| Step |
Hyp |
Ref |
Expression |
| 1 |
|
3re |
⊢ 3 ∈ ℝ |
| 2 |
1
|
a1i |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) → 3 ∈ ℝ ) |
| 3 |
|
0red |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) → 0 ∈ ℝ ) |
| 4 |
3
|
recnd |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) → 0 ∈ ℂ ) |
| 5 |
|
ovexd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ 𝑛 ∈ ℝ+ ) → ( ( log ‘ 𝑛 ) / ( 𝑛 ↑𝑐 ( 𝐵 / if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ) ) ∈ V ) |
| 6 |
|
simpr |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) → 𝐵 ∈ ℝ+ ) |
| 7 |
|
recl |
⊢ ( 𝐴 ∈ ℂ → ( ℜ ‘ 𝐴 ) ∈ ℝ ) |
| 8 |
7
|
adantr |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) → ( ℜ ‘ 𝐴 ) ∈ ℝ ) |
| 9 |
|
1re |
⊢ 1 ∈ ℝ |
| 10 |
|
ifcl |
⊢ ( ( ( ℜ ‘ 𝐴 ) ∈ ℝ ∧ 1 ∈ ℝ ) → if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ∈ ℝ ) |
| 11 |
8 9 10
|
sylancl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) → if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ∈ ℝ ) |
| 12 |
9
|
a1i |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) → 1 ∈ ℝ ) |
| 13 |
|
0lt1 |
⊢ 0 < 1 |
| 14 |
13
|
a1i |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) → 0 < 1 ) |
| 15 |
|
max1 |
⊢ ( ( 1 ∈ ℝ ∧ ( ℜ ‘ 𝐴 ) ∈ ℝ ) → 1 ≤ if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) |
| 16 |
9 8 15
|
sylancr |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) → 1 ≤ if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) |
| 17 |
3 12 11 14 16
|
ltletrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) → 0 < if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) |
| 18 |
11 17
|
elrpd |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) → if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ∈ ℝ+ ) |
| 19 |
6 18
|
rpdivcld |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) → ( 𝐵 / if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ∈ ℝ+ ) |
| 20 |
|
cxploglim |
⊢ ( ( 𝐵 / if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ∈ ℝ+ → ( 𝑛 ∈ ℝ+ ↦ ( ( log ‘ 𝑛 ) / ( 𝑛 ↑𝑐 ( 𝐵 / if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ) ) ) ⇝𝑟 0 ) |
| 21 |
19 20
|
syl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) → ( 𝑛 ∈ ℝ+ ↦ ( ( log ‘ 𝑛 ) / ( 𝑛 ↑𝑐 ( 𝐵 / if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ) ) ) ⇝𝑟 0 ) |
| 22 |
5 21 18
|
rlimcxp |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) → ( 𝑛 ∈ ℝ+ ↦ ( ( ( log ‘ 𝑛 ) / ( 𝑛 ↑𝑐 ( 𝐵 / if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ) ) ↑𝑐 if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ) ⇝𝑟 0 ) |
| 23 |
5 21
|
rlimmptrcl |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ 𝑛 ∈ ℝ+ ) → ( ( log ‘ 𝑛 ) / ( 𝑛 ↑𝑐 ( 𝐵 / if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ) ) ∈ ℂ ) |
| 24 |
11
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ 𝑛 ∈ ℝ+ ) → if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ∈ ℝ ) |
| 25 |
24
|
recnd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ 𝑛 ∈ ℝ+ ) → if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ∈ ℂ ) |
| 26 |
23 25
|
cxpcld |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ 𝑛 ∈ ℝ+ ) → ( ( ( log ‘ 𝑛 ) / ( 𝑛 ↑𝑐 ( 𝐵 / if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ) ) ↑𝑐 if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ∈ ℂ ) |
| 27 |
|
relogcl |
⊢ ( 𝑛 ∈ ℝ+ → ( log ‘ 𝑛 ) ∈ ℝ ) |
| 28 |
27
|
adantl |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ 𝑛 ∈ ℝ+ ) → ( log ‘ 𝑛 ) ∈ ℝ ) |
| 29 |
28
|
