Step | Hyp | Ref
| Expression |
1 | | vex 3203 |
. . . . . 6
⊢ 𝑦 ∈ V |
2 | | stirlinglem13.2 |
. . . . . . 7
⊢ 𝐵 = (𝑛 ∈ ℕ ↦ (log‘(𝐴‘𝑛))) |
3 | 2 | elrnmpt 5372 |
. . . . . 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 477 |
. . . . . . 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 40292 |
. . . . . . . . 9
⊢ (𝑛 ∈ ℕ → (𝐴‘𝑛) ∈
ℝ+) |
8 | 7 | relogcld 24369 |
. . . . . . . 8
⊢ (𝑛 ∈ ℕ →
(log‘(𝐴‘𝑛)) ∈
ℝ) |
9 | 8 | adantr 481 |
. . . . . . 7
⊢ ((𝑛 ∈ ℕ ∧ 𝑦 = (log‘(𝐴‘𝑛))) → (log‘(𝐴‘𝑛)) ∈ ℝ) |
10 | 5, 9 | eqeltrd 2701 |
. . . . . 6
⊢ ((𝑛 ∈ ℕ ∧ 𝑦 = (log‘(𝐴‘𝑛))) → 𝑦 ∈ ℝ) |
11 | 10 | rexlimiva 3028 |
. . . . 5
⊢
(∃𝑛 ∈
ℕ 𝑦 =
(log‘(𝐴‘𝑛)) → 𝑦 ∈ ℝ) |
12 | 4, 11 | sylbi 207 |
. . . 4
⊢ (𝑦 ∈ ran 𝐵 → 𝑦 ∈ ℝ) |
13 | 12 | ssriv 3607 |
. . 3
⊢ ran 𝐵 ⊆
ℝ |
14 | | 1nn 11031 |
. . . . . 6
⊢ 1 ∈
ℕ |
15 | 6 | stirlinglem2 40292 |
. . . . . . . 8
⊢ (1 ∈
ℕ → (𝐴‘1)
∈ ℝ+) |
16 | | relogcl 24322 |
. . . . . . . 8
⊢ ((𝐴‘1) ∈
ℝ+ → (log‘(𝐴‘1)) ∈ ℝ) |
17 | 14, 15, 16 | mp2b 10 |
. . . . . . 7
⊢
(log‘(𝐴‘1)) ∈ ℝ |
18 | | nfcv 2764 |
. . . . . . . 8
⊢
Ⅎ𝑛1 |
19 | | nfcv 2764 |
. . . . . . . . 9
⊢
Ⅎ𝑛log |
20 | | nfmpt1 4747 |
. . . . . . . . . . 11
⊢
Ⅎ𝑛(𝑛 ∈ ℕ ↦ ((!‘𝑛) / ((√‘(2 ·
𝑛)) · ((𝑛 / e)↑𝑛)))) |
21 | 6, 20 | nfcxfr 2762 |
. . . . . . . . . 10
⊢
Ⅎ𝑛𝐴 |
22 | 21, 18 | nffv 6198 |
. . . . . . . . 9
⊢
Ⅎ𝑛(𝐴‘1) |
23 | 19, 22 | nffv 6198 |
. . . . . . . 8
⊢
Ⅎ𝑛(log‘(𝐴‘1)) |
24 | | fveq2 6191 |
. . . . . . . . 9
⊢ (𝑛 = 1 → (𝐴‘𝑛) = (𝐴‘1)) |
25 | 24 | fveq2d 6195 |
. . . . . . . 8
⊢ (𝑛 = 1 → (log‘(𝐴‘𝑛)) = (log‘(𝐴‘1))) |
26 | 18, 23, 25, 2 | fvmptf 6301 |
. . . . . . 7
⊢ ((1
∈ ℕ ∧ (log‘(𝐴‘1)) ∈ ℝ) → (𝐵‘1) = (log‘(𝐴‘1))) |
27 | 14, 17, 26 | mp2an 708 |
. . . . . 6
⊢ (𝐵‘1) = (log‘(𝐴‘1)) |
28 | | fveq2 6191 |
. . . . . . . . 9
