| Step |
Hyp |
Ref |
Expression |
| 1 |
|
fzfid |
⊢ ( 𝐴 ∈ ℕ → ( 2 ... 𝐴 ) ∈ Fin ) |
| 2 |
|
elfzuz |
⊢ ( 𝑛 ∈ ( 2 ... 𝐴 ) → 𝑛 ∈ ( ℤ≥ ‘ 2 ) ) |
| 3 |
|
eluz2nn |
⊢ ( 𝑛 ∈ ( ℤ≥ ‘ 2 ) → 𝑛 ∈ ℕ ) |
| 4 |
2 3
|
syl |
⊢ ( 𝑛 ∈ ( 2 ... 𝐴 ) → 𝑛 ∈ ℕ ) |
| 5 |
4
|
adantl |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → 𝑛 ∈ ℕ ) |
| 6 |
5
|
nnrpd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → 𝑛 ∈ ℝ+ ) |
| 7 |
6
|
relogcld |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( log ‘ 𝑛 ) ∈ ℝ ) |
| 8 |
2
|
adantl |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → 𝑛 ∈ ( ℤ≥ ‘ 2 ) ) |
| 9 |
|
uz2m1nn |
⊢ ( 𝑛 ∈ ( ℤ≥ ‘ 2 ) → ( 𝑛 − 1 ) ∈ ℕ ) |
| 10 |
8 9
|
syl |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( 𝑛 − 1 ) ∈ ℕ ) |
| 11 |
5 10
|
nnmulcld |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( 𝑛 · ( 𝑛 − 1 ) ) ∈ ℕ ) |
| 12 |
7 11
|
nndivred |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( ( log ‘ 𝑛 ) / ( 𝑛 · ( 𝑛 − 1 ) ) ) ∈ ℝ ) |
| 13 |
1 12
|
fsumrecl |
⊢ ( 𝐴 ∈ ℕ → Σ 𝑛 ∈ ( 2 ... 𝐴 ) ( ( log ‘ 𝑛 ) / ( 𝑛 · ( 𝑛 − 1 ) ) ) ∈ ℝ ) |
| 14 |
|
2re |
⊢ 2 ∈ ℝ |
| 15 |
10
|
nnrpd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( 𝑛 − 1 ) ∈ ℝ+ ) |
| 16 |
15
|
rpsqrtcld |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( √ ‘ ( 𝑛 − 1 ) ) ∈ ℝ+ ) |
| 17 |
|
rerpdivcl |
⊢ ( ( 2 ∈ ℝ ∧ ( √ ‘ ( 𝑛 − 1 ) ) ∈ ℝ+ ) → ( 2 / ( √ ‘ ( 𝑛 − 1 ) ) ) ∈ ℝ ) |
| 18 |
14 16 17
|
sylancr |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( 2 / ( √ ‘ ( 𝑛 − 1 ) ) ) ∈ ℝ ) |
| 19 |
6
|
rpsqrtcld |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( √ ‘ 𝑛 ) ∈ ℝ+ ) |
| 20 |
|
rerpdivcl |
⊢ ( ( 2 ∈ ℝ ∧ ( √ ‘ 𝑛 ) ∈ ℝ+ ) → ( 2 / ( √ ‘ 𝑛 ) ) ∈ ℝ ) |
| 21 |
14 19 20
|
sylancr |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( 2 / ( √ ‘ 𝑛 ) ) ∈ ℝ ) |
| 22 |
18 21
|
resubcld |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( ( 2 / ( √ ‘ ( 𝑛 − 1 ) ) ) − ( 2 / ( √ ‘ 𝑛 ) ) ) ∈ ℝ ) |
| 23 |
1 22
|
fsumrecl |
⊢ ( 𝐴 ∈ ℕ → Σ 𝑛 ∈ ( 2 ... 𝐴 ) ( ( 2 / ( √ ‘ ( 𝑛 − 1 ) ) ) − ( 2 / ( √ ‘ 𝑛 ) ) ) ∈ ℝ ) |
| 24 |
14
|
a1i |
⊢ ( 𝐴 ∈ ℕ → 2 ∈ ℝ ) |
| 25 |
16
|
rpred |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( √ ‘ ( 𝑛 − 1 ) ) ∈ ℝ ) |
| 26 |
5
|
nnred |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → 𝑛 ∈ ℝ ) |
| 27 |
|
peano2rem |
⊢ ( 𝑛 ∈ ℝ → ( 𝑛 − 1 ) ∈ ℝ ) |
| 28 |
26 27
|
syl |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( 𝑛 − 1 ) ∈ ℝ ) |
| 29 |
26 28
|
remulcld |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( 𝑛 · ( 𝑛 − 1 ) ) ∈ ℝ ) |
| 30 |
29 22
|
