Step |
Hyp |
Ref |
Expression |
1 |
|
dirkertrigeq.d |
|- D = ( n e. NN |-> ( s e. RR |-> if ( ( s mod ( 2 x. _pi ) ) = 0 , ( ( ( 2 x. n ) + 1 ) / ( 2 x. _pi ) ) , ( ( sin ` ( ( n + ( 1 / 2 ) ) x. s ) ) / ( ( 2 x. _pi ) x. ( sin ` ( s / 2 ) ) ) ) ) ) ) |
2 |
|
dirkertrigeq.n |
|- ( ph -> N e. NN ) |
3 |
|
dirkertrigeq.f |
|- F = ( D ` N ) |
4 |
|
dirkertrigeq.h |
|- H = ( s e. RR |-> ( ( ( 1 / 2 ) + sum_ k e. ( 1 ... N ) ( cos ` ( k x. s ) ) ) / _pi ) ) |
5 |
3
|
a1i |
|- ( ph -> F = ( D ` N ) ) |
6 |
1
|
dirkerval |
|- ( N e. NN -> ( D ` N ) = ( s e. RR |-> if ( ( s mod ( 2 x. _pi ) ) = 0 , ( ( ( 2 x. N ) + 1 ) / ( 2 x. _pi ) ) , ( ( sin ` ( ( N + ( 1 / 2 ) ) x. s ) ) / ( ( 2 x. _pi ) x. ( sin ` ( s / 2 ) ) ) ) ) ) ) |
7 |
2 6
|
syl |
|- ( ph -> ( D ` N ) = ( s e. RR |-> if ( ( s mod ( 2 x. _pi ) ) = 0 , ( ( ( 2 x. N ) + 1 ) / ( 2 x. _pi ) ) , ( ( sin ` ( ( N + ( 1 / 2 ) ) x. s ) ) / ( ( 2 x. _pi ) x. ( sin ` ( s / 2 ) ) ) ) ) ) ) |
8 |
|
2cnd |
|- ( ph -> 2 e. CC ) |
9 |
2
|
nncnd |
|- ( ph -> N e. CC ) |
10 |
8 9
|
mulcld |
|- ( ph -> ( 2 x. N ) e. CC ) |
11 |
|
peano2cn |
|- ( ( 2 x. N ) e. CC -> ( ( 2 x. N ) + 1 ) e. CC ) |
12 |
10 11
|
syl |
|- ( ph -> ( ( 2 x. N ) + 1 ) e. CC ) |
13 |
|
picn |
|- _pi e. CC |
14 |
13
|
a1i |
|- ( ph -> _pi e. CC ) |
15 |
|
2ne0 |
|- 2 =/= 0 |
16 |
15
|
a1i |
|- ( ph -> 2 =/= 0 ) |
17 |
|
pire |
|- _pi e. RR |
18 |
|
pipos |
|- 0 < _pi |
19 |
17 18
|
gt0ne0ii |
|- _pi =/= 0 |
20 |
19
|
a1i |
|- ( ph -> _pi =/= 0 ) |
21 |
12 8 14 16 20
|
divdiv1d |
|- ( ph -> ( ( ( ( 2 x. N ) + 1 ) / 2 ) / _pi ) = ( ( ( 2 x. N ) + 1 ) / ( 2 x. _pi ) ) ) |
22 |
21
|
eqcomd |
|- ( ph -> ( ( ( 2 x. N ) + 1 ) / ( 2 x. _pi ) ) = ( ( ( ( 2 x. N ) + 1 ) / 2 ) / _pi ) ) |
23 |
22
|
ad2antrr |
|- ( ( ( ph /\ s e. RR ) /\ ( s mod ( 2 x. _pi ) ) = 0 ) -> ( ( ( 2 x. N ) + 1 ) / ( 2 x. _pi ) ) = ( ( ( ( 2 x. N ) + 1 ) / 2 ) / _pi ) ) |
24 |
|
iftrue |
|- ( ( s mod ( 2 x. _pi ) ) = 0 -> if ( ( s mod ( 2 x. _pi ) ) = 0 , ( ( ( 2 x. N ) + 1 ) / ( 2 x. _pi ) ) , ( ( sin ` ( ( N + ( 1 / 2 ) ) x. s ) ) / ( ( 2 x. _pi ) x. ( sin ` ( s / 2 ) ) ) ) ) = ( ( ( 2 x. N ) + 1 ) / ( 2 x. _pi ) ) ) |
25 |
24
|
adantl |
|- ( ( ( ph /\ s e. RR ) /\ ( s mod ( 2 x. _pi ) ) = 0 ) -> if ( ( s mod ( 2 x. _pi ) ) = 0 , ( ( ( 2 x. N ) + 1 ) / ( 2 x. _pi ) ) , ( ( sin ` ( ( N + ( 1 / 2 ) ) x. s ) ) / ( ( 2 x. _pi ) x. ( sin ` ( s / 2 ) ) ) ) ) = ( ( ( 2 x. N ) + 1 ) / ( 2 x. _pi ) ) ) |
26 |
|
elfzelz |
|- ( k e. ( 1 ... N ) -> k e. ZZ ) |
27 |
26
|
zcnd |
|- ( k e. ( 1 ... N ) -> k e. CC ) |
28 |
27
|
adantl |
|- ( ( ( s e. RR /\ ( s mod ( 2 x. _pi ) ) = 0 ) /\ k e. ( 1 ... N ) ) -> k e. CC ) |
29 |
|
recn |
|- ( s e. RR -> s e. CC ) |
30 |
29
|
ad2antrr |
|- ( ( ( s e. RR /\ ( s mod ( 2 x. _pi ) ) = 0 ) /\ k e. ( 1 ... N ) ) -> s e. CC ) |
31 |
|
2cn |
|- 2 e. CC |
32 |
31 13
|
mulcli |
|- ( 2 x. _pi ) e. CC |
33 |
32
|
a1i |
|- ( ( ( s e. RR /\ ( s mod ( 2 x. _pi ) ) = 0 ) /\ k e. ( 1 ... N ) ) -> ( 2 x. _pi ) e. CC ) |
34 |
31 13 15 19
|
mulne0i |
|- ( 2 x. _pi ) =/= 0 |
35 |
34
