝑛) ∈ (0[,]1)) |
| 170 | | iirev 24956 |
. . . . . . . . 9
⊢ ((1 /
𝑛) ∈ (0[,]1) → (1
− (1 / 𝑛)) ∈
(0[,]1)) |
| 171 | 169, 170 | syl 17 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑛
∈ ℕ) → (1 − (1 / 𝑛)) ∈ (0[,]1)) |
| 172 | 171 | fmpttd 7135 |
. . . . . . 7
⊢ (⊤
→ (𝑛 ∈ ℕ
↦ (1 − (1 / 𝑛))):ℕ⟶(0[,]1)) |
| 173 | | 1cnd 11256 |
. . . . . . . . 9
⊢ (⊤
→ 1 ∈ ℂ) |
| 174 | | nnex 12272 |
. . . . . . . . . . 11
⊢ ℕ
∈ V |
| 175 | 174 | mptex 7243 |
. . . . . . . . . 10
⊢ (𝑛 ∈ ℕ ↦ (1
− (1 / 𝑛))) ∈
V |
| 176 | 175 | a1i 11 |
. . . . . . . . 9
⊢ (⊤
→ (𝑛 ∈ ℕ
↦ (1 − (1 / 𝑛))) ∈ V) |
| 177 | 89 | recnd 11289 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((𝑛
∈ ℕ ↦ (1 / 𝑛))‘𝑘) ∈ ℂ) |
| 178 | 82 | oveq2d 7447 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑘 → (1 − (1 / 𝑛)) = (1 − (1 / 𝑘))) |
| 179 | | eqid 2737 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ ℕ ↦ (1
− (1 / 𝑛))) = (𝑛 ∈ ℕ ↦ (1
− (1 / 𝑛))) |
| 180 | | ovex 7464 |
. . . . . . . . . . . 12
⊢ (1
− (1 / 𝑘)) ∈
V |
| 181 | 178, 179,
180 | fvmpt 7016 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (1
− (1 / 𝑛)))‘𝑘) = (1 − (1 / 𝑘))) |
| 182 | 85 | oveq2d 7447 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ ℕ → (1
− ((𝑛 ∈ ℕ
↦ (1 / 𝑛))‘𝑘)) = (1 − (1 / 𝑘))) |
| 183 | 181, 182 | eqtr4d 2780 |
. . . . . . . . . 10
⊢ (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (1
− (1 / 𝑛)))‘𝑘) = (1 − ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘))) |
| 184 | 183 | adantl 481 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((𝑛
∈ ℕ ↦ (1 − (1 / 𝑛)))‘𝑘) = (1 − ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘))) |
| 185 | 74, 75, 78, 173, 176, 177, 184 | climsubc2 15675 |
. . . . . . . 8
⊢ (⊤
→ (𝑛 ∈ ℕ
↦ (1 − (1 / 𝑛))) ⇝ (1 − 0)) |
| 186 | | 1m0e1 12387 |
. . . . . . . 8
⊢ (1
− 0) = 1 |
| 187 | 185, 186 | breqtrdi 5184 |
. . . . . . 7
⊢ (⊤
→ (𝑛 ∈ ℕ
↦ (1 − (1 / 𝑛))) ⇝ 1) |
| 188 | | 1elunit 13510 |
. . . . . . . 8
⊢ 1 ∈
(0[,]1) |
| 189 | 188 | a1i 11 |
. . . . . . 7
⊢ (⊤
→ 1 ∈ (0[,]1)) |
| 190 | 74, 75, 149, 172, 187, 189 | climcncf 24926 |
. . . . . 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)) |
| 191 | | eqidd 2738 |
. . . . . . . 8
⊢ (⊤
→ (𝑛 ∈ ℕ
↦ (1 − (1 / 𝑛))) = (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))) |
| 192 | | eqidd 2738 |
. . . . . . . 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)))) |
| 193 | | oveq1 7438 |
. . . . . . . . . 10
⊢ (𝑥 = (1 − (1 / 𝑛)) → (𝑥↑𝑗) = ((1 − (1 / 𝑛))↑𝑗)) |
| 194 | 193 | oveq2d 7447 |
. . . . . . . . 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))) |
| 195 | 194 | sumeq2sdv 15739 |
. . . . . . . 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))) |
| 196 | 171, 191,
192, 195 | fmptco 7149 |
. . . . . . 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)))) |
| 197 | | 0zd 12625 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑛
∈ ℕ) → 0 ∈ ℤ) |
| 198 | 8 | adantll 714 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((⊤ ∧ 𝑘
∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((𝑘 − 1) / 2) ∈
ℕ0) |
| 199 | 6, 198, 9 | sylancr 587 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((⊤ ∧ 𝑘
∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (-1↑((𝑘 − 1) / 2)) ∈
ℝ) |
| 200 | 199 | recnd 11289 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((⊤ ∧ 𝑘
∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (-1↑((𝑘 − 1) / 2)) ∈
ℂ) |
| 201 | 200 | adantllr 719 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (-1↑((𝑘 − 1) / 2)) ∈