remulcld |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( ( 𝑛 · ( 𝑛 − 1 ) ) · ( ( 2 / ( √ ‘ ( 𝑛 − 1 ) ) ) − ( 2 / ( √ ‘ 𝑛 ) ) ) ) ∈ ℝ ) |
| 31 |
5
|
nncnd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → 𝑛 ∈ ℂ ) |
| 32 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
| 33 |
|
npcan |
⊢ ( ( 𝑛 ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( 𝑛 − 1 ) + 1 ) = 𝑛 ) |
| 34 |
31 32 33
|
sylancl |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( ( 𝑛 − 1 ) + 1 ) = 𝑛 ) |
| 35 |
34
|
fveq2d |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( log ‘ ( ( 𝑛 − 1 ) + 1 ) ) = ( log ‘ 𝑛 ) ) |
| 36 |
15
|
rpge0d |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → 0 ≤ ( 𝑛 − 1 ) ) |
| 37 |
|
loglesqrt |
⊢ ( ( ( 𝑛 − 1 ) ∈ ℝ ∧ 0 ≤ ( 𝑛 − 1 ) ) → ( log ‘ ( ( 𝑛 − 1 ) + 1 ) ) ≤ ( √ ‘ ( 𝑛 − 1 ) ) ) |
| 38 |
28 36 37
|
syl2anc |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( log ‘ ( ( 𝑛 − 1 ) + 1 ) ) ≤ ( √ ‘ ( 𝑛 − 1 ) ) ) |
| 39 |
35 38
|
eqbrtrrd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( log ‘ 𝑛 ) ≤ ( √ ‘ ( 𝑛 − 1 ) ) ) |
| 40 |
19
|
rpred |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( √ ‘ 𝑛 ) ∈ ℝ ) |
| 41 |
40 25
|
readdcld |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( ( √ ‘ 𝑛 ) + ( √ ‘ ( 𝑛 − 1 ) ) ) ∈ ℝ ) |
| 42 |
|
remulcl |
⊢ ( ( ( √ ‘ 𝑛 ) ∈ ℝ ∧ 2 ∈ ℝ ) → ( ( √ ‘ 𝑛 ) · 2 ) ∈ ℝ ) |
| 43 |
40 14 42
|
sylancl |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( ( √ ‘ 𝑛 ) · 2 ) ∈ ℝ ) |
| 44 |
40 25
|
resubcld |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( ( √ ‘ 𝑛 ) − ( √ ‘ ( 𝑛 − 1 ) ) ) ∈ ℝ ) |
| 45 |
26
|
lem1d |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( 𝑛 − 1 ) ≤ 𝑛 ) |
| 46 |
6
|
rpge0d |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → 0 ≤ 𝑛 ) |
| 47 |
28 36 26 46
|
sqrtled |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( ( 𝑛 − 1 ) ≤ 𝑛 ↔ ( √ ‘ ( 𝑛 − 1 ) ) ≤ ( √ ‘ 𝑛 ) ) ) |
| 48 |
45 47
|
mpbid |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( √ ‘ ( 𝑛 − 1 ) ) ≤ ( √ ‘ 𝑛 ) ) |
| 49 |
40 25
|
subge0d |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( 0 ≤ ( ( √ ‘ 𝑛 ) − ( √ ‘ ( 𝑛 − 1 ) ) ) ↔ ( √ ‘ ( 𝑛 − 1 ) ) ≤ ( √ ‘ 𝑛 ) ) ) |
| 50 |
48 49
|
mpbird |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → 0 ≤ ( ( √ ‘ 𝑛 ) − ( √ ‘ ( 𝑛 − 1 ) ) ) ) |
| 51 |
25 40 40 48
|
leadd2dd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( ( √ ‘ 𝑛 ) + ( √ ‘ ( 𝑛 − 1 ) ) ) ≤ ( ( √ ‘ 𝑛 ) + ( √ ‘ 𝑛 ) ) ) |
| 52 |
19
|
rpcnd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( √ ‘ 𝑛 ) ∈ ℂ ) |
| 53 |
52
|
times2d |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( ( √ ‘ 𝑛 ) · 2 ) = ( ( √ ‘ 𝑛 ) + ( √ ‘ 𝑛 ) ) ) |
