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Theorem fourierdlem115 38197
Description: Fourier serier convergence, for piecewise smooth functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
fourierdlem115.f  |-  ( ph  ->  F : RR --> RR )
fourierdlem115.t  |-  T  =  ( 2  x.  pi )
fourierdlem115.per  |-  ( (
ph  /\  x  e.  RR )  ->  ( F `
 ( x  +  T ) )  =  ( F `  x
) )
fourierdlem115.g  |-  G  =  ( ( RR  _D  F )  |`  ( -u pi (,) pi ) )
fourierdlem115.dmdv  |-  ( ph  ->  ( ( -u pi (,) pi )  \  dom  G )  e.  Fin )
fourierdlem115.dvcn  |-  ( ph  ->  G  e.  ( dom 
G -cn-> CC ) )
fourierdlem115.rlim  |-  ( (
ph  /\  x  e.  ( ( -u pi [,) pi )  \  dom  G ) )  ->  (
( G  |`  (
x (,) +oo )
) lim CC  x )  =/=  (/) )
fourierdlem115.llim  |-  ( (
ph  /\  x  e.  ( ( -u pi (,] pi )  \  dom  G ) )  ->  (
( G  |`  ( -oo (,) x ) ) lim
CC  x )  =/=  (/) )
fourierdlem115.x  |-  ( ph  ->  X  e.  RR )
fourierdlem115.l  |-  ( ph  ->  L  e.  ( ( F  |`  ( -oo (,) X ) ) lim CC  X ) )
fourierdlem115.r  |-  ( ph  ->  R  e.  ( ( F  |`  ( X (,) +oo ) ) lim CC  X ) )
fourierdlem115.a  |-  A  =  ( n  e.  NN0  |->  ( S. ( -u pi (,) pi ) ( ( F `  x )  x.  ( cos `  (
n  x.  x ) ) )  _d x  /  pi ) )
fourierdlem115.b  |-  B  =  ( n  e.  NN  |->  ( S. ( -u pi (,) pi ) ( ( F `  x )  x.  ( sin `  (
n  x.  x ) ) )  _d x  /  pi ) )
fourierdlem115.s  |-  S  =  ( k  e.  NN  |->  ( ( ( A `
 k )  x.  ( cos `  (
k  x.  X ) ) )  +  ( ( B `  k
)  x.  ( sin `  ( k  x.  X
) ) ) ) )
Assertion
Ref Expression
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 ) ) )
Distinct variable groups:    A, k    B, k    k, F, n, x    k, G, x   
k, L    R, k    T, k, x    k, X, n, x    ph, k, x
Allowed substitution hints:    ph( n)    A( x, n)    B( x, n)    R( x, n)    S( x, k, n)    T( n)    G( n)    L( x, n)

Proof of Theorem fourierdlem115
Dummy variables  z 
f  g  w  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fourierdlem115.f . . . 4  |-  ( ph  ->  F : RR --> RR )
2 fourierdlem115.t . . . 4  |-  T  =  ( 2  x.  pi )
3 fourierdlem115.per . . . 4  |-  ( (