⊢ (𝑗 = 1 → (𝐴‘𝑗) = (𝐴‘1)) |
29 | 28 | fveq2d 6195 |
. . . . . . . 8
⊢ (𝑗 = 1 → (log‘(𝐴‘𝑗)) = (log‘(𝐴‘1))) |
30 | 29 | eqeq2d 2632 |
. . . . . . 7
⊢ (𝑗 = 1 → ((𝐵‘1) = (log‘(𝐴‘𝑗)) ↔ (𝐵‘1) = (log‘(𝐴‘1)))) |
31 | 30 | rspcev 3309 |
. . . . . 6
⊢ ((1
∈ ℕ ∧ (𝐵‘1) = (log‘(𝐴‘1))) → ∃𝑗 ∈ ℕ (𝐵‘1) = (log‘(𝐴‘𝑗))) |
32 | 14, 27, 31 | mp2an 708 |
. . . . 5
⊢
∃𝑗 ∈
ℕ (𝐵‘1) =
(log‘(𝐴‘𝑗)) |
33 | 27, 17 | eqeltri 2697 |
. . . . . 6
⊢ (𝐵‘1) ∈
ℝ |
34 | | nfcv 2764 |
. . . . . . . . 9
⊢
Ⅎ𝑗(log‘(𝐴‘𝑛)) |
35 | | nfcv 2764 |
. . . . . . . . . . 11
⊢
Ⅎ𝑛𝑗 |
36 | 21, 35 | nffv 6198 |
. . . . . . . . . 10
⊢
Ⅎ𝑛(𝐴‘𝑗) |
37 | 19, 36 | nffv 6198 |
. . . . . . . . 9
⊢
Ⅎ𝑛(log‘(𝐴‘𝑗)) |
38 | | fveq2 6191 |
. . . . . . . . . 10
⊢ (𝑛 = 𝑗 → (𝐴‘𝑛) = (𝐴‘𝑗)) |
39 | 38 | fveq2d 6195 |
. . . . . . . . 9
⊢ (𝑛 = 𝑗 → (log‘(𝐴‘𝑛)) = (log‘(𝐴‘𝑗))) |
40 | 34, 37, 39 | cbvmpt 4749 |
. . . . . . . 8
⊢ (𝑛 ∈ ℕ ↦
(log‘(𝐴‘𝑛))) = (𝑗 ∈ ℕ ↦ (log‘(𝐴‘𝑗))) |
41 | 2, 40 | eqtri 2644 |
. . . . . . 7
⊢ 𝐵 = (𝑗 ∈ ℕ ↦ (log‘(𝐴‘𝑗))) |
42 | 41 | elrnmpt 5372 |
. . . . . 6
⊢ ((𝐵‘1) ∈ ℝ →
((𝐵‘1) ∈ ran
𝐵 ↔ ∃𝑗 ∈ ℕ (𝐵‘1) = (log‘(𝐴‘𝑗)))) |
43 | 33, 42 | ax-mp 5 |
. . . . 5
⊢ ((𝐵‘1) ∈ ran 𝐵 ↔ ∃𝑗 ∈ ℕ (𝐵‘1) = (log‘(𝐴‘𝑗))) |
44 | 32, 43 | mpbir 221 |
. . . 4
⊢ (𝐵‘1) ∈ ran 𝐵 |
45 | 44 | ne0ii 3923 |
. . 3
⊢ ran 𝐵 ≠ ∅ |
46 | | 4re 11097 |
. . . . . . 7
⊢ 4 ∈
ℝ |
47 | | 4ne0 11117 |
. . . . . . 7
⊢ 4 ≠
0 |
48 | 46, 47 | rereccli 10790 |
. . . . . 6
⊢ (1 / 4)
∈ ℝ |
49 | 33, 48 | resubcli 10343 |
. . . . 5
⊢ ((𝐵‘1) − (1 / 4))
∈ ℝ |
50 | | eqid 2622 |
. . . . . . 7
⊢ (𝑛 ∈ ℕ ↦ (1 /
(𝑛 · (𝑛 + 1)))) = (𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1)))) |
51 | 6, 2, 50 | stirlinglem12 40302 |
. . . . . 6
⊢ (𝑗 ∈ ℕ → ((𝐵‘1) − (1 / 4)) ≤
(𝐵‘𝑗)) |
52 | 51 | rgen 2922 |
