| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | vex 3484 | . . . . . 6
⊢ 𝑦 ∈ V | 
| 2 |  | stirlinglem13.2 | . . . . . . 7
⊢ 𝐵 = (𝑛 ∈ ℕ ↦ (log‘(𝐴‘𝑛))) | 
| 3 | 2 | elrnmpt 5969 | . . . . . 6
⊢ (𝑦 ∈ V → (𝑦 ∈ ran 𝐵 ↔ ∃𝑛 ∈ ℕ 𝑦 = (log‘(𝐴‘𝑛)))) | 
| 4 | 1, 3 | ax-mp 5 | . . . . 5
⊢ (𝑦 ∈ ran 𝐵 ↔ ∃𝑛 ∈ ℕ 𝑦 = (log‘(𝐴‘𝑛))) | 
| 5 |  | simpr 484 | . . . . . . 7
⊢ ((𝑛 ∈ ℕ ∧ 𝑦 = (log‘(𝐴‘𝑛))) → 𝑦 = (log‘(𝐴‘𝑛))) | 
| 6 |  | stirlinglem13.1 | . . . . . . . . . 10
⊢ 𝐴 = (𝑛 ∈ ℕ ↦ ((!‘𝑛) / ((√‘(2 ·
𝑛)) · ((𝑛 / e)↑𝑛)))) | 
| 7 | 6 | stirlinglem2 46090 | . . . . . . . . 9
⊢ (𝑛 ∈ ℕ → (𝐴‘𝑛) ∈
ℝ+) | 
| 8 | 7 | relogcld 26665 | . . . . . . . 8
⊢ (𝑛 ∈ ℕ →
(log‘(𝐴‘𝑛)) ∈
ℝ) | 
| 9 | 8 | adantr 480 | . . . . . . 7
⊢ ((𝑛 ∈ ℕ ∧ 𝑦 = (log‘(𝐴‘𝑛))) → (log‘(𝐴‘𝑛)) ∈ ℝ) | 
| 10 | 5, 9 | eqeltrd 2841 | . . . . . 6
⊢ ((𝑛 ∈ ℕ ∧ 𝑦 = (log‘(𝐴‘𝑛))) → 𝑦 ∈ ℝ) | 
| 11 | 10 | rexlimiva 3147 | . . . . 5
⊢
(∃𝑛 ∈
ℕ 𝑦 =
(log‘(𝐴‘𝑛)) → 𝑦 ∈ ℝ) | 
| 12 | 4, 11 | sylbi 217 | . . . 4
⊢ (𝑦 ∈ ran 𝐵 → 𝑦 ∈ ℝ) | 
| 13 | 12 | ssriv 3987 | . . 3
⊢ ran 𝐵 ⊆
ℝ | 
| 14 |  | 1nn 12277 | . . . . . 6
⊢ 1 ∈
ℕ | 
| 15 | 6 | stirlinglem2 46090 | . . . . . . . 8
⊢ (1 ∈
ℕ → (𝐴‘1)
∈ ℝ+) | 
| 16 |  | relogcl 26617 | . . . . . . . 8
⊢ ((𝐴‘1) ∈
ℝ+ → (log‘(𝐴‘1)) ∈ ℝ) | 
| 17 | 14, 15, 16 | mp2b 10 | . . . . . . 7
⊢
(log‘(𝐴‘1)) ∈ ℝ | 
| 18 |  | nfcv 2905 | . . . . . . . 8
⊢
Ⅎ𝑛1 | 
| 19 |  | nfcv 2905 | . . . . . . . . 9
⊢
Ⅎ𝑛log | 
| 20 |  | nfmpt1 5250 | . . . . . . . . . . 11
⊢
Ⅎ𝑛(𝑛 ∈ ℕ ↦ ((!‘𝑛) / ((√‘(2 ·
𝑛)) · ((𝑛 / e)↑𝑛)))) | 
| 21 | 6, 20 | nfcxfr 2903 | . . . . . . . . . 10
⊢
Ⅎ𝑛𝐴 | 
| 22 | 21, 18 | nffv 6916 | . . . . . . . . 9
⊢
Ⅎ𝑛(𝐴‘1) | 
| 23 | 19, 22 | nffv 6916 | . . . . . . . 8
⊢
Ⅎ𝑛(log‘(𝐴‘1)) | 
| 24 |  | 2fveq3 6911 | . . . . . . . 8
⊢ (𝑛 = 1 → (log‘(𝐴‘𝑛)) = (log‘(𝐴‘1))) | 