ph  /\  x  e.  RR )  ->  ( F `
 ( x  +  T ) )  =  ( F `  x
) )
4 fourierdlem115.g . . . 4  |-  G  =  ( ( RR  _D  F )  |`  ( -u pi (,) pi ) )
5 fourierdlem115.dmdv . . . 4  |-  ( ph  ->  ( ( -u pi (,) pi )  \  dom  G )  e.  Fin )
6 fourierdlem115.dvcn . . . 4  |-  ( ph  ->  G  e.  ( dom 
G -cn-> CC ) )
7 fourierdlem115.rlim . . . 4  |-  ( (
ph  /\  x  e.  ( ( -u pi [,) pi )  \  dom  G ) )  ->  (
( G  |`  (
x (,) +oo )
) lim CC  x )  =/=  (/) )
8 fourierdlem115.llim . . . 4  |-  ( (
ph  /\  x  e.  ( ( -u pi (,] pi )  \  dom  G ) )  ->  (
( G  |`  ( -oo (,) x ) ) lim
CC  x )  =/=  (/) )
9 fourierdlem115.x . . . 4  |-  ( ph  ->  X  e.  RR )
10 fourierdlem115.l . . . 4  |-  ( ph  ->  L  e.  ( ( F  |`  ( -oo (,) X ) ) lim CC  X ) )
11 fourierdlem115.r . . . 4  |-  ( ph  ->  R  e.  ( ( F  |`  ( X (,) +oo ) ) lim CC  X ) )
12 fourierdlem115.a . . . . 5  |-  A  =  ( n  e.  NN0  |->  ( S. ( -u pi (,) pi ) ( ( F `  x )  x.  ( cos `  (
n  x.  x ) ) )  _d x  /  pi ) )
13 oveq1 6315 . . . . . . . . . . 11  |-  ( n  =  k  ->  (
n  x.  x )  =  ( k  x.  x ) )
1413fveq2d 5883 . . . . . . . . . 10  |-  ( n  =  k  ->  ( cos `  ( n  x.  x ) )  =  ( cos `  (
k  x.  x ) ) )
1514oveq2d 6324 . . . . . . . . 9  |-  ( n  =  k  ->  (
( F `  x
)  x.  ( cos `  ( n  x.  x
) ) )  =  ( ( F `  x )  x.  ( cos `  ( k  x.  x ) ) ) )
1615adantr 472 . . . . . . . 8  |-  ( ( n  =  k  /\  x  e.  ( -u pi (,) pi ) )  -> 
( ( F `  x )  x.  ( cos `  ( n  x.  x ) ) )  =  ( ( F `
 x )  x.  ( cos `  (
k  x.  x ) ) ) )
1716itgeq2dv 22818 . . . . . . 7  |-  ( n  =  k  ->  S. ( -u pi (,) pi ) ( ( F `
 x )  x.  ( cos `  (
n  x.  x ) ) )  _d x  =  S. ( -u pi (,) pi ) ( ( F `  x
)  x.  ( cos `  ( k  x.  x
) ) )  _d x )
1817oveq1d 6323 . . . . . 6  |-  ( n  =  k  ->  ( S. ( -u pi (,) pi ) ( ( F `
 x )  x.  ( cos `  (
n  x.  x ) ) )  _d x  /  pi )  =  ( S. ( -u pi (,) pi ) ( ( F `  x
)  x.  ( cos `  ( k  x.  x
) ) )  _d x  /  pi ) )
1918cbvmptv 4488 . . . . 5  |-  ( n  e.  NN0  |->  ( S. ( -u pi (,) pi ) ( ( F `
 x )  x.  ( cos `  (
n  x.  x ) ) )  _d x  /  pi ) )  =  ( k  e. 
