Step |
Hyp |
Ref |
Expression |
1 |
|
oveq1 |
|- ( N = 0 -> ( N x. x ) = ( 0 x. x ) ) |
2 |
1
|
oveq2d |
|- ( N = 0 -> ( ( _i x. ( 2 x. _pi ) ) x. ( N x. x ) ) = ( ( _i x. ( 2 x. _pi ) ) x. ( 0 x. x ) ) ) |
3 |
2
|
fveq2d |
|- ( N = 0 -> ( exp ` ( ( _i x. ( 2 x. _pi ) ) x. ( N x. x ) ) ) = ( exp ` ( ( _i x. ( 2 x. _pi ) ) x. ( 0 x. x ) ) ) ) |
4 |
|
ioossre |
|- ( 0 (,) 1 ) C_ RR |
5 |
|
ax-resscn |
|- RR C_ CC |
6 |
4 5
|
sstri |
|- ( 0 (,) 1 ) C_ CC |
7 |
6
|
sseli |
|- ( x e. ( 0 (,) 1 ) -> x e. CC ) |
8 |
7
|
mul02d |
|- ( x e. ( 0 (,) 1 ) -> ( 0 x. x ) = 0 ) |
9 |
8
|
oveq2d |
|- ( x e. ( 0 (,) 1 ) -> ( ( _i x. ( 2 x. _pi ) ) x. ( 0 x. x ) ) = ( ( _i x. ( 2 x. _pi ) ) x. 0 ) ) |
10 |
|
ax-icn |
|- _i e. CC |
11 |
|
2cn |
|- 2 e. CC |
12 |
|
picn |
|- _pi e. CC |
13 |
11 12
|
mulcli |
|- ( 2 x. _pi ) e. CC |
14 |
10 13
|
mulcli |
|- ( _i x. ( 2 x. _pi ) ) e. CC |
15 |
14
|
mul01i |
|- ( ( _i x. ( 2 x. _pi ) ) x. 0 ) = 0 |
16 |
9 15
|
eqtrdi |
|- ( x e. ( 0 (,) 1 ) -> ( ( _i x. ( 2 x. _pi ) ) x. ( 0 x. x ) ) = 0 ) |
17 |
16
|
fveq2d |
|- ( x e. ( 0 (,) 1 ) -> ( exp ` ( ( _i x. ( 2 x. _pi ) ) x. ( 0 x. x ) ) ) = ( exp ` 0 ) ) |
18 |
|
ef0 |
|- ( exp ` 0 ) = 1 |
19 |
17 18
|
eqtrdi |
|- ( x e. ( 0 (,) 1 ) -> ( exp ` ( ( _i x. ( 2 x. _pi ) ) x. ( 0 x. x ) ) ) = 1 ) |
20 |
3 19
|
sylan9eq |
|- ( ( N = 0 /\ x e. ( 0 (,) 1 ) ) -> ( exp ` ( ( _i x. ( 2 x. _pi ) ) x. ( N x. x ) ) ) = 1 ) |
21 |
20
|
ralrimiva |
|- ( N = 0 -> A. x e. ( 0 (,) 1 ) ( exp ` ( ( _i x. ( 2 x. _pi ) ) x. ( N x. x ) ) ) = 1 ) |
22 |
|
itgeq2 |
|- ( A. x e. ( 0 (,) 1 ) ( exp ` ( ( _i x. ( 2 x. _pi ) ) x. ( N x. x ) ) ) = 1 -> S. ( 0 (,) 1 ) ( exp ` ( ( _i x. ( 2 x. _pi ) ) x. ( N x. x ) ) ) _d x = S. ( 0 (,) 1 ) 1 _d x ) |
23 |
21 22
|
syl |
|- ( N = 0 -> S. ( 0 (,) 1 ) ( exp ` ( ( _i x. ( 2 x. _pi ) ) x. ( N x. x ) ) ) _d x = S. ( 0 (,) 1 ) 1 _d x ) |
24 |
|
ioombl |
|- ( 0 (,) 1 ) e. dom vol |
25 |
|
0re |
|- 0 e. RR |
26 |
|
1re |
|- 1 e. RR |
27 |
|
ioovolcl |
|- ( ( 0 e. RR /\ 1 e. RR ) -> ( vol ` ( 0 (,) 1 ) ) e. RR ) |
28 |
25 26 27
|
mp2an |
|- ( vol ` ( 0 (,) 1 ) ) e. RR |
29 |
|
ax-1cn |
|- 1 e. CC |
30 |
|
itgconst |
|- ( ( ( 0 (,) 1 ) e. dom vol /\ ( vol ` ( 0 (,) 1 ) ) e. RR /\ 1 e. CC ) -> S. ( 0 (,) 1 ) 1 _d x = ( 1 x. ( vol ` ( 0 (,) 1 ) ) ) ) |
31 |
24 28 29 30
|
mp3an |
|- S. ( 0 (,) 1 ) 1 _d x = ( 1 x. ( vol ` ( 0 (,) 1 ) ) ) |
32 |
|
0le1 |
|- 0 <_ 1 |
33 |
|
volioo |
|- ( ( 0 e. RR /\ 1 e. RR /\ 0 <_ 1 ) -> ( vol ` ( 0 (,) 1 ) ) = ( 1 - 0 ) ) |
34 |
25 26 32 33
|
mp3an |
|- ( vol ` ( 0 (,) 1 ) ) = ( 1 - 0 ) |
35 |
29
|
subid1i |
|- ( 1 - 0 ) = 1 |
36 |
34 35
|
eqtri |
|- ( vol ` ( 0 (,) 1 ) ) = 1 |
37 |
36
|
oveq2i |
|- ( 1 x. ( vol ` ( 0 (,) 1 ) ) ) = ( 1 x. 1 ) |
38 |
29
|
mulid1i |
|- ( 1 x. 1 ) = 1 |
39 |
31 37 38
|
3eqtri |
|- S. ( 0 (,) 1 ) 1 _d x = 1 |
40 |
23 39
|
eqtrdi |
|- ( N = 0 -> S. ( 0 (,) 1 ) ( exp ` ( ( _i x. ( 2 x. _pi ) ) x. ( N x. x ) ) ) _d x = 1 ) |
41 |
40
|
adantl |
|- ( ( N e. ZZ /\ N = 0 ) -> S. ( 0 (,) 1 ) ( exp ` ( ( _i x. ( 2 x. _pi ) ) x. ( N x. x ) ) ) _d x = 1 ) |
42 |
41
|
eqcomd |
|- ( ( N e. ZZ /\ N = 0 ) -> 1 = S. ( 0 (,) 1 ) ( exp ` ( ( _i x. ( 2 x. _pi ) ) x. ( N x. x ) ) ) _d x ) |
43 |
|
ioomax |
|- ( -oo (,) +oo ) = RR |
44 |
43
|
eqcomi |
|- RR = ( -oo (,) +oo ) |
