Step |
Hyp |
Ref |
Expression |
1 |
|
dvcncxp1.d |
⊢ 𝐷 = ( ℂ ∖ ( -∞ (,] 0 ) ) |
2 |
|
cnelprrecn |
⊢ ℂ ∈ { ℝ , ℂ } |
3 |
2
|
a1i |
⊢ ( 𝐴 ∈ ℂ → ℂ ∈ { ℝ , ℂ } ) |
4 |
|
difss |
⊢ ( ℂ ∖ ( -∞ (,] 0 ) ) ⊆ ℂ |
5 |
1 4
|
eqsstri |
⊢ 𝐷 ⊆ ℂ |
6 |
5
|
sseli |
⊢ ( 𝑥 ∈ 𝐷 → 𝑥 ∈ ℂ ) |
7 |
1
|
logdmn0 |
⊢ ( 𝑥 ∈ 𝐷 → 𝑥 ≠ 0 ) |
8 |
6 7
|
logcld |
⊢ ( 𝑥 ∈ 𝐷 → ( log ‘ 𝑥 ) ∈ ℂ ) |
9 |
8
|
adantl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝐷 ) → ( log ‘ 𝑥 ) ∈ ℂ ) |
10 |
6 7
|
reccld |
⊢ ( 𝑥 ∈ 𝐷 → ( 1 / 𝑥 ) ∈ ℂ ) |
11 |
10
|
adantl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝐷 ) → ( 1 / 𝑥 ) ∈ ℂ ) |
12 |
|
mulcl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( 𝐴 · 𝑦 ) ∈ ℂ ) |
13 |
|
efcl |
⊢ ( ( 𝐴 · 𝑦 ) ∈ ℂ → ( exp ‘ ( 𝐴 · 𝑦 ) ) ∈ ℂ ) |
14 |
12 13
|
syl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( exp ‘ ( 𝐴 · 𝑦 ) ) ∈ ℂ ) |
15 |
|
ovexd |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( ( exp ‘ ( 𝐴 · 𝑦 ) ) · 𝐴 ) ∈ V ) |
16 |
1
|
logcn |
⊢ ( log ↾ 𝐷 ) ∈ ( 𝐷 –cn→ ℂ ) |
17 |
|
cncff |
⊢ ( ( log ↾ 𝐷 ) ∈ ( 𝐷 –cn→ ℂ ) → ( log ↾ 𝐷 ) : 𝐷 ⟶ ℂ ) |
18 |
16 17
|
mp1i |
⊢ ( 𝐴 ∈ ℂ → ( log ↾ 𝐷 ) : 𝐷 ⟶ ℂ ) |
19 |
18
|
feqmptd |
⊢ ( 𝐴 ∈ ℂ → ( log ↾ 𝐷 ) = ( 𝑥 ∈ 𝐷 ↦ ( ( log ↾ 𝐷 ) ‘ 𝑥 ) ) ) |
20 |
|
fvres |
⊢ ( 𝑥 ∈ 𝐷 → ( ( log ↾ 𝐷 ) ‘ 𝑥 ) = ( log ‘ 𝑥 ) ) |
21 |
20
|
mpteq2ia |
⊢ ( 𝑥 ∈ 𝐷 ↦ ( ( log ↾ 𝐷 ) ‘ 𝑥 ) ) = ( 𝑥 ∈ 𝐷 ↦ ( log ‘ 𝑥 ) ) |
22 |
19 21
|
eqtrdi |
⊢ ( 𝐴 ∈ ℂ → ( log ↾ 𝐷 ) = ( 𝑥 ∈ 𝐷 ↦ ( log ‘ 𝑥 ) ) ) |
23 |
22
|
oveq2d |
⊢ ( 𝐴 ∈ ℂ → ( ℂ D ( log ↾ 𝐷 ) ) = ( ℂ D ( 𝑥 ∈ 𝐷 ↦ ( log ‘ 𝑥 ) ) ) ) |
24 |
1
|
dvlog |
⊢ ( ℂ D ( log ↾ 𝐷 ) ) = ( 𝑥 ∈ 𝐷 ↦ ( 1 / 𝑥 ) ) |
