Step | Hyp | Ref
| Expression |
1 | | nnuz 12356 |
. . . . 5
⊢ ℕ =
(ℤ≥‘1) |
2 | | 1zzd 12087 |
. . . . 5
⊢ (⊤
→ 1 ∈ ℤ) |
3 | | oveq2 7172 |
. . . . . . . . . 10
⊢ (𝑛 = 𝑘 → (1 / 𝑛) = (1 / 𝑘)) |
4 | 3 | oveq2d 7180 |
. . . . . . . . . . 11
⊢ (𝑛 = 𝑘 → (1 + (1 / 𝑛)) = (1 + (1 / 𝑘))) |
5 | 4 | fveq2d 6672 |
. . . . . . . . . 10
⊢ (𝑛 = 𝑘 → (log‘(1 + (1 / 𝑛))) = (log‘(1 + (1 / 𝑘)))) |
6 | 3, 5 | oveq12d 7182 |
. . . . . . . . 9
⊢ (𝑛 = 𝑘 → ((1 / 𝑛) − (log‘(1 + (1 / 𝑛)))) = ((1 / 𝑘) − (log‘(1 + (1 / 𝑘))))) |
7 | | emcl.4 |
. . . . . . . . 9
⊢ 𝑇 = (𝑛 ∈ ℕ ↦ ((1 / 𝑛) − (log‘(1 + (1 /
𝑛))))) |
8 | | ovex 7197 |
. . . . . . . . 9
⊢ ((1 /
𝑘) − (log‘(1 +
(1 / 𝑘)))) ∈
V |
9 | 6, 7, 8 | fvmpt 6769 |
. . . . . . . 8
⊢ (𝑘 ∈ ℕ → (𝑇‘𝑘) = ((1 / 𝑘) − (log‘(1 + (1 / 𝑘))))) |
10 | 9 | adantl 485 |
. . . . . . 7
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝑇‘𝑘) = ((1 / 𝑘) − (log‘(1 + (1 / 𝑘))))) |
11 | | nnrecre 11751 |
. . . . . . . . . 10
⊢ (𝑘 ∈ ℕ → (1 /
𝑘) ∈
ℝ) |
12 | 11 | adantl 485 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (1 / 𝑘) ∈ ℝ) |
13 | | 1rp 12469 |
. . . . . . . . . . 11
⊢ 1 ∈
ℝ+ |
14 | | nnrp 12476 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ ℕ → 𝑘 ∈
ℝ+) |
15 | 14 | rpreccld 12517 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ ℕ → (1 /
𝑘) ∈
ℝ+) |
16 | 15 | adantl 485 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (1 / 𝑘) ∈
ℝ+) |
17 | | rpaddcl 12487 |
. . . . . . . . . . 11
⊢ ((1
∈ ℝ+ ∧ (1 / 𝑘) ∈ ℝ+) → (1 + (1
/ 𝑘)) ∈
ℝ+) |
18 | 13, 16, 17 | sylancr 590 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (1 + (1 / 𝑘)) ∈
ℝ+) |
19 | 18 | relogcld 25358 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (log‘(1 + (1 / 𝑘))) ∈ ℝ) |
20 | 12, 19 | resubcld 11139 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((1 / 𝑘) − (log‘(1 + (1 / 𝑘)))) ∈
ℝ) |
21 | 20 | recnd 10740 |
. . . . . . 7
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((1 / 𝑘) − (log‘(1 + (1 / 𝑘)))) ∈
ℂ) |
22 | | emcl.1 |
. . . . . . . . . 10
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘𝑛))) |
23 | | emcl.2 |
. . . . . . . . . 10
⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘(𝑛 + 1)))) |
24 | | emcl.3 |
. . . . . . . . . 10
⊢ 𝐻 = (𝑛 ∈ ℕ ↦ (log‘(1 + (1 /
𝑛)))) |
25 | 22, 23, 24, 7 | emcllem5 25729 |
. . . . . . . . 9
⊢ 𝐺 = seq1( + , 𝑇) |
26 | 22, 23 | emcllem1 25725 |
. . . . . . . . . . . 12
⊢ (𝐹:ℕ⟶ℝ ∧
𝐺:ℕ⟶ℝ) |
27 | 26 | simpri 489 |
. . . . . . . . . . 11
⊢ 𝐺:ℕ⟶ℝ |
28 | 27 | a1i 11 |
. . . . . . . . . 10
⊢ (⊤
→ 𝐺:ℕ⟶ℝ) |
29 | 22, 23 | emcllem2 25726 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ ℕ → ((𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘) ∧ (𝐺‘𝑘) ≤ (𝐺‘(𝑘 + 1)))) |