NN0  |->  ( S. (
-u pi (,) pi ) ( ( F `
 x )  x.  ( cos `  (
k  x.  x ) ) )  _d x  /  pi ) )
2012, 19eqtri 2493 . . . 4  |-  A  =  ( k  e.  NN0  |->  ( S. ( -u pi (,) pi ) ( ( F `  x )  x.  ( cos `  (
k  x.  x ) ) )  _d x  /  pi ) )
21 fourierdlem115.b . . . . 5  |-  B  =  ( n  e.  NN  |->  ( S. ( -u pi (,) pi ) ( ( F `  x )  x.  ( sin `  (
n  x.  x ) ) )  _d x  /  pi ) )
2213fveq2d 5883 . . . . . . . . . 10  |-  ( n  =  k  ->  ( sin `  ( n  x.  x ) )  =  ( sin `  (
k  x.  x ) ) )
2322oveq2d 6324 . . . . . . . . 9  |-  ( n  =  k  ->  (
( F `  x
)  x.  ( sin `  ( n  x.  x
) ) )  =  ( ( F `  x )  x.  ( sin `  ( k  x.  x ) ) ) )
2423adantr 472 . . . . . . . 8  |-  ( ( n  =  k  /\  x  e.  ( -u pi (,) pi ) )  -> 
( ( F `  x )  x.  ( sin `  ( n  x.  x ) ) )  =  ( ( F `
 x )  x.  ( sin `  (
k  x.  x ) ) ) )
2524itgeq2dv 22818 . . . . . . 7  |-  ( n  =  k  ->  S. ( -u pi (,) pi ) ( ( F `
 x )  x.  ( sin `  (
n  x.  x ) ) )  _d x  =  S. ( -u pi (,) pi ) ( ( F `  x
)  x.  ( sin `  ( k  x.  x
) ) )  _d x )
2625oveq1d 6323 . . . . . 6  |-  ( n  =  k  ->  ( S. ( -u pi (,) pi ) ( ( F `
 x )  x.  ( sin `  (
n  x.  x ) ) )  _d x  /  pi )  =  ( S. ( -u pi (,) pi ) ( ( F `  x
)  x.  ( sin `  ( k  x.  x
) ) )  _d x  /  pi ) )
2726cbvmptv 4488 . . . . 5  |-  ( n  e.  NN  |->  ( S. ( -u pi (,) pi ) ( ( F `
 x )  x.  ( sin `  (
n  x.  x ) ) )  _d x  /  pi ) )  =  ( k  e.  NN  |->  ( S. (
-u pi (,) pi ) ( ( F `
 x )  x.  ( sin `  (
k  x.  x ) ) )  _d x  /  pi ) )
2821, 27eqtri 2493 . . . 4  |-  B  =  ( k  e.  NN  |->  ( S. ( -u pi (,) pi ) ( ( F `  x )  x.  ( sin `  (
k  x.  x ) ) )  _d x  /  pi ) )
29 fourierdlem115.s . . . 4  |-  S  =  ( k  e.  NN  |->  ( ( ( A `
 k )  x.  ( cos `  (
k  x.  X ) ) )  +  ( ( B `  k
)  x.  ( sin `  ( k  x.  X
) ) ) ) )
30 eqid 2471 . . . 4  |-  ( k  e.  NN  |->  { w  e.  ( RR  ^m  (
0 ... k ) )  |  ( ( ( w `  0 )  =  -u pi  /\  (
w `  k )  =  pi )  /\  A. z  e.  ( 0..^ k ) ( w `
 z )  < 
( w `  (
z  +  1 ) ) ) } )  =  ( k  e.  NN  |->  { w  e.  ( RR  ^m  (
0 ... k ) )  |  ( ( ( w `  0 )  =  -u pi  /\  (
w `  k )  =  pi )  /\  A. z  e.  ( 0..^ k ) ( w `
 z )  < 
( w `  (
z  +  1 ) ) ) } )
31 id 22 . . . . . 6  |-  ( y  =  x  ->  y  =  x )
32 oveq2 6316 . . . . . . . . 9  |-  ( y  =  x  ->  (
pi  -  y )  =  ( pi  -  x ) )