. . . . 5
⊢
∀𝑗 ∈
ℕ ((𝐵‘1)
− (1 / 4)) ≤ (𝐵‘𝑗) |
53 | | breq1 4656 |
. . . . . . 7
⊢ (𝑥 = ((𝐵‘1) − (1 / 4)) → (𝑥 ≤ (𝐵‘𝑗) ↔ ((𝐵‘1) − (1 / 4)) ≤ (𝐵‘𝑗))) |
54 | 53 | ralbidv 2986 |
. . . . . 6
⊢ (𝑥 = ((𝐵‘1) − (1 / 4)) →
(∀𝑗 ∈ ℕ
𝑥 ≤ (𝐵‘𝑗) ↔ ∀𝑗 ∈ ℕ ((𝐵‘1) − (1 / 4)) ≤ (𝐵‘𝑗))) |
55 | 54 | rspcev 3309 |
. . . . 5
⊢ ((((𝐵‘1) − (1 / 4))
∈ ℝ ∧ ∀𝑗 ∈ ℕ ((𝐵‘1) − (1 / 4)) ≤ (𝐵‘𝑗)) → ∃𝑥 ∈ ℝ ∀𝑗 ∈ ℕ 𝑥 ≤ (𝐵‘𝑗)) |
56 | 49, 52, 55 | mp2an 708 |
. . . 4
⊢
∃𝑥 ∈
ℝ ∀𝑗 ∈
ℕ 𝑥 ≤ (𝐵‘𝑗) |
57 | | simpr 477 |
. . . . . . . 8
⊢
((∀𝑗 ∈
ℕ 𝑥 ≤ (𝐵‘𝑗) ∧ 𝑦 ∈ ran 𝐵) → 𝑦 ∈ ran 𝐵) |
58 | 8 | rgen 2922 |
. . . . . . . . 9
⊢
∀𝑛 ∈
ℕ (log‘(𝐴‘𝑛)) ∈ ℝ |
59 | 2 | fnmpt 6020 |
. . . . . . . . 9
⊢
(∀𝑛 ∈
ℕ (log‘(𝐴‘𝑛)) ∈ ℝ → 𝐵 Fn ℕ) |
60 | | fvelrnb 6243 |
. . . . . . . . 9
⊢ (𝐵 Fn ℕ → (𝑦 ∈ ran 𝐵 ↔ ∃𝑗 ∈ ℕ (𝐵‘𝑗) = 𝑦)) |
61 | 58, 59, 60 | mp2b 10 |
. . . . . . . 8
⊢ (𝑦 ∈ ran 𝐵 ↔ ∃𝑗 ∈ ℕ (𝐵‘𝑗) = 𝑦) |
62 | 57, 61 | sylib 208 |
. . . . . . 7
⊢
((∀𝑗 ∈
ℕ 𝑥 ≤ (𝐵‘𝑗) ∧ 𝑦 ∈ ran 𝐵) → ∃𝑗 ∈ ℕ (𝐵‘𝑗) = 𝑦) |
63 | | nfra1 2941 |
. . . . . . . . 9
⊢
Ⅎ𝑗∀𝑗 ∈ ℕ 𝑥 ≤ (𝐵‘𝑗) |
64 | | nfv 1843 |
. . . . . . . . 9
⊢
Ⅎ𝑗 𝑦 ∈ ran 𝐵 |
65 | 63, 64 | nfan 1828 |
. . . . . . . 8
⊢
Ⅎ𝑗(∀𝑗 ∈ ℕ 𝑥 ≤ (𝐵‘𝑗) ∧ 𝑦 ∈ ran 𝐵) |
66 | | nfv 1843 |
. . . . . . . 8
⊢
Ⅎ𝑗 𝑥 ≤ 𝑦 |
67 | | simp1l 1085 |
. . . . . . . . . . 11
⊢
(((∀𝑗 ∈
ℕ 𝑥 ≤ (𝐵‘𝑗) ∧ 𝑦 ∈ ran 𝐵) ∧ 𝑗 ∈ ℕ ∧ (𝐵‘𝑗) = 𝑦) → ∀𝑗 ∈ ℕ 𝑥 ≤ (𝐵‘𝑗)) |
68 | | simp2 1062 |
. . . . . . . . . . 11
⊢
(((∀𝑗 ∈
ℕ 𝑥 ≤ (𝐵‘𝑗) ∧ 𝑦 ∈ ran 𝐵) ∧ 𝑗 ∈ ℕ ∧ (𝐵‘𝑗) = 𝑦) → 𝑗 ∈ ℕ) |
69 | | rsp 2929 |
. . . . . . . . . . 11
⊢
(∀𝑗 ∈
ℕ 𝑥 ≤ (𝐵‘𝑗) → (𝑗 ∈ ℕ → 𝑥 ≤ (𝐵‘𝑗))) |
70 | 67, 68, 69 | sylc 65 |
. . . . . . . . . 10
⊢
(((∀𝑗 ∈
ℕ 𝑥 ≤ (𝐵‘𝑗) ∧ 𝑦 ∈ ran 𝐵) ∧ 𝑗 ∈ ℕ ∧ (𝐵‘𝑗) = 𝑦) → 𝑥 ≤ (𝐵‘𝑗)) |
71 | | simp3 1063 |
. . . . . . . . . 10
⊢
(((∀𝑗 ∈
ℕ 𝑥 ≤ (𝐵‘𝑗) ∧ 𝑦 ∈ ran 𝐵) ∧ 𝑗 ∈ ℕ ∧ (𝐵‘𝑗) = 𝑦) → (𝐵‘𝑗) = 𝑦) |
