Step |
Hyp |
Ref |
Expression |
1 |
|
logcn.d |
|- D = ( CC \ ( -oo (,] 0 ) ) |
2 |
|
logf1o |
|- log : ( CC \ { 0 } ) -1-1-onto-> ran log |
3 |
|
f1of |
|- ( log : ( CC \ { 0 } ) -1-1-onto-> ran log -> log : ( CC \ { 0 } ) --> ran log ) |
4 |
2 3
|
ax-mp |
|- log : ( CC \ { 0 } ) --> ran log |
5 |
1
|
logdmss |
|- D C_ ( CC \ { 0 } ) |
6 |
|
fssres |
|- ( ( log : ( CC \ { 0 } ) --> ran log /\ D C_ ( CC \ { 0 } ) ) -> ( log |` D ) : D --> ran log ) |
7 |
4 5 6
|
mp2an |
|- ( log |` D ) : D --> ran log |
8 |
|
ffn |
|- ( ( log |` D ) : D --> ran log -> ( log |` D ) Fn D ) |
9 |
7 8
|
ax-mp |
|- ( log |` D ) Fn D |
10 |
|
dffn5 |
|- ( ( log |` D ) Fn D <-> ( log |` D ) = ( x e. D |-> ( ( log |` D ) ` x ) ) ) |
11 |
9 10
|
mpbi |
|- ( log |` D ) = ( x e. D |-> ( ( log |` D ) ` x ) ) |
12 |
|
fvres |
|- ( x e. D -> ( ( log |` D ) ` x ) = ( log ` x ) ) |
13 |
1
|
ellogdm |
|- ( x e. D <-> ( x e. CC /\ ( x e. RR -> x e. RR+ ) ) ) |
14 |
13
|
simplbi |
|- ( x e. D -> x e. CC ) |
15 |
1
|
logdmn0 |
|- ( x e. D -> x =/= 0 ) |
16 |
14 15
|
logcld |
|- ( x e. D -> ( log ` x ) e. CC ) |
17 |
16
|
replimd |
|- ( x e. D -> ( log ` x ) = ( ( Re ` ( log ` x ) ) + ( _i x. ( Im ` ( log ` x ) ) ) ) ) |
18 |
|
relog |
|- ( ( x e. CC /\ x =/= 0 ) -> ( Re ` ( log ` x ) ) = ( log ` ( abs ` x ) ) ) |
19 |
14 15 18
|
syl2anc |
|- ( x e. D -> ( Re ` ( log ` x ) ) = ( log ` ( abs ` x ) ) ) |
20 |
14 15
|
absrpcld |
|- ( x e. D -> ( abs ` x ) e. RR+ ) |
21 |
20
|
fvresd |
|- ( x e. D -> ( ( log |` RR+ ) ` ( abs ` x ) ) = ( log ` ( abs ` x ) ) ) |
22 |
19 21
|
eqtr4d |
|- ( x e. D -> ( Re ` ( log ` x ) ) = ( ( log |` RR+ ) ` ( abs ` x ) ) ) |
23 |
22
|
oveq1d |
|- ( x e. D -> ( ( Re ` ( log ` x ) ) + ( _i x. ( Im ` ( log ` x ) ) ) ) = ( ( ( log |` RR+ ) ` ( abs ` x ) ) + ( _i x. ( Im ` ( log ` x ) ) ) ) ) |
24 |
12 17 23
|
3eqtrd |
|- ( x e. D -> ( ( log |` D ) ` x ) = ( ( ( log |` RR+ ) ` ( abs ` x ) ) + ( _i x. ( Im ` ( log ` x ) ) ) ) ) |
25 |
24
|
mpteq2ia |
|- ( x e. D |-> ( ( log |` D ) ` x ) ) = ( x e. D |-> ( ( ( log |` RR+ ) ` ( abs ` x ) ) + ( _i x. ( Im ` ( log ` x ) ) ) ) ) |
