| Step | Hyp | Ref
| Expression |
| 1 | | nnuz 12921 |
. . . . 5
⊢ ℕ =
(ℤ≥‘1) |
| 2 | | 1zzd 12648 |
. . . . 5
⊢ (⊤
→ 1 ∈ ℤ) |
| 3 | | oveq2 7439 |
. . . . . . . . . 10
⊢ (𝑛 = 𝑘 → (1 / 𝑛) = (1 / 𝑘)) |
| 4 | 3 | oveq2d 7447 |
. . . . . . . . . . 11
⊢ (𝑛 = 𝑘 → (1 + (1 / 𝑛)) = (1 + (1 / 𝑘))) |
| 5 | 4 | fveq2d 6910 |
. . . . . . . . . 10
⊢ (𝑛 = 𝑘 → (log‘(1 + (1 / 𝑛))) = (log‘(1 + (1 / 𝑘)))) |
| 6 | 3, 5 | oveq12d 7449 |
. . . . . . . . 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 7464 |
. . . . . . . . 9
⊢ ((1 /
𝑘) − (log‘(1 +
(1 / 𝑘)))) ∈
V |
| 9 | 6, 7, 8 | fvmpt 7016 |
. . . . . . . 8
⊢ (𝑘 ∈ ℕ → (𝑇‘𝑘) = ((1 / 𝑘) − (log‘(1 + (1 / 𝑘))))) |
| 10 | 9 | adantl 481 |
. . . . . . 7
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝑇‘𝑘) = ((1 / 𝑘) − (log‘(1 + (1 / 𝑘))))) |
| 11 | | nnrecre 12308 |
. . . . . . . . . 10
⊢ (𝑘 ∈ ℕ → (1 /
𝑘) ∈
ℝ) |
| 12 | 11 | adantl 481 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (1 / 𝑘) ∈ ℝ) |
| 13 | | 1rp 13038 |
. . . . . . . . . . 11
⊢ 1 ∈
ℝ+ |
| 14 | | nnrp 13046 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ ℕ → 𝑘 ∈
ℝ+) |
| 15 | 14 | rpreccld 13087 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ ℕ → (1 /
𝑘) ∈
ℝ+) |
| 16 | 15 | adantl 481 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (1 / 𝑘) ∈
ℝ+) |
| 17 | | rpaddcl 13057 |
. . . . . . . . . . 11
⊢ ((1
∈ ℝ+ ∧ (1 / 𝑘) ∈ ℝ+) → (1 + (1
/ 𝑘)) ∈
ℝ+) |
| 18 | 13, 16, 17 | sylancr 587 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (1 + (1 / 𝑘)) ∈
ℝ+) |
| 19 | 18 | relogcld 26665 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (log‘(1 + (1 / 𝑘))) ∈ ℝ) |
| 20 | 12, 19 | resubcld 11691 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((1 / 𝑘) − (log‘(1 + (1 / 𝑘)))) ∈
ℝ) |
| 21 | 20 | recnd 11289 |
. . . . . . 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 27043 |
. . . . . . . . 9
⊢ 𝐺 = seq1( + , 𝑇) |
| 26 | 22, 23 | emcllem1 27039 |
. . . . . . . . . . . 12
⊢ (𝐹:ℕ⟶ℝ ∧
𝐺:ℕ⟶ℝ) |
| 27 | 26 | simpri 485 |
. . . . . . . . . . 11
⊢ 𝐺:ℕ⟶ℝ |
| 28 | 27 | a1i 11 |
. . . . . . . . . 10
⊢ (⊤
→ 𝐺:ℕ⟶ℝ) |
| 29 | 22, 23 | emcllem2 27040 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ ℕ → ((𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘) ∧ (𝐺‘𝑘) ≤ (𝐺‘(𝑘 + 1)))) |
| 30 | 29 | simprd 495 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ ℕ → (𝐺‘𝑘) ≤ (𝐺‘(𝑘 + 1))) |
| 31 | 30 | adantl 481 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝐺‘𝑘) ≤ (𝐺‘(𝑘 + 1))) |
| 32 | | 1nn 12277 |
. . . . . . . . . . . 12
⊢ 1 ∈
ℕ |
| 33 | 26 | simpli 483 |
. . . . . . . . . . . . 13
⊢ 𝐹:ℕ⟶ℝ |
| 34 | 33 | ffvelcdmi 7103 |
. . . . . . . . . . . 12
⊢ (1 ∈
ℕ → (𝐹‘1)
∈ ℝ) |
| 35 | 32, 34 | ax-mp 5 |
. . . . . . . . . . 11
⊢ (𝐹‘1) ∈
ℝ |
| 36 | 27 | ffvelcdmi 7103 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ ℕ → (𝐺‘𝑘) ∈ ℝ) |
| 37 | 36 | adantl 481 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝐺‘𝑘) ∈ ℝ) |
| 38 | 33 | ffvelcdmi 7103 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ ℕ → (𝐹‘𝑘) ∈ ℝ) |
| 39 | 38 | adantl 481 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝐹‘𝑘) ∈ ℝ) |