72 | 70, 71 | breqtrd 4679 |
. . . . . . . . 9
⊢
(((∀𝑗 ∈
ℕ 𝑥 ≤ (𝐵‘𝑗) ∧ 𝑦 ∈ ran 𝐵) ∧ 𝑗 ∈ ℕ ∧ (𝐵‘𝑗) = 𝑦) → 𝑥 ≤ 𝑦) |
73 | 72 | 3exp 1264 |
. . . . . . . 8
⊢
((∀𝑗 ∈
ℕ 𝑥 ≤ (𝐵‘𝑗) ∧ 𝑦 ∈ ran 𝐵) → (𝑗 ∈ ℕ → ((𝐵‘𝑗) = 𝑦 → 𝑥 ≤ 𝑦))) |
74 | 65, 66, 73 | rexlimd 3026 |
. . . . . . 7
⊢
((∀𝑗 ∈
ℕ 𝑥 ≤ (𝐵‘𝑗) ∧ 𝑦 ∈ ran 𝐵) → (∃𝑗 ∈ ℕ (𝐵‘𝑗) = 𝑦 → 𝑥 ≤ 𝑦)) |
75 | 62, 74 | mpd 15 |
. . . . . 6
⊢
((∀𝑗 ∈
ℕ 𝑥 ≤ (𝐵‘𝑗) ∧ 𝑦 ∈ ran 𝐵) → 𝑥 ≤ 𝑦) |
76 | 75 | ralrimiva 2966 |
. . . . 5
⊢
(∀𝑗 ∈
ℕ 𝑥 ≤ (𝐵‘𝑗) → ∀𝑦 ∈ ran 𝐵 𝑥 ≤ 𝑦) |
77 | 76 | reximi 3011 |
. . . 4
⊢
(∃𝑥 ∈
ℝ ∀𝑗 ∈
ℕ 𝑥 ≤ (𝐵‘𝑗) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐵 𝑥 ≤ 𝑦) |
78 | 56, 77 | ax-mp 5 |
. . 3
⊢
∃𝑥 ∈
ℝ ∀𝑦 ∈
ran 𝐵 𝑥 ≤ 𝑦 |
79 | | infrecl 11005 |
. . 3
⊢ ((ran
𝐵 ⊆ ℝ ∧ ran
𝐵 ≠ ∅ ∧
∃𝑥 ∈ ℝ
∀𝑦 ∈ ran 𝐵 𝑥 ≤ 𝑦) → inf(ran 𝐵, ℝ, < ) ∈
ℝ) |
80 | 13, 45, 78, 79 | mp3an 1424 |
. 2
⊢ inf(ran
𝐵, ℝ, < ) ∈
ℝ |
81 | | nnuz 11723 |
. . . 4
⊢ ℕ =
(ℤ≥‘1) |
82 | | 1zzd 11408 |
. . . 4
⊢ (⊤
→ 1 ∈ ℤ) |
83 | 2, 8 | fmpti 6383 |
. . . . 5
⊢ 𝐵:ℕ⟶ℝ |
84 | 83 | a1i 11 |
. . . 4
⊢ (⊤
→ 𝐵:ℕ⟶ℝ) |
85 | | peano2nn 11032 |
. . . . . . . 8
⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ∈
ℕ) |
86 | 6 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝑗 ∈ ℕ → 𝐴 = (𝑛 ∈ ℕ ↦ ((!‘𝑛) / ((√‘(2 ·
𝑛)) · ((𝑛 / e)↑𝑛))))) |
87 | | simpr 477 |
. . . . . . . . . . . . 13
⊢ ((𝑗 ∈ ℕ ∧ 𝑛 = (𝑗 + 1)) → 𝑛 = (𝑗 + 1)) |
88 | 87 | fveq2d 6195 |
. . . . . . . . . . . 12
⊢ ((𝑗 ∈ ℕ ∧ 𝑛 = (𝑗 + 1)) → (!‘𝑛) = (!‘(𝑗 + 1))) |
89 | 87 | oveq2d 6666 |
. . . . . . . . . . . . . 14
⊢ ((𝑗 ∈ ℕ ∧ 𝑛 = (𝑗 + 1)) → (2 · 𝑛) = (2 · (𝑗 + 1))) |
90 | 89 | fveq2d 6195 |
. . . . . . . . . . . . 13
⊢ ((𝑗 ∈ ℕ ∧ 𝑛 = (𝑗 + 1)) → (√‘(2 ·
𝑛)) = (√‘(2
· (𝑗 +
1)))) |
91 | 87 | oveq1d 6665 |