25 |
23 24
|
eqtr3di |
⊢ ( 𝐴 ∈ ℂ → ( ℂ D ( 𝑥 ∈ 𝐷 ↦ ( log ‘ 𝑥 ) ) ) = ( 𝑥 ∈ 𝐷 ↦ ( 1 / 𝑥 ) ) ) |
26 |
|
simpl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → 𝐴 ∈ ℂ ) |
27 |
|
efcl |
⊢ ( 𝑥 ∈ ℂ → ( exp ‘ 𝑥 ) ∈ ℂ ) |
28 |
27
|
adantl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ ) → ( exp ‘ 𝑥 ) ∈ ℂ ) |
29 |
|
simpr |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → 𝑦 ∈ ℂ ) |
30 |
|
1cnd |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → 1 ∈ ℂ ) |
31 |
3
|
dvmptid |
⊢ ( 𝐴 ∈ ℂ → ( ℂ D ( 𝑦 ∈ ℂ ↦ 𝑦 ) ) = ( 𝑦 ∈ ℂ ↦ 1 ) ) |
32 |
|
id |
⊢ ( 𝐴 ∈ ℂ → 𝐴 ∈ ℂ ) |
33 |
3 29 30 31 32
|
dvmptcmul |
⊢ ( 𝐴 ∈ ℂ → ( ℂ D ( 𝑦 ∈ ℂ ↦ ( 𝐴 · 𝑦 ) ) ) = ( 𝑦 ∈ ℂ ↦ ( 𝐴 · 1 ) ) ) |
34 |
|
mulid1 |
⊢ ( 𝐴 ∈ ℂ → ( 𝐴 · 1 ) = 𝐴 ) |
35 |
34
|
mpteq2dv |
⊢ ( 𝐴 ∈ ℂ → ( 𝑦 ∈ ℂ ↦ ( 𝐴 · 1 ) ) = ( 𝑦 ∈ ℂ ↦ 𝐴 ) ) |
36 |
33 35
|
eqtrd |
⊢ ( 𝐴 ∈ ℂ → ( ℂ D ( 𝑦 ∈ ℂ ↦ ( 𝐴 · 𝑦 ) ) ) = ( 𝑦 ∈ ℂ ↦ 𝐴 ) ) |
37 |
|
dvef |
⊢ ( ℂ D exp ) = exp |
38 |
|
eff |
⊢ exp : ℂ ⟶ ℂ |
39 |
38
|
a1i |
⊢ ( 𝐴 ∈ ℂ → exp : ℂ ⟶ ℂ ) |
40 |
39
|
feqmptd |
⊢ ( 𝐴 ∈ ℂ → exp = ( 𝑥 ∈ ℂ ↦ ( exp ‘ 𝑥 ) ) ) |
41 |
40
|
oveq2d |
⊢ ( 𝐴 ∈ ℂ → ( ℂ D exp ) = ( ℂ D ( 𝑥 ∈ ℂ ↦ ( exp ‘ 𝑥 ) ) ) ) |
42 |
37 41 40
|
3eqtr3a |
⊢ ( 𝐴 ∈ ℂ → ( ℂ D ( 𝑥 ∈ ℂ ↦ ( exp ‘ 𝑥 ) ) ) = ( 𝑥 ∈ ℂ ↦ ( exp ‘ 𝑥 ) ) ) |
43 |
|
fveq2 |
⊢ ( 𝑥 = ( 𝐴 · 𝑦 ) → ( exp ‘ 𝑥 ) = ( exp ‘ ( 𝐴 · 𝑦 ) ) ) |
44 |
3 3 12 26 28 28 36 42 43 43
|
dvmptco |
⊢ ( 𝐴 ∈ ℂ → ( ℂ D ( 𝑦 ∈ ℂ ↦ ( exp ‘ ( 𝐴 · 𝑦 ) ) ) ) = ( 𝑦 ∈ ℂ ↦ ( ( exp ‘ ( 𝐴 · 𝑦 ) ) · 𝐴 ) ) ) |
45 |
|
oveq2 |
⊢ ( 𝑦 = ( log ‘ 𝑥 ) → ( 𝐴 · 𝑦 ) = ( 𝐴 · ( log ‘ 𝑥 ) ) ) |
46 |
45
|
fveq2d |