ℂ) |
| 202 | | 1re 11261 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ 1 ∈
ℝ |
| 203 | | resubcl 11573 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((1
∈ ℝ ∧ (1 / 𝑛) ∈ ℝ) → (1 − (1 /
𝑛)) ∈
ℝ) |
| 204 | 202, 153,
203 | sylancr 587 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((⊤ ∧ 𝑛
∈ ℕ) → (1 − (1 / 𝑛)) ∈ ℝ) |
| 205 | 204 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (1 − (1 / 𝑛)) ∈ ℝ) |
| 206 | | simplr 769 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → 𝑘 ∈ ℕ0) |
| 207 | 205, 206 | reexpcld 14203 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((1 − (1 / 𝑛))↑𝑘) ∈ ℝ) |
| 208 | 207 | recnd 11289 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((1 − (1 / 𝑛))↑𝑘) ∈ ℂ) |
| 209 | | nn0cn 12536 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 ∈ ℕ0
→ 𝑘 ∈
ℂ) |
| 210 | 209 | ad2antlr 727 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → 𝑘 ∈ ℂ) |
| 211 | 11 | adantll 714 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → 𝑘 ∈ ℕ) |
| 212 | 211 | nnne0d 12316 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → 𝑘 ≠ 0) |
| 213 | 201, 208,
210, 212 | div12d 12079 |
. . . . . . . . . . . . . . . . . 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))) |
| 214 | 13 | adantll 714 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ ℂ) |
| 215 | 208, 214 | mulcomd 11282 |
. . . . . . . . . . . . . . . . . 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))) |
| 216 | 213, 215 | eqtrd 2777 |
. . . . . . . . . . . . . . . . 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))) |
| 217 | 5, 216 | sylan2b 594 |
. . . . . . . . . . . . . . . 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))) |
| 218 | 217 | ifeq2da 4558 |
. . . . . . . . . . . . . . 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)))) |
| 219 | 204 | recnd 11289 |
. . . . . . . . . . . . . . . . . 18
⊢
((⊤ ∧ 𝑛
∈ ℕ) → (1 − (1 / 𝑛)) ∈ ℂ) |
| 220 | | expcl 14120 |
. . . . . . . . . . . . . . . . . 18
⊢ (((1
− (1 / 𝑛)) ∈
ℂ ∧ 𝑘 ∈
ℕ0) → ((1 − (1 / 𝑛))↑𝑘) ∈ ℂ) |
| 221 | 219, 220 | sylan 580 |
. . . . . . . . . . . . . . . . 17
⊢
(((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) → ((1 − (1 / 𝑛))↑𝑘) ∈ ℂ) |
| 222 | 221 | mul02d 11459 |
. . . . . . . . . . . . . . . 16
⊢
(((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) → (0 · ((1 − (1 / 𝑛))↑𝑘)) = 0) |
| 223 | 222 | ifeq1d 4545 |
. . . . . . . . . . . . . . 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)))) |
| 224 | 218, 223 | eqtr4d 2780 |
. . . . . . . . . . . . . 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)))) |
| 225 | | ovif 7531 |
. . . . . . . . . . . . . 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))) |
| 226 | 224, 225 | eqtr4di 2795 |
. . . . . . . . . . . . 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))) |
| 227 | | simpr 484 |
. . . . . . . . . . . . . 14
⊢
(((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) → 𝑘 ∈ ℕ0) |
| 228 | | c0ex 11255 |
. . . . . . . . . . . . . . 15
⊢ 0 ∈
V |
| 229 | | ovex 7464 |
. . . . . . . . . . . . . . 15
⊢
((-1↑((𝑘
− 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)) ∈ V |
| 230 | 228, 229 | ifex 4576 |
. . . . . . . . . . . . . 14
⊢ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))) ∈ V |
| 231 | | eqid 2737 |
. . . . . . . . . . . . . . 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)))) |
| 232 | 231 | fvmpt2 7027 |
. . . . . . . . . . . . . 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)))) |