. . . . . . . . . . . . . 14
⊢ ((𝑗 ∈ ℕ ∧ 𝑛 = (𝑗 + 1)) → (𝑛 / e) = ((𝑗 + 1) / e)) |
92 | 91, 87 | oveq12d 6668 |
. . . . . . . . . . . . 13
⊢ ((𝑗 ∈ ℕ ∧ 𝑛 = (𝑗 + 1)) → ((𝑛 / e)↑𝑛) = (((𝑗 + 1) / e)↑(𝑗 + 1))) |
93 | 90, 92 | oveq12d 6668 |
. . . . . . . . . . . 12
⊢ ((𝑗 ∈ ℕ ∧ 𝑛 = (𝑗 + 1)) → ((√‘(2 ·
𝑛)) · ((𝑛 / e)↑𝑛)) = ((√‘(2 · (𝑗 + 1))) · (((𝑗 + 1) / e)↑(𝑗 + 1)))) |
94 | 88, 93 | oveq12d 6668 |
. . . . . . . . . . 11
⊢ ((𝑗 ∈ ℕ ∧ 𝑛 = (𝑗 + 1)) → ((!‘𝑛) / ((√‘(2 · 𝑛)) · ((𝑛 / e)↑𝑛))) = ((!‘(𝑗 + 1)) / ((√‘(2 · (𝑗 + 1))) · (((𝑗 + 1) / e)↑(𝑗 + 1))))) |
95 | 85 | nnnn0d 11351 |
. . . . . . . . . . . . 13
⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ∈
ℕ0) |
96 | | faccl 13070 |
. . . . . . . . . . . . 13
⊢ ((𝑗 + 1) ∈ ℕ0
→ (!‘(𝑗 + 1))
∈ ℕ) |
97 | | nncn 11028 |
. . . . . . . . . . . . 13
⊢
((!‘(𝑗 + 1))
∈ ℕ → (!‘(𝑗 + 1)) ∈ ℂ) |
98 | 95, 96, 97 | 3syl 18 |
. . . . . . . . . . . 12
⊢ (𝑗 ∈ ℕ →
(!‘(𝑗 + 1)) ∈
ℂ) |
99 | | 2cnd 11093 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 ∈ ℕ → 2 ∈
ℂ) |
100 | | nncn 11028 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 ∈ ℕ → 𝑗 ∈
ℂ) |
101 | | 1cnd 10056 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 ∈ ℕ → 1 ∈
ℂ) |
102 | 100, 101 | addcld 10059 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ∈
ℂ) |
103 | 99, 102 | mulcld 10060 |
. . . . . . . . . . . . . 14
⊢ (𝑗 ∈ ℕ → (2
· (𝑗 + 1)) ∈
ℂ) |
104 | 103 | sqrtcld 14176 |
. . . . . . . . . . . . 13
⊢ (𝑗 ∈ ℕ →
(√‘(2 · (𝑗 + 1))) ∈ ℂ) |
105 | | ere 14819 |
. . . . . . . . . . . . . . . . 17
⊢ e ∈
ℝ |
106 | 105 | recni 10052 |
. . . . . . . . . . . . . . . 16
⊢ e ∈
ℂ |
107 | 106 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 ∈ ℕ → e ∈
ℂ) |
108 | | 0re 10040 |
. . . . . . . . . . . . . . . . 17