| 40 | 35 | a1i 11 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝐹‘1) ∈ ℝ) |
| 41 | | fvex 6919 |
. . . . . . . . . . . . . . . . . . 19
⊢
(log‘(1 + (1 / 𝑘))) ∈ V |
| 42 | 5, 24, 41 | fvmpt 7016 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ ℕ → (𝐻‘𝑘) = (log‘(1 + (1 / 𝑘)))) |
| 43 | 42 | adantl 481 |
. . . . . . . . . . . . . . . . 17
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝐻‘𝑘) = (log‘(1 + (1 / 𝑘)))) |
| 44 | 22, 23, 24 | emcllem3 27041 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ ℕ → (𝐻‘𝑘) = ((𝐹‘𝑘) − (𝐺‘𝑘))) |
| 45 | 44 | adantl 481 |
. . . . . . . . . . . . . . . . 17
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝐻‘𝑘) = ((𝐹‘𝑘) − (𝐺‘𝑘))) |
| 46 | 43, 45 | eqtr3d 2779 |
. . . . . . . . . . . . . . . 16
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (log‘(1 + (1 / 𝑘))) = ((𝐹‘𝑘) − (𝐺‘𝑘))) |
| 47 | | 1re 11261 |
. . . . . . . . . . . . . . . . . 18
⊢ 1 ∈
ℝ |
| 48 | | readdcl 11238 |
. . . . . . . . . . . . . . . . . 18
⊢ ((1
∈ ℝ ∧ (1 / 𝑘) ∈ ℝ) → (1 + (1 / 𝑘)) ∈
ℝ) |
| 49 | 47, 12, 48 | sylancr 587 |
. . . . . . . . . . . . . . . . 17
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (1 + (1 / 𝑘)) ∈ ℝ) |
| 50 | | ltaddrp 13072 |
. . . . . . . . . . . . . . . . . 18
⊢ ((1
∈ ℝ ∧ (1 / 𝑘) ∈ ℝ+) → 1 <
(1 + (1 / 𝑘))) |
| 51 | 47, 16, 50 | sylancr 587 |
. . . . . . . . . . . . . . . . 17
⊢
((⊤ ∧ 𝑘
∈ ℕ) → 1 < (1 + (1 / 𝑘))) |
| 52 | 49, 51 | rplogcld 26671 |
. . . . . . . . . . . . . . . 16
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (log‘(1 + (1 / 𝑘))) ∈
ℝ+) |
| 53 | 46, 52 | eqeltrrd 2842 |
. . . . . . . . . . . . . . 15
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((𝐹‘𝑘) − (𝐺‘𝑘)) ∈
ℝ+) |
| 54 | 53 | rpge0d 13081 |
. . . . . . . . . . . . . 14
⊢
((⊤ ∧ 𝑘
∈ ℕ) → 0 ≤ ((𝐹‘𝑘) − (𝐺‘𝑘))) |
| 55 | 39, 37 | subge0d 11853 |
. . . . . . . . . . . . . 14
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (0 ≤ ((𝐹‘𝑘) − (𝐺‘𝑘)) ↔ (𝐺‘𝑘) ≤ (𝐹‘𝑘))) |
| 56 | 54, 55 | mpbid 232 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝐺‘𝑘) ≤ (𝐹‘𝑘)) |
| 57 | | fveq2 6906 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = 1 → (𝐹‘𝑥) = (𝐹‘1)) |
| 58 | 57 | breq1d 5153 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = 1 → ((𝐹‘𝑥) ≤ (𝐹‘1) ↔ (𝐹‘1) ≤ (𝐹‘1))) |
| 59 | | fveq2 6906 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = 𝑘 → (𝐹‘𝑥) = (𝐹‘𝑘)) |
| 60 | 59 | breq1d 5153 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑘 → ((𝐹‘𝑥) ≤ (𝐹‘1) ↔ (𝐹‘𝑘) ≤ (𝐹‘1))) |
| 61 | | fveq2 6906 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = (𝑘 + 1) → (𝐹‘𝑥) = (𝐹‘(𝑘 + 1))) |
| 62 | 61 | breq1d 5153 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = (𝑘 + 1) → ((𝐹‘𝑥) ≤ (𝐹‘1) ↔ (𝐹‘(𝑘 + 1)) ≤ (𝐹‘1))) |
| 63 | 35 | leidi 11797 |
. . . . . . . . . . . . . . 15
⊢ (𝐹‘1) ≤ (𝐹‘1) |
| 64 | 29 | simpld 494 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈ ℕ → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘)) |
| 65 | | peano2nn 12278 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈
ℕ) |
| 66 | 33 | ffvelcdmi 7103 |
. . . . . . . . . . . . . . . . . 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 11355 |