| 233 | 227, 230,
232 | sylancl 586 |
. . . . . . . . . . . . 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)))) |
| 234 | | ovex 7464 |
. . . . . . . . . . . . . . . 16
⊢
((-1↑((𝑘
− 1) / 2)) / 𝑘)
∈ V |
| 235 | 228, 234 | ifex 4576 |
. . . . . . . . . . . . . . 15
⊢ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) ∈ V |
| 236 | 140 | fvmpt2 7027 |
. . . . . . . . . . . . . . 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))) |
| 237 | 227, 235,
236 | sylancl 586 |
. . . . . . . . . . . . . 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))) |
| 238 | 237 | oveq1d 7446 |
. . . . . . . . . . . . 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))) |
| 239 | 226, 233,
238 | 3eqtr4d 2787 |
. . . . . . . . . . . 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))) |
| 240 | 239 | ralrimiva 3146 |
. . . . . . . . . . 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))) |
| 241 | | nfv 1914 |
. . . . . . . . . . . 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)) |
| 242 | | nffvmpt1 6917 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑘((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗) |
| 243 | | nffvmpt1 6917 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑘((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) |
| 244 | | nfcv 2905 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑘
· |
| 245 | | nfcv 2905 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑘((1
− (1 / 𝑛))↑𝑗) |
| 246 | 243, 244,
245 | nfov 7461 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑘(((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · ((1 − (1 / 𝑛))↑𝑗)) |
| 247 | 242, 246 | nfeq 2919 |
. . . . . . . . . . . 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)) |
| 248 | | fveq2 6906 |
. . . . . . . . . . . . 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)) |
| 249 | | fveq2 6906 |
. . . . . . . . . . . . . 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)) |
| 250 | | oveq2 7439 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 𝑗 → ((1 − (1 / 𝑛))↑𝑘) = ((1 − (1 / 𝑛))↑𝑗)) |
| 251 | 249, 250 | oveq12d 7449 |
. . . . . . . . . . . . 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))) |
| 252 | 248, 251 | eqeq12d 2753 |
. . . . . . . . . . . 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)))) |
| 253 | 241, 247,
252 | cbvralw 3306 |
. . . . . . . . . . 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))) |
| 254 | 240, 253 | sylib 218 |
. . . . . . . . . 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))) |
| 255 | 254 | r19.21bi 3251 |
. . . . . . . . 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))) |
| 256 | | 0cnd 11254 |
. . . . . . . . . . . . 13
⊢
((((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) ∧ (𝑘 = 0 ∨ 2 ∥ 𝑘)) → 0 ∈ ℂ) |
| 257 | 207, 211 | nndivred 12320 |
. . . . . . . . . . . . . . . 16
⊢
((((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (((1 − (1 / 𝑛))↑𝑘) / 𝑘) ∈ ℝ) |
| 258 | 257 | recnd 11289 |
. . . . . . . . . . . . . . 15
⊢
((((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (((1 − (1 / 𝑛))↑𝑘) / 𝑘) ∈ ℂ) |
| 259 | 201, 258 | mulcld 11281 |
. . . . . . . . . . . . . 14
⊢
((((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 /
𝑛))↑𝑘) / 𝑘)) ∈ ℂ) |
| 260 | 5, 259 | sylan2b 594 |
. . . . . . . . . . . . 13
⊢
((((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) ∧ ¬ (𝑘 = 0 ∨ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 /