30 | 29 | simprd 499 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ ℕ → (𝐺‘𝑘) ≤ (𝐺‘(𝑘 + 1))) |
31 | 30 | adantl 485 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝐺‘𝑘) ≤ (𝐺‘(𝑘 + 1))) |
32 | | 1nn 11720 |
. . . . . . . . . . . 12
⊢ 1 ∈
ℕ |
33 | 26 | simpli 487 |
. . . . . . . . . . . . 13
⊢ 𝐹:ℕ⟶ℝ |
34 | 33 | ffvelrni 6854 |
. . . . . . . . . . . 12
⊢ (1 ∈
ℕ → (𝐹‘1)
∈ ℝ) |
35 | 32, 34 | ax-mp 5 |
. . . . . . . . . . 11
⊢ (𝐹‘1) ∈
ℝ |
36 | 27 | ffvelrni 6854 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ ℕ → (𝐺‘𝑘) ∈ ℝ) |
37 | 36 | adantl 485 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝐺‘𝑘) ∈ ℝ) |
38 | 33 | ffvelrni 6854 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ ℕ → (𝐹‘𝑘) ∈ ℝ) |
39 | 38 | adantl 485 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝐹‘𝑘) ∈ ℝ) |
40 | 35 | a1i 11 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝐹‘1) ∈ ℝ) |
41 | | fvex 6681 |
. . . . . . . . . . . . . . . . . . 19
⊢
(log‘(1 + (1 / 𝑘))) ∈ V |
42 | 5, 24, 41 | fvmpt 6769 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ ℕ → (𝐻‘𝑘) = (log‘(1 + (1 / 𝑘)))) |
43 | 42 | adantl 485 |
. . . . . . . . . . . . . . . . 17
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝐻‘𝑘) = (log‘(1 + (1 / 𝑘)))) |
44 | 22, 23, 24 | emcllem3 25727 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ ℕ → (𝐻‘𝑘) = ((𝐹‘𝑘) − (𝐺‘𝑘))) |
45 | 44 | adantl 485 |
. . . . . . . . . . . . . . . . 17
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝐻‘𝑘) = ((𝐹‘𝑘) − (𝐺‘𝑘))) |
46 | 43, 45 | eqtr3d 2775 |
. . . . . . . . . . . . . . . 16
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (log‘(1 + (1 / 𝑘))) = ((𝐹‘𝑘) − (𝐺‘𝑘))) |
47 | | 1re 10712 |
. . . . . . . . . . . . . . . . . 18
⊢ 1 ∈
ℝ |
48 | | readdcl 10691 |
. . . . . . . . . . . . . . . . . 18
⊢ ((1
∈ ℝ ∧ (1 / 𝑘) ∈ ℝ) → (1 + (1 / 𝑘)) ∈
ℝ) |
49 | 47, 12, 48 | sylancr 590 |
. . . . . . . . . . . . . . . . 17
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (1 + (1 / 𝑘)) ∈ ℝ) |
50 | | ltaddrp 12502 |
. . . . . . . . . . . . . . . . . 18
⊢ ((1
∈ ℝ ∧ (1 / 𝑘) ∈ ℝ+) → 1 <
(1 + (1 / 𝑘))) |
51 | 47, 16, 50 | sylancr 590 |
. . . . . . . . . . . . . . . . 17
⊢
((⊤ ∧ 𝑘
∈ ℕ) → 1 < (1 + (1 / 𝑘))) |
52 | 49, 51 | rplogcld 25364 |
. . . . . . . . . . . . . . . 16
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (log‘(1 + (1 / 𝑘))) ∈
ℝ+) |
53 | 46, 52 | eqeltrrd 2834 |
. . . . . . . . . . . . . . 15
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((𝐹‘𝑘) − (𝐺‘𝑘)) ∈
ℝ+) |
54 | 53 | rpge0d 12511 |
. . . . . . . . . . . . . 14
⊢
((⊤ ∧ 𝑘