⊢ ( 𝑦 = ( log ‘ 𝑥 ) → ( exp ‘ ( 𝐴 · 𝑦 ) ) = ( exp ‘ ( 𝐴 · ( log ‘ 𝑥 ) ) ) ) |
47 |
46
|
oveq1d |
⊢ ( 𝑦 = ( log ‘ 𝑥 ) → ( ( exp ‘ ( 𝐴 · 𝑦 ) ) · 𝐴 ) = ( ( exp ‘ ( 𝐴 · ( log ‘ 𝑥 ) ) ) · 𝐴 ) ) |
48 |
3 3 9 11 14 15 25 44 46 47
|
dvmptco |
⊢ ( 𝐴 ∈ ℂ → ( ℂ D ( 𝑥 ∈ 𝐷 ↦ ( exp ‘ ( 𝐴 · ( log ‘ 𝑥 ) ) ) ) ) = ( 𝑥 ∈ 𝐷 ↦ ( ( ( exp ‘ ( 𝐴 · ( log ‘ 𝑥 ) ) ) · 𝐴 ) · ( 1 / 𝑥 ) ) ) ) |
49 |
6
|
adantl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝐷 ) → 𝑥 ∈ ℂ ) |
50 |
7
|
adantl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝐷 ) → 𝑥 ≠ 0 ) |
51 |
|
simpl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝐷 ) → 𝐴 ∈ ℂ ) |
52 |
49 50 51
|
cxpefd |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝐷 ) → ( 𝑥 ↑𝑐 𝐴 ) = ( exp ‘ ( 𝐴 · ( log ‘ 𝑥 ) ) ) ) |
53 |
52
|
mpteq2dva |
⊢ ( 𝐴 ∈ ℂ → ( 𝑥 ∈ 𝐷 ↦ ( 𝑥 ↑𝑐 𝐴 ) ) = ( 𝑥 ∈ 𝐷 ↦ ( exp ‘ ( 𝐴 · ( log ‘ 𝑥 ) ) ) ) ) |
54 |
53
|
oveq2d |
⊢ ( 𝐴 ∈ ℂ → ( ℂ D ( 𝑥 ∈ 𝐷 ↦ ( 𝑥 ↑𝑐 𝐴 ) ) ) = ( ℂ D ( 𝑥 ∈ 𝐷 ↦ ( exp ‘ ( 𝐴 · ( log ‘ 𝑥 ) ) ) ) ) ) |
55 |
|
1cnd |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝐷 ) → 1 ∈ ℂ ) |
56 |
49 50 51 55
|
cxpsubd |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝐷 ) → ( 𝑥 ↑𝑐 ( 𝐴 − 1 ) ) = ( ( 𝑥 ↑𝑐 𝐴 ) / ( 𝑥 ↑𝑐 1 ) ) ) |
57 |
49
|
cxp1d |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝐷 ) → ( 𝑥 ↑𝑐 1 ) = 𝑥 ) |
58 |
57
|
oveq2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝐷 ) → ( ( 𝑥 ↑𝑐 𝐴 ) / ( 𝑥 ↑𝑐 1 ) ) = ( ( 𝑥 ↑𝑐 𝐴 ) / 𝑥 ) ) |
59 |
49 51
|
cxpcld |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝐷 ) → ( 𝑥 ↑𝑐 𝐴 ) ∈ ℂ ) |
60 |
59 49 50
|