45 |
|
0red |
|- ( ( N e. ZZ /\ -. N = 0 ) -> 0 e. RR ) |
46 |
|
1red |
|- ( ( N e. ZZ /\ -. N = 0 ) -> 1 e. RR ) |
47 |
32
|
a1i |
|- ( ( N e. ZZ /\ -. N = 0 ) -> 0 <_ 1 ) |
48 |
5
|
a1i |
|- ( ( N e. ZZ /\ -. N = 0 ) -> RR C_ CC ) |
49 |
48
|
sselda |
|- ( ( ( N e. ZZ /\ -. N = 0 ) /\ y e. RR ) -> y e. CC ) |
50 |
10
|
a1i |
|- ( ( N e. ZZ /\ -. N = 0 ) -> _i e. CC ) |
51 |
|
2cnd |
|- ( ( N e. ZZ /\ -. N = 0 ) -> 2 e. CC ) |
52 |
12
|
a1i |
|- ( ( N e. ZZ /\ -. N = 0 ) -> _pi e. CC ) |
53 |
51 52
|
mulcld |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( 2 x. _pi ) e. CC ) |
54 |
50 53
|
mulcld |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( _i x. ( 2 x. _pi ) ) e. CC ) |
55 |
|
simpl |
|- ( ( N e. ZZ /\ -. N = 0 ) -> N e. ZZ ) |
56 |
55
|
zcnd |
|- ( ( N e. ZZ /\ -. N = 0 ) -> N e. CC ) |
57 |
54 56
|
mulcld |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( ( _i x. ( 2 x. _pi ) ) x. N ) e. CC ) |
58 |
57
|
adantr |
|- ( ( ( N e. ZZ /\ -. N = 0 ) /\ y e. CC ) -> ( ( _i x. ( 2 x. _pi ) ) x. N ) e. CC ) |
59 |
|
simpr |
|- ( ( ( N e. ZZ /\ -. N = 0 ) /\ y e. CC ) -> y e. CC ) |
60 |
58 59
|
mulcld |
|- ( ( ( N e. ZZ /\ -. N = 0 ) /\ y e. CC ) -> ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) e. CC ) |
61 |
60
|
efcld |
|- ( ( ( N e. ZZ /\ -. N = 0 ) /\ y e. CC ) -> ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) e. CC ) |
62 |
49 61
|
syldan |
|- ( ( ( N e. ZZ /\ -. N = 0 ) /\ y e. RR ) -> ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) e. CC ) |
63 |
57
|
adantr |
|- ( ( ( N e. ZZ /\ -. N = 0 ) /\ y e. RR ) -> ( ( _i x. ( 2 x. _pi ) ) x. N ) e. CC ) |
64 |
|
ine0 |
|- _i =/= 0 |
65 |
|
2ne0 |
|- 2 =/= 0 |
66 |
|
pipos |
|- 0 < _pi |
67 |
25 66
|
gtneii |
|- _pi =/= 0 |
68 |
11 12 65 67
|
mulne0i |
|- ( 2 x. _pi ) =/= 0 |
69 |
10 13 64 68
|
mulne0i |
|- ( _i x. ( 2 x. _pi ) ) =/= 0 |
70 |
69
|
a1i |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( _i x. ( 2 x. _pi ) ) =/= 0 ) |
71 |
|
simpr |
|- ( ( N e. ZZ /\ -. N = 0 ) -> -. N = 0 ) |
72 |
71
|
neqned |
|- ( ( N e. ZZ /\ -. N = 0 ) -> N =/= 0 ) |
73 |
54 56 70 72
|
mulne0d |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( ( _i x. ( 2 x. _pi ) ) x. N ) =/= 0 ) |
74 |
73
|
adantr |
|- ( ( ( N e. ZZ /\ -. N = 0 ) /\ y e. RR ) -> ( ( _i x. ( 2 x. _pi ) ) x. N ) =/= 0 ) |
75 |
62 63 74
|
divcld |
|- ( ( ( N e. ZZ /\ -. N = 0 ) /\ y e. RR ) -> ( ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) e. CC ) |
76 |
75
|
fmpttd |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( y e. RR |-> ( ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) ) : RR --> CC ) |
77 |
|
reelprrecn |
|- RR e. { RR , CC } |
78 |
77
|
a1i |
|- ( ( N e. ZZ /\ -. N = 0 ) -> RR e. { RR , CC } ) |
79 |
|
cnelprrecn |
|- CC e. { RR , CC } |
80 |
79
|
a1i |
|- ( ( N e. ZZ /\ -. N = 0 ) -> CC e. { RR , CC } ) |
81 |
63 49
|
mulcld |
|- ( ( ( N e. ZZ /\ -. N = 0 ) /\ y e. RR ) -> ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) e. CC ) |
82 |
|
simpr |
|- ( ( ( N e. ZZ /\ -. N = 0 ) /\ z e. CC ) -> z e. CC ) |
83 |
82
|
efcld |
|- ( ( ( N e. ZZ /\ -. N = 0 ) /\ z e. CC ) -> ( exp ` z ) e. CC ) |
84 |
57
|
adantr |
|- ( ( ( N e. ZZ /\ -. N = 0 ) /\ z e. CC ) -> ( ( _i x. ( 2 x. _pi ) ) x. N ) e. CC ) |