3332oveq1d 6323 . . . . . . . 8  |-  ( y  =  x  ->  (
( pi  -  y
)  /  T )  =  ( ( pi 
-  x )  /  T ) )
3433fveq2d 5883 . . . . . . 7  |-  ( y  =  x  ->  ( |_ `  ( ( pi 
-  y )  /  T ) )  =  ( |_ `  (
( pi  -  x
)  /  T ) ) )
3534oveq1d 6323 . . . . . 6  |-  ( y  =  x  ->  (
( |_ `  (
( pi  -  y
)  /  T ) )  x.  T )  =  ( ( |_
`  ( ( pi 
-  x )  /  T ) )  x.  T ) )
3631, 35oveq12d 6326 . . . . 5  |-  ( y  =  x  ->  (
y  +  ( ( |_ `  ( ( pi  -  y )  /  T ) )  x.  T ) )  =  ( x  +  ( ( |_ `  ( ( pi  -  x )  /  T
) )  x.  T
) ) )
3736cbvmptv 4488 . . . 4  |-  ( y  e.  RR  |->  ( y  +  ( ( |_
`  ( ( pi 
-  y )  /  T ) )  x.  T ) ) )  =  ( x  e.  RR  |->  ( x  +  ( ( |_ `  ( ( pi  -  x )  /  T
) )  x.  T
) ) )
38 eqid 2471 . . . 4  |-  ( {
-u pi ,  pi ,  ( ( y  e.  RR  |->  ( y  +  ( ( |_
`  ( ( pi 
-  y )  /  T ) )  x.  T ) ) ) `
 X ) }  u.  ( ( -u pi [,] pi )  \  dom  G ) )  =  ( { -u pi ,  pi ,  ( ( y  e.  RR  |->  ( y  +  ( ( |_ `  ( ( pi  -  y )  /  T ) )  x.  T ) ) ) `  X ) }  u.  ( (
-u pi [,] pi )  \  dom  G ) )
39 eqid 2471 . . . 4  |-  ( (
# `  ( { -u pi ,  pi , 
( ( y  e.  RR  |->  ( y  +  ( ( |_ `  ( ( pi  -  y )  /  T
) )  x.  T
) ) ) `  X ) }  u.  ( ( -u pi [,] pi )  \  dom  G ) ) )  - 
1 )  =  ( ( # `  ( { -u pi ,  pi ,  ( ( y  e.  RR  |->  ( y  +  ( ( |_
`  ( ( pi 
-  y )  /  T ) )  x.  T ) ) ) `
 X ) }  u.  ( ( -u pi [,] pi )  \  dom  G ) ) )  -  1 )
40 isoeq1 6228 . . . . 5  |-  ( g  =  f  ->  (
g  Isom  <  ,  <  ( ( 0 ... (
( # `  ( {
-u pi ,  pi ,  ( ( y  e.  RR  |->  ( y  +  ( ( |_
`  ( ( pi 
-  y )  /  T ) )  x.  T ) ) ) `
 X ) }  u.  ( ( -u pi [,] pi )  \  dom  G ) ) )  -  1 ) ) ,  ( { -u pi ,  pi , 
( ( y  e.  RR  |->  ( y  +  ( ( |_ `  ( ( pi  -  y )  /  T
) )  x.  T
) ) ) `  X ) }  u.  ( ( -u pi [,] pi )  \  dom  G ) ) )  <->  f  Isom  <  ,  <  ( ( 0 ... ( ( # `  ( { -u pi ,  pi ,  ( ( y  e.  RR  |->  ( y  +  ( ( |_ `  ( ( pi  -  y )  /  T ) )  x.  T ) ) ) `  X ) }  u.  ( (
-u pi [,] pi )  \  dom  G ) ) )  -  1 ) ) ,  ( { -u pi ,  pi ,  ( (
y  e.  RR  |->  ( y  +  ( ( |_ `  ( ( pi  -  y )  /  T ) )  x.  T ) ) ) `  X ) }  u.  ( (
-u pi [,] pi )  \  dom  G ) ) ) ) )
4140cbviotav 5559 . . . 4  |-  ( iota g g  Isom  <  ,  <  ( ( 0 ... ( ( # `  ( { -u pi ,  pi ,  ( ( y  e.  RR  |->  ( y  +  ( ( |_ `  ( ( pi  -  y )  /  T ) )  x.  T ) ) ) `  X ) }  u.  ( (