recnd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ 𝑛 ∈ ℝ+ ) → ( log ‘ 𝑛 ) ∈ ℂ ) |
| 30 |
|
simpll |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ 𝑛 ∈ ℝ+ ) → 𝐴 ∈ ℂ ) |
| 31 |
29 30
|
cxpcld |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ 𝑛 ∈ ℝ+ ) → ( ( log ‘ 𝑛 ) ↑𝑐 𝐴 ) ∈ ℂ ) |
| 32 |
|
simpr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ 𝑛 ∈ ℝ+ ) → 𝑛 ∈ ℝ+ ) |
| 33 |
|
rpre |
⊢ ( 𝐵 ∈ ℝ+ → 𝐵 ∈ ℝ ) |
| 34 |
33
|
ad2antlr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ 𝑛 ∈ ℝ+ ) → 𝐵 ∈ ℝ ) |
| 35 |
32 34
|
rpcxpcld |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ 𝑛 ∈ ℝ+ ) → ( 𝑛 ↑𝑐 𝐵 ) ∈ ℝ+ ) |
| 36 |
35
|
rpcnd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ 𝑛 ∈ ℝ+ ) → ( 𝑛 ↑𝑐 𝐵 ) ∈ ℂ ) |
| 37 |
35
|
rpne0d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ 𝑛 ∈ ℝ+ ) → ( 𝑛 ↑𝑐 𝐵 ) ≠ 0 ) |
| 38 |
31 36 37
|
divcld |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ 𝑛 ∈ ℝ+ ) → ( ( ( log ‘ 𝑛 ) ↑𝑐 𝐴 ) / ( 𝑛 ↑𝑐 𝐵 ) ) ∈ ℂ ) |
| 39 |
38
|
adantrr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → ( ( ( log ‘ 𝑛 ) ↑𝑐 𝐴 ) / ( 𝑛 ↑𝑐 𝐵 ) ) ∈ ℂ ) |
| 40 |
39
|
abscld |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → ( abs ‘ ( ( ( log ‘ 𝑛 ) ↑𝑐 𝐴 ) / ( 𝑛 ↑𝑐 𝐵 ) ) ) ∈ ℝ ) |
| 41 |
|
rpre |
⊢ ( 𝑛 ∈ ℝ+ → 𝑛 ∈ ℝ ) |
| 42 |
41
|
ad2antrl |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → 𝑛 ∈ ℝ ) |
| 43 |
9
|
a1i |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → 1 ∈ ℝ ) |
| 44 |
1
|
a1i |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → 3 ∈ ℝ ) |
| 45 |
|
1lt3 |
⊢ 1 < 3 |
| 46 |
45
|
a1i |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → 1 < 3 ) |
| 47 |
|
simprr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → 3 ≤ 𝑛 ) |
| 48 |
43 44 42 46 47
|
ltletrd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → 1 < 𝑛 ) |
| 49 |
42 48
|
rplogcld |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → ( log ‘ 𝑛 ) ∈ ℝ+ ) |
| 50 |
32
|
adantrr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → 𝑛 ∈ ℝ+ ) |
| 51 |
33
|
ad2antlr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → 𝐵 ∈ ℝ ) |
| 52 |
18
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ∈ ℝ+ ) |
| 53 |
51 52
|
rerpdivcld |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → ( 𝐵 / if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ∈ ℝ ) |
| 54 |
50 53
|
rpcxpcld |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → ( 𝑛 ↑𝑐 ( 𝐵 / if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ) ∈ ℝ+ ) |
| 55 |
49 54
|
rpdivcld |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → ( ( log ‘ 𝑛 ) / ( 𝑛 ↑𝑐 ( 𝐵 / if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ) ) ∈ ℝ+ ) |
| 56 |
11
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ∈ ℝ ) |
| 57 |
55 56
|
rpcxpcld |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → ( ( ( log ‘ 𝑛 ) / ( 𝑛 ↑𝑐 ( 𝐵 / if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ) ) ↑𝑐 if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ∈ ℝ+ ) |
| 58 |
57
|
rpred |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → ( ( ( log ‘ 𝑛 ) / ( 𝑛 ↑𝑐 ( 𝐵 / if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ) ) ↑𝑐 if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ∈ ℝ ) |
| 59 |
26
|
adantrr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → ( ( ( log ‘ 𝑛 ) / ( 𝑛 ↑𝑐 ( 𝐵 / if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ) ) ↑𝑐 if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ∈ ℂ ) |
| 60 |
59
|
