Step |
Hyp |
Ref |
Expression |
1 |
|
relogcl |
⊢ ( 𝐴 ∈ ℝ+ → ( log ‘ 𝐴 ) ∈ ℝ ) |
2 |
|
rprege0 |
⊢ ( 𝐴 ∈ ℝ+ → ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ) |
3 |
|
flge0nn0 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → ( ⌊ ‘ 𝐴 ) ∈ ℕ0 ) |
4 |
2 3
|
syl |
⊢ ( 𝐴 ∈ ℝ+ → ( ⌊ ‘ 𝐴 ) ∈ ℕ0 ) |
5 |
|
nn0p1nn |
⊢ ( ( ⌊ ‘ 𝐴 ) ∈ ℕ0 → ( ( ⌊ ‘ 𝐴 ) + 1 ) ∈ ℕ ) |
6 |
4 5
|
syl |
⊢ ( 𝐴 ∈ ℝ+ → ( ( ⌊ ‘ 𝐴 ) + 1 ) ∈ ℕ ) |
7 |
6
|
nnrpd |
⊢ ( 𝐴 ∈ ℝ+ → ( ( ⌊ ‘ 𝐴 ) + 1 ) ∈ ℝ+ ) |
8 |
|
relogcl |
⊢ ( ( ( ⌊ ‘ 𝐴 ) + 1 ) ∈ ℝ+ → ( log ‘ ( ( ⌊ ‘ 𝐴 ) + 1 ) ) ∈ ℝ ) |
9 |
7 8
|
syl |
⊢ ( 𝐴 ∈ ℝ+ → ( log ‘ ( ( ⌊ ‘ 𝐴 ) + 1 ) ) ∈ ℝ ) |
10 |
|
fzfid |
⊢ ( 𝐴 ∈ ℝ+ → ( 1 ... ( ⌊ ‘ 𝐴 ) ) ∈ Fin ) |
11 |
|
elfznn |
⊢ ( 𝑚 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) → 𝑚 ∈ ℕ ) |
12 |
11
|
adantl |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → 𝑚 ∈ ℕ ) |
13 |
12
|
nnrecred |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → ( 1 / 𝑚 ) ∈ ℝ ) |
14 |
10 13
|
fsumrecl |
⊢ ( 𝐴 ∈ ℝ+ → Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( 1 / 𝑚 ) ∈ ℝ ) |
15 |
|
rpre |
⊢ ( 𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ ) |
16 |
|
fllep1 |
⊢ ( 𝐴 ∈ ℝ → 𝐴 ≤ ( ( ⌊ ‘ 𝐴 ) + 1 ) ) |
17 |
15 16
|
syl |
⊢ ( 𝐴 ∈ ℝ+ → 𝐴 ≤ ( ( ⌊ ‘ 𝐴 ) + 1 ) ) |
18 |
|
id |
⊢ ( 𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ+ ) |
19 |
18 7
|
logled |
⊢ ( 𝐴 ∈ ℝ+ → ( 𝐴 ≤ ( ( ⌊ ‘ 𝐴 ) + 1 ) ↔ ( log ‘ 𝐴 ) ≤ ( log ‘ ( ( ⌊ ‘ 𝐴 ) + 1 ) ) ) ) |
20 |
17 19
|
mpbid |
⊢ ( 𝐴 ∈ ℝ+ → ( log ‘ 𝐴 ) ≤ ( log ‘ ( ( ⌊ ‘ 𝐴 ) + 1 ) ) ) |
21 |
|
harmonicbnd3 |
⊢ ( ( ⌊ ‘ 𝐴 ) ∈ ℕ0 → ( Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( 1 / 𝑚 ) − ( log ‘ ( ( ⌊ ‘ 𝐴 ) + 1 ) ) ) ∈ ( 0 [,] γ ) ) |
22 |
4 21
|
syl |
⊢ ( 𝐴 ∈ ℝ+ → ( Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( 1 / 𝑚 ) − ( log ‘ ( ( ⌊ ‘ 𝐴 ) + 1 ) ) ) ∈ ( 0 [,] γ ) ) |
23 |
|
0re |
⊢ 0 ∈ ℝ |
24 |
|
emre |
⊢ γ ∈ ℝ |
25 |
23 24
|
elicc2i |
⊢ ( ( Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( 1 / 𝑚 ) − ( log ‘ ( ( ⌊ ‘ 𝐴 ) + 1 ) ) ) ∈ ( 0 [,] γ ) ↔ ( ( Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( 1 / 𝑚 ) − ( log ‘ ( ( ⌊ ‘ 𝐴 ) + 1 ) ) ) ∈ ℝ ∧ 0 ≤ ( Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( 1 / 𝑚 ) − ( log ‘ ( ( ⌊ ‘ 𝐴 ) + 1 ) ) ) ∧ ( Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( 1 / 𝑚 ) − ( log ‘ ( ( ⌊ ‘ 𝐴 ) + 1 ) ) ) ≤ γ ) ) |
26 |
25
|
simp2bi |
⊢ ( ( Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( 1 / 𝑚 ) − ( log ‘ ( ( ⌊ ‘ 𝐴 ) + 1 ) ) ) ∈ ( 0 [,] γ ) → 0 ≤ ( Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( 1 / 𝑚 ) − ( log ‘ ( ( ⌊ ‘ 𝐴 ) + 1 ) ) ) ) |
27 |
22 26
|
syl |
⊢ ( 𝐴 ∈ ℝ+ → 0 ≤ ( Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( 1 / 𝑚 ) − ( log ‘ ( ( ⌊ ‘ 𝐴 ) + 1 ) ) ) ) |
28 |
14 9
|
subge0d |
⊢ ( 𝐴 ∈ ℝ+ → ( 0 ≤ ( Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( 1 / 𝑚 ) − ( log ‘ ( ( ⌊ ‘ 𝐴 ) + 1 ) ) ) ↔ ( log ‘ ( ( ⌊ ‘ 𝐴 ) + 1 ) ) ≤ Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( 1 / 𝑚 ) ) ) |
29 |
27 28
|
mpbid |
⊢ ( 𝐴 ∈ ℝ+ → ( log ‘ ( ( ⌊ ‘ 𝐴 ) + 1 ) ) ≤ Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( 1 / 𝑚 ) ) |
30 |
1 9 14 20 29
|
letrd |
⊢ ( 𝐴 ∈ ℝ+ → ( log ‘ 𝐴 ) ≤ Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( 1 / 𝑚 ) ) |