divrecd |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝐷 ) → ( ( 𝑥 ↑𝑐 𝐴 ) / 𝑥 ) = ( ( 𝑥 ↑𝑐 𝐴 ) · ( 1 / 𝑥 ) ) ) |
61 |
56 58 60
|
3eqtrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝐷 ) → ( 𝑥 ↑𝑐 ( 𝐴 − 1 ) ) = ( ( 𝑥 ↑𝑐 𝐴 ) · ( 1 / 𝑥 ) ) ) |
62 |
61
|
oveq2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝐷 ) → ( 𝐴 · ( 𝑥 ↑𝑐 ( 𝐴 − 1 ) ) ) = ( 𝐴 · ( ( 𝑥 ↑𝑐 𝐴 ) · ( 1 / 𝑥 ) ) ) ) |
63 |
51 59 11
|
mul12d |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝐷 ) → ( 𝐴 · ( ( 𝑥 ↑𝑐 𝐴 ) · ( 1 / 𝑥 ) ) ) = ( ( 𝑥 ↑𝑐 𝐴 ) · ( 𝐴 · ( 1 / 𝑥 ) ) ) ) |
64 |
59 51 11
|
mulassd |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝐷 ) → ( ( ( 𝑥 ↑𝑐 𝐴 ) · 𝐴 ) · ( 1 / 𝑥 ) ) = ( ( 𝑥 ↑𝑐 𝐴 ) · ( 𝐴 · ( 1 / 𝑥 ) ) ) ) |
65 |
63 64
|
eqtr4d |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝐷 ) → ( 𝐴 · ( ( 𝑥 ↑𝑐 𝐴 ) · ( 1 / 𝑥 ) ) ) = ( ( ( 𝑥 ↑𝑐 𝐴 ) · 𝐴 ) · ( 1 / 𝑥 ) ) ) |
66 |
52
|
oveq1d |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝐷 ) → ( ( 𝑥 ↑𝑐 𝐴 ) · 𝐴 ) = ( ( exp ‘ ( 𝐴 · ( log ‘ 𝑥 ) ) ) · 𝐴 ) ) |
67 |
66
|
oveq1d |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝐷 ) → ( ( ( 𝑥 ↑𝑐 𝐴 ) · 𝐴 ) · ( 1 / 𝑥 ) ) = ( ( ( exp ‘ ( 𝐴 · ( log ‘ 𝑥 ) ) ) · 𝐴 ) · ( 1 / 𝑥 ) ) ) |
68 |
62 65 67
|
3eqtrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝐷 ) → ( 𝐴 · ( 𝑥 ↑𝑐 ( 𝐴 − 1 ) ) ) = ( ( ( exp ‘ ( 𝐴 · ( log ‘ 𝑥 ) ) ) · 𝐴 ) · ( 1 / 𝑥 ) ) ) |
69 |
68
|
mpteq2dva |
⊢ ( 𝐴 ∈ ℂ → ( 𝑥 ∈ 𝐷 ↦ ( 𝐴 · ( 𝑥 ↑𝑐 ( 𝐴 − 1 ) ) ) ) = ( 𝑥 ∈ 𝐷 ↦ ( ( ( exp ‘ ( 𝐴 · ( log ‘ 𝑥 ) ) ) · 𝐴 ) · ( 1 / 𝑥 ) ) ) ) |
70 |
48 54 69
|
3eqtr4d |
⊢ ( 𝐴 ∈ ℂ → ( ℂ D ( 𝑥 ∈ 𝐷 ↦ ( 𝑥 ↑𝑐 𝐴 ) ) ) = ( 𝑥 ∈ 𝐷 ↦ ( 𝐴 · ( 𝑥 ↑𝑐 ( 𝐴 − 1 ) ) ) ) ) |