Step |
Hyp |
Ref |
Expression |
1 |
|
emcl.1 |
⊢ 𝐹 = ( 𝑛 ∈ ℕ ↦ ( Σ 𝑚 ∈ ( 1 ... 𝑛 ) ( 1 / 𝑚 ) − ( log ‘ 𝑛 ) ) ) |
2 |
|
emcl.2 |
⊢ 𝐺 = ( 𝑛 ∈ ℕ ↦ ( Σ 𝑚 ∈ ( 1 ... 𝑛 ) ( 1 / 𝑚 ) − ( log ‘ ( 𝑛 + 1 ) ) ) ) |
3 |
|
emcl.3 |
⊢ 𝐻 = ( 𝑛 ∈ ℕ ↦ ( log ‘ ( 1 + ( 1 / 𝑛 ) ) ) ) |
4 |
|
emcl.4 |
⊢ 𝑇 = ( 𝑛 ∈ ℕ ↦ ( ( 1 / 𝑛 ) − ( log ‘ ( 1 + ( 1 / 𝑛 ) ) ) ) ) |
5 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
6 |
|
1zzd |
⊢ ( ⊤ → 1 ∈ ℤ ) |
7 |
|
oveq2 |
⊢ ( 𝑛 = 𝑘 → ( 1 / 𝑛 ) = ( 1 / 𝑘 ) ) |
8 |
7
|
oveq2d |
⊢ ( 𝑛 = 𝑘 → ( 1 + ( 1 / 𝑛 ) ) = ( 1 + ( 1 / 𝑘 ) ) ) |
9 |
8
|
fveq2d |
⊢ ( 𝑛 = 𝑘 → ( log ‘ ( 1 + ( 1 / 𝑛 ) ) ) = ( log ‘ ( 1 + ( 1 / 𝑘 ) ) ) ) |
10 |
7 9
|
oveq12d |
⊢ ( 𝑛 = 𝑘 → ( ( 1 / 𝑛 ) − ( log ‘ ( 1 + ( 1 / 𝑛 ) ) ) ) = ( ( 1 / 𝑘 ) − ( log ‘ ( 1 + ( 1 / 𝑘 ) ) ) ) ) |
11 |
|
ovex |
⊢ ( ( 1 / 𝑘 ) − ( log ‘ ( 1 + ( 1 / 𝑘 ) ) ) ) ∈ V |
12 |
10 4 11
|
fvmpt |
⊢ ( 𝑘 ∈ ℕ → ( 𝑇 ‘ 𝑘 ) = ( ( 1 / 𝑘 ) − ( log ‘ ( 1 + ( 1 / 𝑘 ) ) ) ) ) |
13 |
12
|
adantl |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( 𝑇 ‘ 𝑘 ) = ( ( 1 / 𝑘 ) − ( log ‘ ( 1 + ( 1 / 𝑘 ) ) ) ) ) |
14 |
|
nnrecre |
⊢ ( 𝑘 ∈ ℕ → ( 1 / 𝑘 ) ∈ ℝ ) |
15 |
14
|
adantl |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( 1 / 𝑘 ) ∈ ℝ ) |
16 |
|
1rp |
⊢ 1 ∈ ℝ+ |
17 |
|
nnrp |
⊢ ( 𝑘 ∈ ℕ → 𝑘 ∈ ℝ+ ) |
18 |
17
|
rpreccld |
⊢ ( 𝑘 ∈ ℕ → ( 1 / 𝑘 ) ∈ ℝ+ ) |
19 |
18
|
adantl |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( 1 / 𝑘 ) ∈ ℝ+ ) |
20 |
|
rpaddcl |
⊢ ( ( 1 ∈ ℝ+ ∧ ( 1 / 𝑘 ) ∈ ℝ+ ) → ( 1 + ( 1 / 𝑘 ) ) ∈ ℝ+ ) |
21 |
16 19 20
|
sylancr |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( 1 + ( 1 / 𝑘 ) ) ∈ ℝ+ ) |
22 |
21
|
relogcld |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( log ‘ ( 1 + ( 1 / 𝑘 ) ) ) ∈ ℝ ) |
23 |
15 22
|
resubcld |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( ( 1 / 𝑘 ) − ( log ‘ ( 1 + ( 1 / 𝑘 ) ) ) ) ∈ ℝ ) |
24 |
23
|
recnd |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( ( 1 / 𝑘 ) − ( log ‘ ( 1 + ( 1 / 𝑘 ) ) ) ) ∈ ℂ ) |
25 |
1 2 3 4
|
emcllem5 |
⊢ 𝐺 = seq 1 ( + , 𝑇 ) |
26 |
1 2
|
emcllem1 |
⊢ ( 𝐹 : ℕ ⟶ ℝ ∧ 𝐺 : ℕ ⟶ ℝ ) |
27 |
26
|
simpri |
⊢ 𝐺 : ℕ ⟶ ℝ |
28 |
27
|
a1i |
⊢ ( ⊤ → 𝐺 : ℕ ⟶ ℝ ) |
