Step |
Hyp |
Ref |
Expression |
1 |
|
1re |
⊢ 1 ∈ ℝ |
2 |
|
elicopnf |
⊢ ( 1 ∈ ℝ → ( 𝑛 ∈ ( 1 [,) +∞ ) ↔ ( 𝑛 ∈ ℝ ∧ 1 ≤ 𝑛 ) ) ) |
3 |
1 2
|
ax-mp |
⊢ ( 𝑛 ∈ ( 1 [,) +∞ ) ↔ ( 𝑛 ∈ ℝ ∧ 1 ≤ 𝑛 ) ) |
4 |
3
|
simplbi |
⊢ ( 𝑛 ∈ ( 1 [,) +∞ ) → 𝑛 ∈ ℝ ) |
5 |
|
0red |
⊢ ( 𝑛 ∈ ( 1 [,) +∞ ) → 0 ∈ ℝ ) |
6 |
|
1red |
⊢ ( 𝑛 ∈ ( 1 [,) +∞ ) → 1 ∈ ℝ ) |
7 |
|
0lt1 |
⊢ 0 < 1 |
8 |
7
|
a1i |
⊢ ( 𝑛 ∈ ( 1 [,) +∞ ) → 0 < 1 ) |
9 |
3
|
simprbi |
⊢ ( 𝑛 ∈ ( 1 [,) +∞ ) → 1 ≤ 𝑛 ) |
10 |
5 6 4 8 9
|
ltletrd |
⊢ ( 𝑛 ∈ ( 1 [,) +∞ ) → 0 < 𝑛 ) |
11 |
4 10
|
elrpd |
⊢ ( 𝑛 ∈ ( 1 [,) +∞ ) → 𝑛 ∈ ℝ+ ) |
12 |
11
|
ssriv |
⊢ ( 1 [,) +∞ ) ⊆ ℝ+ |
13 |
|
resmpt |
⊢ ( ( 1 [,) +∞ ) ⊆ ℝ+ → ( ( 𝑛 ∈ ℝ+ ↦ ( ( 𝑛 ↑𝑐 𝐴 ) / ( 𝐵 ↑𝑐 𝑛 ) ) ) ↾ ( 1 [,) +∞ ) ) = ( 𝑛 ∈ ( 1 [,) +∞ ) ↦ ( ( 𝑛 ↑𝑐 𝐴 ) / ( 𝐵 ↑𝑐 𝑛 ) ) ) ) |
14 |
12 13
|
ax-mp |
⊢ ( ( 𝑛 ∈ ℝ+ ↦ ( ( 𝑛 ↑𝑐 𝐴 ) / ( 𝐵 ↑𝑐 𝑛 ) ) ) ↾ ( 1 [,) +∞ ) ) = ( 𝑛 ∈ ( 1 [,) +∞ ) ↦ ( ( 𝑛 ↑𝑐 𝐴 ) / ( 𝐵 ↑𝑐 𝑛 ) ) ) |
15 |
|
0red |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) → 0 ∈ ℝ ) |
16 |
12
|
a1i |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) → ( 1 [,) +∞ ) ⊆ ℝ+ ) |
17 |
|
rpre |
⊢ ( 𝑛 ∈ ℝ+ → 𝑛 ∈ ℝ ) |
18 |
17
|
adantl |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ℝ+ ) → 𝑛 ∈ ℝ ) |
19 |
|
rpge0 |
⊢ ( 𝑛 ∈ ℝ+ → 0 ≤ 𝑛 ) |
20 |
19
|
adantl |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ℝ+ ) → 0 ≤ 𝑛 ) |
21 |
|
simpl2 |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ℝ+ ) → 𝐵 ∈ ℝ ) |
22 |
|
0red |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ℝ+ ) → 0 ∈ ℝ ) |
23 |
|
1red |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ℝ+ ) → 1 ∈ ℝ ) |
24 |
7
|
a1i |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ℝ+ ) → 0 < 1 ) |
25 |
|
simpl3 |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ℝ+ ) → 1 < 𝐵 ) |
26 |
22 23 21 24 25
|
lttrd |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ℝ+ ) → 0 < 𝐵 ) |
27 |
21 26
|
elrpd |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ℝ+ ) → 𝐵 ∈ ℝ+ ) |
28 |
27 18
|
rpcxpcld |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ℝ+ ) → ( 𝐵 ↑𝑐 𝑛 ) ∈ ℝ+ ) |
29 |
|
simp1 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) → 𝐴 ∈ ℝ ) |
30 |
|
ifcl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 1 ∈ ℝ ) → if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ∈ ℝ ) |
31 |
29 1 30
|
sylancl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) → if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ∈ ℝ ) |
32 |
|
1red |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) → 1 ∈ ℝ ) |
33 |
7
|
a1i |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) → 0 < 1 ) |
34 |
|
max1 |
⊢ ( ( 1 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → 1 ≤ if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) |
35 |
1 29 34
|
sylancr |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) → 1 ≤ if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) |
36 |
15 32 31 33 35
|
ltletrd |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) → 0 < if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) |
37 |
31 36
|
elrpd |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) → if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ∈ ℝ+ ) |
38 |
37
|
rprecred |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) → ( 1 / if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ∈ ℝ ) |
39 |
38
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ℝ+ ) → ( 1 / if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ∈ ℝ ) |
