Step |
Hyp |
Ref |
Expression |
1 |
|
fourierdlem58.k |
|- K = ( s e. A |-> ( s / ( 2 x. ( sin ` ( s / 2 ) ) ) ) ) |
2 |
|
fourierdlem58.ass |
|- ( ph -> A C_ ( -u _pi [,] _pi ) ) |
3 |
|
fourierdlem58.0nA |
|- ( ph -> -. 0 e. A ) |
4 |
|
fourierdlem58.4 |
|- ( ph -> A e. ( topGen ` ran (,) ) ) |
5 |
|
pire |
|- _pi e. RR |
6 |
5
|
a1i |
|- ( ( ph /\ s e. A ) -> _pi e. RR ) |
7 |
6
|
renegcld |
|- ( ( ph /\ s e. A ) -> -u _pi e. RR ) |
8 |
7 6
|
iccssred |
|- ( ( ph /\ s e. A ) -> ( -u _pi [,] _pi ) C_ RR ) |
9 |
2
|
sselda |
|- ( ( ph /\ s e. A ) -> s e. ( -u _pi [,] _pi ) ) |
10 |
8 9
|
sseldd |
|- ( ( ph /\ s e. A ) -> s e. RR ) |
11 |
|
2re |
|- 2 e. RR |
12 |
11
|
a1i |
|- ( ( ph /\ s e. A ) -> 2 e. RR ) |
13 |
10
|
rehalfcld |
|- ( ( ph /\ s e. A ) -> ( s / 2 ) e. RR ) |
14 |
13
|
resincld |
|- ( ( ph /\ s e. A ) -> ( sin ` ( s / 2 ) ) e. RR ) |
15 |
12 14
|
remulcld |
|- ( ( ph /\ s e. A ) -> ( 2 x. ( sin ` ( s / 2 ) ) ) e. RR ) |
16 |
|
2cnd |
|- ( ( ph /\ s e. A ) -> 2 e. CC ) |
17 |
10
|
recnd |
|- ( ( ph /\ s e. A ) -> s e. CC ) |
18 |
17
|
halfcld |
|- ( ( ph /\ s e. A ) -> ( s / 2 ) e. CC ) |
19 |
18
|
sincld |
|- ( ( ph /\ s e. A ) -> ( sin ` ( s / 2 ) ) e. CC ) |
20 |
|
2ne0 |
|- 2 =/= 0 |
21 |
20
|
a1i |
|- ( ( ph /\ s e. A ) -> 2 =/= 0 ) |
22 |
|
eqcom |
|- ( s = 0 <-> 0 = s ) |
23 |
22
|
biimpi |
|- ( s = 0 -> 0 = s ) |
24 |
23
|
adantl |
|- ( ( s e. A /\ s = 0 ) -> 0 = s ) |
25 |
|
simpl |
|- ( ( s e. A /\ s = 0 ) -> s e. A ) |
26 |
24 25
|
eqeltrd |
|- ( ( s e. A /\ s = 0 ) -> 0 e. A ) |
27 |
26
|
adantll |
|- ( ( ( ph /\ s e. A ) /\ s = 0 ) -> 0 e. A ) |
28 |
3
|
ad2antrr |
|- ( ( ( ph /\ s e. A ) /\ s = 0 ) -> -. 0 e. A ) |
29 |
27 28
|
pm2.65da |
|- ( ( ph /\ s e. A ) -> -. s = 0 ) |
30 |
29
|
neqned |
|- ( ( ph /\ s e. A ) -> s =/= 0 ) |
31 |
|
fourierdlem44 |
|- ( ( s e. ( -u _pi [,] _pi ) /\ s =/= 0 ) -> ( sin ` ( s / 2 ) ) =/= 0 ) |
32 |
9 30 31
|
syl2anc |
|- ( ( ph /\ s e. A ) -> ( sin ` ( s / 2 ) ) =/= 0 ) |
33 |
16 19 21 32
|
mulne0d |
|- ( ( ph /\ s e. A ) -> ( 2 x. ( sin ` ( s / 2 ) ) ) =/= 0 ) |
34 |
10 15 33
|
redivcld |
|- ( ( ph /\ s e. A ) -> ( s / ( 2 x. ( sin ` ( s / 2 ) ) ) ) e. RR ) |
35 |
34 1
|
fmptd |
|- ( ph -> K : A --> RR ) |
36 |
5
|
a1i |
|- ( ph -> _pi e. RR ) |
37 |
36
|
renegcld |
|- ( ph -> -u _pi e. RR ) |
38 |
37 36
|
iccssred |
|- ( ph -> ( -u _pi [,] _pi ) C_ RR ) |
39 |
2 38
|
sstrd |
|- ( ph -> A C_ RR ) |
40 |
|
dvfre |
|- ( ( K : A --> RR /\ A C_ RR ) -> ( RR _D K ) : dom ( RR _D K ) --> RR ) |
41 |
35 39 40
|
syl2anc |
|- ( ph -> ( RR _D K ) : dom ( RR _D K ) --> RR ) |
42 |
|
eqidd |