29 |
1 2
|
emcllem2 |
⊢ ( 𝑘 ∈ ℕ → ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) ≤ ( 𝐹 ‘ 𝑘 ) ∧ ( 𝐺 ‘ 𝑘 ) ≤ ( 𝐺 ‘ ( 𝑘 + 1 ) ) ) ) |
30 |
29
|
simprd |
⊢ ( 𝑘 ∈ ℕ → ( 𝐺 ‘ 𝑘 ) ≤ ( 𝐺 ‘ ( 𝑘 + 1 ) ) ) |
31 |
30
|
adantl |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( 𝐺 ‘ 𝑘 ) ≤ ( 𝐺 ‘ ( 𝑘 + 1 ) ) ) |
32 |
|
1nn |
⊢ 1 ∈ ℕ |
33 |
26
|
simpli |
⊢ 𝐹 : ℕ ⟶ ℝ |
34 |
33
|
ffvelrni |
⊢ ( 1 ∈ ℕ → ( 𝐹 ‘ 1 ) ∈ ℝ ) |
35 |
32 34
|
ax-mp |
⊢ ( 𝐹 ‘ 1 ) ∈ ℝ |
36 |
27
|
ffvelrni |
⊢ ( 𝑘 ∈ ℕ → ( 𝐺 ‘ 𝑘 ) ∈ ℝ ) |
37 |
36
|
adantl |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( 𝐺 ‘ 𝑘 ) ∈ ℝ ) |
38 |
33
|
ffvelrni |
⊢ ( 𝑘 ∈ ℕ → ( 𝐹 ‘ 𝑘 ) ∈ ℝ ) |
39 |
38
|
adantl |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( 𝐹 ‘ 𝑘 ) ∈ ℝ ) |
40 |
35
|
a1i |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( 𝐹 ‘ 1 ) ∈ ℝ ) |
41 |
|
fvex |
⊢ ( log ‘ ( 1 + ( 1 / 𝑘 ) ) ) ∈ V |
42 |
9 3 41
|
fvmpt |
⊢ ( 𝑘 ∈ ℕ → ( 𝐻 ‘ 𝑘 ) = ( log ‘ ( 1 + ( 1 / 𝑘 ) ) ) ) |
43 |
42
|
adantl |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( 𝐻 ‘ 𝑘 ) = ( log ‘ ( 1 + ( 1 / 𝑘 ) ) ) ) |
44 |
1 2 3
|
emcllem3 |
⊢ ( 𝑘 ∈ ℕ → ( 𝐻 ‘ 𝑘 ) = ( ( 𝐹 ‘ 𝑘 ) − ( 𝐺 ‘ 𝑘 ) ) ) |
45 |
44
|
adantl |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( 𝐻 ‘ 𝑘 ) = ( ( 𝐹 ‘ 𝑘 ) − ( 𝐺 ‘ 𝑘 ) ) ) |
46 |
43 45
|
eqtr3d |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( log ‘ ( 1 + ( 1 / 𝑘 ) ) ) = ( ( 𝐹 ‘ 𝑘 ) − ( 𝐺 ‘ 𝑘 ) ) ) |
47 |
|
1re |
⊢ 1 ∈ ℝ |
48 |
|
readdcl |
⊢ ( ( 1 ∈ ℝ ∧ ( 1 / 𝑘 ) ∈ ℝ ) → ( 1 + ( 1 / 𝑘 ) ) ∈ ℝ ) |
49 |
47 15 48
|
sylancr |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( 1 + ( 1 / 𝑘 ) ) ∈ ℝ ) |
50 |
|
ltaddrp |
⊢ ( ( 1 ∈ ℝ ∧ ( 1 / 𝑘 ) ∈ ℝ+ ) → 1 < ( 1 + ( 1 / 𝑘 ) ) ) |
51 |
47 19 50
|
sylancr |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → 1 < ( 1 + ( 1 / 𝑘 ) ) ) |
52 |
49 51
|
rplogcld |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( log ‘ ( 1 + ( 1 / 𝑘 ) ) ) ∈ ℝ+ ) |
53 |
46 52
|
eqeltrrd |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( ( 𝐹 ‘ 𝑘 ) − ( 𝐺 ‘ 𝑘 ) ) ∈ ℝ+ ) |
54 |
53
|
rpge0d |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → 0 ≤ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐺 ‘ 𝑘 ) ) ) |