| 25 | 18, 23, 24, 2 | fvmptf 7037 | . . . . . . 7
⊢ ((1
∈ ℕ ∧ (log‘(𝐴‘1)) ∈ ℝ) → (𝐵‘1) = (log‘(𝐴‘1))) | 
| 26 | 14, 17, 25 | mp2an 692 | . . . . . 6
⊢ (𝐵‘1) = (log‘(𝐴‘1)) | 
| 27 |  | 2fveq3 6911 | . . . . . . 7
⊢ (𝑗 = 1 → (log‘(𝐴‘𝑗)) = (log‘(𝐴‘1))) | 
| 28 | 27 | rspceeqv 3645 | . . . . . 6
⊢ ((1
∈ ℕ ∧ (𝐵‘1) = (log‘(𝐴‘1))) → ∃𝑗 ∈ ℕ (𝐵‘1) = (log‘(𝐴‘𝑗))) | 
| 29 | 14, 26, 28 | mp2an 692 | . . . . 5
⊢
∃𝑗 ∈
ℕ (𝐵‘1) =
(log‘(𝐴‘𝑗)) | 
| 30 | 26, 17 | eqeltri 2837 | . . . . . 6
⊢ (𝐵‘1) ∈
ℝ | 
| 31 |  | nfcv 2905 | . . . . . . . . 9
⊢
Ⅎ𝑗(log‘(𝐴‘𝑛)) | 
| 32 |  | nfcv 2905 | . . . . . . . . . . 11
⊢
Ⅎ𝑛𝑗 | 
| 33 | 21, 32 | nffv 6916 | . . . . . . . . . 10
⊢
Ⅎ𝑛(𝐴‘𝑗) | 
| 34 | 19, 33 | nffv 6916 | . . . . . . . . 9
⊢
Ⅎ𝑛(log‘(𝐴‘𝑗)) | 
| 35 |  | 2fveq3 6911 | . . . . . . . . 9
⊢ (𝑛 = 𝑗 → (log‘(𝐴‘𝑛)) = (log‘(𝐴‘𝑗))) | 
| 36 | 31, 34, 35 | cbvmpt 5253 | . . . . . . . 8
⊢ (𝑛 ∈ ℕ ↦
(log‘(𝐴‘𝑛))) = (𝑗 ∈ ℕ ↦ (log‘(𝐴‘𝑗))) | 
| 37 | 2, 36 | eqtri 2765 | . . . . . . 7
⊢ 𝐵 = (𝑗 ∈ ℕ ↦ (log‘(𝐴‘𝑗))) | 
| 38 | 37 | elrnmpt 5969 | . . . . . 6
⊢ ((𝐵‘1) ∈ ℝ →
((𝐵‘1) ∈ ran
𝐵 ↔ ∃𝑗 ∈ ℕ (𝐵‘1) = (log‘(𝐴‘𝑗)))) | 
| 39 | 30, 38 | ax-mp 5 | . . . . 5
⊢ ((𝐵‘1) ∈ ran 𝐵 ↔ ∃𝑗 ∈ ℕ (𝐵‘1) = (log‘(𝐴‘𝑗))) | 
| 40 | 29, 39 | mpbir 231 | . . . 4
⊢ (𝐵‘1) ∈ ran 𝐵 | 
| 41 | 40 | ne0ii 4344 | . . 3
⊢ ran 𝐵 ≠ ∅ | 
| 42 |  | 4re 12350 | . . . . . . 7
⊢ 4 ∈
ℝ | 
| 43 |  | 4ne0 12374 | . . . . . . 7
⊢ 4 ≠
0 | 
| 44 | 42, 43 | rereccli 12032 | . . . . . 6
⊢ (1 / 4)
∈ ℝ | 
| 45 | 30, 44 | resubcli 11571 | . . . . 5
⊢ ((𝐵‘1) − (1 / 4))
∈ ℝ | 
| 46 |  | eqid 2737 | . . . . . . 7
⊢ (𝑛 ∈ ℕ ↦ (1 /
(𝑛 · (𝑛 + 1)))) = (𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1)))) | 
| 47 | 6, 2, 46 | stirlinglem12 46100 | . . . . . 6
⊢ (𝑗 ∈ ℕ → ((𝐵‘1) − (1 / 4)) ≤
(𝐵‘𝑗)) | 
| 48 | 47 | rgen 3063 | . . . . 5
⊢
∀𝑗 ∈
ℕ ((𝐵‘1)