85 |
73
|
adantr |
|- ( ( ( N e. ZZ /\ -. N = 0 ) /\ z e. CC ) -> ( ( _i x. ( 2 x. _pi ) ) x. N ) =/= 0 ) |
86 |
83 84 85
|
divcld |
|- ( ( ( N e. ZZ /\ -. N = 0 ) /\ z e. CC ) -> ( ( exp ` z ) / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) e. CC ) |
87 |
26
|
a1i |
|- ( ( ( N e. ZZ /\ -. N = 0 ) /\ y e. RR ) -> 1 e. RR ) |
88 |
78
|
dvmptid |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( RR _D ( y e. RR |-> y ) ) = ( y e. RR |-> 1 ) ) |
89 |
78 49 87 88 57
|
dvmptcmul |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( RR _D ( y e. RR |-> ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) ) = ( y e. RR |-> ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. 1 ) ) ) |
90 |
63
|
mulid1d |
|- ( ( ( N e. ZZ /\ -. N = 0 ) /\ y e. RR ) -> ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. 1 ) = ( ( _i x. ( 2 x. _pi ) ) x. N ) ) |
91 |
90
|
mpteq2dva |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( y e. RR |-> ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. 1 ) ) = ( y e. RR |-> ( ( _i x. ( 2 x. _pi ) ) x. N ) ) ) |
92 |
89 91
|
eqtrd |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( RR _D ( y e. RR |-> ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) ) = ( y e. RR |-> ( ( _i x. ( 2 x. _pi ) ) x. N ) ) ) |
93 |
|
dvef |
|- ( CC _D exp ) = exp |
94 |
|
eff |
|- exp : CC --> CC |
95 |
94
|
a1i |
|- ( ( N e. ZZ /\ -. N = 0 ) -> exp : CC --> CC ) |
96 |
95
|
feqmptd |
|- ( ( N e. ZZ /\ -. N = 0 ) -> exp = ( z e. CC |-> ( exp ` z ) ) ) |
97 |
96
|
oveq2d |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( CC _D exp ) = ( CC _D ( z e. CC |-> ( exp ` z ) ) ) ) |
98 |
93 97 96
|
3eqtr3a |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( CC _D ( z e. CC |-> ( exp ` z ) ) ) = ( z e. CC |-> ( exp ` z ) ) ) |
99 |
80 83 83 98 57 73
|
dvmptdivc |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( CC _D ( z e. CC |-> ( ( exp ` z ) / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) ) ) = ( z e. CC |-> ( ( exp ` z ) / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) ) ) |
100 |
|
fveq2 |
|- ( z = ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) -> ( exp ` z ) = ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) ) |
101 |
100
|
oveq1d |
|- ( z = ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) -> ( ( exp ` z ) / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) = ( ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) ) |
102 |
78 80 81 63 86 86 92 99 101 101
|
dvmptco |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( RR _D ( y e. RR |-> ( ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) ) ) = ( y e. RR |-> ( ( ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) x. ( ( _i x. ( 2 x. _pi ) ) x. N ) ) ) ) |
103 |
62 63 74
|
divcan1d |
|- ( ( ( N e. ZZ /\ -. N = 0 ) /\ y e. RR ) -> ( ( ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) x. ( ( _i x. ( 2 x. _pi ) ) x. N ) ) = ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) ) |
104 |
103
|
mpteq2dva |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( y e. RR |-> ( ( ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) x. ( ( _i x. ( 2 x. _pi ) ) x. N ) ) ) = ( y e. RR |-> ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) ) ) |