26 |
11 25
|
eqtri |
|- ( log |` D ) = ( x e. D |-> ( ( ( log |` RR+ ) ` ( abs ` x ) ) + ( _i x. ( Im ` ( log ` x ) ) ) ) ) |
27 |
|
eqid |
|- ( TopOpen ` CCfld ) = ( TopOpen ` CCfld ) |
28 |
27
|
addcn |
|- + e. ( ( ( TopOpen ` CCfld ) tX ( TopOpen ` CCfld ) ) Cn ( TopOpen ` CCfld ) ) |
29 |
28
|
a1i |
|- ( T. -> + e. ( ( ( TopOpen ` CCfld ) tX ( TopOpen ` CCfld ) ) Cn ( TopOpen ` CCfld ) ) ) |
30 |
27
|
cnfldtopon |
|- ( TopOpen ` CCfld ) e. ( TopOn ` CC ) |
31 |
14
|
ssriv |
|- D C_ CC |
32 |
|
resttopon |
|- ( ( ( TopOpen ` CCfld ) e. ( TopOn ` CC ) /\ D C_ CC ) -> ( ( TopOpen ` CCfld ) |`t D ) e. ( TopOn ` D ) ) |
33 |
30 31 32
|
mp2an |
|- ( ( TopOpen ` CCfld ) |`t D ) e. ( TopOn ` D ) |
34 |
33
|
a1i |
|- ( T. -> ( ( TopOpen ` CCfld ) |`t D ) e. ( TopOn ` D ) ) |
35 |
|
absf |
|- abs : CC --> RR |
36 |
|
fssres |
|- ( ( abs : CC --> RR /\ D C_ CC ) -> ( abs |` D ) : D --> RR ) |
37 |
35 31 36
|
mp2an |
|- ( abs |` D ) : D --> RR |
38 |
37
|
a1i |
|- ( T. -> ( abs |` D ) : D --> RR ) |
39 |
38
|
feqmptd |
|- ( T. -> ( abs |` D ) = ( x e. D |-> ( ( abs |` D ) ` x ) ) ) |
40 |
|
fvres |
|- ( x e. D -> ( ( abs |` D ) ` x ) = ( abs ` x ) ) |
41 |
40
|
mpteq2ia |
|- ( x e. D |-> ( ( abs |` D ) ` x ) ) = ( x e. D |-> ( abs ` x ) ) |
42 |
39 41
|
eqtrdi |
|- ( T. -> ( abs |` D ) = ( x e. D |-> ( abs ` x ) ) ) |
43 |
|
ffn |
|- ( ( abs |` D ) : D --> RR -> ( abs |` D ) Fn D ) |
44 |
37 43
|
ax-mp |
|- ( abs |` D ) Fn D |
45 |
40 20
|
eqeltrd |
|- ( x e. D -> ( ( abs |` D ) ` x ) e. RR+ ) |
46 |
45
|
rgen |
|- A. x e. D ( ( abs |` D ) ` x ) e. RR+ |
47 |
|
ffnfv |
|- ( ( abs |` D ) : D --> RR+ <-> ( ( abs |` D ) Fn D /\ A. x e. D ( ( abs |` D ) ` x ) e. RR+ ) ) |
48 |
44 46 47
|
mpbir2an |
|- ( abs |` D ) : D --> RR+ |
49 |
|
rpssre |
|- RR+ C_ RR |
50 |
|
ax-resscn |
|- RR C_ CC |
51 |
49 50
|
sstri |
|- RR+ C_ CC |
52 |
|
abscncf |
|- abs e. ( CC -cn-> RR ) |
53 |
|
rescncf |
|- ( D C_ CC -> ( abs e. ( CC -cn-> RR ) -> ( abs |` D ) e. ( D -cn-> RR ) ) ) |
54 |
31 52 53
|
mp2 |
|- ( abs |` D ) e. ( D -cn-> RR ) |