40 |
28 39
|
rpcxpcld |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ℝ+ ) → ( ( 𝐵 ↑𝑐 𝑛 ) ↑𝑐 ( 1 / if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ) ∈ ℝ+ ) |
41 |
31
|
recnd |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) → if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ∈ ℂ ) |
42 |
41
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ℝ+ ) → if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ∈ ℂ ) |
43 |
18 20 40 42
|
divcxpd |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ℝ+ ) → ( ( 𝑛 / ( ( 𝐵 ↑𝑐 𝑛 ) ↑𝑐 ( 1 / if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ) ) ↑𝑐 if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) = ( ( 𝑛 ↑𝑐 if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) / ( ( ( 𝐵 ↑𝑐 𝑛 ) ↑𝑐 ( 1 / if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ) ↑𝑐 if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ) ) |
44 |
37
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ℝ+ ) → if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ∈ ℝ+ ) |
45 |
44
|
rpne0d |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ℝ+ ) → if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ≠ 0 ) |
46 |
42 45
|
recid2d |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ℝ+ ) → ( ( 1 / if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) · if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) = 1 ) |
47 |
46
|
oveq2d |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ℝ+ ) → ( ( 𝐵 ↑𝑐 𝑛 ) ↑𝑐 ( ( 1 / if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) · if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ) = ( ( 𝐵 ↑𝑐 𝑛 ) ↑𝑐 1 ) ) |
48 |
28 39 42
|
cxpmuld |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ℝ+ ) → ( ( 𝐵 ↑𝑐 𝑛 ) ↑𝑐 ( ( 1 / if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) · if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ) = ( ( ( 𝐵 ↑𝑐 𝑛 ) ↑𝑐 ( 1 / if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ) ↑𝑐 if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ) |
49 |
28
|
rpcnd |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ℝ+ ) → ( 𝐵 ↑𝑐 𝑛 ) ∈ ℂ ) |
50 |
49
|
cxp1d |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ℝ+ ) → ( ( 𝐵 ↑𝑐 𝑛 ) ↑𝑐 1 ) = ( 𝐵 ↑𝑐 𝑛 ) ) |
51 |
47 48 50
|
3eqtr3d |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ℝ+ ) → ( ( ( 𝐵 ↑𝑐 𝑛 ) ↑𝑐 ( 1 / if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ) ↑𝑐 if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) = ( 𝐵 ↑𝑐 𝑛 ) ) |
52 |
51
|
oveq2d |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ℝ+ ) → ( ( 𝑛 ↑𝑐 if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) / ( ( ( 𝐵 ↑𝑐 𝑛 ) ↑𝑐 ( 1 / if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ) ↑𝑐 if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ) = ( ( 𝑛 ↑𝑐 if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) / ( 𝐵 ↑𝑐 𝑛 ) ) ) |
53 |
43 52
|
eqtrd |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ℝ+ ) → ( ( 𝑛 / ( ( 𝐵 ↑𝑐 𝑛 ) ↑𝑐 ( 1 / if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ) ) ↑𝑐 if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) = ( ( 𝑛 ↑𝑐 if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) / ( 𝐵 ↑𝑐 𝑛 ) ) ) |
54 |
53
|
mpteq2dva |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) → ( 𝑛 ∈ ℝ+ ↦ ( ( 𝑛 / ( ( 𝐵 ↑𝑐 𝑛 ) ↑𝑐 ( 1 / if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ) ) ↑𝑐 if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ) = ( 𝑛 ∈ ℝ+ ↦ ( ( 𝑛 ↑𝑐 if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) / ( 𝐵 ↑𝑐 𝑛 ) ) ) ) |