abscld |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → ( abs ‘ ( ( ( log ‘ 𝑛 ) / ( 𝑛 ↑𝑐 ( 𝐵 / if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ) ) ↑𝑐 if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ) ∈ ℝ ) |
| 61 |
31
|
adantrr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → ( ( log ‘ 𝑛 ) ↑𝑐 𝐴 ) ∈ ℂ ) |
| 62 |
61
|
abscld |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → ( abs ‘ ( ( log ‘ 𝑛 ) ↑𝑐 𝐴 ) ) ∈ ℝ ) |
| 63 |
49 56
|
rpcxpcld |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → ( ( log ‘ 𝑛 ) ↑𝑐 if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ∈ ℝ+ ) |
| 64 |
63
|
rpred |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → ( ( log ‘ 𝑛 ) ↑𝑐 if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ∈ ℝ ) |
| 65 |
35
|
adantrr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → ( 𝑛 ↑𝑐 𝐵 ) ∈ ℝ+ ) |
| 66 |
|
simpll |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → 𝐴 ∈ ℂ ) |
| 67 |
|
abscxp |
⊢ ( ( ( log ‘ 𝑛 ) ∈ ℝ+ ∧ 𝐴 ∈ ℂ ) → ( abs ‘ ( ( log ‘ 𝑛 ) ↑𝑐 𝐴 ) ) = ( ( log ‘ 𝑛 ) ↑𝑐 ( ℜ ‘ 𝐴 ) ) ) |
| 68 |
49 66 67
|
syl2anc |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → ( abs ‘ ( ( log ‘ 𝑛 ) ↑𝑐 𝐴 ) ) = ( ( log ‘ 𝑛 ) ↑𝑐 ( ℜ ‘ 𝐴 ) ) ) |
| 69 |
66
|
recld |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → ( ℜ ‘ 𝐴 ) ∈ ℝ ) |
| 70 |
|
max2 |
⊢ ( ( 1 ∈ ℝ ∧ ( ℜ ‘ 𝐴 ) ∈ ℝ ) → ( ℜ ‘ 𝐴 ) ≤ if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) |
| 71 |
9 69 70
|
sylancr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → ( ℜ ‘ 𝐴 ) ≤ if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) |
| 72 |
27
|
ad2antrl |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → ( log ‘ 𝑛 ) ∈ ℝ ) |
| 73 |
|
loge |
⊢ ( log ‘ e ) = 1 |
| 74 |
|
ere |
⊢ e ∈ ℝ |
| 75 |
74
|
a1i |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → e ∈ ℝ ) |
| 76 |
|
egt2lt3 |
⊢ ( 2 < e ∧ e < 3 ) |
| 77 |
76
|
simpri |
⊢ e < 3 |
| 78 |
77
|
a1i |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → e < 3 ) |
| 79 |
75 44 42 78 47
|
ltletrd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → e < 𝑛 ) |
| 80 |
|
epr |
⊢ e ∈ ℝ+ |
| 81 |
|
logltb |
⊢ ( ( e ∈ ℝ+ ∧ 𝑛 ∈ ℝ+ ) → ( e < 𝑛 ↔ ( log ‘ e ) < ( log ‘ 𝑛 ) ) ) |
| 82 |
80 50 81
|
sylancr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → ( e < 𝑛 ↔ ( log ‘ e ) < ( log ‘ 𝑛 ) ) ) |
| 83 |
79 82
|
mpbid |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → ( log ‘ e ) < ( log ‘ 𝑛 ) ) |
| 84 |
73 83
|
eqbrtrrid |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → 1 < ( log ‘ 𝑛 ) ) |
| 85 |
72 84 69 56
|
cxpled |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → ( ( ℜ ‘ 𝐴 ) ≤ if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ↔ ( ( log ‘ 𝑛 ) ↑𝑐 ( ℜ ‘ 𝐴 ) ) ≤ ( ( log ‘ 𝑛 ) ↑𝑐 if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ) ) |
| 86 |
71 85
|
mpbid |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → ( ( log ‘ 𝑛 ) ↑𝑐 ( ℜ ‘ 𝐴 ) ) ≤ ( ( log ‘ 𝑛 ) ↑𝑐 if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ) |
| 87 |
68 86
|
eqbrtrd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → ( abs ‘ ( ( log ‘ 𝑛 ) ↑𝑐 𝐴 ) ) ≤ ( ( log ‘ 𝑛 ) ↑𝑐 if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ) |
| 88 |
62 64 65 87
|
lediv1dd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → ( ( abs ‘ ( ( log ‘ 𝑛 ) ↑𝑐 𝐴 ) ) / ( 𝑛 ↑𝑐 𝐵 ) ) ≤ ( ( ( log ‘ 𝑛 ) ↑𝑐 if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) / ( 𝑛 ↑𝑐 𝐵 ) ) ) |
| 89 |
31 36 37
|