55 |
|
cncffvrn |
|- ( ( RR+ C_ CC /\ ( abs |` D ) e. ( D -cn-> RR ) ) -> ( ( abs |` D ) e. ( D -cn-> RR+ ) <-> ( abs |` D ) : D --> RR+ ) ) |
56 |
51 54 55
|
mp2an |
|- ( ( abs |` D ) e. ( D -cn-> RR+ ) <-> ( abs |` D ) : D --> RR+ ) |
57 |
48 56
|
mpbir |
|- ( abs |` D ) e. ( D -cn-> RR+ ) |
58 |
42 57
|
eqeltrrdi |
|- ( T. -> ( x e. D |-> ( abs ` x ) ) e. ( D -cn-> RR+ ) ) |
59 |
|
eqid |
|- ( ( TopOpen ` CCfld ) |`t D ) = ( ( TopOpen ` CCfld ) |`t D ) |
60 |
|
eqid |
|- ( ( TopOpen ` CCfld ) |`t RR+ ) = ( ( TopOpen ` CCfld ) |`t RR+ ) |
61 |
27 59 60
|
cncfcn |
|- ( ( D C_ CC /\ RR+ C_ CC ) -> ( D -cn-> RR+ ) = ( ( ( TopOpen ` CCfld ) |`t D ) Cn ( ( TopOpen ` CCfld ) |`t RR+ ) ) ) |
62 |
31 51 61
|
mp2an |
|- ( D -cn-> RR+ ) = ( ( ( TopOpen ` CCfld ) |`t D ) Cn ( ( TopOpen ` CCfld ) |`t RR+ ) ) |
63 |
58 62
|
eleqtrdi |
|- ( T. -> ( x e. D |-> ( abs ` x ) ) e. ( ( ( TopOpen ` CCfld ) |`t D ) Cn ( ( TopOpen ` CCfld ) |`t RR+ ) ) ) |
64 |
|
ssid |
|- CC C_ CC |
65 |
|
cncfss |
|- ( ( RR C_ CC /\ CC C_ CC ) -> ( RR+ -cn-> RR ) C_ ( RR+ -cn-> CC ) ) |
66 |
50 64 65
|
mp2an |
|- ( RR+ -cn-> RR ) C_ ( RR+ -cn-> CC ) |
67 |
|
relogcn |
|- ( log |` RR+ ) e. ( RR+ -cn-> RR ) |
68 |
66 67
|
sselii |
|- ( log |` RR+ ) e. ( RR+ -cn-> CC ) |
69 |
68
|
a1i |
|- ( T. -> ( log |` RR+ ) e. ( RR+ -cn-> CC ) ) |
70 |
30
|
toponrestid |
|- ( TopOpen ` CCfld ) = ( ( TopOpen ` CCfld ) |`t CC ) |
71 |
27 60 70
|
cncfcn |
|- ( ( RR+ C_ CC /\ CC C_ CC ) -> ( RR+ -cn-> CC ) = ( ( ( TopOpen ` CCfld ) |`t RR+ ) Cn ( TopOpen ` CCfld ) ) ) |
72 |
51 64 71
|
mp2an |
|- ( RR+ -cn-> CC ) = ( ( ( TopOpen ` CCfld ) |`t RR+ ) Cn ( TopOpen ` CCfld ) ) |
73 |
69 72
|
eleqtrdi |
|- ( T. -> ( log |` RR+ ) e. ( ( ( TopOpen ` CCfld ) |`t RR+ ) Cn ( TopOpen ` CCfld ) ) ) |
74 |
34 63 73
|
cnmpt11f |
|- ( T. -> ( x e. D |-> ( ( log |` RR+ ) ` ( abs ` x ) ) ) e. ( ( ( TopOpen ` CCfld ) |`t D ) Cn ( TopOpen ` CCfld ) ) ) |
75 |
27 59 70
|
cncfcn |
|- ( ( D C_ CC /\ CC C_ CC ) -> ( D -cn-> CC ) = ( ( ( TopOpen ` CCfld ) |`t D ) Cn ( TopOpen ` CCfld ) ) ) |
76 |
31 64 75
|
mp2an |