|- ( ph -> ( s e. A |-> s ) = ( s e. A |-> s ) ) |
43 |
|
eqidd |
|- ( ph -> ( s e. A |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) = ( s e. A |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) ) |
44 |
4 10 15 42 43
|
offval2 |
|- ( ph -> ( ( s e. A |-> s ) oF / ( s e. A |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) ) = ( s e. A |-> ( s / ( 2 x. ( sin ` ( s / 2 ) ) ) ) ) ) |
45 |
1 44
|
eqtr4id |
|- ( ph -> K = ( ( s e. A |-> s ) oF / ( s e. A |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) ) ) |
46 |
45
|
oveq2d |
|- ( ph -> ( RR _D K ) = ( RR _D ( ( s e. A |-> s ) oF / ( s e. A |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) ) ) ) |
47 |
|
reelprrecn |
|- RR e. { RR , CC } |
48 |
47
|
a1i |
|- ( ph -> RR e. { RR , CC } ) |
49 |
|
eqid |
|- ( s e. A |-> s ) = ( s e. A |-> s ) |
50 |
17 49
|
fmptd |
|- ( ph -> ( s e. A |-> s ) : A --> CC ) |
51 |
16 19
|
mulcld |
|- ( ( ph /\ s e. A ) -> ( 2 x. ( sin ` ( s / 2 ) ) ) e. CC ) |
52 |
33
|
neneqd |
|- ( ( ph /\ s e. A ) -> -. ( 2 x. ( sin ` ( s / 2 ) ) ) = 0 ) |
53 |
|
elsng |
|- ( ( 2 x. ( sin ` ( s / 2 ) ) ) e. RR -> ( ( 2 x. ( sin ` ( s / 2 ) ) ) e. { 0 } <-> ( 2 x. ( sin ` ( s / 2 ) ) ) = 0 ) ) |
54 |
15 53
|
syl |
|- ( ( ph /\ s e. A ) -> ( ( 2 x. ( sin ` ( s / 2 ) ) ) e. { 0 } <-> ( 2 x. ( sin ` ( s / 2 ) ) ) = 0 ) ) |
55 |
52 54
|
mtbird |
|- ( ( ph /\ s e. A ) -> -. ( 2 x. ( sin ` ( s / 2 ) ) ) e. { 0 } ) |
56 |
51 55
|
eldifd |
|- ( ( ph /\ s e. A ) -> ( 2 x. ( sin ` ( s / 2 ) ) ) e. ( CC \ { 0 } ) ) |
57 |
|
eqid |
|- ( s e. A |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) = ( s e. A |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) |
58 |
56 57
|
fmptd |
|- ( ph -> ( s e. A |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) : A --> ( CC \ { 0 } ) ) |
59 |
|
eqid |
|- ( TopOpen ` CCfld ) = ( TopOpen ` CCfld ) |
60 |
59
|
tgioo2 |
|- ( topGen ` ran (,) ) = ( ( TopOpen ` CCfld ) |`t RR ) |
61 |
4 60
|
eleqtrdi |
|- ( ph -> A e. ( ( TopOpen ` CCfld ) |`t RR ) ) |
62 |
48 61
|
dvmptidg |
|- ( ph -> ( RR _D ( s e. A |-> s ) ) = ( s e. A |-> 1 ) ) |
63 |
|
ax-resscn |
|- RR C_ CC |
64 |
63
|
a1i |
|- ( ph -> RR C_ CC ) |
65 |
39 64
|
sstrd |
|- ( ph -> A C_ CC ) |
66 |
|
1cnd |
|- ( ph -> 1 e. CC ) |
67 |
|
ssid |
|- CC C_ CC |
68 |
67
|
a1i |
|- ( ph -> CC C_ CC ) |
69 |
65 66 68
|
constcncfg |
|- ( ph -> ( s e. A |-> 1 ) e. ( A -cn-> CC ) ) |
70 |
62 69
|
eqeltrd |
|- ( ph -> ( RR _D ( s e. A |-> s ) ) e. ( A -cn-> CC ) ) |
71 |
39
|
resmptd |
|- ( ph -> ( ( s e. RR |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) |` A ) = ( s e. A |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) ) |