55 |
|
ovexd |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ℝ+ ) → ( 𝑛 / ( ( 𝐵 ↑𝑐 𝑛 ) ↑𝑐 ( 1 / if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ) ) ∈ V ) |
56 |
18
|
recnd |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ℝ+ ) → 𝑛 ∈ ℂ ) |
57 |
38
|
recnd |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) → ( 1 / if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ∈ ℂ ) |
58 |
57
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ℝ+ ) → ( 1 / if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ∈ ℂ ) |
59 |
56 58
|
mulcomd |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ℝ+ ) → ( 𝑛 · ( 1 / if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ) = ( ( 1 / if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) · 𝑛 ) ) |
60 |
59
|
oveq2d |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ℝ+ ) → ( 𝐵 ↑𝑐 ( 𝑛 · ( 1 / if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ) ) = ( 𝐵 ↑𝑐 ( ( 1 / if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) · 𝑛 ) ) ) |
61 |
27 18 58
|
cxpmuld |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ℝ+ ) → ( 𝐵 ↑𝑐 ( 𝑛 · ( 1 / if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ) ) = ( ( 𝐵 ↑𝑐 𝑛 ) ↑𝑐 ( 1 / if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ) ) |
62 |
27 39 56
|
cxpmuld |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ℝ+ ) → ( 𝐵 ↑𝑐 ( ( 1 / if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) · 𝑛 ) ) = ( ( 𝐵 ↑𝑐 ( 1 / if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ) ↑𝑐 𝑛 ) ) |
63 |
60 61 62
|
3eqtr3d |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ℝ+ ) → ( ( 𝐵 ↑𝑐 𝑛 ) ↑𝑐 ( 1 / if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ) = ( ( 𝐵 ↑𝑐 ( 1 / if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ) ↑𝑐 𝑛 ) ) |
64 |
63
|
oveq2d |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ℝ+ ) → ( 𝑛 / ( ( 𝐵 ↑𝑐 𝑛 ) ↑𝑐 ( 1 / if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ) ) = ( 𝑛 / ( ( 𝐵 ↑𝑐 ( 1 / if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ) ↑𝑐 𝑛 ) ) ) |
65 |
64
|
mpteq2dva |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) → ( 𝑛 ∈ ℝ+ ↦ ( 𝑛 / ( ( 𝐵 ↑𝑐 𝑛 ) ↑𝑐 ( 1 / if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ) ) ) = ( 𝑛 ∈ ℝ+ ↦ ( 𝑛 / ( ( 𝐵 ↑𝑐 ( 1 / if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ) ↑𝑐 𝑛 ) ) ) ) |
66 |
|
simp2 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) → 𝐵 ∈ ℝ ) |
67 |
|
simp3 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) → 1 < 𝐵 ) |
68 |
15 32 66 33 67
|
lttrd |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) → 0 < 𝐵 ) |
69 |
66 68
|
elrpd |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) → 𝐵 ∈ ℝ+ ) |
70 |
69 38
|
rpcxpcld |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) → ( 𝐵 ↑𝑐 ( 1 / if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ) ∈ ℝ+ ) |
71 |
70
|
rpred |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) → ( 𝐵 ↑𝑐 ( 1 / if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ) ∈ ℝ ) |
72 |
57
|
1cxpd |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) → ( 1 ↑𝑐 ( 1 / if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ) = 1 ) |
73 |
|
0le1 |
⊢ 0 ≤ 1 |
74 |
73
|
a1i |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) → 0 ≤ 1 ) |
75 |
69
|
rpge0d |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) → 0 ≤ 𝐵 ) |