| 54 |
51 53
|
breqtrrd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( ( √ ‘ 𝑛 ) + ( √ ‘ ( 𝑛 − 1 ) ) ) ≤ ( ( √ ‘ 𝑛 ) · 2 ) ) |
| 55 |
41 43 44 50 54
|
lemul1ad |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( ( ( √ ‘ 𝑛 ) + ( √ ‘ ( 𝑛 − 1 ) ) ) · ( ( √ ‘ 𝑛 ) − ( √ ‘ ( 𝑛 − 1 ) ) ) ) ≤ ( ( ( √ ‘ 𝑛 ) · 2 ) · ( ( √ ‘ 𝑛 ) − ( √ ‘ ( 𝑛 − 1 ) ) ) ) ) |
| 56 |
31
|
sqsqrtd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( ( √ ‘ 𝑛 ) ↑ 2 ) = 𝑛 ) |
| 57 |
|
subcl |
⊢ ( ( 𝑛 ∈ ℂ ∧ 1 ∈ ℂ ) → ( 𝑛 − 1 ) ∈ ℂ ) |
| 58 |
31 32 57
|
sylancl |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( 𝑛 − 1 ) ∈ ℂ ) |
| 59 |
58
|
sqsqrtd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( ( √ ‘ ( 𝑛 − 1 ) ) ↑ 2 ) = ( 𝑛 − 1 ) ) |
| 60 |
56 59
|
oveq12d |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( ( ( √ ‘ 𝑛 ) ↑ 2 ) − ( ( √ ‘ ( 𝑛 − 1 ) ) ↑ 2 ) ) = ( 𝑛 − ( 𝑛 − 1 ) ) ) |
| 61 |
16
|
rpcnd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( √ ‘ ( 𝑛 − 1 ) ) ∈ ℂ ) |
| 62 |
|
subsq |
⊢ ( ( ( √ ‘ 𝑛 ) ∈ ℂ ∧ ( √ ‘ ( 𝑛 − 1 ) ) ∈ ℂ ) → ( ( ( √ ‘ 𝑛 ) ↑ 2 ) − ( ( √ ‘ ( 𝑛 − 1 ) ) ↑ 2 ) ) = ( ( ( √ ‘ 𝑛 ) + ( √ ‘ ( 𝑛 − 1 ) ) ) · ( ( √ ‘ 𝑛 ) − ( √ ‘ ( 𝑛 − 1 ) ) ) ) ) |
| 63 |
52 61 62
|
syl2anc |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( ( ( √ ‘ 𝑛 ) ↑ 2 ) − ( ( √ ‘ ( 𝑛 − 1 ) ) ↑ 2 ) ) = ( ( ( √ ‘ 𝑛 ) + ( √ ‘ ( 𝑛 − 1 ) ) ) · ( ( √ ‘ 𝑛 ) − ( √ ‘ ( 𝑛 − 1 ) ) ) ) ) |
| 64 |
|
nncan |
⊢ ( ( 𝑛 ∈ ℂ ∧ 1 ∈ ℂ ) → ( 𝑛 − ( 𝑛 − 1 ) ) = 1 ) |
| 65 |
31 32 64
|
sylancl |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( 𝑛 − ( 𝑛 − 1 ) ) = 1 ) |
| 66 |
60 63 65
|
3eqtr3d |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( ( ( √ ‘ 𝑛 ) + ( √ ‘ ( 𝑛 − 1 ) ) ) · ( ( √ ‘ 𝑛 ) − ( √ ‘ ( 𝑛 − 1 ) ) ) ) = 1 ) |
| 67 |
|
2cn |
⊢ 2 ∈ ℂ |
| 68 |
67
|
a1i |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → 2 ∈ ℂ ) |
| 69 |
44
|
recnd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( ( √ ‘ 𝑛 ) − ( √ ‘ ( 𝑛 − 1 ) ) ) ∈ ℂ ) |
| 70 |
52 68 69
|
mulassd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( ( ( √ ‘ 𝑛 ) · 2 ) · ( ( √ ‘ 𝑛 ) − ( √ ‘ ( 𝑛 − 1 ) ) ) ) = ( ( √ ‘ 𝑛 ) · ( 2 · ( ( √ ‘ 𝑛 ) − ( √ ‘ ( 𝑛 − 1 ) ) ) ) ) ) |
| 71 |
55 66 70
|
3brtr3d |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → 1 ≤ ( ( √ ‘ 𝑛 ) · ( 2 · ( ( √ ‘ 𝑛 ) − ( √ ‘ ( 𝑛 − 1 ) ) ) ) ) ) |
| 72 |
|
1red |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → 1 ∈ ℝ ) |
| 73 |
|
remulcl |
⊢ ( ( 2 ∈ ℝ ∧ ( ( √ ‘ 𝑛 ) − ( √ ‘ ( 𝑛 − 1 ) ) ) ∈ ℝ ) → ( 2 · ( ( √ ‘ 𝑛 ) − ( √ ‘ ( 𝑛 − 1 ) ) ) ) ∈ ℝ ) |
| 74 |
14 44 73
|