105 |
102 104
|
eqtrd |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( RR _D ( y e. RR |-> ( ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) ) ) = ( y e. RR |-> ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) ) ) |
106 |
|
efcn |
|- exp e. ( CC -cn-> CC ) |
107 |
106
|
a1i |
|- ( ( N e. ZZ /\ -. N = 0 ) -> exp e. ( CC -cn-> CC ) ) |
108 |
|
resmpt |
|- ( RR C_ CC -> ( ( y e. CC |-> ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) |` RR ) = ( y e. RR |-> ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) ) |
109 |
5 108
|
mp1i |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( ( y e. CC |-> ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) |` RR ) = ( y e. RR |-> ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) ) |
110 |
|
eqid |
|- ( y e. CC |-> ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) = ( y e. CC |-> ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) |
111 |
110
|
mulc1cncf |
|- ( ( ( _i x. ( 2 x. _pi ) ) x. N ) e. CC -> ( y e. CC |-> ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) e. ( CC -cn-> CC ) ) |
112 |
57 111
|
syl |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( y e. CC |-> ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) e. ( CC -cn-> CC ) ) |
113 |
|
rescncf |
|- ( RR C_ CC -> ( ( y e. CC |-> ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) e. ( CC -cn-> CC ) -> ( ( y e. CC |-> ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) |` RR ) e. ( RR -cn-> CC ) ) ) |
114 |
5 113
|
mp1i |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( ( y e. CC |-> ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) e. ( CC -cn-> CC ) -> ( ( y e. CC |-> ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) |` RR ) e. ( RR -cn-> CC ) ) ) |
115 |
112 114
|
mpd |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( ( y e. CC |-> ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) |` RR ) e. ( RR -cn-> CC ) ) |
116 |
109 115
|
eqeltrrd |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( y e. RR |-> ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) e. ( RR -cn-> CC ) ) |
117 |
107 116
|
cncfmpt1f |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( y e. RR |-> ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) ) e. ( RR -cn-> CC ) ) |
118 |
105 117
|
eqeltrd |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( RR _D ( y e. RR |-> ( ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) ) ) e. ( RR -cn-> CC ) ) |
119 |
44 45 46 47 76 118
|
ftc2re |
|- ( ( N e. ZZ /\ -. N = 0 ) -> S. ( 0 (,) 1 ) ( ( RR _D ( y e. RR |-> ( ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) ) ) ` x ) _d x = ( ( ( y e. RR |-> ( ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) ) ` 1 ) - ( ( y e. RR |-> ( ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) ) ` 0 ) ) ) |
120 |
4
|
sseli |
|- ( x e. ( 0 (,) 1 ) -> x e. RR ) |
121 |
105
|
adantr |
|- ( ( ( N e. ZZ /\ -. N = 0 ) /\ x e. RR ) -> ( RR _D ( y e. RR |-> ( ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) ) ) = ( y e. RR |-> ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) ) ) |
122 |
121
|
fveq1d |
|- ( ( ( N e. ZZ /\ -. N = 0 ) /\ x e. RR ) -> ( ( RR _D ( y e. RR |-> ( ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) ) ) ` x ) = ( ( y e. RR |-> ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) ) ` x ) ) |