𝑛))↑𝑘) / 𝑘)) ∈ ℂ) |
| 261 | 256, 260 | ifclda 4561 |
. . . . . . . . . . . 12
⊢
(((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑘
∈ ℕ0) → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 /
𝑛))↑𝑘) / 𝑘))) ∈ ℂ) |
| 262 | 261 | fmpttd 7135 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑛
∈ ℕ) → (𝑘
∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 /
𝑛))↑𝑘) / 𝑘)))):ℕ0⟶ℂ) |
| 263 | 262 | ffvelcdmda 7104 |
. . . . . . . . . 10
⊢
(((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑗
∈ ℕ0) → ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗) ∈ ℂ) |
| 264 | 255, 263 | eqeltrrd 2842 |
. . . . . . . . 9
⊢
(((⊤ ∧ 𝑛
∈ ℕ) ∧ 𝑗
∈ ℕ0) → (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · ((1 − (1 / 𝑛))↑𝑗)) ∈ ℂ) |
| 265 | | 0nn0 12541 |
. . . . . . . . . . . 12
⊢ 0 ∈
ℕ0 |
| 266 | 265 | a1i 11 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑛
∈ ℕ) → 0 ∈ ℕ0) |
| 267 | | 0p1e1 12388 |
. . . . . . . . . . . . 13
⊢ (0 + 1) =
1 |
| 268 | | seqeq1 14045 |
. . . . . . . . . . . . 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)))))) |
| 269 | 267, 268 | 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))))) |
| 270 | | 1zzd 12648 |
. . . . . . . . . . . . . 14
⊢
((⊤ ∧ 𝑛
∈ ℕ) → 1 ∈ ℤ) |
| 271 | | elnnuz 12922 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 ∈ ℕ ↔ 𝑗 ∈
(ℤ≥‘1)) |
| 272 | | nnne0 12300 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑘 ∈ ℕ → 𝑘 ≠ 0) |
| 273 | 272 | neneqd 2945 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑘 ∈ ℕ → ¬
𝑘 = 0) |
| 274 | | biorf 937 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (¬
𝑘 = 0 → (2 ∥
𝑘 ↔ (𝑘 = 0 ∨ 2 ∥ 𝑘))) |
| 275 | 273, 274 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 ∈ ℕ → (2
∥ 𝑘 ↔ (𝑘 = 0 ∨ 2 ∥ 𝑘))) |
| 276 | 275 | bicomd 223 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 ∈ ℕ → ((𝑘 = 0 ∨ 2 ∥ 𝑘) ↔ 2 ∥ 𝑘)) |
| 277 | 276 | ifbid 4549 |
. . . . . . . . . . . . . . . . . . . 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)))) |
| 278 | 90, 230, 232 | sylancl 586 |
. . . . . . . . . . . . . . . . . . . 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)))) |
| 279 | 228, 229 | ifex 4576 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ if(2
∥ 𝑘, 0,
((-1↑((𝑘 − 1) /
2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) ∈ V |
| 280 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . . . . 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)))) |
| 281 | 280 | fvmpt2 7027 |
. . . . . . . . . . . . . . . . . . . . 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)))) |
| 282 | 279, 281 | mpan2 691 |
. . . . . . . . . . . . . . . . . . . 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)))) |
| 283 | 277, 278,
282 | 3eqtr4d 2787 |
. . . . . . . . . . . . . . . . . . 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)) |
| 284 | 283 | rgen 3063 |
. . . . . . . . . . . . . . . . . 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) |
| 285 | 284 | 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)) |
| 286 | | nfv 1914 |
. . . . . . . . . . . . . . . . . 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) |
| 287 | | nffvmpt1 6917 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑘((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗) |
| 288 | 242, 287 | nfeq 2919 |
. . . . . . . . . . . . . . . . . 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) |
