Step |
Hyp |
Ref |
Expression |
1 |
|
fourierclimd.f |
|- ( ph -> F : RR --> RR ) |
2 |
|
fourierclimd.t |
|- T = ( 2 x. _pi ) |
3 |
|
fourierclimd.per |
|- ( ( ph /\ x e. RR ) -> ( F ` ( x + T ) ) = ( F ` x ) ) |
4 |
|
fourierclimd.g |
|- G = ( ( RR _D F ) |` ( -u _pi (,) _pi ) ) |
5 |
|
fourierclimd.dmdv |
|- ( ph -> ( ( -u _pi (,) _pi ) \ dom G ) e. Fin ) |
6 |
|
fourierclimd.dvcn |
|- ( ph -> G e. ( dom G -cn-> CC ) ) |
7 |
|
fourierclimd.rlim |
|- ( ( ph /\ x e. ( ( -u _pi [,) _pi ) \ dom G ) ) -> ( ( G |` ( x (,) +oo ) ) limCC x ) =/= (/) ) |
8 |
|
fourierclimd.llim |
|- ( ( ph /\ x e. ( ( -u _pi (,] _pi ) \ dom G ) ) -> ( ( G |` ( -oo (,) x ) ) limCC x ) =/= (/) ) |
9 |
|
fourierclimd.x |
|- ( ph -> X e. RR ) |
10 |
|
fourierclimd.l |
|- ( ph -> L e. ( ( F |` ( -oo (,) X ) ) limCC X ) ) |
11 |
|
fourierclimd.r |
|- ( ph -> R e. ( ( F |` ( X (,) +oo ) ) limCC X ) ) |
12 |
|
fourierclimd.a |
|- A = ( n e. NN0 |-> ( S. ( -u _pi (,) _pi ) ( ( F ` x ) x. ( cos ` ( n x. x ) ) ) _d x / _pi ) ) |
13 |
|
fourierclimd.b |
|- B = ( n e. NN |-> ( S. ( -u _pi (,) _pi ) ( ( F ` x ) x. ( sin ` ( n x. x ) ) ) _d x / _pi ) ) |
14 |
|
fourierclimd.s |
|- S = ( n e. NN |-> ( ( ( A ` n ) x. ( cos ` ( n x. X ) ) ) + ( ( B ` n ) x. ( sin ` ( n x. X ) ) ) ) ) |
15 |
|
nfcv |
|- F/_ k ( ( ( A ` n ) x. ( cos ` ( n x. X ) ) ) + ( ( B ` n ) x. ( sin ` ( n x. X ) ) ) ) |
16 |
|
nfmpt1 |
|- F/_ n ( n e. NN0 |-> ( S. ( -u _pi (,) _pi ) ( ( F ` x ) x. ( cos ` ( n x. x ) ) ) _d x / _pi ) ) |
17 |
12 16
|
nfcxfr |
|- F/_ n A |
18 |
|
nfcv |
|- F/_ n k |
19 |
17 18
|
nffv |
|- F/_ n ( A ` k ) |
20 |
|
nfcv |
|- F/_ n x. |
21 |
|
nfcv |
|- F/_ n ( cos ` ( k x. X ) ) |
22 |
19 20 21
|
nfov |
|- F/_ n ( ( A ` k ) x. ( cos ` ( k x. X ) ) ) |
23 |
|
nfcv |
|- F/_ n + |
24 |
|
nfmpt1 |
|- F/_ n ( n e. NN |-> ( S. ( -u _pi (,) _pi ) ( ( F ` x ) x. ( sin ` ( n x. x ) ) ) _d x / _pi ) ) |
25 |
13 24
|
nfcxfr |
|- F/_ n B |
26 |
25 18
|
nffv |
|- F/_ n ( B ` k ) |
27 |
|
nfcv |
|- F/_ n ( sin ` ( k x. X ) ) |
28 |
26 20 27
|
nfov |
|- F/_ n ( ( B ` k ) x. ( sin ` ( k x. X ) ) ) |
29 |
22 23 28
|
nfov |
|- F/_ n ( ( ( A ` k ) x. ( cos ` ( k x. X ) ) ) + ( ( B ` k ) x. ( sin ` ( k x. X ) ) ) ) |
30 |
|
fveq2 |
|- ( n = k -> ( A ` n ) = ( A ` k ) ) |
31 |
|
oveq1 |
|- ( n = k -> ( n x. X ) = ( k x. X ) ) |
32 |
31
|
fveq2d |
|- ( n = k -> ( cos ` ( n x. X ) ) = ( cos ` ( k x. X ) ) ) |
33 |
30 32
|
oveq12d |
|- ( n = k -> ( ( A ` n ) x. ( cos ` ( n x. X ) ) ) = ( ( A ` k ) x. ( cos ` ( k x. X ) ) ) ) |
34 |
|
fveq2 |
|- ( n = k -> ( B ` n ) = ( B ` k ) ) |
35 |
31
|
fveq2d |
|- ( n = k -> ( sin ` ( n x. X ) ) = ( sin ` ( k x. X ) ) ) |
36 |
34 35
|
oveq12d |
|- ( n = k -> ( ( B ` n ) x. ( sin ` ( n x. X ) ) ) = ( ( B ` k ) x. ( sin ` ( k x. X ) ) ) ) |
37 |
33 36
|
oveq12d |
|- ( n = k -> ( ( ( A ` n ) x. ( cos ` ( n x. X ) ) ) + ( ( B ` n ) x. ( sin ` ( n x. X ) ) ) ) = ( ( ( A ` k ) x. ( cos ` ( k x. X ) ) ) + ( ( B ` k ) x. ( sin ` ( k x. X ) ) ) ) ) |
38 |
15 29 37
|
cbvmpt |
|- ( n e. NN |-> ( ( ( A ` n ) x. ( cos ` ( n x. X ) ) ) + ( ( B ` n ) x. ( sin ` ( n x. X ) ) ) ) ) = ( k e. NN |-> ( ( ( A ` k ) x. ( cos ` ( k x. X ) ) ) + ( ( B ` k ) x. ( sin ` ( k x. X ) ) ) ) ) |
39 |
14 38
|
eqtri |
|- S = ( k e. NN |-> ( ( ( A ` k ) x. ( cos ` ( k x. X ) ) ) + ( ( B ` k ) x. ( sin ` ( k x. X ) ) ) ) ) |
40 |
1 2 3 4 5 6 7 8 9 10 11 12 13 39
|
fourierdlem115 |
|- ( ph -> ( seq 1 ( + , S ) ~~> ( ( ( L + R ) / 2 ) - ( ( A ` 0 ) / 2 ) ) /\ ( ( ( A ` 0 ) / 2 ) + sum_ n e. NN ( ( ( A ` n ) x. ( cos ` ( n x. X ) ) ) + ( ( B ` n ) x. ( sin ` ( n x. X ) ) ) ) ) = ( ( L + R ) / 2 ) ) ) |
41 |
40
|
simpld |
|- ( ph -> seq 1 ( + , S ) ~~> ( ( ( L + R ) / 2 ) - ( ( A ` 0 ) / 2 ) ) ) |