|
a1i |
|- ( ( ( s e. RR /\ ( s mod ( 2 x. _pi ) ) = 0 ) /\ k e. ( 1 ... N ) ) -> ( 2 x. _pi ) =/= 0 ) |
36 |
28 30 33 35
|
divassd |
|- ( ( ( s e. RR /\ ( s mod ( 2 x. _pi ) ) = 0 ) /\ k e. ( 1 ... N ) ) -> ( ( k x. s ) / ( 2 x. _pi ) ) = ( k x. ( s / ( 2 x. _pi ) ) ) ) |
37 |
26
|
adantl |
|- ( ( ( s e. RR /\ ( s mod ( 2 x. _pi ) ) = 0 ) /\ k e. ( 1 ... N ) ) -> k e. ZZ ) |
38 |
|
simpr |
|- ( ( s e. RR /\ ( s mod ( 2 x. _pi ) ) = 0 ) -> ( s mod ( 2 x. _pi ) ) = 0 ) |
39 |
|
simpl |
|- ( ( s e. RR /\ ( s mod ( 2 x. _pi ) ) = 0 ) -> s e. RR ) |
40 |
|
2rp |
|- 2 e. RR+ |
41 |
|
pirp |
|- _pi e. RR+ |
42 |
|
rpmulcl |
|- ( ( 2 e. RR+ /\ _pi e. RR+ ) -> ( 2 x. _pi ) e. RR+ ) |
43 |
40 41 42
|
mp2an |
|- ( 2 x. _pi ) e. RR+ |
44 |
|
mod0 |
|- ( ( s e. RR /\ ( 2 x. _pi ) e. RR+ ) -> ( ( s mod ( 2 x. _pi ) ) = 0 <-> ( s / ( 2 x. _pi ) ) e. ZZ ) ) |
45 |
39 43 44
|
sylancl |
|- ( ( s e. RR /\ ( s mod ( 2 x. _pi ) ) = 0 ) -> ( ( s mod ( 2 x. _pi ) ) = 0 <-> ( s / ( 2 x. _pi ) ) e. ZZ ) ) |
46 |
38 45
|
mpbid |
|- ( ( s e. RR /\ ( s mod ( 2 x. _pi ) ) = 0 ) -> ( s / ( 2 x. _pi ) ) e. ZZ ) |
47 |
46
|
adantr |
|- ( ( ( s e. RR /\ ( s mod ( 2 x. _pi ) ) = 0 ) /\ k e. ( 1 ... N ) ) -> ( s / ( 2 x. _pi ) ) e. ZZ ) |
48 |
37 47
|
zmulcld |
|- ( ( ( s e. RR /\ ( s mod ( 2 x. _pi ) ) = 0 ) /\ k e. ( 1 ... N ) ) -> ( k x. ( s / ( 2 x. _pi ) ) ) e. ZZ ) |
49 |
36 48
|
eqeltrd |
|- ( ( ( s e. RR /\ ( s mod ( 2 x. _pi ) ) = 0 ) /\ k e. ( 1 ... N ) ) -> ( ( k x. s ) / ( 2 x. _pi ) ) e. ZZ ) |
50 |
27
|
adantl |
|- ( ( s e. RR /\ k e. ( 1 ... N ) ) -> k e. CC ) |
51 |
29
|
adantr |
|- ( ( s e. RR /\ k e. ( 1 ... N ) ) -> s e. CC ) |
52 |
50 51
|
mulcld |
|- ( ( s e. RR /\ k e. ( 1 ... N ) ) -> ( k x. s ) e. CC ) |
53 |
|
coseq1 |
|- ( ( k x. s ) e. CC -> ( ( cos ` ( k x. s ) ) = 1 <-> ( ( k x. s ) / ( 2 x. _pi ) ) e. ZZ ) ) |
54 |
52 53
|
syl |
|- ( ( s e. RR /\ k e. ( 1 ... N ) ) -> ( ( cos ` ( k x. s ) ) = 1 <-> ( ( k x. s ) / ( 2 x. _pi ) ) e. ZZ ) ) |
55 |
54
|
adantlr |
|- ( ( ( s e. RR /\ ( s mod ( 2 x. _pi ) ) = 0 ) /\ k e. ( 1 ... N ) ) -> ( ( cos ` ( k x. s ) ) = 1 <-> ( ( k x. s ) / ( 2 x. _pi ) ) e. ZZ ) ) |
56 |
49 55
|
mpbird |
|- ( ( ( s e. RR /\ ( s mod ( 2 x. _pi ) ) = 0 ) /\ k e. ( 1 ... N ) ) -> ( cos ` ( k x. s ) ) = 1 ) |
57 |
56
|
ralrimiva |
|- ( ( s e. RR /\ ( s mod ( 2 x. _pi ) ) = 0 ) -> A. k e. ( 1 ... N ) ( cos ` ( k x. s ) ) = 1 ) |
58 |
57
|
adantll |
|- ( ( ( ph /\ s e. RR ) /\ ( s mod ( 2 x. _pi ) ) = 0 ) -> A. k e. ( 1 ... N ) ( cos ` ( k x. s ) ) = 1 ) |
59 |
58
|
sumeq2d |
|- ( ( ( ph /\ s e. RR ) /\ ( s mod ( 2 x. _pi ) ) = 0 ) -> sum_ k e. ( 1 ... N ) ( cos ` ( k x. s ) ) = sum_ k e. ( 1 ... N ) 1 ) |
60 |
|
fzfid |
|- ( ( ( ph /\ s e. RR ) /\ ( s mod ( 2 x. _pi ) ) = 0 ) -> ( 1 ... N ) e. Fin ) |
61 |
|
1cnd |
|- ( ( ( ph /\ s e. RR ) /\ ( s mod ( 2 x. _pi ) ) = 0 ) -> 1 e. CC ) |
62 |
|
fsumconst |
|- ( ( ( 1 ... N ) e. Fin /\ 1 e. CC ) -> sum_ k e. ( 1 ... N ) 1 = ( ( # ` ( 1 ... N ) ) x. 1 ) ) |
63 |
60 61 62
|
syl2anc |