− (1 / 4)) ≤ (𝐵‘𝑗) | 
| 49 |  | breq1 5146 | . . . . . . 7
⊢ (𝑥 = ((𝐵‘1) − (1 / 4)) → (𝑥 ≤ (𝐵‘𝑗) ↔ ((𝐵‘1) − (1 / 4)) ≤ (𝐵‘𝑗))) | 
| 50 | 49 | ralbidv 3178 | . . . . . 6
⊢ (𝑥 = ((𝐵‘1) − (1 / 4)) →
(∀𝑗 ∈ ℕ
𝑥 ≤ (𝐵‘𝑗) ↔ ∀𝑗 ∈ ℕ ((𝐵‘1) − (1 / 4)) ≤ (𝐵‘𝑗))) | 
| 51 | 50 | rspcev 3622 | . . . . 5
⊢ ((((𝐵‘1) − (1 / 4))
∈ ℝ ∧ ∀𝑗 ∈ ℕ ((𝐵‘1) − (1 / 4)) ≤ (𝐵‘𝑗)) → ∃𝑥 ∈ ℝ ∀𝑗 ∈ ℕ 𝑥 ≤ (𝐵‘𝑗)) | 
| 52 | 45, 48, 51 | mp2an 692 | . . . 4
⊢
∃𝑥 ∈
ℝ ∀𝑗 ∈
ℕ 𝑥 ≤ (𝐵‘𝑗) | 
| 53 |  | simpr 484 | . . . . . . . 8
⊢
((∀𝑗 ∈
ℕ 𝑥 ≤ (𝐵‘𝑗) ∧ 𝑦 ∈ ran 𝐵) → 𝑦 ∈ ran 𝐵) | 
| 54 | 8 | rgen 3063 | . . . . . . . . 9
⊢
∀𝑛 ∈
ℕ (log‘(𝐴‘𝑛)) ∈ ℝ | 
| 55 | 2 | fnmpt 6708 | . . . . . . . . 9
⊢
(∀𝑛 ∈
ℕ (log‘(𝐴‘𝑛)) ∈ ℝ → 𝐵 Fn ℕ) | 
| 56 |  | fvelrnb 6969 | . . . . . . . . 9
⊢ (𝐵 Fn ℕ → (𝑦 ∈ ran 𝐵 ↔ ∃𝑗 ∈ ℕ (𝐵‘𝑗) = 𝑦)) | 
| 57 | 54, 55, 56 | mp2b 10 | . . . . . . . 8
⊢ (𝑦 ∈ ran 𝐵 ↔ ∃𝑗 ∈ ℕ (𝐵‘𝑗) = 𝑦) | 
| 58 | 53, 57 | sylib 218 | . . . . . . 7
⊢
((∀𝑗 ∈
ℕ 𝑥 ≤ (𝐵‘𝑗) ∧ 𝑦 ∈ ran 𝐵) → ∃𝑗 ∈ ℕ (𝐵‘𝑗) = 𝑦) | 
| 59 |  | nfra1 3284 | . . . . . . . . 9
⊢
Ⅎ𝑗∀𝑗 ∈ ℕ 𝑥 ≤ (𝐵‘𝑗) | 
| 60 |  | nfv 1914 | . . . . . . . . 9
⊢
Ⅎ𝑗 𝑦 ∈ ran 𝐵 | 
| 61 | 59, 60 | nfan 1899 | . . . . . . . 8
⊢
Ⅎ𝑗(∀𝑗 ∈ ℕ 𝑥 ≤ (𝐵‘𝑗) ∧ 𝑦 ∈ ran 𝐵) | 
| 62 |  | nfv 1914 | . . . . . . . 8
⊢
Ⅎ𝑗 𝑥 ≤ 𝑦 | 
| 63 |  | simp1l 1198 | . . . . . . . . . . 11
⊢
(((∀𝑗 ∈
ℕ 𝑥 ≤ (𝐵‘𝑗) ∧ 𝑦 ∈ ran 𝐵) ∧ 𝑗 ∈ ℕ ∧ (𝐵‘𝑗) = 𝑦) → ∀𝑗 ∈ ℕ 𝑥 ≤ (𝐵‘𝑗)) | 
| 64 |  | simp2 1138 | . . . . . . . . . . 11
⊢
(((∀𝑗 ∈
ℕ 𝑥 ≤ (𝐵‘𝑗) ∧ 𝑦 ∈ ran 𝐵) ∧ 𝑗 ∈ ℕ ∧ (𝐵‘𝑗) = 𝑦) → 𝑗 ∈ ℕ) | 
| 65 |  | rsp 3247 | . . . . . . . . . . 11
⊢
(∀𝑗 ∈
ℕ 𝑥 ≤ (𝐵‘𝑗) → (𝑗 ∈ ℕ → 𝑥 ≤ (𝐵‘𝑗))) | 
| 66 | 63, 64, 65 | sylc 65 | . . . . . . . . . 10
⊢
(((∀𝑗 ∈
ℕ 𝑥 ≤ (𝐵‘𝑗) ∧ 𝑦 ∈ ran 𝐵) ∧ 𝑗 ∈ ℕ ∧ (𝐵‘𝑗) = 𝑦) → 𝑥 ≤ (𝐵‘𝑗)) | 
| 67 |  | simp3 1139 | . . . . . . . . . 10
⊢
(((∀𝑗 ∈
ℕ 𝑥 ≤ (𝐵‘𝑗) ∧ 𝑦 ∈ ran 𝐵) ∧ 𝑗 ∈ ℕ ∧ (𝐵‘𝑗) = 𝑦) → (𝐵‘𝑗) = 𝑦) | 