⊢ 0 ∈
ℝ |
109 | | epos 14935 |
. . . . . . . . . . . . . . . . 17
⊢ 0 <
e |
110 | 108, 109 | gtneii 10149 |
. . . . . . . . . . . . . . . 16
⊢ e ≠
0 |
111 | 110 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 ∈ ℕ → e ≠
0) |
112 | 102, 107,
111 | divcld 10801 |
. . . . . . . . . . . . . 14
⊢ (𝑗 ∈ ℕ → ((𝑗 + 1) / e) ∈
ℂ) |
113 | 112, 95 | expcld 13008 |
. . . . . . . . . . . . 13
⊢ (𝑗 ∈ ℕ → (((𝑗 + 1) / e)↑(𝑗 + 1)) ∈
ℂ) |
114 | 104, 113 | mulcld 10060 |
. . . . . . . . . . . 12
⊢ (𝑗 ∈ ℕ →
((√‘(2 · (𝑗 + 1))) · (((𝑗 + 1) / e)↑(𝑗 + 1))) ∈ ℂ) |
115 | | 2rp 11837 |
. . . . . . . . . . . . . . . . 17
⊢ 2 ∈
ℝ+ |
116 | 115 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 ∈ ℕ → 2 ∈
ℝ+) |
117 | | nnre 11027 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 ∈ ℕ → 𝑗 ∈
ℝ) |
118 | 108 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 ∈ ℕ → 0 ∈
ℝ) |
119 | | 1red 10055 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 ∈ ℕ → 1 ∈
ℝ) |
120 | | 0le1 10551 |
. . . . . . . . . . . . . . . . . . 19
⊢ 0 ≤
1 |
121 | 120 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 ∈ ℕ → 0 ≤
1) |
122 | | nnge1 11046 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 ∈ ℕ → 1 ≤
𝑗) |
123 | 118, 119,
117, 121, 122 | letrd 10194 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 ∈ ℕ → 0 ≤
𝑗) |
124 | 117, 123 | ge0p1rpd 11902 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ∈
ℝ+) |
125 | 116, 124 | rpmulcld 11888 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 ∈ ℕ → (2
· (𝑗 + 1)) ∈
ℝ+) |
126 | 125 | sqrtgt0d 14151 |
. . . . . . . . . . . . . 14
⊢ (𝑗 ∈ ℕ → 0 <
(√‘(2 · (𝑗 + 1)))) |
127 | 126 | gt0ne0d 10592 |
. . . . . . . . . . . . 13
⊢ (𝑗 ∈ ℕ →
(√‘(2 · (𝑗 + 1))) ≠ 0) |