| 289 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . . 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)) |
| 290 | 248, 289 | eqeq12d 2753 |
. . . . . . . . . . . . . . . . . 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))) |
| 291 | 286, 288,
290 | cbvralw 3306 |
. . . . . . . . . . . . . . . . 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)) |
| 292 | 285, 291 | sylib 218 |
. . . . . . . . . . . . . . . 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)) |
| 293 | 292 | r19.21bi 3251 |
. . . . . . . . . . . . . . 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)) |
| 294 | 271, 293 | sylan2br 595 |
. . . . . . . . . . . . . 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)) |
| 295 | 270, 294 | seqfeq 14068 |
. . . . . . . . . . . . 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)))))) |
| 296 | 153, 162,
167 | abssubge0d 15470 |
. . . . . . . . . . . . . . 15
⊢
((⊤ ∧ 𝑛
∈ ℕ) → (abs‘(1 − (1 / 𝑛))) = (1 − (1 / 𝑛))) |
| 297 | | ltsubrp 13071 |
. . . . . . . . . . . . . . . 16
⊢ ((1
∈ ℝ ∧ (1 / 𝑛) ∈ ℝ+) → (1
− (1 / 𝑛)) <
1) |
| 298 | 202, 152,
297 | sylancr 587 |
. . . . . . . . . . . . . . 15
⊢
((⊤ ∧ 𝑛
∈ ℕ) → (1 − (1 / 𝑛)) < 1) |
| 299 | 296, 298 | eqbrtrd 5165 |
. . . . . . . . . . . . . 14
⊢
((⊤ ∧ 𝑛
∈ ℕ) → (abs‘(1 − (1 / 𝑛))) < 1) |
| 300 | 280 | atantayl2 26981 |
. . . . . . . . . . . . . 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)))) |
| 301 | 219, 299,
300 | syl2anc 584 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑛
∈ ℕ) → seq1( + , (𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))) ⇝ (arctan‘(1 − (1 /
𝑛)))) |
| 302 | 295, 301 | eqbrtrd 5165 |
. . . . . . . . . . . 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)))) |
| 303 | 269, 302 | eqbrtrid 5178 |
. . . . . . . . . . 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)))) |
| 304 | 1, 266, 263, 303 | clim2ser2 15692 |
. . . . . . . . . 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))) |
| 305 | | 0z 12624 |
. . . . . . . . . . . . . 14
⊢ 0 ∈
ℤ |
| 306 | | seq1 14055 |
. . . . . . . . . . . . . 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)) |
| 307 | 305, 306 | 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) |
| 308 | | iftrue 4531 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑘 = 0 ∨ 2 ∥ 𝑘) → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 /
𝑛))↑𝑘) / 𝑘))) = 0) |
| 309 | 308 | orcs 876 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 0 → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))) = 0) |
| 310 | 309, 231,
228 | fvmpt 7016 |
. . . . . . . . . . . . . 14
⊢ (0 ∈
ℕ0 → ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1
− (1 / 𝑛))↑𝑘) / 𝑘))))‘0) = 0) |
| 311 | 265, 310 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢ ((𝑘 ∈ ℕ0
↦ if((𝑘 = 0 ∨ 2
∥ 𝑘), 0,
((-1↑((𝑘 − 1) /
2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘0) = 0 |
| 312 | 307, 311 | eqtri 2765 |
. . . . . . . . . . . 12
⊢ (seq0( +
, (𝑘 ∈
ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 /
𝑛))↑𝑘) / 𝑘)))))‘0) = 0 |
| 313 | 312 | oveq2i 7442 |
. . . . . . . . . . 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) |
| 314 | | atanrecl 26954 |
. . . . . . . . . . . . . 14
⊢ ((1
− (1 / 𝑛)) ∈
ℝ → (arctan‘(1 − (1 / 𝑛))) ∈ ℝ) |
| 315 | 204, 314 | syl 17 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑛
∈ ℕ) → (arctan‘(1 − (1 / 𝑛))) ∈ ℝ) |
| 316 | 315 | recnd 11289 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑛
∈ ℕ) → (arctan‘(1 − (1 / 𝑛))) ∈ ℂ) |