| 68 | 66, 67 | breqtrd 5169 | . . . . . . . . 9
⊢
(((∀𝑗 ∈
ℕ 𝑥 ≤ (𝐵‘𝑗) ∧ 𝑦 ∈ ran 𝐵) ∧ 𝑗 ∈ ℕ ∧ (𝐵‘𝑗) = 𝑦) → 𝑥 ≤ 𝑦) | 
| 69 | 68 | 3exp 1120 | . . . . . . . 8
⊢
((∀𝑗 ∈
ℕ 𝑥 ≤ (𝐵‘𝑗) ∧ 𝑦 ∈ ran 𝐵) → (𝑗 ∈ ℕ → ((𝐵‘𝑗) = 𝑦 → 𝑥 ≤ 𝑦))) | 
| 70 | 61, 62, 69 | rexlimd 3266 | . . . . . . 7
⊢
((∀𝑗 ∈
ℕ 𝑥 ≤ (𝐵‘𝑗) ∧ 𝑦 ∈ ran 𝐵) → (∃𝑗 ∈ ℕ (𝐵‘𝑗) = 𝑦 → 𝑥 ≤ 𝑦)) | 
| 71 | 58, 70 | mpd 15 | . . . . . 6
⊢
((∀𝑗 ∈
ℕ 𝑥 ≤ (𝐵‘𝑗) ∧ 𝑦 ∈ ran 𝐵) → 𝑥 ≤ 𝑦) | 
| 72 | 71 | ralrimiva 3146 | . . . . 5
⊢
(∀𝑗 ∈
ℕ 𝑥 ≤ (𝐵‘𝑗) → ∀𝑦 ∈ ran 𝐵 𝑥 ≤ 𝑦) | 
| 73 | 72 | reximi 3084 | . . . 4
⊢
(∃𝑥 ∈
ℝ ∀𝑗 ∈
ℕ 𝑥 ≤ (𝐵‘𝑗) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐵 𝑥 ≤ 𝑦) | 
| 74 | 52, 73 | ax-mp 5 | . . 3
⊢
∃𝑥 ∈
ℝ ∀𝑦 ∈
ran 𝐵 𝑥 ≤ 𝑦 | 
| 75 |  | infrecl 12250 | . . 3
⊢ ((ran
𝐵 ⊆ ℝ ∧ ran
𝐵 ≠ ∅ ∧
∃𝑥 ∈ ℝ
∀𝑦 ∈ ran 𝐵 𝑥 ≤ 𝑦) → inf(ran 𝐵, ℝ, < ) ∈
ℝ) | 
| 76 | 13, 41, 74, 75 | mp3an 1463 | . 2
⊢ inf(ran
𝐵, ℝ, < ) ∈
ℝ | 
| 77 |  | nnuz 12921 | . . . 4
⊢ ℕ =
(ℤ≥‘1) | 
| 78 |  | 1zzd 12648 | . . . 4
⊢ (⊤
→ 1 ∈ ℤ) | 
| 79 | 2, 8 | fmpti 7132 | . . . . 5
⊢ 𝐵:ℕ⟶ℝ | 
| 80 | 79 | a1i 11 | . . . 4
⊢ (⊤
→ 𝐵:ℕ⟶ℝ) | 
| 81 |  | peano2nn 12278 | . . . . . . . 8
⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ∈
ℕ) | 
| 82 | 6 | a1i 11 | . . . . . . . . . . 11
⊢ (𝑗 ∈ ℕ → 𝐴 = (𝑛 ∈ ℕ ↦ ((!‘𝑛) / ((√‘(2 ·
𝑛)) · ((𝑛 / e)↑𝑛))))) | 
| 83 |  | simpr 484 | . . . . . . . . . . . . 13
⊢ ((𝑗 ∈ ℕ ∧ 𝑛 = (𝑗 + 1)) → 𝑛 = (𝑗 + 1)) | 
| 84 | 83 | fveq2d 6910 | . . . . . . . . . . . 12
⊢ ((𝑗 ∈ ℕ ∧ 𝑛 = (𝑗 + 1)) → (!‘𝑛) = (!‘(𝑗 + 1))) | 
| 85 | 83 | oveq2d 7447 | . . . . . . . . . . . . . 14
⊢ ((𝑗 ∈ ℕ ∧ 𝑛 = (𝑗 + 1)) → (2 · 𝑛) = (2 · (𝑗 + 1))) | 
| 86 | 85 | fveq2d 6910 | . . . . . . . . . . . . 13
⊢ ((𝑗 ∈ ℕ ∧ 𝑛 = (𝑗 + 1)) → (√‘(2 ·
𝑛)) = (√‘(2
· (𝑗 +
1)))) | 
| 87 | 83 | oveq1d 7446 | . . . . . . . . . . . . . 14