-u pi [,] pi )  \  dom  G ) ) )  -  1 ) ) ,  ( { -u pi ,  pi ,  ( (
y  e.  RR  |->  ( y  +  ( ( |_ `  ( ( pi  -  y )  /  T ) )  x.  T ) ) ) `  X ) }  u.  ( (
-u pi [,] pi )  \  dom  G ) ) ) )  =  ( iota f f 
Isom  <  ,  <  (
( 0 ... (
( # `  ( {
-u pi ,  pi ,  ( ( y  e.  RR  |->  ( y  +  ( ( |_
`  ( ( pi 
-  y )  /  T ) )  x.  T ) ) ) `
 X ) }  u.  ( ( -u pi [,] pi )  \  dom  G ) ) )  -  1 ) ) ,  ( { -u pi ,  pi , 
( ( y  e.  RR  |->  ( y  +  ( ( |_ `  ( ( pi  -  y )  /  T
) )  x.  T
) ) ) `  X ) }  u.  ( ( -u pi [,] pi )  \  dom  G ) ) ) )
421, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 20, 28, 29, 30, 37, 38, 39, 41fourierdlem114 38196 . . 3  |-  ( ph  ->  (  seq 1 (  +  ,  S )  ~~>  ( ( ( L  +  R )  / 
2 )  -  (
( A `  0
)  /  2 ) )  /\  ( ( ( A `  0
)  /  2 )  +  sum_ k  e.  NN  ( ( ( A `
 k )  x.  ( cos `  (
k  x.  X ) ) )  +  ( ( B `  k
)  x.  ( sin `  ( k  x.  X
) ) ) ) )  =  ( ( L  +  R )  /  2 ) ) )
4342simpld 466 . 2  |-  ( ph  ->  seq 1 (  +  ,  S )  ~~>  ( ( ( L  +  R
)  /  2 )  -  ( ( A `
 0 )  / 
2 ) ) )
44 nfcv 2612 . . . . 5  |-  F/_ k
( ( ( A `
 n )  x.  ( cos `  (
n  x.  X ) ) )  +  ( ( B `  n
)  x.  ( sin `  ( n  x.  X
) ) ) )
45 nfmpt1 4485 . . . . . . . . 9  |-  F/_ n
( n  e.  NN0  |->  ( S. ( -u pi (,) pi ) ( ( F `  x )  x.  ( cos `  (
n  x.  x ) ) )  _d x  /  pi ) )
4612, 45nfcxfr 2610 . . . . . . . 8  |-  F/_ n A
47 nfcv 2612 . . . . . . . 8  |-  F/_ n
k
4846, 47nffv 5886 . . . . . . 7  |-  F/_ n
( A `  k
)
49 nfcv 2612 . . . . . . 7  |-  F/_ n  x.
50 nfcv 2612 . . . . . . 7  |-  F/_ n
( cos `  (
k  x.  X ) )
5148, 49, 50nfov 6334 . . . . . 6  |-  F/_ n
( ( A `  k )  x.  ( cos `  ( k  x.  X ) ) )
52 nfcv 2612 . . . . . 6  |-  F/_ n  +
53 nfmpt1 4485 . . . . . . . . 9  |-  F/_ n
( n  e.  NN  |->  ( S. ( -u pi (,) pi ) ( ( F `  x )  x.  ( sin `  (
n  x.  x ) ) )  _d x  /  pi ) )
5421, 53nfcxfr 2610 . . . . . . . 8  |-  F/_ n B
5554, 47nffv 5886 . . . . . . 7  |-  F/_ n
( B `  k
)
56 nfcv 2612 . . . . . . 7  |-  F/_ n
( sin `  (
k  x.  X ) )
5755, 49, 56nfov 6334 . . . . . 6  |-  F/_ n
( ( B `  k )  x.  ( sin `  ( k  x.  X ) ) )
5851, 52, 57nfov 6334 . . . . 5  |-  F/_ n
( ( ( A `
 k )  x.  ( cos `  (
k  x.  X ) ) )  +  ( ( B `  k
)  x.  ( sin `  ( k  x.  X
) ) ) )