123 |
|
oveq2 |
|- ( y = x -> ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) = ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. x ) ) |
124 |
123
|
fveq2d |
|- ( y = x -> ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) = ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. x ) ) ) |
125 |
124
|
cbvmptv |
|- ( y e. RR |-> ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) ) = ( x e. RR |-> ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. x ) ) ) |
126 |
125
|
a1i |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( y e. RR |-> ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) ) = ( x e. RR |-> ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. x ) ) ) ) |
127 |
57
|
adantr |
|- ( ( ( N e. ZZ /\ -. N = 0 ) /\ x e. RR ) -> ( ( _i x. ( 2 x. _pi ) ) x. N ) e. CC ) |
128 |
48
|
sselda |
|- ( ( ( N e. ZZ /\ -. N = 0 ) /\ x e. RR ) -> x e. CC ) |
129 |
127 128
|
mulcld |
|- ( ( ( N e. ZZ /\ -. N = 0 ) /\ x e. RR ) -> ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. x ) e. CC ) |
130 |
129
|
efcld |
|- ( ( ( N e. ZZ /\ -. N = 0 ) /\ x e. RR ) -> ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. x ) ) e. CC ) |
131 |
126 130
|
fvmpt2d |
|- ( ( ( N e. ZZ /\ -. N = 0 ) /\ x e. RR ) -> ( ( y e. RR |-> ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) ) ` x ) = ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. x ) ) ) |
132 |
14
|
a1i |
|- ( ( ( N e. ZZ /\ -. N = 0 ) /\ x e. RR ) -> ( _i x. ( 2 x. _pi ) ) e. CC ) |
133 |
56
|
adantr |
|- ( ( ( N e. ZZ /\ -. N = 0 ) /\ x e. RR ) -> N e. CC ) |
134 |
132 133 128
|
mulassd |
|- ( ( ( N e. ZZ /\ -. N = 0 ) /\ x e. RR ) -> ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. x ) = ( ( _i x. ( 2 x. _pi ) ) x. ( N x. x ) ) ) |
135 |
134
|
fveq2d |
|- ( ( ( N e. ZZ /\ -. N = 0 ) /\ x e. RR ) -> ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. x ) ) = ( exp ` ( ( _i x. ( 2 x. _pi ) ) x. ( N x. x ) ) ) ) |
136 |
131 135
|
eqtrd |
|- ( ( ( N e. ZZ /\ -. N = 0 ) /\ x e. RR ) -> ( ( y e. RR |-> ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) ) ` x ) = ( exp ` ( ( _i x. ( 2 x. _pi ) ) x. ( N x. x ) ) ) ) |
137 |
122 136
|
eqtrd |
|- ( ( ( N e. ZZ /\ -. N = 0 ) /\ x e. RR ) -> ( ( RR _D ( y e. RR |-> ( ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) ) ) ` x ) = ( exp ` ( ( _i x. ( 2 x. _pi ) ) x. ( N x. x ) ) ) ) |
138 |
120 137
|
sylan2 |
|- ( ( ( N e. ZZ /\ -. N = 0 ) /\ x e. ( 0 (,) 1 ) ) -> ( ( RR _D ( y e. RR |-> ( ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) ) ) ` x ) = ( exp ` ( ( _i x. ( 2 x. _pi ) ) x. ( N x. x ) ) ) ) |
139 |
138
|
ralrimiva |
|- ( ( N e. ZZ /\ -. N = 0 ) -> A. x e. ( 0 (,) 1 ) ( ( RR _D ( y e. RR |-> ( ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) ) ) ` x ) = ( exp ` ( ( _i x. ( 2 x. _pi ) ) x. ( N x. x ) ) ) ) |
140 |
|
itgeq2 |
|- ( A. x e. ( 0 (,) 1 ) ( ( RR _D ( y e. RR |-> ( ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) ) ) ` x ) = ( exp ` ( ( _i x. ( 2 x. _pi ) ) x. ( N x. x ) ) ) -> S. ( 0 (,) 1 ) ( ( RR _D ( y e. RR |-> ( ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) ) ) ` x ) _d x = S. ( 0 (,) 1 ) ( exp ` ( ( _i x. ( 2 x. _pi ) ) x. ( N x. x ) ) ) _d x ) |