sylancr |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( 2 · ( ( √ ‘ 𝑛 ) − ( √ ‘ ( 𝑛 − 1 ) ) ) ) ∈ ℝ ) |
| 75 |
40 74
|
remulcld |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( ( √ ‘ 𝑛 ) · ( 2 · ( ( √ ‘ 𝑛 ) − ( √ ‘ ( 𝑛 − 1 ) ) ) ) ) ∈ ℝ ) |
| 76 |
72 75 16
|
lemul1d |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( 1 ≤ ( ( √ ‘ 𝑛 ) · ( 2 · ( ( √ ‘ 𝑛 ) − ( √ ‘ ( 𝑛 − 1 ) ) ) ) ) ↔ ( 1 · ( √ ‘ ( 𝑛 − 1 ) ) ) ≤ ( ( ( √ ‘ 𝑛 ) · ( 2 · ( ( √ ‘ 𝑛 ) − ( √ ‘ ( 𝑛 − 1 ) ) ) ) ) · ( √ ‘ ( 𝑛 − 1 ) ) ) ) ) |
| 77 |
71 76
|
mpbid |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( 1 · ( √ ‘ ( 𝑛 − 1 ) ) ) ≤ ( ( ( √ ‘ 𝑛 ) · ( 2 · ( ( √ ‘ 𝑛 ) − ( √ ‘ ( 𝑛 − 1 ) ) ) ) ) · ( √ ‘ ( 𝑛 − 1 ) ) ) ) |
| 78 |
61
|
mullidd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( 1 · ( √ ‘ ( 𝑛 − 1 ) ) ) = ( √ ‘ ( 𝑛 − 1 ) ) ) |
| 79 |
74
|
recnd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( 2 · ( ( √ ‘ 𝑛 ) − ( √ ‘ ( 𝑛 − 1 ) ) ) ) ∈ ℂ ) |
| 80 |
52 79 61
|
mul32d |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( ( ( √ ‘ 𝑛 ) · ( 2 · ( ( √ ‘ 𝑛 ) − ( √ ‘ ( 𝑛 − 1 ) ) ) ) ) · ( √ ‘ ( 𝑛 − 1 ) ) ) = ( ( ( √ ‘ 𝑛 ) · ( √ ‘ ( 𝑛 − 1 ) ) ) · ( 2 · ( ( √ ‘ 𝑛 ) − ( √ ‘ ( 𝑛 − 1 ) ) ) ) ) ) |
| 81 |
77 78 80
|
3brtr3d |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( √ ‘ ( 𝑛 − 1 ) ) ≤ ( ( ( √ ‘ 𝑛 ) · ( √ ‘ ( 𝑛 − 1 ) ) ) · ( 2 · ( ( √ ‘ 𝑛 ) − ( √ ‘ ( 𝑛 − 1 ) ) ) ) ) ) |
| 82 |
|
remsqsqrt |
⊢ ( ( 𝑛 ∈ ℝ ∧ 0 ≤ 𝑛 ) → ( ( √ ‘ 𝑛 ) · ( √ ‘ 𝑛 ) ) = 𝑛 ) |
| 83 |
26 46 82
|
syl2anc |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( ( √ ‘ 𝑛 ) · ( √ ‘ 𝑛 ) ) = 𝑛 ) |
| 84 |
|
remsqsqrt |
⊢ ( ( ( 𝑛 − 1 ) ∈ ℝ ∧ 0 ≤ ( 𝑛 − 1 ) ) → ( ( √ ‘ ( 𝑛 − 1 ) ) · ( √ ‘ ( 𝑛 − 1 ) ) ) = ( 𝑛 − 1 ) ) |
| 85 |
28 36 84
|
syl2anc |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( ( √ ‘ ( 𝑛 − 1 ) ) · ( √ ‘ ( 𝑛 − 1 ) ) ) = ( 𝑛 − 1 ) ) |
| 86 |
83 85
|
oveq12d |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( ( ( √ ‘ 𝑛 ) · ( √ ‘ 𝑛 ) ) · ( ( √ ‘ ( 𝑛 − 1 ) ) · ( √ ‘ ( 𝑛 − 1 ) ) ) ) = ( 𝑛 · ( 𝑛 − 1 ) ) ) |
| 87 |
52 52 61 61
|
mul4d |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( ( ( √ ‘ 𝑛 ) · ( √ ‘ 𝑛 ) ) · ( ( √ ‘ ( 𝑛 − 1 ) ) · ( √ ‘ ( 𝑛 − 1 ) ) ) ) = ( ( ( √ ‘ 𝑛 ) · ( √ ‘ ( 𝑛 − 1 ) ) ) · ( ( √ ‘ 𝑛 ) · ( √ ‘ ( 𝑛 − 1 ) ) ) ) ) |
| 88 |
86 87
|
eqtr3d |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( 𝑛 · ( 𝑛 − 1 ) ) = ( ( ( √ ‘ 𝑛 ) · ( √ ‘ ( 𝑛 − 1 ) ) ) · ( ( √ ‘ 𝑛 ) · ( √ ‘ ( 𝑛 − 1 ) ) ) ) ) |
| 89 |
16
|
rpcnne0d |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( ( √ ‘ ( 𝑛 − 1 ) ) ∈ ℂ ∧ ( √ ‘ ( 𝑛 − 1 ) ) ≠ 0 ) ) |