|- ( ( ( ph /\ s e. RR ) /\ ( s mod ( 2 x. _pi ) ) = 0 ) -> sum_ k e. ( 1 ... N ) 1 = ( ( # ` ( 1 ... N ) ) x. 1 ) ) |
64 |
2
|
nnnn0d |
|- ( ph -> N e. NN0 ) |
65 |
|
hashfz1 |
|- ( N e. NN0 -> ( # ` ( 1 ... N ) ) = N ) |
66 |
64 65
|
syl |
|- ( ph -> ( # ` ( 1 ... N ) ) = N ) |
67 |
66
|
oveq1d |
|- ( ph -> ( ( # ` ( 1 ... N ) ) x. 1 ) = ( N x. 1 ) ) |
68 |
9
|
mulid1d |
|- ( ph -> ( N x. 1 ) = N ) |
69 |
67 68
|
eqtrd |
|- ( ph -> ( ( # ` ( 1 ... N ) ) x. 1 ) = N ) |
70 |
69
|
ad2antrr |
|- ( ( ( ph /\ s e. RR ) /\ ( s mod ( 2 x. _pi ) ) = 0 ) -> ( ( # ` ( 1 ... N ) ) x. 1 ) = N ) |
71 |
59 63 70
|
3eqtrd |
|- ( ( ( ph /\ s e. RR ) /\ ( s mod ( 2 x. _pi ) ) = 0 ) -> sum_ k e. ( 1 ... N ) ( cos ` ( k x. s ) ) = N ) |
72 |
71
|
oveq2d |
|- ( ( ( ph /\ s e. RR ) /\ ( s mod ( 2 x. _pi ) ) = 0 ) -> ( ( 1 / 2 ) + sum_ k e. ( 1 ... N ) ( cos ` ( k x. s ) ) ) = ( ( 1 / 2 ) + N ) ) |
73 |
9
|
div1d |
|- ( ph -> ( N / 1 ) = N ) |
74 |
73
|
eqcomd |
|- ( ph -> N = ( N / 1 ) ) |
75 |
74
|
oveq2d |
|- ( ph -> ( ( 1 / 2 ) + N ) = ( ( 1 / 2 ) + ( N / 1 ) ) ) |
76 |
|
1cnd |
|- ( ph -> 1 e. CC ) |
77 |
|
ax-1ne0 |
|- 1 =/= 0 |
78 |
77
|
a1i |
|- ( ph -> 1 =/= 0 ) |
79 |
76 8 9 76 16 78
|
divadddivd |
|- ( ph -> ( ( 1 / 2 ) + ( N / 1 ) ) = ( ( ( 1 x. 1 ) + ( N x. 2 ) ) / ( 2 x. 1 ) ) ) |
80 |
76 76
|
mulcld |
|- ( ph -> ( 1 x. 1 ) e. CC ) |
81 |
9 8
|
mulcld |
|- ( ph -> ( N x. 2 ) e. CC ) |
82 |
80 81
|
addcomd |
|- ( ph -> ( ( 1 x. 1 ) + ( N x. 2 ) ) = ( ( N x. 2 ) + ( 1 x. 1 ) ) ) |
83 |
9 8
|
mulcomd |
|- ( ph -> ( N x. 2 ) = ( 2 x. N ) ) |
84 |
76
|
mulid1d |
|- ( ph -> ( 1 x. 1 ) = 1 ) |
85 |
83 84
|
oveq12d |
|- ( ph -> ( ( N x. 2 ) + ( 1 x. 1 ) ) = ( ( 2 x. N ) + 1 ) ) |
86 |
82 85
|
eqtrd |
|- ( ph -> ( ( 1 x. 1 ) + ( N x. 2 ) ) = ( ( 2 x. N ) + 1 ) ) |
87 |
8
|
mulid1d |
|- ( ph -> ( 2 x. 1 ) = 2 ) |
88 |
86 87
|
oveq12d |
|- ( ph -> ( ( ( 1 x. 1 ) + ( N x. 2 ) ) / ( 2 x. 1 ) ) = ( ( ( 2 x. N ) + 1 ) / 2 ) ) |
89 |
75 79 88
|
3eqtrd |
|- ( ph -> ( ( 1 / 2 ) + N ) = ( ( ( 2 x. N ) + 1 ) / 2 ) ) |
90 |
89
|
ad2antrr |
|- ( ( ( ph /\ s e. RR ) /\ ( s mod ( 2 x. _pi ) ) = 0 ) -> ( ( 1 / 2 ) + N ) = ( ( ( 2 x. N ) + 1 ) / 2 ) ) |
91 |
72 90
|
eqtrd |
|- ( ( ( ph /\ s e. RR ) /\ ( s mod ( 2 x. _pi ) ) = 0 ) -> ( ( 1 / 2 ) + sum_ k e. ( 1 ... N ) ( cos ` ( k x. s ) ) ) = ( ( ( 2 x. N ) + 1 ) / 2 ) ) |
92 |
91
|
oveq1d |
|- ( ( ( ph /\ s e. RR ) /\ ( s mod ( 2 x. _pi ) ) = 0 ) -> ( ( ( 1 / 2 ) + sum_ k e. ( 1 ... N ) ( cos ` ( k x. s ) ) ) / _pi ) = ( ( ( ( 2 x. N ) + 1 ) / 2 ) / _pi ) ) |
93 |
23 25 92
|
3eqtr4rd |
|- ( ( ( ph /\ s e. RR ) /\ ( s mod ( 2 x. _pi ) ) = 0 ) -> ( ( ( 1 / 2 ) + sum_ k e. ( 1 ... N ) ( cos ` ( k x. s ) ) ) / _pi ) = if ( ( s mod ( 2 x. _pi ) ) = 0 , ( ( ( 2 x. N ) + 1 ) / ( 2 x. _pi ) ) , ( ( sin ` ( ( N + ( 1 / 2 ) ) x. s ) ) / ( ( 2 x. _pi ) x. ( sin ` ( s / 2 ) ) ) ) ) ) |
94 |
|
iffalse |
|- ( -. ( s mod ( 2 x. _pi ) ) = 0 -> if ( ( s mod ( 2 x. _pi ) ) = 0 , ( ( ( 2 x. N ) + 1 ) / ( 2 x. _pi ) ) , ( ( sin ` ( ( N + ( 1 / 2 ) ) x. s ) ) / ( ( 2 x. _pi ) x. ( sin ` ( s / 2 ) ) ) ) ) = ( ( sin ` ( ( N + ( 1 / 2 ) ) x. s ) ) / ( ( 2 x. _pi ) x. ( sin ` ( s / 2 ) ) ) ) ) |