72 |
71
|
eqcomd |
|- ( ph -> ( s e. A |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) = ( ( s e. RR |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) |` A ) ) |
73 |
72
|
oveq2d |
|- ( ph -> ( RR _D ( s e. A |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) ) = ( RR _D ( ( s e. RR |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) |` A ) ) ) |
74 |
|
eqid |
|- ( s e. RR |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) = ( s e. RR |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) |
75 |
|
2cnd |
|- ( s e. RR -> 2 e. CC ) |
76 |
|
recn |
|- ( s e. RR -> s e. CC ) |
77 |
76
|
halfcld |
|- ( s e. RR -> ( s / 2 ) e. CC ) |
78 |
77
|
sincld |
|- ( s e. RR -> ( sin ` ( s / 2 ) ) e. CC ) |
79 |
75 78
|
mulcld |
|- ( s e. RR -> ( 2 x. ( sin ` ( s / 2 ) ) ) e. CC ) |
80 |
74 79
|
fmpti |
|- ( s e. RR |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) : RR --> CC |
81 |
80
|
a1i |
|- ( ph -> ( s e. RR |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) : RR --> CC ) |
82 |
|
ssid |
|- RR C_ RR |
83 |
82
|
a1i |
|- ( ph -> RR C_ RR ) |
84 |
59 60
|
dvres |
|- ( ( ( RR C_ CC /\ ( s e. RR |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) : RR --> CC ) /\ ( RR C_ RR /\ A C_ RR ) ) -> ( RR _D ( ( s e. RR |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) |` A ) ) = ( ( RR _D ( s e. RR |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) ) |` ( ( int ` ( topGen ` ran (,) ) ) ` A ) ) ) |
85 |
64 81 83 39 84
|
syl22anc |
|- ( ph -> ( RR _D ( ( s e. RR |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) |` A ) ) = ( ( RR _D ( s e. RR |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) ) |` ( ( int ` ( topGen ` ran (,) ) ) ` A ) ) ) |
86 |
|
retop |
|- ( topGen ` ran (,) ) e. Top |
87 |
86
|
a1i |
|- ( ph -> ( topGen ` ran (,) ) e. Top ) |
88 |
|
uniretop |
|- RR = U. ( topGen ` ran (,) ) |
89 |
88
|
isopn3 |
|- ( ( ( topGen ` ran (,) ) e. Top /\ A C_ RR ) -> ( A e. ( topGen ` ran (,) ) <-> ( ( int ` ( topGen ` ran (,) ) ) ` A ) = A ) ) |
90 |
87 39 89
|
syl2anc |
|- ( ph -> ( A e. ( topGen ` ran (,) ) <-> ( ( int ` ( topGen ` ran (,) ) ) ` A ) = A ) ) |
91 |
4 90
|
mpbid |
|- ( ph -> ( ( int ` ( topGen ` ran (,) ) ) ` A ) = A ) |
92 |
91
|
reseq2d |
|- ( ph -> ( ( RR _D ( s e. RR |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) ) |` ( ( int ` ( topGen ` ran (,) ) ) ` A ) ) = ( ( RR _D ( s e. RR |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) ) |` A ) ) |
93 |
|
resmpt |