| 90 |
19
|
rpcnne0d |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( ( √ ‘ 𝑛 ) ∈ ℂ ∧ ( √ ‘ 𝑛 ) ≠ 0 ) ) |
| 91 |
|
divsubdiv |
⊢ ( ( ( 2 ∈ ℂ ∧ 2 ∈ ℂ ) ∧ ( ( ( √ ‘ ( 𝑛 − 1 ) ) ∈ ℂ ∧ ( √ ‘ ( 𝑛 − 1 ) ) ≠ 0 ) ∧ ( ( √ ‘ 𝑛 ) ∈ ℂ ∧ ( √ ‘ 𝑛 ) ≠ 0 ) ) ) → ( ( 2 / ( √ ‘ ( 𝑛 − 1 ) ) ) − ( 2 / ( √ ‘ 𝑛 ) ) ) = ( ( ( 2 · ( √ ‘ 𝑛 ) ) − ( 2 · ( √ ‘ ( 𝑛 − 1 ) ) ) ) / ( ( √ ‘ ( 𝑛 − 1 ) ) · ( √ ‘ 𝑛 ) ) ) ) |
| 92 |
68 68 89 90 91
|
syl22anc |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( ( 2 / ( √ ‘ ( 𝑛 − 1 ) ) ) − ( 2 / ( √ ‘ 𝑛 ) ) ) = ( ( ( 2 · ( √ ‘ 𝑛 ) ) − ( 2 · ( √ ‘ ( 𝑛 − 1 ) ) ) ) / ( ( √ ‘ ( 𝑛 − 1 ) ) · ( √ ‘ 𝑛 ) ) ) ) |
| 93 |
68 52 61
|
subdid |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( 2 · ( ( √ ‘ 𝑛 ) − ( √ ‘ ( 𝑛 − 1 ) ) ) ) = ( ( 2 · ( √ ‘ 𝑛 ) ) − ( 2 · ( √ ‘ ( 𝑛 − 1 ) ) ) ) ) |
| 94 |
52 61
|
mulcomd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( ( √ ‘ 𝑛 ) · ( √ ‘ ( 𝑛 − 1 ) ) ) = ( ( √ ‘ ( 𝑛 − 1 ) ) · ( √ ‘ 𝑛 ) ) ) |
| 95 |
93 94
|
oveq12d |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( ( 2 · ( ( √ ‘ 𝑛 ) − ( √ ‘ ( 𝑛 − 1 ) ) ) ) / ( ( √ ‘ 𝑛 ) · ( √ ‘ ( 𝑛 − 1 ) ) ) ) = ( ( ( 2 · ( √ ‘ 𝑛 ) ) − ( 2 · ( √ ‘ ( 𝑛 − 1 ) ) ) ) / ( ( √ ‘ ( 𝑛 − 1 ) ) · ( √ ‘ 𝑛 ) ) ) ) |
| 96 |
92 95
|
eqtr4d |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( ( 2 / ( √ ‘ ( 𝑛 − 1 ) ) ) − ( 2 / ( √ ‘ 𝑛 ) ) ) = ( ( 2 · ( ( √ ‘ 𝑛 ) − ( √ ‘ ( 𝑛 − 1 ) ) ) ) / ( ( √ ‘ 𝑛 ) · ( √ ‘ ( 𝑛 − 1 ) ) ) ) ) |
| 97 |
88 96
|
oveq12d |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( ( 𝑛 · ( 𝑛 − 1 ) ) · ( ( 2 / ( √ ‘ ( 𝑛 − 1 ) ) ) − ( 2 / ( √ ‘ 𝑛 ) ) ) ) = ( ( ( ( √ ‘ 𝑛 ) · ( √ ‘ ( 𝑛 − 1 ) ) ) · ( ( √ ‘ 𝑛 ) · ( √ ‘ ( 𝑛 − 1 ) ) ) ) · ( ( 2 · ( ( √ ‘ 𝑛 ) − ( √ ‘ ( 𝑛 − 1 ) ) ) ) / ( ( √ ‘ 𝑛 ) · ( √ ‘ ( 𝑛 − 1 ) ) ) ) ) ) |
| 98 |
52 61
|
mulcld |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( ( √ ‘ 𝑛 ) · ( √ ‘ ( 𝑛 − 1 ) ) ) ∈ ℂ ) |
| 99 |
19 16
|
rpmulcld |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( ( √ ‘ 𝑛 ) · ( √ ‘ ( 𝑛 − 1 ) ) ) ∈ ℝ+ ) |
| 100 |
74 99
|
rerpdivcld |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( ( 2 · ( ( √ ‘ 𝑛 ) − ( √ ‘ ( 𝑛 − 1 ) ) ) ) / ( ( √ ‘ 𝑛 ) · ( √ ‘ ( 𝑛 − 1 ) ) ) ) ∈ ℝ ) |
| 101 |
100
|
recnd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( ( 2 · ( ( √ ‘ 𝑛 ) − ( √ ‘ ( 𝑛 − 1 ) ) ) ) / ( ( √ ‘ 𝑛 ) · ( √ ‘ ( 𝑛 − 1 ) ) ) ) ∈ ℂ ) |
| 102 |
98 98 101
|
mulassd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( ( ( ( √ ‘ 𝑛 ) · ( √ ‘ ( 𝑛 − 1 ) ) ) · ( ( √ ‘ 𝑛 ) · ( √ ‘ ( 𝑛 − 1 ) ) ) ) · ( ( 2 · ( ( √ ‘ 𝑛 ) − ( √ ‘ ( 𝑛 − 1 ) ) ) ) / ( ( √ ‘ 𝑛 ) · ( √ ‘ ( 𝑛 − 1 ) ) ) ) ) = ( ( ( √ ‘ 𝑛 ) · ( √ ‘ ( 𝑛 − 1 ) ) ) · ( ( ( √ ‘ 𝑛 ) · ( √ ‘ ( 𝑛 − 1 ) ) ) · ( ( 2 · ( ( √ ‘ 𝑛 ) − ( √ ‘ ( 𝑛 − 1 ) ) ) ) / ( ( √ ‘ 𝑛 ) · ( √ ‘ ( 𝑛 − 1 ) ) ) ) ) ) ) |