128 | 85 | nnne0d 11065 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ≠ 0) |
129 | 102, 107,
128, 111 | divne0d 10817 |
. . . . . . . . . . . . . 14
⊢ (𝑗 ∈ ℕ → ((𝑗 + 1) / e) ≠
0) |
130 | | nnz 11399 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 ∈ ℕ → 𝑗 ∈
ℤ) |
131 | 130 | peano2zd 11485 |
. . . . . . . . . . . . . 14
⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ∈
ℤ) |
132 | 112, 129,
131 | expne0d 13014 |
. . . . . . . . . . . . 13
⊢ (𝑗 ∈ ℕ → (((𝑗 + 1) / e)↑(𝑗 + 1)) ≠ 0) |
133 | 104, 113,
127, 132 | mulne0d 10679 |
. . . . . . . . . . . 12
⊢ (𝑗 ∈ ℕ →
((√‘(2 · (𝑗 + 1))) · (((𝑗 + 1) / e)↑(𝑗 + 1))) ≠ 0) |
134 | 98, 114, 133 | divcld 10801 |
. . . . . . . . . . 11
⊢ (𝑗 ∈ ℕ →
((!‘(𝑗 + 1)) /
((√‘(2 · (𝑗 + 1))) · (((𝑗 + 1) / e)↑(𝑗 + 1)))) ∈ ℂ) |
135 | 86, 94, 85, 134 | fvmptd 6288 |
. . . . . . . . . 10
⊢ (𝑗 ∈ ℕ → (𝐴‘(𝑗 + 1)) = ((!‘(𝑗 + 1)) / ((√‘(2 · (𝑗 + 1))) · (((𝑗 + 1) / e)↑(𝑗 + 1))))) |
136 | | nnrp 11842 |
. . . . . . . . . . . 12
⊢
((!‘(𝑗 + 1))
∈ ℕ → (!‘(𝑗 + 1)) ∈
ℝ+) |
137 | 95, 96, 136 | 3syl 18 |
. . . . . . . . . . 11
⊢ (𝑗 ∈ ℕ →
(!‘(𝑗 + 1)) ∈
ℝ+) |
138 | 125 | rpsqrtcld 14150 |
. . . . . . . . . . . 12
⊢ (𝑗 ∈ ℕ →
(√‘(2 · (𝑗 + 1))) ∈
ℝ+) |
139 | | epr 14936 |
. . . . . . . . . . . . . . 15
⊢ e ∈
ℝ+ |
140 | 139 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝑗 ∈ ℕ → e ∈
ℝ+) |
141 | 124, 140 | rpdivcld 11889 |
. . . . . . . . . . . . 13
⊢ (𝑗 ∈ ℕ → ((𝑗 + 1) / e) ∈
ℝ+) |
142 | 141, 131 | rpexpcld 13032 |
. . . . . . . . . . . 12
⊢ (𝑗 ∈ ℕ → (((𝑗 + 1) / e)↑(𝑗 + 1)) ∈
ℝ+) |
143 | 138, 142 | rpmulcld 11888 |
. . . . . . . . . . 11
⊢ (𝑗 ∈ ℕ →
((√‘(2 · (𝑗 + 1))) · (((𝑗 + 1) / e)↑(𝑗 + 1))) ∈
ℝ+) |
144 | 137, 143 | rpdivcld 11889 |
. . . . . . . . . 10
⊢ (𝑗 ∈ ℕ →
((!‘(𝑗 + 1)) /