|- ( RR C_ CC -> ( ( s e. CC |-> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) ) |` RR ) = ( s e. RR |-> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) ) ) |
94 |
63 93
|
ax-mp |
|- ( ( s e. CC |-> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) ) |` RR ) = ( s e. RR |-> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) ) |
95 |
|
id |
|- ( s e. CC -> s e. CC ) |
96 |
|
2cnd |
|- ( s e. CC -> 2 e. CC ) |
97 |
20
|
a1i |
|- ( s e. CC -> 2 =/= 0 ) |
98 |
95 96 97
|
divrec2d |
|- ( s e. CC -> ( s / 2 ) = ( ( 1 / 2 ) x. s ) ) |
99 |
98
|
eqcomd |
|- ( s e. CC -> ( ( 1 / 2 ) x. s ) = ( s / 2 ) ) |
100 |
76 99
|
syl |
|- ( s e. RR -> ( ( 1 / 2 ) x. s ) = ( s / 2 ) ) |
101 |
100
|
fveq2d |
|- ( s e. RR -> ( sin ` ( ( 1 / 2 ) x. s ) ) = ( sin ` ( s / 2 ) ) ) |
102 |
101
|
oveq2d |
|- ( s e. RR -> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) = ( 2 x. ( sin ` ( s / 2 ) ) ) ) |
103 |
102
|
mpteq2ia |
|- ( s e. RR |-> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) ) = ( s e. RR |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) |
104 |
94 103
|
eqtr2i |
|- ( s e. RR |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) = ( ( s e. CC |-> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) ) |` RR ) |
105 |
104
|
oveq2i |
|- ( RR _D ( s e. RR |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) ) = ( RR _D ( ( s e. CC |-> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) ) |` RR ) ) |
106 |
|
eqid |
|- ( s e. CC |-> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) ) = ( s e. CC |-> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) ) |
107 |
|
halfcn |
|- ( 1 / 2 ) e. CC |
108 |
107
|
a1i |
|- ( s e. CC -> ( 1 / 2 ) e. CC ) |
109 |
108 95
|
mulcld |
|- ( s e. CC -> ( ( 1 / 2 ) x. s ) e. CC ) |
110 |
109
|
sincld |
|- ( s e. CC -> ( sin ` ( ( 1 / 2 ) x. s ) ) e. CC ) |
111 |
96 110
|
mulcld |
|- ( s e. CC -> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) e. CC ) |
112 |
106 111
|
fmpti |
|- ( s e. CC |-> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) ) : CC --> CC |
113 |
|
2cn |
|- 2 e. CC |
114 |
|
dvasinbx |
|- ( ( 2 e. CC /\ ( 1 / 2 ) e. CC ) -> ( CC _D ( s e. CC |-> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) ) ) = ( s e. CC |-> ( ( 2 x. ( 1 / 2 ) ) x. ( cos ` ( ( 1 / 2 ) x. s ) ) ) ) ) |
115 |
113 107 114
|
mp2an |
|- ( CC _D ( s e. CC |-> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) ) ) = ( s e. CC |-> ( ( 2 x. ( 1 / 2 ) ) x. ( cos ` ( ( 1 / 2 ) x. s ) ) ) ) |
116 |
113 20
|
recidi |
|- ( 2 x. ( 1 / 2 ) ) = 1 |
117 |
116
|
a1i |
|- ( s e. CC -> ( 2 x. ( 1 / 2 ) ) = 1 ) |
118 |
99
|
fveq2d |
|- ( s e. CC -> ( cos ` ( ( 1 / 2 ) x. s ) ) = ( cos ` ( s / 2 ) ) ) |