95 |
94
|
adantl |
|- ( ( ( ph /\ s e. RR ) /\ -. ( s mod ( 2 x. _pi ) ) = 0 ) -> if ( ( s mod ( 2 x. _pi ) ) = 0 , ( ( ( 2 x. N ) + 1 ) / ( 2 x. _pi ) ) , ( ( sin ` ( ( N + ( 1 / 2 ) ) x. s ) ) / ( ( 2 x. _pi ) x. ( sin ` ( s / 2 ) ) ) ) ) = ( ( sin ` ( ( N + ( 1 / 2 ) ) x. s ) ) / ( ( 2 x. _pi ) x. ( sin ` ( s / 2 ) ) ) ) ) |
96 |
13
|
a1i |
|- ( s e. RR -> _pi e. CC ) |
97 |
19
|
a1i |
|- ( s e. RR -> _pi =/= 0 ) |
98 |
29 96 97
|
divcan1d |
|- ( s e. RR -> ( ( s / _pi ) x. _pi ) = s ) |
99 |
98
|
eqcomd |
|- ( s e. RR -> s = ( ( s / _pi ) x. _pi ) ) |
100 |
99
|
ad2antrr |
|- ( ( ( s e. RR /\ -. ( s mod ( 2 x. _pi ) ) = 0 ) /\ ( s mod _pi ) = 0 ) -> s = ( ( s / _pi ) x. _pi ) ) |
101 |
|
simpr |
|- ( ( s e. RR /\ ( s mod _pi ) = 0 ) -> ( s mod _pi ) = 0 ) |
102 |
|
simpl |
|- ( ( s e. RR /\ ( s mod _pi ) = 0 ) -> s e. RR ) |
103 |
|
mod0 |
|- ( ( s e. RR /\ _pi e. RR+ ) -> ( ( s mod _pi ) = 0 <-> ( s / _pi ) e. ZZ ) ) |
104 |
102 41 103
|
sylancl |
|- ( ( s e. RR /\ ( s mod _pi ) = 0 ) -> ( ( s mod _pi ) = 0 <-> ( s / _pi ) e. ZZ ) ) |
105 |
101 104
|
mpbid |
|- ( ( s e. RR /\ ( s mod _pi ) = 0 ) -> ( s / _pi ) e. ZZ ) |
106 |
105
|
adantlr |
|- ( ( ( s e. RR /\ -. ( s mod ( 2 x. _pi ) ) = 0 ) /\ ( s mod _pi ) = 0 ) -> ( s / _pi ) e. ZZ ) |
107 |
|
rpreccl |
|- ( _pi e. RR+ -> ( 1 / _pi ) e. RR+ ) |
108 |
41 107
|
ax-mp |
|- ( 1 / _pi ) e. RR+ |
109 |
|
moddi |
|- ( ( ( 1 / _pi ) e. RR+ /\ s e. RR /\ ( 2 x. _pi ) e. RR+ ) -> ( ( 1 / _pi ) x. ( s mod ( 2 x. _pi ) ) ) = ( ( ( 1 / _pi ) x. s ) mod ( ( 1 / _pi ) x. ( 2 x. _pi ) ) ) ) |
110 |
108 43 109
|
mp3an13 |
|- ( s e. RR -> ( ( 1 / _pi ) x. ( s mod ( 2 x. _pi ) ) ) = ( ( ( 1 / _pi ) x. s ) mod ( ( 1 / _pi ) x. ( 2 x. _pi ) ) ) ) |
111 |
29 96 97
|
divrec2d |
|- ( s e. RR -> ( s / _pi ) = ( ( 1 / _pi ) x. s ) ) |
112 |
111
|
eqcomd |
|- ( s e. RR -> ( ( 1 / _pi ) x. s ) = ( s / _pi ) ) |
113 |
96 97
|
reccld |
|- ( s e. RR -> ( 1 / _pi ) e. CC ) |
114 |
32
|
a1i |
|- ( s e. RR -> ( 2 x. _pi ) e. CC ) |
115 |
113 114
|
mulcomd |
|- ( s e. RR -> ( ( 1 / _pi ) x. ( 2 x. _pi ) ) = ( ( 2 x. _pi ) x. ( 1 / _pi ) ) ) |
116 |
|
2cnd |
|- ( s e. RR -> 2 e. CC ) |
117 |
116 96 113
|
mulassd |
|- ( s e. RR -> ( ( 2 x. _pi ) x. ( 1 / _pi ) ) = ( 2 x. ( _pi x. ( 1 / _pi ) ) ) ) |
118 |
13 19
|
recidi |
|- ( _pi x. ( 1 / _pi ) ) = 1 |
119 |
118
|
oveq2i |
|- ( 2 x. ( _pi x. ( 1 / _pi ) ) ) = ( 2 x. 1 ) |
120 |
116
|
mulid1d |
|- ( s e. RR -> ( 2 x. 1 ) = 2 ) |
121 |
119 120
|
eqtrid |
|- ( s e. RR -> ( 2 x. ( _pi x. ( 1 / _pi ) ) ) = 2 ) |
122 |
115 117 121
|
3eqtrd |
|- ( s e. RR -> ( ( 1 / _pi ) x. ( 2 x. _pi ) ) = 2 ) |
123 |
112 122
|
oveq12d |
|- ( s e. RR -> ( ( ( 1 / _pi ) x. s ) mod ( ( 1 / _pi ) x. ( 2 x. _pi ) ) ) = ( ( s / _pi ) mod 2 ) ) |
124 |
110 123
|
eqtr2d |
|- ( s e. RR -> ( ( s / _pi ) mod 2 ) = ( ( 1 / _pi ) x. ( s mod ( 2 x. _pi ) ) ) ) |
125 |
124
|
adantr |