| 103 |
99
|
rpne0d |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( ( √ ‘ 𝑛 ) · ( √ ‘ ( 𝑛 − 1 ) ) ) ≠ 0 ) |
| 104 |
79 98 103
|
divcan2d |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( ( ( √ ‘ 𝑛 ) · ( √ ‘ ( 𝑛 − 1 ) ) ) · ( ( 2 · ( ( √ ‘ 𝑛 ) − ( √ ‘ ( 𝑛 − 1 ) ) ) ) / ( ( √ ‘ 𝑛 ) · ( √ ‘ ( 𝑛 − 1 ) ) ) ) ) = ( 2 · ( ( √ ‘ 𝑛 ) − ( √ ‘ ( 𝑛 − 1 ) ) ) ) ) |
| 105 |
104
|
oveq2d |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( ( ( √ ‘ 𝑛 ) · ( √ ‘ ( 𝑛 − 1 ) ) ) · ( ( ( √ ‘ 𝑛 ) · ( √ ‘ ( 𝑛 − 1 ) ) ) · ( ( 2 · ( ( √ ‘ 𝑛 ) − ( √ ‘ ( 𝑛 − 1 ) ) ) ) / ( ( √ ‘ 𝑛 ) · ( √ ‘ ( 𝑛 − 1 ) ) ) ) ) ) = ( ( ( √ ‘ 𝑛 ) · ( √ ‘ ( 𝑛 − 1 ) ) ) · ( 2 · ( ( √ ‘ 𝑛 ) − ( √ ‘ ( 𝑛 − 1 ) ) ) ) ) ) |
| 106 |
97 102 105
|
3eqtrd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( ( 𝑛 · ( 𝑛 − 1 ) ) · ( ( 2 / ( √ ‘ ( 𝑛 − 1 ) ) ) − ( 2 / ( √ ‘ 𝑛 ) ) ) ) = ( ( ( √ ‘ 𝑛 ) · ( √ ‘ ( 𝑛 − 1 ) ) ) · ( 2 · ( ( √ ‘ 𝑛 ) − ( √ ‘ ( 𝑛 − 1 ) ) ) ) ) ) |
| 107 |
81 106
|
breqtrrd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( √ ‘ ( 𝑛 − 1 ) ) ≤ ( ( 𝑛 · ( 𝑛 − 1 ) ) · ( ( 2 / ( √ ‘ ( 𝑛 − 1 ) ) ) − ( 2 / ( √ ‘ 𝑛 ) ) ) ) ) |
| 108 |
7 25 30 39 107
|
letrd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( log ‘ 𝑛 ) ≤ ( ( 𝑛 · ( 𝑛 − 1 ) ) · ( ( 2 / ( √ ‘ ( 𝑛 − 1 ) ) ) − ( 2 / ( √ ‘ 𝑛 ) ) ) ) ) |
| 109 |
11
|
nngt0d |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → 0 < ( 𝑛 · ( 𝑛 − 1 ) ) ) |
| 110 |
|
ledivmul |
⊢ ( ( ( log ‘ 𝑛 ) ∈ ℝ ∧ ( ( 2 / ( √ ‘ ( 𝑛 − 1 ) ) ) − ( 2 / ( √ ‘ 𝑛 ) ) ) ∈ ℝ ∧ ( ( 𝑛 · ( 𝑛 − 1 ) ) ∈ ℝ ∧ 0 < ( 𝑛 · ( 𝑛 − 1 ) ) ) ) → ( ( ( log ‘ 𝑛 ) / ( 𝑛 · ( 𝑛 − 1 ) ) ) ≤ ( ( 2 / ( √ ‘ ( 𝑛 − 1 ) ) ) − ( 2 / ( √ ‘ 𝑛 ) ) ) ↔ ( log ‘ 𝑛 ) ≤ ( ( 𝑛 · ( 𝑛 − 1 ) ) · ( ( 2 / ( √ ‘ ( 𝑛 − 1 ) ) ) − ( 2 / ( √ ‘ 𝑛 ) ) ) ) ) ) |
| 111 |
7 22 29 109 110
|
syl112anc |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( ( ( log ‘ 𝑛 ) / ( 𝑛 · ( 𝑛 − 1 ) ) ) ≤ ( ( 2 / ( √ ‘ ( 𝑛 − 1 ) ) ) − ( 2 / ( √ ‘ 𝑛 ) ) ) ↔ ( log ‘ 𝑛 ) ≤ ( ( 𝑛 · ( 𝑛 − 1 ) ) · ( ( 2 / ( √ ‘ ( 𝑛 − 1 ) ) ) − ( 2 / ( √ ‘ 𝑛 ) ) ) ) ) ) |
| 112 |
108 111
|
mpbird |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( ( log ‘ 𝑛 ) / ( 𝑛 · ( 𝑛 − 1 ) ) ) ≤ ( ( 2 / ( √ ‘ ( 𝑛 − 1 ) ) ) − ( 2 / ( √ ‘ 𝑛 ) ) ) ) |
| 113 |
1 12 22 112
|
fsumle |
⊢ ( 𝐴 ∈ ℕ → Σ 𝑛 ∈ ( 2 ... 𝐴 ) ( ( log ‘ 𝑛 ) / ( 𝑛 · ( 𝑛 − 1 ) ) ) ≤ Σ 𝑛 ∈ ( 2 ... 𝐴 ) ( ( 2 / ( √ ‘ ( 𝑛 − 1 ) ) ) − ( 2 / ( √ ‘ 𝑛 ) ) ) ) |
| 114 |