⊢ ((𝑗 ∈ ℕ ∧ 𝑛 = (𝑗 + 1)) → (𝑛 / e) = ((𝑗 + 1) / e)) | 
| 88 | 87, 83 | oveq12d 7449 | . . . . . . . . . . . . 13
⊢ ((𝑗 ∈ ℕ ∧ 𝑛 = (𝑗 + 1)) → ((𝑛 / e)↑𝑛) = (((𝑗 + 1) / e)↑(𝑗 + 1))) | 
| 89 | 86, 88 | oveq12d 7449 | . . . . . . . . . . . 12
⊢ ((𝑗 ∈ ℕ ∧ 𝑛 = (𝑗 + 1)) → ((√‘(2 ·
𝑛)) · ((𝑛 / e)↑𝑛)) = ((√‘(2 · (𝑗 + 1))) · (((𝑗 + 1) / e)↑(𝑗 + 1)))) | 
| 90 | 84, 89 | oveq12d 7449 | . . . . . . . . . . 11
⊢ ((𝑗 ∈ ℕ ∧ 𝑛 = (𝑗 + 1)) → ((!‘𝑛) / ((√‘(2 · 𝑛)) · ((𝑛 / e)↑𝑛))) = ((!‘(𝑗 + 1)) / ((√‘(2 · (𝑗 + 1))) · (((𝑗 + 1) / e)↑(𝑗 + 1))))) | 
| 91 | 81 | nnnn0d 12587 | . . . . . . . . . . . . 13
⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ∈
ℕ0) | 
| 92 |  | faccl 14322 | . . . . . . . . . . . . 13
⊢ ((𝑗 + 1) ∈ ℕ0
→ (!‘(𝑗 + 1))
∈ ℕ) | 
| 93 |  | nncn 12274 | . . . . . . . . . . . . 13
⊢
((!‘(𝑗 + 1))
∈ ℕ → (!‘(𝑗 + 1)) ∈ ℂ) | 
| 94 | 91, 92, 93 | 3syl 18 | . . . . . . . . . . . 12
⊢ (𝑗 ∈ ℕ →
(!‘(𝑗 + 1)) ∈
ℂ) | 
| 95 |  | 2cnd 12344 | . . . . . . . . . . . . . . 15
⊢ (𝑗 ∈ ℕ → 2 ∈
ℂ) | 
| 96 |  | nncn 12274 | . . . . . . . . . . . . . . . 16
⊢ (𝑗 ∈ ℕ → 𝑗 ∈
ℂ) | 
| 97 |  | 1cnd 11256 | . . . . . . . . . . . . . . . 16
⊢ (𝑗 ∈ ℕ → 1 ∈
ℂ) | 
| 98 | 96, 97 | addcld 11280 | . . . . . . . . . . . . . . 15
⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ∈
ℂ) | 
| 99 | 95, 98 | mulcld 11281 | . . . . . . . . . . . . . 14
⊢ (𝑗 ∈ ℕ → (2
· (𝑗 + 1)) ∈
ℂ) | 
| 100 | 99 | sqrtcld 15476 | . . . . . . . . . . . . 13
⊢ (𝑗 ∈ ℕ →
(√‘(2 · (𝑗 + 1))) ∈ ℂ) | 
| 101 |  | ere 16125 | . . . . . . . . . . . . . . . . 17
⊢ e ∈
ℝ | 
| 102 | 101 | recni 11275 | . . . . . . . . . . . . . . . 16
⊢ e ∈
ℂ | 
| 103 | 102 | a1i 11 | . . . . . . . . . . . . . . 15
⊢ (𝑗 ∈ ℕ → e ∈
ℂ) | 
| 104 |  | 0re 11263 | . . . . . . . . . . . . . . . . 17
⊢ 0 ∈
ℝ | 