119 |
117 118
|
oveq12d |
|- ( s e. CC -> ( ( 2 x. ( 1 / 2 ) ) x. ( cos ` ( ( 1 / 2 ) x. s ) ) ) = ( 1 x. ( cos ` ( s / 2 ) ) ) ) |
120 |
|
halfcl |
|- ( s e. CC -> ( s / 2 ) e. CC ) |
121 |
120
|
coscld |
|- ( s e. CC -> ( cos ` ( s / 2 ) ) e. CC ) |
122 |
121
|
mulid2d |
|- ( s e. CC -> ( 1 x. ( cos ` ( s / 2 ) ) ) = ( cos ` ( s / 2 ) ) ) |
123 |
119 122
|
eqtrd |
|- ( s e. CC -> ( ( 2 x. ( 1 / 2 ) ) x. ( cos ` ( ( 1 / 2 ) x. s ) ) ) = ( cos ` ( s / 2 ) ) ) |
124 |
123
|
mpteq2ia |
|- ( s e. CC |-> ( ( 2 x. ( 1 / 2 ) ) x. ( cos ` ( ( 1 / 2 ) x. s ) ) ) ) = ( s e. CC |-> ( cos ` ( s / 2 ) ) ) |
125 |
115 124
|
eqtri |
|- ( CC _D ( s e. CC |-> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) ) ) = ( s e. CC |-> ( cos ` ( s / 2 ) ) ) |
126 |
125
|
dmeqi |
|- dom ( CC _D ( s e. CC |-> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) ) ) = dom ( s e. CC |-> ( cos ` ( s / 2 ) ) ) |
127 |
|
dmmptg |
|- ( A. s e. CC ( cos ` ( s / 2 ) ) e. CC -> dom ( s e. CC |-> ( cos ` ( s / 2 ) ) ) = CC ) |
128 |
127 121
|
mprg |
|- dom ( s e. CC |-> ( cos ` ( s / 2 ) ) ) = CC |
129 |
126 128
|
eqtri |
|- dom ( CC _D ( s e. CC |-> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) ) ) = CC |
130 |
63 129
|
sseqtrri |
|- RR C_ dom ( CC _D ( s e. CC |-> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) ) ) |
131 |
|
dvres3 |
|- ( ( ( RR e. { RR , CC } /\ ( s e. CC |-> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) ) : CC --> CC ) /\ ( CC C_ CC /\ RR C_ dom ( CC _D ( s e. CC |-> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) ) ) ) ) -> ( RR _D ( ( s e. CC |-> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) ) |` RR ) ) = ( ( CC _D ( s e. CC |-> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) ) ) |` RR ) ) |
132 |
47 112 67 130 131
|
mp4an |
|- ( RR _D ( ( s e. CC |-> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) ) |` RR ) ) = ( ( CC _D ( s e. CC |-> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) ) ) |` RR ) |
133 |
125
|
reseq1i |
|- ( ( CC _D ( s e. CC |-> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) ) ) |` RR ) = ( ( s e. CC |-> ( cos ` ( s / 2 ) ) ) |` RR ) |
134 |
105 132 133
|
3eqtri |
|- ( RR _D ( s e. RR |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) ) = ( ( s e. CC |-> ( cos ` ( s / 2 ) ) ) |` RR ) |
135 |
134
|
reseq1i |
|- ( ( RR _D ( s e. RR |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) ) |` A ) = ( ( ( s e. CC |-> ( cos ` ( s / 2 ) ) ) |` RR ) |` A ) |
136 |
135
|
a1i |
|- ( ph -> ( ( RR _D ( s e. RR |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) ) |` A ) = ( ( ( s e. CC |-> ( cos ` ( s / 2 ) ) ) |` RR ) |` A ) ) |