55 |
39 37
|
subge0d |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( 0 ≤ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐺 ‘ 𝑘 ) ) ↔ ( 𝐺 ‘ 𝑘 ) ≤ ( 𝐹 ‘ 𝑘 ) ) ) |
56 |
54 55
|
mpbid |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( 𝐺 ‘ 𝑘 ) ≤ ( 𝐹 ‘ 𝑘 ) ) |
57 |
|
fveq2 |
⊢ ( 𝑥 = 1 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 1 ) ) |
58 |
57
|
breq1d |
⊢ ( 𝑥 = 1 → ( ( 𝐹 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 1 ) ↔ ( 𝐹 ‘ 1 ) ≤ ( 𝐹 ‘ 1 ) ) ) |
59 |
|
fveq2 |
⊢ ( 𝑥 = 𝑘 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑘 ) ) |
60 |
59
|
breq1d |
⊢ ( 𝑥 = 𝑘 → ( ( 𝐹 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 1 ) ↔ ( 𝐹 ‘ 𝑘 ) ≤ ( 𝐹 ‘ 1 ) ) ) |
61 |
|
fveq2 |
⊢ ( 𝑥 = ( 𝑘 + 1 ) → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) |
62 |
61
|
breq1d |
⊢ ( 𝑥 = ( 𝑘 + 1 ) → ( ( 𝐹 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 1 ) ↔ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ≤ ( 𝐹 ‘ 1 ) ) ) |
63 |
35
|
leidi |
⊢ ( 𝐹 ‘ 1 ) ≤ ( 𝐹 ‘ 1 ) |
64 |
29
|
simpld |
⊢ ( 𝑘 ∈ ℕ → ( 𝐹 ‘ ( 𝑘 + 1 ) ) ≤ ( 𝐹 ‘ 𝑘 ) ) |
65 |
|
peano2nn |
⊢ ( 𝑘 ∈ ℕ → ( 𝑘 + 1 ) ∈ ℕ ) |
66 |
33
|
ffvelrni |
⊢ ( ( 𝑘 + 1 ) ∈ ℕ → ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ ℝ ) |
67 |
65 66
|
syl |
⊢ ( 𝑘 ∈ ℕ → ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ ℝ ) |
68 |
35
|
a1i |
⊢ ( 𝑘 ∈ ℕ → ( 𝐹 ‘ 1 ) ∈ ℝ ) |
69 |
|
letr |
⊢ ( ( ( 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 |
⊢ ( 𝑘 ∈ ℕ → ( ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) ≤ ( 𝐹 ‘ 𝑘 ) ∧ ( 𝐹 ‘ 𝑘 ) ≤ ( 𝐹 ‘ 1 ) ) → ( 𝐹 ‘ ( 𝑘 + 1 ) ) ≤ ( 𝐹 ‘ 1 ) ) ) |
71 |
64 70
|
mpand |
⊢ ( 𝑘 ∈ ℕ → ( ( 𝐹 ‘ 𝑘 ) ≤ ( 𝐹 ‘ 1 ) → ( 𝐹 ‘ ( 𝑘 + 1 ) ) ≤ ( 𝐹 ‘ 1 ) ) ) |
72 |
58 60 62 60 63 71
|
nnind |
⊢ ( 𝑘 ∈ ℕ → ( 𝐹 ‘ 𝑘 ) ≤ ( 𝐹 ‘ 1 ) ) |
73 |
72
|
adantl |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( 𝐹 ‘ 𝑘 ) ≤ ( 𝐹 ‘ 1 ) ) |
74 |
37 39 40 56 73
|
letrd |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( 𝐺 ‘ 𝑘 ) ≤ ( 𝐹 ‘ 1 ) ) |
75 |
74
|
ralrimiva |
⊢ ( ⊤ → ∀ 𝑘 ∈ ℕ ( 𝐺 ‘ 𝑘 ) ≤ ( 𝐹 ‘ 1 ) ) |
76 |
|
brralrspcev |