((√‘(2 · (𝑗 + 1))) · (((𝑗 + 1) / e)↑(𝑗 + 1)))) ∈
ℝ+) |
145 | 135, 144 | eqeltrd 2701 |
. . . . . . . . 9
⊢ (𝑗 ∈ ℕ → (𝐴‘(𝑗 + 1)) ∈
ℝ+) |
146 | 145 | relogcld 24369 |
. . . . . . . 8
⊢ (𝑗 ∈ ℕ →
(log‘(𝐴‘(𝑗 + 1))) ∈
ℝ) |
147 | | nfcv 2764 |
. . . . . . . . 9
⊢
Ⅎ𝑛(𝑗 + 1) |
148 | 21, 147 | nffv 6198 |
. . . . . . . . . 10
⊢
Ⅎ𝑛(𝐴‘(𝑗 + 1)) |
149 | 19, 148 | nffv 6198 |
. . . . . . . . 9
⊢
Ⅎ𝑛(log‘(𝐴‘(𝑗 + 1))) |
150 | | fveq2 6191 |
. . . . . . . . . 10
⊢ (𝑛 = (𝑗 + 1) → (𝐴‘𝑛) = (𝐴‘(𝑗 + 1))) |
151 | 150 | fveq2d 6195 |
. . . . . . . . 9
⊢ (𝑛 = (𝑗 + 1) → (log‘(𝐴‘𝑛)) = (log‘(𝐴‘(𝑗 + 1)))) |
152 | 147, 149,
151, 2 | fvmptf 6301 |
. . . . . . . 8
⊢ (((𝑗 + 1) ∈ ℕ ∧
(log‘(𝐴‘(𝑗 + 1))) ∈ ℝ) →
(𝐵‘(𝑗 + 1)) = (log‘(𝐴‘(𝑗 + 1)))) |
153 | 85, 146, 152 | syl2anc 693 |
. . . . . . 7
⊢ (𝑗 ∈ ℕ → (𝐵‘(𝑗 + 1)) = (log‘(𝐴‘(𝑗 + 1)))) |
154 | 153, 146 | eqeltrd 2701 |
. . . . . 6
⊢ (𝑗 ∈ ℕ → (𝐵‘(𝑗 + 1)) ∈ ℝ) |
155 | 83 | ffvelrni 6358 |
. . . . . 6
⊢ (𝑗 ∈ ℕ → (𝐵‘𝑗) ∈ ℝ) |
156 | | eqid 2622 |
. . . . . . 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)))) |
157 | 6, 2, 156 | stirlinglem11 40301 |
. . . . . 6
⊢ (𝑗 ∈ ℕ → (𝐵‘(𝑗 + 1)) < (𝐵‘𝑗)) |
158 | 154, 155,
157 | ltled 10185 |
. . . . 5
⊢ (𝑗 ∈ ℕ → (𝐵‘(𝑗 + 1)) ≤ (𝐵‘𝑗)) |
159 | 158 | adantl 482 |
. . . 4
⊢
((⊤ ∧ 𝑗
∈ ℕ) → (𝐵‘(𝑗 + 1)) ≤ (𝐵‘𝑗)) |
160 | 56 | a1i 11 |
. . . 4
⊢ (⊤
→ ∃𝑥 ∈
ℝ ∀𝑗 ∈
ℕ 𝑥 ≤ (𝐵‘𝑗)) |
161 | 81, 82, 84, 159, 160 | climinf 39838 |
. . 3
⊢ (⊤
→ 𝐵 ⇝ inf(ran
𝐵, ℝ, <
)) |
162 | 161 | trud 1493 |
. 2
⊢ 𝐵 ⇝ inf(ran 𝐵, ℝ, <
) |
163 | | breq2 4657 |
. . 3
⊢ (𝑑 = inf(ran 𝐵, ℝ, < ) → (𝐵 ⇝ 𝑑 ↔ 𝐵 ⇝ inf(ran 𝐵, ℝ, < ))) |
164 | 163 | rspcev 3309 |
. 2
⊢ ((inf(ran
𝐵, ℝ, < ) ∈
ℝ ∧ 𝐵 ⇝
inf(ran 𝐵, ℝ, < ))
→ ∃𝑑 ∈
ℝ 𝐵 ⇝ 𝑑) |
165 | 80, 162, 164 | mp2an 708 |
1
⊢
∃𝑑 ∈
ℝ 𝐵 ⇝ 𝑑 |