137 |
39
|
resabs1d |
|- ( ph -> ( ( ( s e. CC |-> ( cos ` ( s / 2 ) ) ) |` RR ) |` A ) = ( ( s e. CC |-> ( cos ` ( s / 2 ) ) ) |` A ) ) |
138 |
65
|
resmptd |
|- ( ph -> ( ( s e. CC |-> ( cos ` ( s / 2 ) ) ) |` A ) = ( s e. A |-> ( cos ` ( s / 2 ) ) ) ) |
139 |
137 138
|
eqtrd |
|- ( ph -> ( ( ( s e. CC |-> ( cos ` ( s / 2 ) ) ) |` RR ) |` A ) = ( s e. A |-> ( cos ` ( s / 2 ) ) ) ) |
140 |
92 136 139
|
3eqtrd |
|- ( ph -> ( ( RR _D ( s e. RR |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) ) |` ( ( int ` ( topGen ` ran (,) ) ) ` A ) ) = ( s e. A |-> ( cos ` ( s / 2 ) ) ) ) |
141 |
73 85 140
|
3eqtrd |
|- ( ph -> ( RR _D ( s e. A |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) ) = ( s e. A |-> ( cos ` ( s / 2 ) ) ) ) |
142 |
|
coscn |
|- cos e. ( CC -cn-> CC ) |
143 |
142
|
a1i |
|- ( ph -> cos e. ( CC -cn-> CC ) ) |
144 |
65 68
|
idcncfg |
|- ( ph -> ( s e. A |-> s ) e. ( A -cn-> CC ) ) |
145 |
|
2cnd |
|- ( ph -> 2 e. CC ) |
146 |
20
|
a1i |
|- ( ph -> 2 =/= 0 ) |
147 |
|
eldifsn |
|- ( 2 e. ( CC \ { 0 } ) <-> ( 2 e. CC /\ 2 =/= 0 ) ) |
148 |
145 146 147
|
sylanbrc |
|- ( ph -> 2 e. ( CC \ { 0 } ) ) |
149 |
|
difssd |
|- ( ph -> ( CC \ { 0 } ) C_ CC ) |
150 |
65 148 149
|
constcncfg |
|- ( ph -> ( s e. A |-> 2 ) e. ( A -cn-> ( CC \ { 0 } ) ) ) |
151 |
144 150
|
divcncf |
|- ( ph -> ( s e. A |-> ( s / 2 ) ) e. ( A -cn-> CC ) ) |
152 |
143 151
|
cncfmpt1f |
|- ( ph -> ( s e. A |-> ( cos ` ( s / 2 ) ) ) e. ( A -cn-> CC ) ) |
153 |
141 152
|
eqeltrd |
|- ( ph -> ( RR _D ( s e. A |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) ) e. ( A -cn-> CC ) ) |
154 |
48 50 58 70 153
|
dvdivcncf |
|- ( ph -> ( RR _D ( ( s e. A |-> s ) oF / ( s e. A |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) ) ) e. ( A -cn-> CC ) ) |
155 |
46 154
|
eqeltrd |
|- ( ph -> ( RR _D K ) e. ( A -cn-> CC ) ) |
156 |
|
cncff |
|- ( ( RR _D K ) e. ( A -cn-> CC ) -> ( RR _D K ) : A --> CC ) |
157 |
|
fdm |
|- ( ( RR _D K ) : A --> CC -> dom ( RR _D K ) = A ) |
158 |
155 156 157
|
3syl |
|- ( ph -> dom ( RR _D K ) = A ) |
159 |
158
|
feq2d |
|- ( ph -> ( ( RR _D K ) : dom ( RR _D K ) --> RR <-> ( RR _D K ) : A --> RR ) ) |
160 |
41 159
|
mpbid |
|- ( ph -> ( RR _D K ) : A --> RR ) |
161 |
|
cncffvrn |
|- ( ( RR C_ CC /\ ( RR _D K ) e. ( A -cn-> CC ) ) -> ( ( RR _D K ) e. ( A -cn-> RR ) <-> ( RR _D K ) : A --> RR ) ) |
162 |
64 155 161
|
syl2anc |
|- ( ph -> ( ( RR _D K ) e. ( A -cn-> RR ) <-> ( RR _D K ) : A --> RR ) ) |
163 |
160 162
|
mpbird |
|- ( ph -> ( RR _D K ) e. ( A -cn-> RR ) ) |