76 |
37
|
rpreccld |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) → ( 1 / if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ∈ ℝ+ ) |
77 |
32 74 66 75 76
|
cxplt2d |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) → ( 1 < 𝐵 ↔ ( 1 ↑𝑐 ( 1 / if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ) < ( 𝐵 ↑𝑐 ( 1 / if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ) ) ) |
78 |
67 77
|
mpbid |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) → ( 1 ↑𝑐 ( 1 / if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ) < ( 𝐵 ↑𝑐 ( 1 / if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ) ) |
79 |
72 78
|
eqbrtrrd |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) → 1 < ( 𝐵 ↑𝑐 ( 1 / if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ) ) |
80 |
|
cxp2limlem |
⊢ ( ( ( 𝐵 ↑𝑐 ( 1 / if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ) ∈ ℝ ∧ 1 < ( 𝐵 ↑𝑐 ( 1 / if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ) ) → ( 𝑛 ∈ ℝ+ ↦ ( 𝑛 / ( ( 𝐵 ↑𝑐 ( 1 / if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ) ↑𝑐 𝑛 ) ) ) ⇝𝑟 0 ) |
81 |
71 79 80
|
syl2anc |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) → ( 𝑛 ∈ ℝ+ ↦ ( 𝑛 / ( ( 𝐵 ↑𝑐 ( 1 / if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ) ↑𝑐 𝑛 ) ) ) ⇝𝑟 0 ) |
82 |
65 81
|
eqbrtrd |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) → ( 𝑛 ∈ ℝ+ ↦ ( 𝑛 / ( ( 𝐵 ↑𝑐 𝑛 ) ↑𝑐 ( 1 / if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ) ) ) ⇝𝑟 0 ) |
83 |
55 82 37
|
rlimcxp |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) → ( 𝑛 ∈ ℝ+ ↦ ( ( 𝑛 / ( ( 𝐵 ↑𝑐 𝑛 ) ↑𝑐 ( 1 / if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ) ) ↑𝑐 if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ) ⇝𝑟 0 ) |
84 |
54 83
|
eqbrtrrd |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) → ( 𝑛 ∈ ℝ+ ↦ ( ( 𝑛 ↑𝑐 if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) / ( 𝐵 ↑𝑐 𝑛 ) ) ) ⇝𝑟 0 ) |
85 |
16 84
|
rlimres2 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) → ( 𝑛 ∈ ( 1 [,) +∞ ) ↦ ( ( 𝑛 ↑𝑐 if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) / ( 𝐵 ↑𝑐 𝑛 ) ) ) ⇝𝑟 0 ) |
86 |
|
simpr |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ℝ+ ) → 𝑛 ∈ ℝ+ ) |
87 |
31
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ℝ+ ) → if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ∈ ℝ ) |
88 |
86 87
|
rpcxpcld |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ℝ+ ) → ( 𝑛 ↑𝑐 if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ∈ ℝ+ ) |
89 |
88 28
|
rpdivcld |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ℝ+ ) → ( ( 𝑛 ↑𝑐 if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) / ( 𝐵 ↑𝑐 𝑛 ) ) ∈ ℝ+ ) |
90 |
89
|
rpred |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ℝ+ ) → ( ( 𝑛 ↑𝑐 if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) / ( 𝐵 ↑𝑐 𝑛 ) ) ∈ ℝ ) |
91 |
11 90
|
sylan2 |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ( 1 [,) +∞ ) ) → ( ( 𝑛 ↑𝑐 if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) / ( 𝐵 ↑𝑐 𝑛 ) ) ∈ ℝ ) |
92 |
|
simpl1 |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ℝ+ ) → 𝐴 ∈ ℝ ) |
93 |
86 92
|
rpcxpcld |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ℝ+ ) → ( 𝑛 ↑𝑐 𝐴 ) ∈ ℝ+ ) |
94 |
93 28
|
rpdivcld |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ℝ+ ) → ( ( 𝑛 ↑𝑐 𝐴 ) / ( 𝐵 ↑𝑐 𝑛 ) ) ∈ ℝ+ ) |
95 |
11 94
|
sylan2 |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ( 1 [,) +∞ ) ) → ( ( 𝑛 ↑𝑐 𝐴 ) / ( 𝐵 ↑𝑐 𝑛 ) ) ∈ ℝ+ ) |