| 105 |  | epos 16243 | . . . . . . . . . . . . . . . . 17
⊢ 0 <
e | 
| 106 | 104, 105 | gtneii 11373 | . . . . . . . . . . . . . . . 16
⊢ e ≠
0 | 
| 107 | 106 | a1i 11 | . . . . . . . . . . . . . . 15
⊢ (𝑗 ∈ ℕ → e ≠
0) | 
| 108 | 98, 103, 107 | divcld 12043 | . . . . . . . . . . . . . 14
⊢ (𝑗 ∈ ℕ → ((𝑗 + 1) / e) ∈
ℂ) | 
| 109 | 108, 91 | expcld 14186 | . . . . . . . . . . . . 13
⊢ (𝑗 ∈ ℕ → (((𝑗 + 1) / e)↑(𝑗 + 1)) ∈
ℂ) | 
| 110 | 100, 109 | mulcld 11281 | . . . . . . . . . . . 12
⊢ (𝑗 ∈ ℕ →
((√‘(2 · (𝑗 + 1))) · (((𝑗 + 1) / e)↑(𝑗 + 1))) ∈ ℂ) | 
| 111 |  | 2rp 13039 | . . . . . . . . . . . . . . . . 17
⊢ 2 ∈
ℝ+ | 
| 112 | 111 | a1i 11 | . . . . . . . . . . . . . . . 16
⊢ (𝑗 ∈ ℕ → 2 ∈
ℝ+) | 
| 113 |  | nnre 12273 | . . . . . . . . . . . . . . . . 17
⊢ (𝑗 ∈ ℕ → 𝑗 ∈
ℝ) | 
| 114 | 104 | a1i 11 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑗 ∈ ℕ → 0 ∈
ℝ) | 
| 115 |  | 1red 11262 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑗 ∈ ℕ → 1 ∈
ℝ) | 
| 116 |  | 0le1 11786 | . . . . . . . . . . . . . . . . . . 19
⊢ 0 ≤
1 | 
| 117 | 116 | a1i 11 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑗 ∈ ℕ → 0 ≤
1) | 
| 118 |  | nnge1 12294 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑗 ∈ ℕ → 1 ≤
𝑗) | 
| 119 | 114, 115,
113, 117, 118 | letrd 11418 | . . . . . . . . . . . . . . . . 17
⊢ (𝑗 ∈ ℕ → 0 ≤
𝑗) | 
| 120 | 113, 119 | ge0p1rpd 13107 | . . . . . . . . . . . . . . . 16
⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ∈
ℝ+) | 
| 121 | 112, 120 | rpmulcld 13093 | . . . . . . . . . . . . . . 15
⊢ (𝑗 ∈ ℕ → (2
· (𝑗 + 1)) ∈
ℝ+) | 
| 122 | 121 | sqrtgt0d 15451 | . . . . . . . . . . . . . 14
⊢ (𝑗 ∈ ℕ → 0 <
(√‘(2 · (𝑗 + 1)))) | 
| 123 | 122 | gt0ne0d 11827 | . . . . . . . . . . . . 13
⊢ (𝑗 ∈ ℕ →
(√‘(2 · (𝑗 + 1))) ≠ 0) | 
| 124 | 81 | nnne0d 12316 | . . . . . . . . . . . . . . 15
⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ≠ 0) | 
| 125 | 98, 103, 124, 107 | divne0d 12059 | . . . . . . . . . . . . . 14
⊢ (𝑗 ∈ ℕ → ((𝑗 + 1) / e) ≠
0) | 
| 126 |  | nnz 12634 | . . . . . . . . . . . . . . 15
⊢ (𝑗 ∈ ℕ → 𝑗 ∈
ℤ) | 
| 127 | 126 | peano2zd 12725 | . . . . . . . . . . . . . 14
⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ∈
ℤ) | 
| 128 | 108, 125,
127 | expne0d 14192 | . . . . . . . . . . . . 13
⊢ (𝑗 ∈ ℕ → (((𝑗 + 1) / e)↑(𝑗 + 1)) ≠ 0) | 
| 129 | 100, 109,
123, 128 | mulne0d 11915 | . . . . . . . . . . . 12
⊢ (𝑗 ∈ ℕ →
((√‘(2 · (𝑗 + 1))) · (((𝑗 + 1) / e)↑(𝑗 + 1))) ≠ 0) | 
| 130 | 94, 110, 129 | divcld 12043 | . . . . . . . . . . 11
⊢ (𝑗 ∈ ℕ →
((!‘(𝑗 + 1)) /
((√‘(2 · (𝑗 + 1))) · (((𝑗 + 1) / e)↑(𝑗 + 1)))) ∈ ℂ) | 
| 131 | 82, 90, 81, 130 | fvmptd 7023 | . . . . . . . . . 10
⊢ (𝑗 ∈ ℕ → (𝐴‘(𝑗 + 1)) = ((!‘(𝑗 + 1)) / ((√‘(2 · (𝑗 + 1))) · (((𝑗 + 1) / e)↑(𝑗 + 1))))) | 
| 132 |  | nnrp 13046 | . . . . . . . . . . . 12
⊢
((!‘(𝑗 + 1))
∈ ℕ → (!‘(𝑗 + 1)) ∈
ℝ+) | 
| 133 | 91, 92, 132 | 3syl 18 | . . . . . . . . . . 11
⊢ (𝑗 ∈ ℕ →
(!‘(𝑗 + 1)) ∈
ℝ+) | 
| 134 | 121 | rpsqrtcld 15450 | . . . . . . . . . . . 12
⊢ (𝑗 ∈ ℕ →
(√‘(2 · (𝑗 + 1))) ∈
ℝ+) | 
| 135 |  | epr 16244 | . . . . . . . . . . . . . . 15
⊢ e ∈
ℝ+ | 
| 136 | 135 | a1i 11 | . . . . . . . . . . . . . 14
⊢ (𝑗 ∈ ℕ → e ∈
ℝ+) | 
| 137 | 120, 136 | rpdivcld 13094 | . . . . . . . . . . . . 13
⊢ (𝑗 ∈ ℕ → ((𝑗 + 1) / e) ∈
ℝ+) | 
| 138 | 137, 127 | rpexpcld 14286 | . . . . . . . . . . . 12
⊢ (𝑗 ∈ ℕ → (((𝑗 + 1) / e)↑(𝑗 + 1)) ∈
ℝ+) | 
| 139 | 134, 138 | rpmulcld 13093 | . . . . . . . . . . 11
⊢ (𝑗 ∈ ℕ →
((√‘(2 · (𝑗 + 1))) · (((𝑗 + 1) / e)↑(𝑗 + 1))) ∈
ℝ+) | 
| 140 | 133, 139 | rpdivcld 13094 | . . . . . . . . . 10
⊢ (𝑗 ∈ ℕ →
((!‘(𝑗 + 1)) /