absdivd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ 𝑛 ∈ ℝ+ ) → ( abs ‘ ( ( ( log ‘ 𝑛 ) ↑𝑐 𝐴 ) / ( 𝑛 ↑𝑐 𝐵 ) ) ) = ( ( abs ‘ ( ( log ‘ 𝑛 ) ↑𝑐 𝐴 ) ) / ( abs ‘ ( 𝑛 ↑𝑐 𝐵 ) ) ) ) |
| 90 |
89
|
adantrr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → ( abs ‘ ( ( ( log ‘ 𝑛 ) ↑𝑐 𝐴 ) / ( 𝑛 ↑𝑐 𝐵 ) ) ) = ( ( abs ‘ ( ( log ‘ 𝑛 ) ↑𝑐 𝐴 ) ) / ( abs ‘ ( 𝑛 ↑𝑐 𝐵 ) ) ) ) |
| 91 |
65
|
rprege0d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → ( ( 𝑛 ↑𝑐 𝐵 ) ∈ ℝ ∧ 0 ≤ ( 𝑛 ↑𝑐 𝐵 ) ) ) |
| 92 |
|
absid |
⊢ ( ( ( 𝑛 ↑𝑐 𝐵 ) ∈ ℝ ∧ 0 ≤ ( 𝑛 ↑𝑐 𝐵 ) ) → ( abs ‘ ( 𝑛 ↑𝑐 𝐵 ) ) = ( 𝑛 ↑𝑐 𝐵 ) ) |
| 93 |
91 92
|
syl |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → ( abs ‘ ( 𝑛 ↑𝑐 𝐵 ) ) = ( 𝑛 ↑𝑐 𝐵 ) ) |
| 94 |
93
|
oveq2d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → ( ( abs ‘ ( ( log ‘ 𝑛 ) ↑𝑐 𝐴 ) ) / ( abs ‘ ( 𝑛 ↑𝑐 𝐵 ) ) ) = ( ( abs ‘ ( ( log ‘ 𝑛 ) ↑𝑐 𝐴 ) ) / ( 𝑛 ↑𝑐 𝐵 ) ) ) |
| 95 |
90 94
|
eqtrd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → ( abs ‘ ( ( ( log ‘ 𝑛 ) ↑𝑐 𝐴 ) / ( 𝑛 ↑𝑐 𝐵 ) ) ) = ( ( abs ‘ ( ( log ‘ 𝑛 ) ↑𝑐 𝐴 ) ) / ( 𝑛 ↑𝑐 𝐵 ) ) ) |
| 96 |
49
|
rprege0d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → ( ( log ‘ 𝑛 ) ∈ ℝ ∧ 0 ≤ ( log ‘ 𝑛 ) ) ) |
| 97 |
11
|
recnd |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) → if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ∈ ℂ ) |
| 98 |
97
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ∈ ℂ ) |
| 99 |
|
divcxp |
⊢ ( ( ( ( log ‘ 𝑛 ) ∈ ℝ ∧ 0 ≤ ( log ‘ 𝑛 ) ) ∧ ( 𝑛 ↑𝑐 ( 𝐵 / if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ) ∈ ℝ+ ∧ if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ∈ ℂ ) → ( ( ( log ‘ 𝑛 ) / ( 𝑛 ↑𝑐 ( 𝐵 / if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ) ) ↑𝑐 if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) = ( ( ( log ‘ 𝑛 ) ↑𝑐 if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) / ( ( 𝑛 ↑𝑐 ( 𝐵 / if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ) ↑𝑐 if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ) ) |
| 100 |
96 54 98 99
|
syl3anc |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → ( ( ( log ‘ 𝑛 ) / ( 𝑛 ↑𝑐 ( 𝐵 / if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ) ) ↑𝑐 if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) = ( ( ( log ‘ 𝑛 ) ↑𝑐 if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) / ( ( 𝑛 ↑𝑐 ( 𝐵 / if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ) ↑𝑐 if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ) ) |
| 101 |
50 53 98
|
cxpmuld |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → ( 𝑛 ↑𝑐 ( ( 𝐵 / if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) · if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ) = ( ( 𝑛 ↑𝑐 ( 𝐵 / if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ) ↑𝑐 if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ) |
| 102 |
51
|
recnd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → 𝐵 ∈ ℂ ) |
| 103 |
52
|
rpne0d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ≠ 0 ) |
| 104 |
102 98 103
|
divcan1d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → ( ( 𝐵 / if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) · if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) = 𝐵 ) |
| 105 |
104
|
oveq2d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → ( 𝑛 ↑𝑐 ( ( 𝐵 / if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) · if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ) = ( 𝑛 ↑𝑐 𝐵 ) ) |
| 106 |
101 105
|