|- ( ( s e. RR /\ -. ( s mod ( 2 x. _pi ) ) = 0 ) -> ( ( s / _pi ) mod 2 ) = ( ( 1 / _pi ) x. ( s mod ( 2 x. _pi ) ) ) ) |
126 |
113
|
adantr |
|- ( ( s e. RR /\ -. ( s mod ( 2 x. _pi ) ) = 0 ) -> ( 1 / _pi ) e. CC ) |
127 |
|
id |
|- ( s e. RR -> s e. RR ) |
128 |
43
|
a1i |
|- ( s e. RR -> ( 2 x. _pi ) e. RR+ ) |
129 |
127 128
|
modcld |
|- ( s e. RR -> ( s mod ( 2 x. _pi ) ) e. RR ) |
130 |
129
|
recnd |
|- ( s e. RR -> ( s mod ( 2 x. _pi ) ) e. CC ) |
131 |
130
|
adantr |
|- ( ( s e. RR /\ -. ( s mod ( 2 x. _pi ) ) = 0 ) -> ( s mod ( 2 x. _pi ) ) e. CC ) |
132 |
|
ax-1cn |
|- 1 e. CC |
133 |
132 13 77 19
|
divne0i |
|- ( 1 / _pi ) =/= 0 |
134 |
133
|
a1i |
|- ( ( s e. RR /\ -. ( s mod ( 2 x. _pi ) ) = 0 ) -> ( 1 / _pi ) =/= 0 ) |
135 |
|
neqne |
|- ( -. ( s mod ( 2 x. _pi ) ) = 0 -> ( s mod ( 2 x. _pi ) ) =/= 0 ) |
136 |
135
|
adantl |
|- ( ( s e. RR /\ -. ( s mod ( 2 x. _pi ) ) = 0 ) -> ( s mod ( 2 x. _pi ) ) =/= 0 ) |
137 |
126 131 134 136
|
mulne0d |
|- ( ( s e. RR /\ -. ( s mod ( 2 x. _pi ) ) = 0 ) -> ( ( 1 / _pi ) x. ( s mod ( 2 x. _pi ) ) ) =/= 0 ) |
138 |
125 137
|
eqnetrd |
|- ( ( s e. RR /\ -. ( s mod ( 2 x. _pi ) ) = 0 ) -> ( ( s / _pi ) mod 2 ) =/= 0 ) |
139 |
138
|
adantr |
|- ( ( ( s e. RR /\ -. ( s mod ( 2 x. _pi ) ) = 0 ) /\ ( s mod _pi ) = 0 ) -> ( ( s / _pi ) mod 2 ) =/= 0 ) |
140 |
|
oddfl |
|- ( ( ( s / _pi ) e. ZZ /\ ( ( s / _pi ) mod 2 ) =/= 0 ) -> ( s / _pi ) = ( ( 2 x. ( |_ ` ( ( s / _pi ) / 2 ) ) ) + 1 ) ) |
141 |
106 139 140
|
syl2anc |
|- ( ( ( s e. RR /\ -. ( s mod ( 2 x. _pi ) ) = 0 ) /\ ( s mod _pi ) = 0 ) -> ( s / _pi ) = ( ( 2 x. ( |_ ` ( ( s / _pi ) / 2 ) ) ) + 1 ) ) |
142 |
141
|
oveq1d |
|- ( ( ( s e. RR /\ -. ( s mod ( 2 x. _pi ) ) = 0 ) /\ ( s mod _pi ) = 0 ) -> ( ( s / _pi ) x. _pi ) = ( ( ( 2 x. ( |_ ` ( ( s / _pi ) / 2 ) ) ) + 1 ) x. _pi ) ) |
143 |
100 142
|
eqtrd |
|- ( ( ( s e. RR /\ -. ( s mod ( 2 x. _pi ) ) = 0 ) /\ ( s mod _pi ) = 0 ) -> s = ( ( ( 2 x. ( |_ ` ( ( s / _pi ) / 2 ) ) ) + 1 ) x. _pi ) ) |
144 |
143
|
oveq2d |
|- ( ( ( s e. RR /\ -. ( s mod ( 2 x. _pi ) ) = 0 ) /\ ( s mod _pi ) = 0 ) -> ( k x. s ) = ( k x. ( ( ( 2 x. ( |_ ` ( ( s / _pi ) / 2 ) ) ) + 1 ) x. _pi ) ) ) |
145 |
144
|
fveq2d |
|- ( ( ( s e. RR /\ -. ( s mod ( 2 x. _pi ) ) = 0 ) /\ ( s mod _pi ) = 0 ) -> ( cos ` ( k x. s ) ) = ( cos ` ( k x. ( ( ( 2 x. ( |_ ` ( ( s / _pi ) / 2 ) ) ) + 1 ) x. _pi ) ) ) ) |
146 |
145
|
sumeq2sdv |
|- ( ( ( s e. RR /\ -. ( s mod ( 2 x. _pi ) ) = 0 ) /\ ( s mod _pi ) = 0 ) -> sum_ k e. ( 1 ... N ) ( cos ` ( k x. s ) ) = sum_ k e. ( 1 ... N ) ( cos ` ( k x. ( ( ( 2 x. ( |_ ` ( ( s / _pi ) / 2 ) ) ) + 1 ) x. _pi ) ) ) ) |
147 |
146
|
oveq2d |
|- ( ( ( s e. RR /\ -. ( s mod ( 2 x. _pi ) ) = 0 ) /\ ( s mod _pi ) = 0 ) -> ( ( 1 / 2 ) + sum_ k e. ( 1 ... N ) ( cos ` ( k x. s ) ) ) = ( ( 1 / 2 ) + sum_ k e. ( 1 ... N ) ( cos ` ( k x. ( ( ( 2 x. ( |_ ` ( ( s / _pi ) / 2 ) ) ) + 1 ) x. _pi ) ) ) ) ) |
148 |
147
|
oveq1d |
|- ( ( ( s e. RR /\ -. ( s mod ( 2 x. _pi ) ) = 0 ) /\ ( s mod _pi ) = 0 ) -> ( ( ( 1 / 2 ) + sum_ k e. ( 1 ... N ) ( cos ` ( k x. s ) ) ) / _pi ) = ( ( ( 1 / 2 ) + sum_ k e. ( 1 ... N ) ( cos ` ( k x. ( ( ( 2 x. ( |_ ` ( ( s / _pi ) / 2 ) ) ) + 1 ) x. _pi ) ) ) ) / _pi ) ) |