|
fvoveq1 |
⊢ ( 𝑘 = 𝑛 → ( √ ‘ ( 𝑘 − 1 ) ) = ( √ ‘ ( 𝑛 − 1 ) ) ) |
| 115 |
114
|
oveq2d |
⊢ ( 𝑘 = 𝑛 → ( 2 / ( √ ‘ ( 𝑘 − 1 ) ) ) = ( 2 / ( √ ‘ ( 𝑛 − 1 ) ) ) ) |
| 116 |
|
fvoveq1 |
⊢ ( 𝑘 = ( 𝑛 + 1 ) → ( √ ‘ ( 𝑘 − 1 ) ) = ( √ ‘ ( ( 𝑛 + 1 ) − 1 ) ) ) |
| 117 |
116
|
oveq2d |
⊢ ( 𝑘 = ( 𝑛 + 1 ) → ( 2 / ( √ ‘ ( 𝑘 − 1 ) ) ) = ( 2 / ( √ ‘ ( ( 𝑛 + 1 ) − 1 ) ) ) ) |
| 118 |
|
oveq1 |
⊢ ( 𝑘 = 2 → ( 𝑘 − 1 ) = ( 2 − 1 ) ) |
| 119 |
|
2m1e1 |
⊢ ( 2 − 1 ) = 1 |
| 120 |
118 119
|
eqtrdi |
⊢ ( 𝑘 = 2 → ( 𝑘 − 1 ) = 1 ) |
| 121 |
120
|
fveq2d |
⊢ ( 𝑘 = 2 → ( √ ‘ ( 𝑘 − 1 ) ) = ( √ ‘ 1 ) ) |
| 122 |
|
sqrt1 |
⊢ ( √ ‘ 1 ) = 1 |
| 123 |
121 122
|
eqtrdi |
⊢ ( 𝑘 = 2 → ( √ ‘ ( 𝑘 − 1 ) ) = 1 ) |
| 124 |
123
|
oveq2d |
⊢ ( 𝑘 = 2 → ( 2 / ( √ ‘ ( 𝑘 − 1 ) ) ) = ( 2 / 1 ) ) |
| 125 |
67
|
div1i |
⊢ ( 2 / 1 ) = 2 |
| 126 |
124 125
|
eqtrdi |
⊢ ( 𝑘 = 2 → ( 2 / ( √ ‘ ( 𝑘 − 1 ) ) ) = 2 ) |
| 127 |
|
fvoveq1 |
⊢ ( 𝑘 = ( 𝐴 + 1 ) → ( √ ‘ ( 𝑘 − 1 ) ) = ( √ ‘ ( ( 𝐴 + 1 ) − 1 ) ) ) |
| 128 |
127
|
oveq2d |
⊢ ( 𝑘 = ( 𝐴 + 1 ) → ( 2 / ( √ ‘ ( 𝑘 − 1 ) ) ) = ( 2 / ( √ ‘ ( ( 𝐴 + 1 ) − 1 ) ) ) ) |
| 129 |
|
nnz |
⊢ ( 𝐴 ∈ ℕ → 𝐴 ∈ ℤ ) |
| 130 |
|
eluzp1p1 |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 1 ) → ( 𝐴 + 1 ) ∈ ( ℤ≥ ‘ ( 1 + 1 ) ) ) |
| 131 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
| 132 |
130 131
|
eleq2s |
⊢ ( 𝐴 ∈ ℕ → ( 𝐴 + 1 ) ∈ ( ℤ≥ ‘ ( 1 + 1 ) ) ) |
| 133 |
|
df-2 |
⊢ 2 = ( 1 + 1 ) |
| 134 |
133
|
fveq2i |
⊢ ( ℤ≥ ‘ 2 ) = ( ℤ≥ ‘ ( 1 + 1 ) ) |
| 135 |
132 134
|
eleqtrrdi |
⊢ ( 𝐴 ∈ ℕ → ( 𝐴 + 1 ) ∈ ( ℤ≥ ‘ 2 ) ) |
| 136 |
|
elfzuz |
⊢ ( 𝑘 ∈ ( 2 ... ( 𝐴 + 1 ) ) → 𝑘 ∈ ( ℤ≥ ‘ 2 ) ) |
| 137 |
|
uz2m1nn |
⊢ ( 𝑘 ∈ ( ℤ≥ ‘ 2 ) → ( 𝑘 − 1 ) ∈ ℕ ) |
| 138 |
136 137
|
syl |
⊢ ( 𝑘 ∈ ( 2 ... ( 𝐴 + 1 ) ) → ( 𝑘 − 1 ) ∈ ℕ ) |
| 139 |
138
|
adantl |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑘 ∈ ( 2 ... ( 𝐴 + 1 ) ) ) → ( 𝑘 − 1 ) ∈ ℕ ) |
| 140 |
139
|
nnrpd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑘 ∈ ( 2 ... ( 𝐴 + 1 ) ) ) → ( 𝑘 − 1 ) ∈ ℝ+ ) |
| 141 |
140
|
rpsqrtcld |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑘 ∈ ( 2 ... ( 𝐴 + 1 ) ) ) → ( √ ‘ ( 𝑘 − 1 ) ) ∈ ℝ+ ) |
| 142 |
|
rerpdivcl |
⊢ ( ( 2 ∈ ℝ ∧ ( √ ‘ ( 𝑘 − 1 ) ) ∈ ℝ+ ) → ( 2 / ( √ ‘ ( 𝑘 − 1 ) ) ) ∈ ℝ ) |
| 143 |
14 141 142
|
sylancr |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑘 ∈ ( 2 ... ( 𝐴 + 1 ) ) ) → ( 2 / ( √ ‘ ( 𝑘 − 1 ) ) ) ∈ ℝ ) |
| 144 |
143
|
recnd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑘 ∈ ( 2 ... ( 𝐴 + 1 ) ) ) → ( 2 / ( √ ‘ ( 𝑘 − 1 ) ) ) ∈ ℂ ) |
| 145 |
115 117 126 128 129 135 144
|
telfsum |
