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