149 |
148
|
adantlll |
|- ( ( ( ( ph /\ s e. RR ) /\ -. ( s mod ( 2 x. _pi ) ) = 0 ) /\ ( s mod _pi ) = 0 ) -> ( ( ( 1 / 2 ) + sum_ k e. ( 1 ... N ) ( cos ` ( k x. s ) ) ) / _pi ) = ( ( ( 1 / 2 ) + sum_ k e. ( 1 ... N ) ( cos ` ( k x. ( ( ( 2 x. ( |_ ` ( ( s / _pi ) / 2 ) ) ) + 1 ) x. _pi ) ) ) ) / _pi ) ) |
150 |
2
|
ad2antrr |
|- ( ( ( ph /\ s e. RR ) /\ ( s mod _pi ) = 0 ) -> N e. NN ) |
151 |
17
|
a1i |
|- ( s e. RR -> _pi e. RR ) |
152 |
127 151 97
|
redivcld |
|- ( s e. RR -> ( s / _pi ) e. RR ) |
153 |
152
|
rehalfcld |
|- ( s e. RR -> ( ( s / _pi ) / 2 ) e. RR ) |
154 |
153
|
flcld |
|- ( s e. RR -> ( |_ ` ( ( s / _pi ) / 2 ) ) e. ZZ ) |
155 |
154
|
ad2antlr |
|- ( ( ( ph /\ s e. RR ) /\ ( s mod _pi ) = 0 ) -> ( |_ ` ( ( s / _pi ) / 2 ) ) e. ZZ ) |
156 |
|
eqid |
|- ( ( ( 2 x. ( |_ ` ( ( s / _pi ) / 2 ) ) ) + 1 ) x. _pi ) = ( ( ( 2 x. ( |_ ` ( ( s / _pi ) / 2 ) ) ) + 1 ) x. _pi ) |
157 |
150 155 156
|
dirkertrigeqlem3 |
|- ( ( ( ph /\ s e. RR ) /\ ( s mod _pi ) = 0 ) -> ( ( ( 1 / 2 ) + sum_ k e. ( 1 ... N ) ( cos ` ( k x. ( ( ( 2 x. ( |_ ` ( ( s / _pi ) / 2 ) ) ) + 1 ) x. _pi ) ) ) ) / _pi ) = ( ( sin ` ( ( N + ( 1 / 2 ) ) x. ( ( ( 2 x. ( |_ ` ( ( s / _pi ) / 2 ) ) ) + 1 ) x. _pi ) ) ) / ( ( 2 x. _pi ) x. ( sin ` ( ( ( ( 2 x. ( |_ ` ( ( s / _pi ) / 2 ) ) ) + 1 ) x. _pi ) / 2 ) ) ) ) ) |
158 |
157
|
adantlr |
|- ( ( ( ( ph /\ s e. RR ) /\ -. ( s mod ( 2 x. _pi ) ) = 0 ) /\ ( s mod _pi ) = 0 ) -> ( ( ( 1 / 2 ) + sum_ k e. ( 1 ... N ) ( cos ` ( k x. ( ( ( 2 x. ( |_ ` ( ( s / _pi ) / 2 ) ) ) + 1 ) x. _pi ) ) ) ) / _pi ) = ( ( sin ` ( ( N + ( 1 / 2 ) ) x. ( ( ( 2 x. ( |_ ` ( ( s / _pi ) / 2 ) ) ) + 1 ) x. _pi ) ) ) / ( ( 2 x. _pi ) x. ( sin ` ( ( ( ( 2 x. ( |_ ` ( ( s / _pi ) / 2 ) ) ) + 1 ) x. _pi ) / 2 ) ) ) ) ) |
159 |
141
|
adantlll |
|- ( ( ( ( ph /\ s e. RR ) /\ -. ( s mod ( 2 x. _pi ) ) = 0 ) /\ ( s mod _pi ) = 0 ) -> ( s / _pi ) = ( ( 2 x. ( |_ ` ( ( s / _pi ) / 2 ) ) ) + 1 ) ) |
160 |
159
|
eqcomd |
|- ( ( ( ( ph /\ s e. RR ) /\ -. ( s mod ( 2 x. _pi ) ) = 0 ) /\ ( s mod _pi ) = 0 ) -> ( ( 2 x. ( |_ ` ( ( s / _pi ) / 2 ) ) ) + 1 ) = ( s / _pi ) ) |
161 |
160
|
oveq1d |
|- ( ( ( ( ph /\ s e. RR ) /\ -. ( s mod ( 2 x. _pi ) ) = 0 ) /\ ( s mod _pi ) = 0 ) -> ( ( ( 2 x. ( |_ ` ( ( s / _pi ) / 2 ) ) ) + 1 ) x. _pi ) = ( ( s / _pi ) x. _pi ) ) |
162 |
161
|
oveq2d |
|- ( ( ( ( ph /\ s e. RR ) /\ -. ( s mod ( 2 x. _pi ) ) = 0 ) /\ ( s mod _pi ) = 0 ) -> ( ( N + ( 1 / 2 ) ) x. ( ( ( 2 x. ( |_ ` ( ( s / _pi ) / 2 ) ) ) + 1 ) x. _pi ) ) = ( ( N + ( 1 / 2 ) ) x. ( ( s / _pi ) x. _pi ) ) ) |
163 |
162
|
fveq2d |
|- ( ( ( ( ph /\ s e. RR ) /\ -. ( s mod ( 2 x. _pi ) ) = 0 ) /\ ( s mod _pi ) = 0 ) -> ( sin ` ( ( N + ( 1 / 2 ) ) x. ( ( ( 2 x. ( |_ ` ( ( s / _pi ) / 2 ) ) ) + 1 ) x. _pi ) ) ) = ( sin ` ( ( N + ( 1 / 2 ) ) x. ( ( s / _pi ) x. _pi ) ) ) ) |
164 |
161
|
fvoveq1d |