((√‘(2 · (𝑗 + 1))) · (((𝑗 + 1) / e)↑(𝑗 + 1)))) ∈
ℝ+) | 
| 141 | 131, 140 | eqeltrd 2841 | . . . . . . . . 9
⊢ (𝑗 ∈ ℕ → (𝐴‘(𝑗 + 1)) ∈
ℝ+) | 
| 142 | 141 | relogcld 26665 | . . . . . . . 8
⊢ (𝑗 ∈ ℕ →
(log‘(𝐴‘(𝑗 + 1))) ∈
ℝ) | 
| 143 |  | nfcv 2905 | . . . . . . . . 9
⊢
Ⅎ𝑛(𝑗 + 1) | 
| 144 | 21, 143 | nffv 6916 | . . . . . . . . . 10
⊢
Ⅎ𝑛(𝐴‘(𝑗 + 1)) | 
| 145 | 19, 144 | nffv 6916 | . . . . . . . . 9
⊢
Ⅎ𝑛(log‘(𝐴‘(𝑗 + 1))) | 
| 146 |  | 2fveq3 6911 | . . . . . . . . 9
⊢ (𝑛 = (𝑗 + 1) → (log‘(𝐴‘𝑛)) = (log‘(𝐴‘(𝑗 + 1)))) | 
| 147 | 143, 145,
146, 2 | fvmptf 7037 | . . . . . . . 8
⊢ (((𝑗 + 1) ∈ ℕ ∧
(log‘(𝐴‘(𝑗 + 1))) ∈ ℝ) →
(𝐵‘(𝑗 + 1)) = (log‘(𝐴‘(𝑗 + 1)))) | 
| 148 | 81, 142, 147 | syl2anc 584 | . . . . . . 7
⊢ (𝑗 ∈ ℕ → (𝐵‘(𝑗 + 1)) = (log‘(𝐴‘(𝑗 + 1)))) | 
| 149 | 148, 142 | eqeltrd 2841 | . . . . . 6
⊢ (𝑗 ∈ ℕ → (𝐵‘(𝑗 + 1)) ∈ ℝ) | 
| 150 | 79 | ffvelcdmi 7103 | . . . . . 6
⊢ (𝑗 ∈ ℕ → (𝐵‘𝑗) ∈ ℝ) | 
| 151 |  | eqid 2737 | . . . . . . 7
⊢ (𝑧 ∈ ℕ ↦ ((1 /
((2 · 𝑧) + 1))
· ((1 / ((2 · 𝑗) + 1))↑(2 · 𝑧)))) = (𝑧 ∈ ℕ ↦ ((1 / ((2 ·
𝑧) + 1)) · ((1 / ((2
· 𝑗) + 1))↑(2
· 𝑧)))) | 
| 152 | 6, 2, 151 | stirlinglem11 46099 | . . . . . 6
⊢ (𝑗 ∈ ℕ → (𝐵‘(𝑗 + 1)) < (𝐵‘𝑗)) | 
| 153 | 149, 150,
152 | ltled 11409 | . . . . 5
⊢ (𝑗 ∈ ℕ → (𝐵‘(𝑗 + 1)) ≤ (𝐵‘𝑗)) | 
| 154 | 153 | adantl 481 | . . . 4
⊢
((⊤ ∧ 𝑗
∈ ℕ) → (𝐵‘(𝑗 + 1)) ≤ (𝐵‘𝑗)) | 
| 155 | 52 | a1i 11 | . . . 4
⊢ (⊤
→ ∃𝑥 ∈
ℝ ∀𝑗 ∈
ℕ 𝑥 ≤ (𝐵‘𝑗)) | 
| 156 | 77, 78, 80, 154, 155 | climinf 45621 | . . 3
⊢ (⊤
→ 𝐵 ⇝ inf(ran
𝐵, ℝ, <
)) | 
| 157 | 156 | mptru 1547 | . 2
⊢ 𝐵 ⇝ inf(ran 𝐵, ℝ, <
) | 
| 158 |  | breq2 5147 | . . 3
⊢ (𝑑 = inf(ran 𝐵, ℝ, < ) → (𝐵 ⇝ 𝑑 ↔ 𝐵 ⇝ inf(ran 𝐵, ℝ, < ))) | 
| 159 | 158 | rspcev 3622 | . 2
⊢ ((inf(ran
𝐵, ℝ, < ) ∈
ℝ ∧ 𝐵 ⇝
inf(ran 𝐵, ℝ, < ))
→ ∃𝑑 ∈
ℝ 𝐵 ⇝ 𝑑) | 
| 160 | 76, 157, 159 | mp2an 692 | 1
⊢
∃𝑑 ∈
ℝ 𝐵 ⇝ 𝑑 |