. . . . . . . . . . . . . . . . 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 1373 |
. . . . . . . . . . . . . . . 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 12284 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ ℕ → (𝐹‘𝑘) ≤ (𝐹‘1)) |
| 73 | 72 | adantl 481 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝐹‘𝑘) ≤ (𝐹‘1)) |
| 74 | 37, 39, 40, 56, 73 | letrd 11418 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝐺‘𝑘) ≤ (𝐹‘1)) |
| 75 | 74 | ralrimiva 3146 |
. . . . . . . . . . 11
⊢ (⊤
→ ∀𝑘 ∈
ℕ (𝐺‘𝑘) ≤ (𝐹‘1)) |
| 76 | | brralrspcev 5203 |
. . . . . . . . . . 11
⊢ (((𝐹‘1) ∈ ℝ ∧
∀𝑘 ∈ ℕ
(𝐺‘𝑘) ≤ (𝐹‘1)) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝐺‘𝑘) ≤ 𝑥) |
| 77 | 35, 75, 76 | sylancr 587 |
. . . . . . . . . 10
⊢ (⊤
→ ∃𝑥 ∈
ℝ ∀𝑘 ∈
ℕ (𝐺‘𝑘) ≤ 𝑥) |
| 78 | 1, 2, 28, 31, 77 | climsup 15706 |
. . . . . . . . 9
⊢ (⊤
→ 𝐺 ⇝ sup(ran
𝐺, ℝ, <
)) |
| 79 | 25, 78 | eqbrtrrid 5179 |
. . . . . . . 8
⊢ (⊤
→ seq1( + , 𝑇) ⇝
sup(ran 𝐺, ℝ, <
)) |
| 80 | | climrel 15528 |
. . . . . . . . 9
⊢ Rel
⇝ |
| 81 | 80 | releldmi 5959 |
. . . . . . . 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 15794 |
. . . . . 6
⊢ (⊤
→ seq1( + , 𝑇) ⇝
Σ𝑘 ∈ ℕ ((1
/ 𝑘) − (log‘(1
+ (1 / 𝑘))))) |
| 84 | | df-em 27036 |
. . . . . 6
⊢ γ =
Σ𝑘 ∈ ℕ ((1
/ 𝑘) − (log‘(1
+ (1 / 𝑘)))) |
| 85 | 83, 25, 84 | 3brtr4g 5177 |
. . . . 5
⊢ (⊤
→ 𝐺 ⇝
γ) |
| 86 | | nnex 12272 |
. . . . . . . 8
⊢ ℕ
∈ V |
| 87 | 86 | mptex 7243 |
. . . . . . 7
⊢ (𝑛 ∈ ℕ ↦
(Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘𝑛))) ∈ V |
| 88 | 22, 87 | eqeltri 2837 |
. . . . . 6
⊢ 𝐹 ∈ V |
| 89 | 88 | a1i 11 |
. . . . 5
⊢ (⊤
→ 𝐹 ∈
V) |
| 90 | 22, 23, 24 | emcllem4 27042 |
. . . . . 6
⊢ 𝐻 ⇝ 0 |
| 91 | 90 | a1i 11 |
. . . . 5
⊢ (⊤
→ 𝐻 ⇝
0) |
| 92 | 37 | recnd 11289 |
. . . . 5
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝐺‘𝑘) ∈ ℂ) |
| 93 | 39, 37 | resubcld 11691 |
. . . . . . 7
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((𝐹‘𝑘) − (𝐺‘𝑘)) ∈ ℝ) |
| 94 | 45, 93 | eqeltrd 2841 |
. . . . . 6
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝐻‘𝑘) ∈ ℝ) |
| 95 | 94 | recnd 11289 |
. . . . 5
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝐻‘𝑘) ∈ ℂ) |
| 96 | 45 | oveq2d 7447 |
. . . . . 6
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((𝐺‘𝑘) + (𝐻‘𝑘)) = ((𝐺‘𝑘) + ((𝐹‘𝑘) − (𝐺‘𝑘)))) |
| 97 | 39 | recnd 11289 |
. . . . . . 7
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝐹‘𝑘) ∈ ℂ) |
| 98 | 92, 97 | pncan3d 11623 |
. . . . . 6
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((𝐺‘𝑘) + ((𝐹‘𝑘) − (𝐺‘𝑘))) = (𝐹‘𝑘)) |
| 99 | 96, 98 | eqtr2d 2778 |
. . . . 5
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝐹‘𝑘) = ((𝐺‘𝑘) + (𝐻‘𝑘))) |
| 100 | 1, 2, 85, 89, 91, 92, 95, 99 | climadd 15668 |
. . . 4
⊢ (⊤
→ 𝐹 ⇝ (γ +
0)) |
| 101 | 85 | mptru 1547 |
. . . . . 6
⊢ 𝐺 ⇝
γ |
| 102 | | climcl 15535 |
. . . . . 6
⊢ (𝐺 ⇝ γ → γ
∈ ℂ) |
| 103 | 101, 102 | ax-mp 5 |
. . . . 5
⊢ γ
∈ ℂ |
| 104 | 103 | addridi 11448 |
. . . 4
⊢ (γ
+ 0) = γ |
| 105 | 100, 104 | breqtrdi 5184 |
. . 3
⊢ (⊤
→ 𝐹 ⇝
γ) |
| 106 | 105 | mptru 1547 |
. 2
⊢ 𝐹 ⇝
γ |
| 107 | 106, 101 | pm3.2i 470 |
1
⊢ (𝐹 ⇝ γ ∧ 𝐺 ⇝
γ) |