| 317 | 316 | addridd 11461 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑛
∈ ℕ) → ((arctan‘(1 − (1 / 𝑛))) + 0) = (arctan‘(1 − (1 /
𝑛)))) |
| 318 | 313, 317 | eqtrid 2789 |
. . . . . . . . . 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)))) |
| 319 | 304, 318 | breqtrd 5169 |
. . . . . . . . 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)))) |
| 320 | 1, 197, 255, 264, 319 | isumclim 15793 |
. . . . . . . 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)))) |
| 321 | 320 | mpteq2dva 5242 |
. . . . . . 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))))) |
| 322 | 196, 321 | eqtrd 2777 |
. . . . . 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))))) |
| 323 | | oveq1 7438 |
. . . . . . . . . . . 12
⊢ (𝑥 = 1 → (𝑥↑𝑗) = (1↑𝑗)) |
| 324 | | nn0z 12638 |
. . . . . . . . . . . . 13
⊢ (𝑗 ∈ ℕ0
→ 𝑗 ∈
ℤ) |
| 325 | | 1exp 14132 |
. . . . . . . . . . . . 13
⊢ (𝑗 ∈ ℤ →
(1↑𝑗) =
1) |
| 326 | 324, 325 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝑗 ∈ ℕ0
→ (1↑𝑗) =
1) |
| 327 | 323, 326 | sylan9eq 2797 |
. . . . . . . . . . 11
⊢ ((𝑥 = 1 ∧ 𝑗 ∈ ℕ0) → (𝑥↑𝑗) = 1) |
| 328 | 327 | oveq2d 7447 |
. . . . . . . . . 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)) |
| 329 | 17 | mptru 1547 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ ℕ0
↦ if((𝑘 = 0 ∨ 2
∥ 𝑘), 0,
((-1↑((𝑘 − 1) /
2)) / 𝑘))):ℕ0⟶ℂ |
| 330 | 329 | ffvelcdmi 7103 |
. . . . . . . . . . . 12
⊢ (𝑗 ∈ ℕ0
→ ((𝑘 ∈
ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) ∈ ℂ) |
| 331 | 330 | mulridd 11278 |
. . . . . . . . . . 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)) |
| 332 | 331 | 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)) |
| 333 | 328, 332 | eqtrd 2777 |
. . . . . . . . 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)) |
| 334 | 333 | sumeq2dv 15738 |
. . . . . . . 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)) |
| 335 | | sumex 15724 |
. . . . . . . 8
⊢
Σ𝑗 ∈
ℕ0 ((𝑘
∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) ∈ V |
| 336 | 334, 148,
335 | fvmpt 7016 |
. . . . . . 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)) |
| 337 | 188, 336 | 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)) |
| 338 | 190, 322,
337 | 3brtr3d 5174 |
. . . . 5
⊢ (⊤
→ (𝑛 ∈ ℕ
↦ (arctan‘(1 − (1 / 𝑛)))) ⇝ Σ𝑗 ∈ ℕ0 ((𝑘 ∈ ℕ0
↦ if((𝑘 = 0 ∨ 2
∥ 𝑘), 0,
((-1↑((𝑘 − 1) /
2)) / 𝑘)))‘𝑗)) |
| 339 | | eqid 2737 |
. . . . . . . . 9
⊢ (ℂ
∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0)) |
| 340 | | eqid 2737 |
. . . . . . . . 9
⊢ {𝑥 ∈ ℂ ∣ (1 +
(𝑥↑2)) ∈ (ℂ
∖ (-∞(,]0))} = {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ
∖ (-∞(,]0))} |
| 341 | 339, 340 | atancn 26979 |
. . . . . . . 8
⊢ (arctan
↾ {𝑥 ∈ ℂ
∣ (1 + (𝑥↑2))
∈ (ℂ ∖ (-∞(,]0))}) ∈ ({𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ
∖ (-∞(,]0))}–cn→ℂ) |
| 342 | 341 | a1i 11 |
. . . . . . 7
⊢ (⊤
→ (arctan ↾ {𝑥
∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖
(-∞(,]0))}) ∈ ({𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ
∖ (-∞(,]0))}–cn→ℂ)) |
| 343 | | unitssre 13539 |
. . . . . . . . 9
⊢ (0[,]1)
⊆ ℝ |
| 344 | 339, 340 | ressatans 26977 |
. . . . . . . . 9
⊢ ℝ
⊆ {𝑥 ∈ ℂ
∣ (1 + (𝑥↑2))
∈ (ℂ ∖ (-∞(,]0))} |
| 345 | 343, 344 | sstri 3993 |
. . . . . . . 8
⊢ (0[,]1)
⊆ {𝑥 ∈ ℂ
∣ (1 + (𝑥↑2))
∈ (ℂ ∖ (-∞(,]0))} |
| 346 | | fss 6752 |
. . . . . . . 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))}) |
| 347 | 172, 345,
346 | sylancl 586 |
. . . . . . 7
⊢ (⊤
→ (𝑛 ∈ ℕ