|- ( ( ( ( ph /\ s e. RR ) /\ -. ( s mod ( 2 x. _pi ) ) = 0 ) /\ ( s mod _pi ) = 0 ) -> ( sin ` ( ( ( ( 2 x. ( |_ ` ( ( s / _pi ) / 2 ) ) ) + 1 ) x. _pi ) / 2 ) ) = ( sin ` ( ( ( s / _pi ) x. _pi ) / 2 ) ) ) |
165 |
164
|
oveq2d |
|- ( ( ( ( ph /\ s e. RR ) /\ -. ( s mod ( 2 x. _pi ) ) = 0 ) /\ ( s mod _pi ) = 0 ) -> ( ( 2 x. _pi ) x. ( sin ` ( ( ( ( 2 x. ( |_ ` ( ( s / _pi ) / 2 ) ) ) + 1 ) x. _pi ) / 2 ) ) ) = ( ( 2 x. _pi ) x. ( sin ` ( ( ( s / _pi ) x. _pi ) / 2 ) ) ) ) |
166 |
163 165
|
oveq12d |
|- ( ( ( ( ph /\ s e. RR ) /\ -. ( s mod ( 2 x. _pi ) ) = 0 ) /\ ( s mod _pi ) = 0 ) -> ( ( sin ` ( ( N + ( 1 / 2 ) ) x. ( ( ( 2 x. ( |_ ` ( ( s / _pi ) / 2 ) ) ) + 1 ) x. _pi ) ) ) / ( ( 2 x. _pi ) x. ( sin ` ( ( ( ( 2 x. ( |_ ` ( ( s / _pi ) / 2 ) ) ) + 1 ) x. _pi ) / 2 ) ) ) ) = ( ( sin ` ( ( N + ( 1 / 2 ) ) x. ( ( s / _pi ) x. _pi ) ) ) / ( ( 2 x. _pi ) x. ( sin ` ( ( ( s / _pi ) x. _pi ) / 2 ) ) ) ) ) |
167 |
98
|
oveq2d |
|- ( s e. RR -> ( ( N + ( 1 / 2 ) ) x. ( ( s / _pi ) x. _pi ) ) = ( ( N + ( 1 / 2 ) ) x. s ) ) |
168 |
167
|
fveq2d |
|- ( s e. RR -> ( sin ` ( ( N + ( 1 / 2 ) ) x. ( ( s / _pi ) x. _pi ) ) ) = ( sin ` ( ( N + ( 1 / 2 ) ) x. s ) ) ) |
169 |
98
|
fvoveq1d |
|- ( s e. RR -> ( sin ` ( ( ( s / _pi ) x. _pi ) / 2 ) ) = ( sin ` ( s / 2 ) ) ) |
170 |
169
|
oveq2d |
|- ( s e. RR -> ( ( 2 x. _pi ) x. ( sin ` ( ( ( s / _pi ) x. _pi ) / 2 ) ) ) = ( ( 2 x. _pi ) x. ( sin ` ( s / 2 ) ) ) ) |
171 |
168 170
|
oveq12d |
|- ( s e. RR -> ( ( sin ` ( ( N + ( 1 / 2 ) ) x. ( ( s / _pi ) x. _pi ) ) ) / ( ( 2 x. _pi ) x. ( sin ` ( ( ( s / _pi ) x. _pi ) / 2 ) ) ) ) = ( ( sin ` ( ( N + ( 1 / 2 ) ) x. s ) ) / ( ( 2 x. _pi ) x. ( sin ` ( s / 2 ) ) ) ) ) |
172 |
171
|
adantl |
|- ( ( ph /\ s e. RR ) -> ( ( sin ` ( ( N + ( 1 / 2 ) ) x. ( ( s / _pi ) x. _pi ) ) ) / ( ( 2 x. _pi ) x. ( sin ` ( ( ( s / _pi ) x. _pi ) / 2 ) ) ) ) = ( ( sin ` ( ( N + ( 1 / 2 ) ) x. s ) ) / ( ( 2 x. _pi ) x. ( sin ` ( s / 2 ) ) ) ) ) |
173 |
172
|
ad2antrr |
|- ( ( ( ( ph /\ s e. RR ) /\ -. ( s mod ( 2 x. _pi ) ) = 0 ) /\ ( s mod _pi ) = 0 ) -> ( ( sin ` ( ( N + ( 1 / 2 ) ) x. ( ( s / _pi ) x. _pi ) ) ) / ( ( 2 x. _pi ) x. ( sin ` ( ( ( s / _pi ) x. _pi ) / 2 ) ) ) ) = ( ( sin ` ( ( N + ( 1 / 2 ) ) x. s ) ) / ( ( 2 x. _pi ) x. ( sin ` ( s / 2 ) ) ) ) ) |
174 |
166 173
|
eqtrd |
|- ( ( ( ( ph /\ s e. RR ) /\ -. ( s mod ( 2 x. _pi ) ) = 0 ) /\ ( s mod _pi ) = 0 ) -> ( ( sin ` ( ( N + ( 1 / 2 ) ) x. ( ( ( 2 x. ( |_ ` ( ( s / _pi ) / 2 ) ) ) + 1 ) x. _pi ) ) ) / ( ( 2 x. _pi ) x. ( sin ` ( ( ( ( 2 x. ( |_ ` ( ( s / _pi ) / 2 ) ) ) + 1 ) x. _pi ) / 2 ) ) ) ) = ( ( sin ` ( ( N + ( 1 / 2 ) ) x. s ) ) / ( ( 2 x. _pi ) x. ( sin ` ( s / 2 ) ) ) ) ) |
175 |
149 158 174
|
3eqtrrd |
|- ( ( ( ( ph /\ s e. RR ) /\ -. ( s mod ( 2 x. _pi ) ) = 0 ) /\ ( s mod _pi ) = 0 ) -> ( ( sin ` ( ( N + ( 1 / 2 ) ) x. s ) ) / ( ( 2 x. _pi ) x. ( sin ` ( s / 2 ) ) ) ) = ( ( ( 1 / 2 ) + sum_ k e. ( 1 ... N ) ( cos ` ( k x. s ) ) ) / _pi ) ) |
176 |
|
simplr |
|- ( ( ( ph /\ s e. RR ) /\ -. ( s mod _pi ) = 0 ) -> s e. RR ) |
177 |
|
simpr |
|- ( ( ( ph /\ s e. RR ) /\ -. ( s mod _pi ) = 0 ) -> -. ( s mod _pi ) = 0 ) |