141 |
139 140
|
syl |
|- ( ( N e. ZZ /\ -. N = 0 ) -> S. ( 0 (,) 1 ) ( ( RR _D ( y e. RR |-> ( ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) ) ) ` x ) _d x = S. ( 0 (,) 1 ) ( exp ` ( ( _i x. ( 2 x. _pi ) ) x. ( N x. x ) ) ) _d x ) |
142 |
|
eqidd |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( y e. RR |-> ( ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) ) = ( y e. RR |-> ( ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) ) ) |
143 |
|
simpr |
|- ( ( ( N e. ZZ /\ -. N = 0 ) /\ y = 1 ) -> y = 1 ) |
144 |
143
|
oveq2d |
|- ( ( ( N e. ZZ /\ -. N = 0 ) /\ y = 1 ) -> ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) = ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. 1 ) ) |
145 |
144
|
fveq2d |
|- ( ( ( N e. ZZ /\ -. N = 0 ) /\ y = 1 ) -> ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) = ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. 1 ) ) ) |
146 |
145
|
oveq1d |
|- ( ( ( N e. ZZ /\ -. N = 0 ) /\ y = 1 ) -> ( ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) = ( ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. 1 ) ) / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) ) |
147 |
29
|
a1i |
|- ( ( N e. ZZ /\ -. N = 0 ) -> 1 e. CC ) |
148 |
57 147
|
mulcld |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. 1 ) e. CC ) |
149 |
148
|
efcld |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. 1 ) ) e. CC ) |
150 |
149 57 73
|
divcld |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. 1 ) ) / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) e. CC ) |
151 |
142 146 46 150
|
fvmptd |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( ( y e. RR |-> ( ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) ) ` 1 ) = ( ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. 1 ) ) / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) ) |
152 |
57
|
mulid1d |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. 1 ) = ( ( _i x. ( 2 x. _pi ) ) x. N ) ) |
153 |
152
|
fveq2d |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. 1 ) ) = ( exp ` ( ( _i x. ( 2 x. _pi ) ) x. N ) ) ) |
154 |
|
ef2kpi |
|- ( N e. ZZ -> ( exp ` ( ( _i x. ( 2 x. _pi ) ) x. N ) ) = 1 ) |
155 |
55 154
|
syl |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( exp ` ( ( _i x. ( 2 x. _pi ) ) x. N ) ) = 1 ) |
156 |
153 155
|
eqtrd |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. 1 ) ) = 1 ) |
157 |
156
|
oveq1d |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. 1 ) ) / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) = ( 1 / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) ) |
158 |
151 157
|
eqtrd |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( ( y e. RR |-> ( ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) ) ` 1 ) = ( 1 / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) ) |
159 |
|
simpr |
|- ( ( ( N e. ZZ /\ -. N = 0 ) /\ y = 0 ) -> y = 0 ) |
160 |
159
|
oveq2d |
|- ( ( ( N e. ZZ /\ -. N = 0 ) /\ y = 0 ) -> ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) = ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. 0 ) ) |
161 |
160
|
fveq2d |
|- ( ( ( N e. ZZ /\ -. N = 0 ) /\ y = 0 ) -> ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) = ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. 0 ) ) ) |