96 |
95
|
rpred |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ( 1 [,) +∞ ) ) → ( ( 𝑛 ↑𝑐 𝐴 ) / ( 𝐵 ↑𝑐 𝑛 ) ) ∈ ℝ ) |
97 |
11 93
|
sylan2 |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ( 1 [,) +∞ ) ) → ( 𝑛 ↑𝑐 𝐴 ) ∈ ℝ+ ) |
98 |
97
|
rpred |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ( 1 [,) +∞ ) ) → ( 𝑛 ↑𝑐 𝐴 ) ∈ ℝ ) |
99 |
11 88
|
sylan2 |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ( 1 [,) +∞ ) ) → ( 𝑛 ↑𝑐 if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ∈ ℝ+ ) |
100 |
99
|
rpred |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ( 1 [,) +∞ ) ) → ( 𝑛 ↑𝑐 if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ∈ ℝ ) |
101 |
11 28
|
sylan2 |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ( 1 [,) +∞ ) ) → ( 𝐵 ↑𝑐 𝑛 ) ∈ ℝ+ ) |
102 |
4
|
adantl |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ( 1 [,) +∞ ) ) → 𝑛 ∈ ℝ ) |
103 |
9
|
adantl |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ( 1 [,) +∞ ) ) → 1 ≤ 𝑛 ) |
104 |
|
simpl1 |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ( 1 [,) +∞ ) ) → 𝐴 ∈ ℝ ) |
105 |
31
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ( 1 [,) +∞ ) ) → if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ∈ ℝ ) |
106 |
|
max2 |
⊢ ( ( 1 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → 𝐴 ≤ if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) |
107 |
1 104 106
|
sylancr |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ( 1 [,) +∞ ) ) → 𝐴 ≤ if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) |
108 |
102 103 104 105 107
|
cxplead |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ( 1 [,) +∞ ) ) → ( 𝑛 ↑𝑐 𝐴 ) ≤ ( 𝑛 ↑𝑐 if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) ) |
109 |
98 100 101 108
|
lediv1dd |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ( 1 [,) +∞ ) ) → ( ( 𝑛 ↑𝑐 𝐴 ) / ( 𝐵 ↑𝑐 𝑛 ) ) ≤ ( ( 𝑛 ↑𝑐 if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) / ( 𝐵 ↑𝑐 𝑛 ) ) ) |
110 |
109
|
adantrr |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ ( 𝑛 ∈ ( 1 [,) +∞ ) ∧ 0 ≤ 𝑛 ) ) → ( ( 𝑛 ↑𝑐 𝐴 ) / ( 𝐵 ↑𝑐 𝑛 ) ) ≤ ( ( 𝑛 ↑𝑐 if ( 1 ≤ 𝐴 , 𝐴 , 1 ) ) / ( 𝐵 ↑𝑐 𝑛 ) ) ) |
111 |
95
|
rpge0d |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ( 1 [,) +∞ ) ) → 0 ≤ ( ( 𝑛 ↑𝑐 𝐴 ) / ( 𝐵 ↑𝑐 𝑛 ) ) ) |
112 |
111
|
adantrr |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ ( 𝑛 ∈ ( 1 [,) +∞ ) ∧ 0 ≤ 𝑛 ) ) → 0 ≤ ( ( 𝑛 ↑𝑐 𝐴 ) / ( 𝐵 ↑𝑐 𝑛 ) ) ) |
113 |
15 15 85 91 96 110 112
|
rlimsqz2 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) → ( 𝑛 ∈ ( 1 [,) +∞ ) ↦ ( ( 𝑛 ↑𝑐 𝐴 ) / ( 𝐵 ↑𝑐 𝑛 ) ) ) ⇝𝑟 0 ) |
114 |
14 113
|
eqbrtrid |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) → ( ( 𝑛 ∈ ℝ+ ↦ ( ( 𝑛 ↑𝑐 𝐴 ) / ( 𝐵 ↑𝑐 𝑛 ) ) ) ↾ ( 1 [,) +∞ ) ) ⇝𝑟 0 ) |
115 |
94
|
rpcnd |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) ∧ 𝑛 ∈ ℝ+ ) → ( ( 𝑛 ↑𝑐 𝐴 ) / ( 𝐵 ↑𝑐 𝑛 ) ) ∈ ℂ ) |
116 |
115
|
fmpttd |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) → ( 𝑛 ∈ ℝ+ ↦ ( ( 𝑛 ↑𝑐 𝐴 ) / ( 𝐵 ↑𝑐 𝑛 ) ) ) : ℝ+ ⟶ ℂ ) |
117 |
|
rpssre |
⊢ ℝ+ ⊆ ℝ |
118 |
117
|
a1i |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) → ℝ+ ⊆ ℝ ) |
119 |
116 118 32
|
rlimresb |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) → ( ( 𝑛 ∈ ℝ+ ↦ ( ( 𝑛 ↑𝑐 𝐴 ) / ( 𝐵 ↑𝑐 𝑛 ) ) ) ⇝𝑟 0 ↔ ( ( 𝑛 ∈ ℝ+ ↦ ( ( 𝑛 ↑𝑐 𝐴 ) / ( 𝐵 ↑𝑐 𝑛 ) ) ) ↾ ( 1 [,) +∞ ) ) ⇝𝑟 0 ) ) |
120 |
114 119
|
mpbird |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵 ) → ( 𝑛 ∈ ℝ+ ↦ ( ( 𝑛 ↑𝑐 𝐴 ) / ( 𝐵 ↑𝑐 𝑛 ) ) ) ⇝𝑟 0 ) |