↦ (1 − (1 / 𝑛))):ℕ⟶{𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ
∖ (-∞(,]0))}) |
| 348 | 344, 202 | sselii 3980 |
. . . . . . . 8
⊢ 1 ∈
{𝑥 ∈ ℂ ∣
(1 + (𝑥↑2)) ∈
(ℂ ∖ (-∞(,]0))} |
| 349 | 348 | a1i 11 |
. . . . . . 7
⊢ (⊤
→ 1 ∈ {𝑥 ∈
ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖
(-∞(,]0))}) |
| 350 | 74, 75, 342, 347, 187, 349 | climcncf 24926 |
. . . . . 6
⊢ (⊤
→ ((arctan ↾ {𝑥
∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖
(-∞(,]0))}) ∘ (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))) ⇝ ((arctan ↾
{𝑥 ∈ ℂ ∣
(1 + (𝑥↑2)) ∈
(ℂ ∖ (-∞(,]0))})‘1)) |
| 351 | 345, 171 | sselid 3981 |
. . . . . . 7
⊢
((⊤ ∧ 𝑛
∈ ℕ) → (1 − (1 / 𝑛)) ∈ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ
∖ (-∞(,]0))}) |
| 352 | | cncff 24919 |
. . . . . . . . . 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) |
| 353 | 341, 352 | mp1i 13 |
. . . . . . . . 9
⊢ (⊤
→ (arctan ↾ {𝑥
∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖
(-∞(,]0))}):{𝑥 ∈
ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖
(-∞(,]0))}⟶ℂ) |
| 354 | 353 | feqmptd 6977 |
. . . . . . . 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))) |
| 355 | | fvres 6925 |
. . . . . . . . 9
⊢ (𝑘 ∈ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ
∖ (-∞(,]0))} → ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ
∖ (-∞(,]0))})‘𝑘) = (arctan‘𝑘)) |
| 356 | 355 | mpteq2ia 5245 |
. . . . . . . 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)) |
| 357 | 354, 356 | eqtrdi 2793 |
. . . . . . 7
⊢ (⊤
→ (arctan ↾ {𝑥
∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖
(-∞(,]0))}) = (𝑘
∈ {𝑥 ∈ ℂ
∣ (1 + (𝑥↑2))
∈ (ℂ ∖ (-∞(,]0))} ↦ (arctan‘𝑘))) |
| 358 | | fveq2 6906 |
. . . . . . 7
⊢ (𝑘 = (1 − (1 / 𝑛)) → (arctan‘𝑘) = (arctan‘(1 − (1
/ 𝑛)))) |
| 359 | 351, 191,
357, 358 | fmptco 7149 |
. . . . . 6
⊢ (⊤
→ ((arctan ↾ {𝑥
∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖
(-∞(,]0))}) ∘ (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))) = (𝑛 ∈ ℕ ↦ (arctan‘(1
− (1 / 𝑛))))) |
| 360 | | fvres 6925 |
. . . . . . . 8
⊢ (1 ∈
{𝑥 ∈ ℂ ∣
(1 + (𝑥↑2)) ∈
(ℂ ∖ (-∞(,]0))} → ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ
∖ (-∞(,]0))})‘1) = (arctan‘1)) |
| 361 | 348, 360 | mp1i 13 |
. . . . . . 7
⊢ (⊤
→ ((arctan ↾ {𝑥
∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖
(-∞(,]0))})‘1) = (arctan‘1)) |
| 362 | | atan1 26971 |
. . . . . . 7
⊢
(arctan‘1) = (π / 4) |
| 363 | 361, 362 | eqtrdi 2793 |
. . . . . 6
⊢ (⊤
→ ((arctan ↾ {𝑥
∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖
(-∞(,]0))})‘1) = (π / 4)) |
| 364 | 350, 359,
363 | 3brtr3d 5174 |
. . . . 5
⊢ (⊤
→ (𝑛 ∈ ℕ
↦ (arctan‘(1 − (1 / 𝑛)))) ⇝ (π / 4)) |
| 365 | | climuni 15588 |
. . . . 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)) |
| 366 | 338, 364,
365 | syl2anc 584 |
. . . 4
⊢ (⊤
→ Σ𝑗 ∈
ℕ0 ((𝑘
∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) = (π / 4)) |
| 367 | 147, 366 | breqtrd 5169 |
. . 3
⊢ (⊤
→ seq0( + , (𝑘 ∈
ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ⇝ (π / 4)) |
| 368 | 367 | mptru 1547 |
. 2
⊢ seq0( + ,
(𝑘 ∈
ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ⇝ (π / 4) |
| 369 | | leibpi.1 |
. . 3
⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦
((-1↑𝑛) / ((2 ·
𝑛) + 1))) |
| 370 | | ovex 7464 |
. . 3
⊢ (π /
4) ∈ V |
| 371 | 369, 140,
370 | leibpilem2 26984 |
. 2
⊢ (seq0( +
, 𝐹) ⇝ (π / 4)
↔ seq0( + , (𝑘 ∈
ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ⇝ (π / 4)) |
| 372 | 368, 371 | mpbir 231 |
1
⊢ seq0( + ,
𝐹) ⇝ (π /
4) |