162 |
161
|
oveq1d |
|- ( ( ( N e. ZZ /\ -. N = 0 ) /\ y = 0 ) -> ( ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) = ( ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. 0 ) ) / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) ) |
163 |
5 45
|
sselid |
|- ( ( N e. ZZ /\ -. N = 0 ) -> 0 e. CC ) |
164 |
57 163
|
mulcld |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. 0 ) e. CC ) |
165 |
164
|
efcld |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. 0 ) ) e. CC ) |
166 |
165 57 73
|
divcld |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. 0 ) ) / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) e. CC ) |
167 |
142 162 45 166
|
fvmptd |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( ( y e. RR |-> ( ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) ) ` 0 ) = ( ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. 0 ) ) / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) ) |
168 |
57
|
mul01d |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. 0 ) = 0 ) |
169 |
168
|
fveq2d |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. 0 ) ) = ( exp ` 0 ) ) |
170 |
169 18
|
eqtrdi |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. 0 ) ) = 1 ) |
171 |
170
|
oveq1d |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. 0 ) ) / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) = ( 1 / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) ) |
172 |
167 171
|
eqtrd |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( ( y e. RR |-> ( ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) ) ` 0 ) = ( 1 / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) ) |
173 |
158 172
|
oveq12d |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( ( ( y e. RR |-> ( ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) ) ` 1 ) - ( ( y e. RR |-> ( ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) ) ` 0 ) ) = ( ( 1 / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) - ( 1 / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) ) ) |
174 |
157 150
|
eqeltrrd |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( 1 / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) e. CC ) |
175 |
174
|
subidd |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( ( 1 / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) - ( 1 / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) ) = 0 ) |
176 |
173 175
|
eqtrd |
|- ( ( N e. ZZ /\ -. N = 0 ) -> ( ( ( y e. RR |-> ( ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) ) ` 1 ) - ( ( y e. RR |-> ( ( exp ` ( ( ( _i x. ( 2 x. _pi ) ) x. N ) x. y ) ) / ( ( _i x. ( 2 x. _pi ) ) x. N ) ) ) ` 0 ) ) = 0 ) |
177 |
119 141 176
|
3eqtr3d |
|- ( ( N e. ZZ /\ -. N = 0 ) -> S. ( 0 (,) 1 ) ( exp ` ( ( _i x. ( 2 x. _pi ) ) x. ( N x. x ) ) ) _d x = 0 ) |
178 |
177
|
eqcomd |
|- ( ( N e. ZZ /\ -. N = 0 ) -> 0 = S. ( 0 (,) 1 ) ( exp ` ( ( _i x. ( 2 x. _pi ) ) x. ( N x. x ) ) ) _d x ) |
179 |
42 178
|
ifeqda |
|- ( N e. ZZ -> if ( N = 0 , 1 , 0 ) = S. ( 0 (,) 1 ) ( exp ` ( ( _i x. ( 2 x. _pi ) ) x. ( N x. x ) ) ) _d x ) |
180 |
179
|
eqcomd |
|- ( N e. ZZ -> S. ( 0 (,) 1 ) ( exp ` ( ( _i x. ( 2 x. _pi ) ) x. ( N x. x ) ) ) _d x = if ( N = 0 , 1 , 0 ) ) |