178 |
176 41 103
|
sylancl |
|- ( ( ( ph /\ s e. RR ) /\ -. ( s mod _pi ) = 0 ) -> ( ( s mod _pi ) = 0 <-> ( s / _pi ) e. ZZ ) ) |
179 |
177 178
|
mtbid |
|- ( ( ( ph /\ s e. RR ) /\ -. ( s mod _pi ) = 0 ) -> -. ( s / _pi ) e. ZZ ) |
180 |
176
|
recnd |
|- ( ( ( ph /\ s e. RR ) /\ -. ( s mod _pi ) = 0 ) -> s e. CC ) |
181 |
|
sineq0 |
|- ( s e. CC -> ( ( sin ` s ) = 0 <-> ( s / _pi ) e. ZZ ) ) |
182 |
180 181
|
syl |
|- ( ( ( ph /\ s e. RR ) /\ -. ( s mod _pi ) = 0 ) -> ( ( sin ` s ) = 0 <-> ( s / _pi ) e. ZZ ) ) |
183 |
179 182
|
mtbird |
|- ( ( ( ph /\ s e. RR ) /\ -. ( s mod _pi ) = 0 ) -> -. ( sin ` s ) = 0 ) |
184 |
183
|
neqned |
|- ( ( ( ph /\ s e. RR ) /\ -. ( s mod _pi ) = 0 ) -> ( sin ` s ) =/= 0 ) |
185 |
2
|
ad2antrr |
|- ( ( ( ph /\ s e. RR ) /\ -. ( s mod _pi ) = 0 ) -> N e. NN ) |
186 |
176 184 185
|
dirkertrigeqlem2 |
|- ( ( ( ph /\ s e. RR ) /\ -. ( s mod _pi ) = 0 ) -> ( ( ( 1 / 2 ) + sum_ k e. ( 1 ... N ) ( cos ` ( k x. s ) ) ) / _pi ) = ( ( sin ` ( ( N + ( 1 / 2 ) ) x. s ) ) / ( ( 2 x. _pi ) x. ( sin ` ( s / 2 ) ) ) ) ) |
187 |
186
|
eqcomd |
|- ( ( ( ph /\ s e. RR ) /\ -. ( s mod _pi ) = 0 ) -> ( ( sin ` ( ( N + ( 1 / 2 ) ) x. s ) ) / ( ( 2 x. _pi ) x. ( sin ` ( s / 2 ) ) ) ) = ( ( ( 1 / 2 ) + sum_ k e. ( 1 ... N ) ( cos ` ( k x. s ) ) ) / _pi ) ) |
188 |
187
|
adantlr |
|- ( ( ( ( ph /\ s e. RR ) /\ -. ( s mod ( 2 x. _pi ) ) = 0 ) /\ -. ( s mod _pi ) = 0 ) -> ( ( sin ` ( ( N + ( 1 / 2 ) ) x. s ) ) / ( ( 2 x. _pi ) x. ( sin ` ( s / 2 ) ) ) ) = ( ( ( 1 / 2 ) + sum_ k e. ( 1 ... N ) ( cos ` ( k x. s ) ) ) / _pi ) ) |
189 |
175 188
|
pm2.61dan |
|- ( ( ( ph /\ s e. RR ) /\ -. ( s mod ( 2 x. _pi ) ) = 0 ) -> ( ( sin ` ( ( N + ( 1 / 2 ) ) x. s ) ) / ( ( 2 x. _pi ) x. ( sin ` ( s / 2 ) ) ) ) = ( ( ( 1 / 2 ) + sum_ k e. ( 1 ... N ) ( cos ` ( k x. s ) ) ) / _pi ) ) |
190 |
95 189
|
eqtr2d |
|- ( ( ( ph /\ s e. RR ) /\ -. ( s mod ( 2 x. _pi ) ) = 0 ) -> ( ( ( 1 / 2 ) + sum_ k e. ( 1 ... N ) ( cos ` ( k x. s ) ) ) / _pi ) = if ( ( s mod ( 2 x. _pi ) ) = 0 , ( ( ( 2 x. N ) + 1 ) / ( 2 x. _pi ) ) , ( ( sin ` ( ( N + ( 1 / 2 ) ) x. s ) ) / ( ( 2 x. _pi ) x. ( sin ` ( s / 2 ) ) ) ) ) ) |
191 |
93 190
|
pm2.61dan |
|- ( ( ph /\ s e. RR ) -> ( ( ( 1 / 2 ) + sum_ k e. ( 1 ... N ) ( cos ` ( k x. s ) ) ) / _pi ) = if ( ( s mod ( 2 x. _pi ) ) = 0 , ( ( ( 2 x. N ) + 1 ) / ( 2 x. _pi ) ) , ( ( sin ` ( ( N + ( 1 / 2 ) ) x. s ) ) / ( ( 2 x. _pi ) x. ( sin ` ( s / 2 ) ) ) ) ) ) |
192 |
191
|
mpteq2dva |
|- ( ph -> ( s e. RR |-> ( ( ( 1 / 2 ) + sum_ k e. ( 1 ... N ) ( cos ` ( k x. s ) ) ) / _pi ) ) = ( s e. RR |-> if ( ( s mod ( 2 x. _pi ) ) = 0 , ( ( ( 2 x. N ) + 1 ) / ( 2 x. _pi ) ) , ( ( sin ` ( ( N + ( 1 / 2 ) ) x. s ) ) / ( ( 2 x. _pi ) x. ( sin ` ( s / 2 ) ) ) ) ) ) ) |
193 |
4 192
|
eqtr2id |
|- ( ph -> ( s e. RR |-> if ( ( s mod ( 2 x. _pi ) ) = 0 , ( ( ( 2 x. N ) + 1 ) / ( 2 x. _pi ) ) , ( ( sin ` ( ( N + ( 1 / 2 ) ) x. s ) ) / ( ( 2 x. _pi ) x. ( sin ` ( s / 2 ) ) ) ) ) ) = H ) |
194 |
5 7 193
|
3eqtrd |
|- ( ph -> F = H ) |