∈ ℕ) → 0 ≤ ((𝐹‘𝑘) − (𝐺‘𝑘))) |
55 | 39, 37 | subge0d 11301 |
. . . . . . . . . . . . . 14
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (0 ≤ ((𝐹‘𝑘) − (𝐺‘𝑘)) ↔ (𝐺‘𝑘) ≤ (𝐹‘𝑘))) |
56 | 54, 55 | mpbid 235 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝐺‘𝑘) ≤ (𝐹‘𝑘)) |
57 | | fveq2 6668 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = 1 → (𝐹‘𝑥) = (𝐹‘1)) |
58 | 57 | breq1d 5037 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = 1 → ((𝐹‘𝑥) ≤ (𝐹‘1) ↔ (𝐹‘1) ≤ (𝐹‘1))) |
59 | | fveq2 6668 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = 𝑘 → (𝐹‘𝑥) = (𝐹‘𝑘)) |
60 | 59 | breq1d 5037 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑘 → ((𝐹‘𝑥) ≤ (𝐹‘1) ↔ (𝐹‘𝑘) ≤ (𝐹‘1))) |
61 | | fveq2 6668 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = (𝑘 + 1) → (𝐹‘𝑥) = (𝐹‘(𝑘 + 1))) |
62 | 61 | breq1d 5037 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = (𝑘 + 1) → ((𝐹‘𝑥) ≤ (𝐹‘1) ↔ (𝐹‘(𝑘 + 1)) ≤ (𝐹‘1))) |
63 | 35 | leidi 11245 |
. . . . . . . . . . . . . . 15
⊢ (𝐹‘1) ≤ (𝐹‘1) |
64 | 29 | simpld 498 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈ ℕ → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘)) |
65 | | peano2nn 11721 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈
ℕ) |
66 | 33 | ffvelrni 6854 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑘 + 1) ∈ ℕ →
(𝐹‘(𝑘 + 1)) ∈
ℝ) |
67 | 65, 66 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ ℕ → (𝐹‘(𝑘 + 1)) ∈ ℝ) |
68 | 35 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ ℕ → (𝐹‘1) ∈
ℝ) |
69 | | letr 10805 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐹‘(𝑘 + 1)) ∈ ℝ ∧ (𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘1) ∈ ℝ) → (((𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘) ∧ (𝐹‘𝑘) ≤ (𝐹‘1)) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘1))) |
70 | 67, 38, 68, 69 | syl3anc 1372 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈ ℕ → (((𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘) ∧ (𝐹‘𝑘) ≤ (𝐹‘1)) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘1))) |
71 | 64, 70 | mpand 695 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ ℕ → ((𝐹‘𝑘) ≤ (𝐹‘1) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘1))) |
72 | 58, 60, 62, 60, 63, 71 | nnind 11727 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ ℕ → (𝐹‘𝑘) ≤ (𝐹‘1)) |
73 | 72 | adantl 485 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝐹‘𝑘) ≤ (𝐹‘1)) |
74 | 37, 39, 40, 56, 73 | letrd 10868 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝐺‘𝑘) ≤ (𝐹‘1)) |
75 | 74 | ralrimiva 3096 |
. . . . . . . . . . 11
⊢ (⊤
→ ∀𝑘 ∈
ℕ (𝐺‘𝑘) ≤ (𝐹‘1)) |
76 | | brralrspcev 5087 |