|- ( D -cn-> CC ) = ( ( ( TopOpen ` CCfld ) |`t D ) Cn ( TopOpen ` CCfld ) ) |
77 |
74 76
|
eleqtrrdi |
|- ( T. -> ( x e. D |-> ( ( log |` RR+ ) ` ( abs ` x ) ) ) e. ( D -cn-> CC ) ) |
78 |
16
|
imcld |
|- ( x e. D -> ( Im ` ( log ` x ) ) e. RR ) |
79 |
78
|
recnd |
|- ( x e. D -> ( Im ` ( log ` x ) ) e. CC ) |
80 |
79
|
adantl |
|- ( ( T. /\ x e. D ) -> ( Im ` ( log ` x ) ) e. CC ) |
81 |
|
eqidd |
|- ( T. -> ( x e. D |-> ( Im ` ( log ` x ) ) ) = ( x e. D |-> ( Im ` ( log ` x ) ) ) ) |
82 |
|
eqidd |
|- ( T. -> ( y e. CC |-> ( _i x. y ) ) = ( y e. CC |-> ( _i x. y ) ) ) |
83 |
|
oveq2 |
|- ( y = ( Im ` ( log ` x ) ) -> ( _i x. y ) = ( _i x. ( Im ` ( log ` x ) ) ) ) |
84 |
80 81 82 83
|
fmptco |
|- ( T. -> ( ( y e. CC |-> ( _i x. y ) ) o. ( x e. D |-> ( Im ` ( log ` x ) ) ) ) = ( x e. D |-> ( _i x. ( Im ` ( log ` x ) ) ) ) ) |
85 |
|
cncfss |
|- ( ( RR C_ CC /\ CC C_ CC ) -> ( D -cn-> RR ) C_ ( D -cn-> CC ) ) |
86 |
50 64 85
|
mp2an |
|- ( D -cn-> RR ) C_ ( D -cn-> CC ) |
87 |
1
|
logcnlem5 |
|- ( x e. D |-> ( Im ` ( log ` x ) ) ) e. ( D -cn-> RR ) |
88 |
86 87
|
sselii |
|- ( x e. D |-> ( Im ` ( log ` x ) ) ) e. ( D -cn-> CC ) |
89 |
88
|
a1i |
|- ( T. -> ( x e. D |-> ( Im ` ( log ` x ) ) ) e. ( D -cn-> CC ) ) |
90 |
|
ax-icn |
|- _i e. CC |
91 |
|
eqid |
|- ( y e. CC |-> ( _i x. y ) ) = ( y e. CC |-> ( _i x. y ) ) |
92 |
91
|
mulc1cncf |
|- ( _i e. CC -> ( y e. CC |-> ( _i x. y ) ) e. ( CC -cn-> CC ) ) |
93 |
90 92
|
mp1i |
|- ( T. -> ( y e. CC |-> ( _i x. y ) ) e. ( CC -cn-> CC ) ) |
94 |
89 93
|
cncfco |
|- ( T. -> ( ( y e. CC |-> ( _i x. y ) ) o. ( x e. D |-> ( Im ` ( log ` x ) ) ) ) e. ( D -cn-> CC ) ) |
95 |
84 94
|
eqeltrrd |
|- ( T. -> ( x e. D |-> ( _i x. ( Im ` ( log ` x ) ) ) ) e. ( D -cn-> CC ) ) |
96 |
27 29 77 95
|
cncfmpt2f |
|- ( T. -> ( x e. D |-> ( ( ( log |` RR+ ) ` ( abs ` x ) ) + ( _i x. ( Im ` ( log ` x ) ) ) ) ) e. ( D -cn-> CC ) ) |
97 |
96
|
mptru |
|- ( x e. D |-> ( ( ( log |` RR+ ) ` ( abs ` x ) ) + ( _i x. ( Im ` ( log ` x ) ) ) ) ) e. ( D -cn-> CC ) |
98 |
26 97
|
eqeltri |
|- ( log |` D ) e. ( D -cn-> CC ) |