⊢ ( 𝐴 ∈ ℕ → Σ 𝑛 ∈ ( 2 ... 𝐴 ) ( ( 2 / ( √ ‘ ( 𝑛 − 1 ) ) ) − ( 2 / ( √ ‘ ( ( 𝑛 + 1 ) − 1 ) ) ) ) = ( 2 − ( 2 / ( √ ‘ ( ( 𝐴 + 1 ) − 1 ) ) ) ) ) |
| 146 |
|
pncan |
⊢ ( ( 𝑛 ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( 𝑛 + 1 ) − 1 ) = 𝑛 ) |
| 147 |
31 32 146
|
sylancl |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( ( 𝑛 + 1 ) − 1 ) = 𝑛 ) |
| 148 |
147
|
fveq2d |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( √ ‘ ( ( 𝑛 + 1 ) − 1 ) ) = ( √ ‘ 𝑛 ) ) |
| 149 |
148
|
oveq2d |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( 2 / ( √ ‘ ( ( 𝑛 + 1 ) − 1 ) ) ) = ( 2 / ( √ ‘ 𝑛 ) ) ) |
| 150 |
149
|
oveq2d |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑛 ∈ ( 2 ... 𝐴 ) ) → ( ( 2 / ( √ ‘ ( 𝑛 − 1 ) ) ) − ( 2 / ( √ ‘ ( ( 𝑛 + 1 ) − 1 ) ) ) ) = ( ( 2 / ( √ ‘ ( 𝑛 − 1 ) ) ) − ( 2 / ( √ ‘ 𝑛 ) ) ) ) |
| 151 |
150
|
sumeq2dv |
⊢ ( 𝐴 ∈ ℕ → Σ 𝑛 ∈ ( 2 ... 𝐴 ) ( ( 2 / ( √ ‘ ( 𝑛 − 1 ) ) ) − ( 2 / ( √ ‘ ( ( 𝑛 + 1 ) − 1 ) ) ) ) = Σ 𝑛 ∈ ( 2 ... 𝐴 ) ( ( 2 / ( √ ‘ ( 𝑛 − 1 ) ) ) − ( 2 / ( √ ‘ 𝑛 ) ) ) ) |
| 152 |
|
nncn |
⊢ ( 𝐴 ∈ ℕ → 𝐴 ∈ ℂ ) |
| 153 |
|
pncan |
⊢ ( ( 𝐴 ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( 𝐴 + 1 ) − 1 ) = 𝐴 ) |
| 154 |
152 32 153
|
sylancl |
⊢ ( 𝐴 ∈ ℕ → ( ( 𝐴 + 1 ) − 1 ) = 𝐴 ) |
| 155 |
154
|
fveq2d |
⊢ ( 𝐴 ∈ ℕ → ( √ ‘ ( ( 𝐴 + 1 ) − 1 ) ) = ( √ ‘ 𝐴 ) ) |
| 156 |
155
|
oveq2d |
⊢ ( 𝐴 ∈ ℕ → ( 2 / ( √ ‘ ( ( 𝐴 + 1 ) − 1 ) ) ) = ( 2 / ( √ ‘ 𝐴 ) ) ) |
| 157 |
156
|
oveq2d |
⊢ ( 𝐴 ∈ ℕ → ( 2 − ( 2 / ( √ ‘ ( ( 𝐴 + 1 ) − 1 ) ) ) ) = ( 2 − ( 2 / ( √ ‘ 𝐴 ) ) ) ) |
| 158 |
145 151 157
|
3eqtr3d |
⊢ ( 𝐴 ∈ ℕ → Σ 𝑛 ∈ ( 2 ... 𝐴 ) ( ( 2 / ( √ ‘ ( 𝑛 − 1 ) ) ) − ( 2 / ( √ ‘ 𝑛 ) ) ) = ( 2 − ( 2 / ( √ ‘ 𝐴 ) ) ) ) |
| 159 |
|
2rp |
⊢ 2 ∈ ℝ+ |
| 160 |
|
nnrp |
⊢ ( 𝐴 ∈ ℕ → 𝐴 ∈ ℝ+ ) |
| 161 |
160
|
rpsqrtcld |
⊢ ( 𝐴 ∈ ℕ → ( √ ‘ 𝐴 ) ∈ ℝ+ ) |
| 162 |
|
rpdivcl |
⊢ ( ( 2 ∈ ℝ+ ∧ ( √ ‘ 𝐴 ) ∈ ℝ+ ) → ( 2 / ( √ ‘ 𝐴 ) ) ∈ ℝ+ ) |
| 163 |
159 161 162
|
sylancr |
⊢ ( 𝐴 ∈ ℕ → ( 2 / ( √ ‘ 𝐴 ) ) ∈ ℝ+ ) |
| 164 |
163
|
rpge0d |
⊢ ( 𝐴 ∈ ℕ → 0 ≤ ( 2 / ( √ ‘ 𝐴 ) ) ) |
| 165 |
163
|
rpred |
⊢ ( 𝐴 ∈ ℕ → ( 2 / ( √ ‘ 𝐴 ) ) ∈ ℝ ) |
| 166 |
|
subge02 |
⊢ ( ( 2 ∈ ℝ ∧ ( 2 / ( √ ‘ 𝐴 ) ) ∈ ℝ ) → ( 0 ≤ ( 2 / ( √ ‘ 𝐴 ) ) ↔ ( 2 − ( 2 / ( √ ‘ 𝐴 ) ) ) ≤ 2 ) ) |
| 167 |
14 165 166
|
sylancr |
⊢ ( 𝐴 ∈ ℕ → ( 0 ≤ ( 2 / ( √ ‘ 𝐴 ) ) ↔ ( 2 − ( 2 / ( √ ‘ 𝐴 ) ) ) ≤ 2 ) ) |
| 168 |
164 167
|
mpbid |
⊢ ( 𝐴 ∈ ℕ → ( 2 − ( 2 / ( √ ‘ 𝐴 ) ) ) ≤ 2 ) |
| 169 |
158 168
|
eqbrtrd |
⊢ ( 𝐴 ∈ ℕ → Σ 𝑛 ∈ ( 2 ... 𝐴 ) ( ( 2 / ( √ ‘ ( 𝑛 − 1 ) ) ) − ( 2 / ( √ ‘ 𝑛 ) ) ) ≤ 2 ) |
| 170 |
13 23 24 113 169
|
letrd |
⊢ ( 𝐴 ∈ ℕ → Σ 𝑛 ∈ ( 2 ... 𝐴 ) ( ( log ‘ 𝑛 ) / ( 𝑛 · ( 𝑛 − 1 ) ) ) ≤ 2 ) |