. . . . . . . . . . 11
⊢ (((𝐹‘1) ∈ ℝ ∧
∀𝑘 ∈ ℕ
(𝐺‘𝑘) ≤ (𝐹‘1)) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝐺‘𝑘) ≤ 𝑥) |
77 | 35, 75, 76 | sylancr 590 |
. . . . . . . . . 10
⊢ (⊤
→ ∃𝑥 ∈
ℝ ∀𝑘 ∈
ℕ (𝐺‘𝑘) ≤ 𝑥) |
78 | 1, 2, 28, 31, 77 | climsup 15112 |
. . . . . . . . 9
⊢ (⊤
→ 𝐺 ⇝ sup(ran
𝐺, ℝ, <
)) |
79 | 25, 78 | eqbrtrrid 5063 |
. . . . . . . 8
⊢ (⊤
→ seq1( + , 𝑇) ⇝
sup(ran 𝐺, ℝ, <
)) |
80 | | climrel 14932 |
. . . . . . . . 9
⊢ Rel
⇝ |
81 | 80 | releldmi 5785 |
. . . . . . . 8
⊢ (seq1( +
, 𝑇) ⇝ sup(ran 𝐺, ℝ, < ) → seq1( +
, 𝑇) ∈ dom ⇝
) |
82 | 79, 81 | syl 17 |
. . . . . . 7
⊢ (⊤
→ seq1( + , 𝑇) ∈
dom ⇝ ) |
83 | 1, 2, 10, 21, 82 | isumclim2 15199 |
. . . . . 6
⊢ (⊤
→ seq1( + , 𝑇) ⇝
Σ𝑘 ∈ ℕ ((1
/ 𝑘) − (log‘(1
+ (1 / 𝑘))))) |
84 | | df-em 25722 |
. . . . . 6
⊢ γ =
Σ𝑘 ∈ ℕ ((1
/ 𝑘) − (log‘(1
+ (1 / 𝑘)))) |
85 | 83, 25, 84 | 3brtr4g 5061 |
. . . . 5
⊢ (⊤
→ 𝐺 ⇝
γ) |
86 | | nnex 11715 |
. . . . . . . 8
⊢ ℕ
∈ V |
87 | 86 | mptex 6990 |
. . . . . . 7
⊢ (𝑛 ∈ ℕ ↦
(Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘𝑛))) ∈ V |
88 | 22, 87 | eqeltri 2829 |
. . . . . 6
⊢ 𝐹 ∈ V |
89 | 88 | a1i 11 |
. . . . 5
⊢ (⊤
→ 𝐹 ∈
V) |
90 | 22, 23, 24 | emcllem4 25728 |
. . . . . 6
⊢ 𝐻 ⇝ 0 |
91 | 90 | a1i 11 |
. . . . 5
⊢ (⊤
→ 𝐻 ⇝
0) |
92 | 37 | recnd 10740 |
. . . . 5
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝐺‘𝑘) ∈ ℂ) |
93 | 39, 37 | resubcld 11139 |
. . . . . . 7
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((𝐹‘𝑘) − (𝐺‘𝑘)) ∈ ℝ) |
94 | 45, 93 | eqeltrd 2833 |
. . . . . 6
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝐻‘𝑘) ∈ ℝ) |
95 | 94 | recnd 10740 |
. . . . 5
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝐻‘𝑘) ∈ ℂ) |
96 | 45 | oveq2d 7180 |
. . . . . 6
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((𝐺‘𝑘) + (𝐻‘𝑘)) = ((𝐺‘𝑘) + ((𝐹‘𝑘) − (𝐺‘𝑘)))) |
97 | 39 | recnd 10740 |
. . . . . . 7
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝐹‘𝑘) ∈ ℂ) |
98 | 92, 97 | pncan3d 11071 |
. . . . . 6
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((𝐺‘𝑘) + ((𝐹‘𝑘) − (𝐺‘𝑘))) = (𝐹‘𝑘)) |
99 | 96, 98 | eqtr2d 2774 |
. . . . 5
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝐹‘𝑘) = ((𝐺‘𝑘) + (𝐻‘𝑘))) |
100 | 1, 2, 85, 89, 91, 92, 95, 99 | climadd 15072 |
. . . 4
⊢ (⊤
→ 𝐹 ⇝ (γ +
0)) |
101 | 85 | mptru 1549 |
. . . . . 6
⊢ 𝐺 ⇝
γ |
102 | | climcl 14939 |
. . . . . 6
⊢ (𝐺 ⇝ γ → γ
∈ ℂ) |
103 | 101, 102 | ax-mp 5 |
. . . . 5
⊢ γ
∈ ℂ |
104 | 103 | addid1i 10898 |
. . . 4
⊢ (γ
+ 0) = γ |
105 | 100, 104 | breqtrdi 5068 |
. . 3
⊢ (⊤
→ 𝐹 ⇝
γ) |
106 | 105 | mptru 1549 |
. 2
⊢ 𝐹 ⇝
γ |
107 | 106, 101 | pm3.2i 474 |
1
⊢ (𝐹 ⇝ γ ∧ 𝐺 ⇝
γ) |