eqtr3d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → ( ( 𝑛 ↑𝑐 ( 𝐵 / if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ) ↑𝑐 if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) = ( 𝑛 ↑𝑐 𝐵 ) ) |
| 107 |
106
|
oveq2d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → ( ( ( log ‘ 𝑛 ) ↑𝑐 if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) / ( ( 𝑛 ↑𝑐 ( 𝐵 / if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ) ↑𝑐 if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ) = ( ( ( log ‘ 𝑛 ) ↑𝑐 if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) / ( 𝑛 ↑𝑐 𝐵 ) ) ) |
| 108 |
100 107
|
eqtrd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → ( ( ( log ‘ 𝑛 ) / ( 𝑛 ↑𝑐 ( 𝐵 / if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ) ) ↑𝑐 if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) = ( ( ( log ‘ 𝑛 ) ↑𝑐 if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) / ( 𝑛 ↑𝑐 𝐵 ) ) ) |
| 109 |
88 95 108
|
3brtr4d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → ( abs ‘ ( ( ( log ‘ 𝑛 ) ↑𝑐 𝐴 ) / ( 𝑛 ↑𝑐 𝐵 ) ) ) ≤ ( ( ( log ‘ 𝑛 ) / ( 𝑛 ↑𝑐 ( 𝐵 / if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ) ) ↑𝑐 if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ) |
| 110 |
58
|
leabsd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → ( ( ( log ‘ 𝑛 ) / ( 𝑛 ↑𝑐 ( 𝐵 / if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ) ) ↑𝑐 if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ≤ ( abs ‘ ( ( ( log ‘ 𝑛 ) / ( 𝑛 ↑𝑐 ( 𝐵 / if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ) ) ↑𝑐 if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ) ) |
| 111 |
40 58 60 109 110
|
letrd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → ( abs ‘ ( ( ( log ‘ 𝑛 ) ↑𝑐 𝐴 ) / ( 𝑛 ↑𝑐 𝐵 ) ) ) ≤ ( abs ‘ ( ( ( log ‘ 𝑛 ) / ( 𝑛 ↑𝑐 ( 𝐵 / if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ) ) ↑𝑐 if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ) ) |
| 112 |
39
|
subid1d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → ( ( ( ( log ‘ 𝑛 ) ↑𝑐 𝐴 ) / ( 𝑛 ↑𝑐 𝐵 ) ) − 0 ) = ( ( ( log ‘ 𝑛 ) ↑𝑐 𝐴 ) / ( 𝑛 ↑𝑐 𝐵 ) ) ) |
| 113 |
112
|
fveq2d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → ( abs ‘ ( ( ( ( log ‘ 𝑛 ) ↑𝑐 𝐴 ) / ( 𝑛 ↑𝑐 𝐵 ) ) − 0 ) ) = ( abs ‘ ( ( ( log ‘ 𝑛 ) ↑𝑐 𝐴 ) / ( 𝑛 ↑𝑐 𝐵 ) ) ) ) |
| 114 |
59
|
subid1d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → ( ( ( ( log ‘ 𝑛 ) / ( 𝑛 ↑𝑐 ( 𝐵 / if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ) ) ↑𝑐 if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) − 0 ) = ( ( ( log ‘ 𝑛 ) / ( 𝑛 ↑𝑐 ( 𝐵 / if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ) ) ↑𝑐 if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ) |
| 115 |
114
|
fveq2d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → ( abs ‘ ( ( ( ( log ‘ 𝑛 ) / ( 𝑛 ↑𝑐 ( 𝐵 / if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ) ) ↑𝑐 if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) − 0 ) ) = ( abs ‘ ( ( ( log ‘ 𝑛 ) / ( 𝑛 ↑𝑐 ( 𝐵 / if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ) ) ↑𝑐 if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ) ) |
| 116 |
111 113 115
|
3brtr4d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) ∧ ( 𝑛 ∈ ℝ+ ∧ 3 ≤ 𝑛 ) ) → ( abs ‘ ( ( ( ( log ‘ 𝑛 ) ↑𝑐 𝐴 ) / ( 𝑛 ↑𝑐 𝐵 ) ) − 0 ) ) ≤ ( abs ‘ ( ( ( ( log ‘ 𝑛 ) / ( 𝑛 ↑𝑐 ( 𝐵 / if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) ) ) ↑𝑐 if ( 1 ≤ ( ℜ ‘ 𝐴 ) , ( ℜ ‘ 𝐴 ) , 1 ) ) − 0 ) ) ) |
| 117 |
2 4 22 26 38 116
|
rlimsqzlem |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+ ) → ( 𝑛 ∈ ℝ+ ↦ ( ( ( log ‘ 𝑛 ) ↑𝑐 𝐴 ) / ( 𝑛 ↑𝑐 𝐵 ) ) ) ⇝𝑟 0 ) |