⊢ ( ( ( 𝐹 ‘ 1 ) ∈ ℝ ∧ ∀ 𝑘 ∈ ℕ ( 𝐺 ‘ 𝑘 ) ≤ ( 𝐹 ‘ 1 ) ) → ∃ 𝑥 ∈ ℝ ∀ 𝑘 ∈ ℕ ( 𝐺 ‘ 𝑘 ) ≤ 𝑥 ) |
77 |
35 75 76
|
sylancr |
⊢ ( ⊤ → ∃ 𝑥 ∈ ℝ ∀ 𝑘 ∈ ℕ ( 𝐺 ‘ 𝑘 ) ≤ 𝑥 ) |
78 |
5 6 28 31 77
|
climsup |
⊢ ( ⊤ → 𝐺 ⇝ sup ( ran 𝐺 , ℝ , < ) ) |
79 |
25 78
|
eqbrtrrid |
⊢ ( ⊤ → seq 1 ( + , 𝑇 ) ⇝ sup ( ran 𝐺 , ℝ , < ) ) |
80 |
|
climrel |
⊢ Rel ⇝ |
81 |
80
|
releldmi |
⊢ ( seq 1 ( + , 𝑇 ) ⇝ sup ( ran 𝐺 , ℝ , < ) → seq 1 ( + , 𝑇 ) ∈ dom ⇝ ) |
82 |
79 81
|
syl |
⊢ ( ⊤ → seq 1 ( + , 𝑇 ) ∈ dom ⇝ ) |
83 |
5 6 13 24 82
|
isumclim2 |
⊢ ( ⊤ → seq 1 ( + , 𝑇 ) ⇝ Σ 𝑘 ∈ ℕ ( ( 1 / 𝑘 ) − ( log ‘ ( 1 + ( 1 / 𝑘 ) ) ) ) ) |
84 |
|
df-em |
⊢ γ = Σ 𝑘 ∈ ℕ ( ( 1 / 𝑘 ) − ( log ‘ ( 1 + ( 1 / 𝑘 ) ) ) ) |
85 |
83 25 84
|
3brtr4g |
⊢ ( ⊤ → 𝐺 ⇝ γ ) |
86 |
|
nnex |
⊢ ℕ ∈ V |
87 |
86
|
mptex |
⊢ ( 𝑛 ∈ ℕ ↦ ( Σ 𝑚 ∈ ( 1 ... 𝑛 ) ( 1 / 𝑚 ) − ( log ‘ 𝑛 ) ) ) ∈ V |
88 |
1 87
|
eqeltri |
⊢ 𝐹 ∈ V |
89 |
88
|
a1i |
⊢ ( ⊤ → 𝐹 ∈ V ) |
90 |
1 2 3
|
emcllem4 |
⊢ 𝐻 ⇝ 0 |
91 |
90
|
a1i |
⊢ ( ⊤ → 𝐻 ⇝ 0 ) |
92 |
37
|
recnd |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( 𝐺 ‘ 𝑘 ) ∈ ℂ ) |
93 |
39 37
|
resubcld |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( ( 𝐹 ‘ 𝑘 ) − ( 𝐺 ‘ 𝑘 ) ) ∈ ℝ ) |
94 |
45 93
|
eqeltrd |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( 𝐻 ‘ 𝑘 ) ∈ ℝ ) |
95 |
94
|
recnd |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( 𝐻 ‘ 𝑘 ) ∈ ℂ ) |
96 |
45
|
oveq2d |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( ( 𝐺 ‘ 𝑘 ) + ( 𝐻 ‘ 𝑘 ) ) = ( ( 𝐺 ‘ 𝑘 ) + ( ( 𝐹 ‘ 𝑘 ) − ( 𝐺 ‘ 𝑘 ) ) ) ) |
97 |
39
|
recnd |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
98 |
92 97
|
pncan3d |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( ( 𝐺 ‘ 𝑘 ) + ( ( 𝐹 ‘ 𝑘 ) − ( 𝐺 ‘ 𝑘 ) ) ) = ( 𝐹 ‘ 𝑘 ) ) |
99 |
96 98
|
eqtr2d |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ ) → ( 𝐹 ‘ 𝑘 ) = ( ( 𝐺 ‘ 𝑘 ) + ( 𝐻 ‘ 𝑘 ) ) ) |
100 |
5 6 85 89 91 92 95 99
|
climadd |
⊢ ( ⊤ → 𝐹 ⇝ ( γ + 0 ) ) |
101 |
85
|
mptru |
⊢ 𝐺 ⇝ γ |
102 |
|
climcl |
⊢ ( 𝐺 ⇝ γ → γ ∈ ℂ ) |
103 |
101 102
|
ax-mp |
⊢ γ ∈ ℂ |
104 |
103
|
addid1i |
⊢ ( γ + 0 ) = γ |
105 |
100 104
|
breqtrdi |
⊢ ( ⊤ → 𝐹 ⇝ γ ) |
106 |
105
|
mptru |
⊢ 𝐹 ⇝ γ |
107 |
106 101
|
pm3.2i |
⊢ ( 𝐹 ⇝ γ ∧ 𝐺 ⇝ γ ) |