59 fveq2 5879 . . . . . . 7  |-  ( n  =  k  ->  ( A `  n )  =  ( A `  k ) )
60 oveq1 6315 . . . . . . . 8  |-  ( n  =  k  ->  (
n  x.  X )  =  ( k  x.  X ) )
6160fveq2d 5883 . . . . . . 7  |-  ( n  =  k  ->  ( cos `  ( n  x.  X ) )  =  ( cos `  (
k  x.  X ) ) )
6259, 61oveq12d 6326 . . . . . 6  |-  ( n  =  k  ->  (
( A `  n
)  x.  ( cos `  ( n  x.  X
) ) )  =  ( ( A `  k )  x.  ( cos `  ( k  x.  X ) ) ) )
63 fveq2 5879 . . . . . . 7  |-  ( n  =  k  ->  ( B `  n )  =  ( B `  k ) )
6460fveq2d 5883 . . . . . . 7  |-  ( n  =  k  ->  ( sin `  ( n  x.  X ) )  =  ( sin `  (
k  x.  X ) ) )
6563, 64oveq12d 6326 . . . . . 6  |-  ( n  =  k  ->  (
( B `  n
)  x.  ( sin `  ( n  x.  X
) ) )  =  ( ( B `  k )  x.  ( sin `  ( k  x.  X ) ) ) )
6662, 65oveq12d 6326 . . . . 5  |-  ( 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
) ) ) ) )
6744, 58, 66cbvsumi 13840 . . . 4  |-  sum_ n  e.  NN  ( ( ( A `  n )  x.  ( cos `  (
n  x.  X ) ) )  +  ( ( B `  n
)  x.  ( sin `  ( n  x.  X
) ) ) )  =  sum_ k  e.  NN  ( ( ( A `
 k )  x.  ( cos `  (
k  x.  X ) ) )  +  ( ( B `  k
)  x.  ( sin `  ( k  x.  X
) ) ) )
6867oveq2i 6319 . . 3  |-  ( ( ( A `  0
)  /  2 )  +  sum_ n  e.  NN  ( ( ( A `
 n )  x.  ( cos `  (
n  x.  X ) ) )  +  ( ( B `  n
)  x.  ( sin `  ( n  x.  X
) ) ) ) )  =  ( ( ( A `  0
)  /  2 )  +  sum_ k  e.  NN  ( ( ( A `
 k )  x.  ( cos `  (
k  x.  X ) ) )  +  ( ( B `  k
)  x.  ( sin `  ( k  x.  X
) ) ) ) )
6942simprd 470 . . 3  |-  ( ph  ->  ( ( ( A `
 0 )  / 
2 )  +  sum_ k  e.  NN  (
( ( A `  k )  x.  ( cos `  ( k  x.  X ) ) )  +  ( ( B `
 k )  x.  ( sin `  (
k  x.  X ) ) ) ) )  =  ( ( L  +  R )  / 
2 ) )
7068, 69syl5eq 2517 . 2  |-  ( ph  ->  ( ( ( A `
 0 )  / 
2 )  +  sum_ n  e.  NN  ( ( ( A `  n
)  x.  ( cos `  ( n  x.  X
) ) )  +  ( ( B `  n )  x.  ( sin `  ( n  x.  X ) ) ) ) )  =  ( ( L  +  R
)  /  2 ) )
7143, 70jca 541 1  |-  ( 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 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   {crab 2760    \ cdif 3387    u. cun 3388   (/)c0 3722   {ctp 3963   class class class wbr 4395    |-> cmpt 4454   dom cdm 4839    |` cres 4841   iotacio 5551   -->wf 5585   ` cfv 5589    Isom wiso 5590  (class class class)co 6308    ^m cmap 7490   Fincfn 7587   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560    x. cmul 9562   +oocpnf 9690   -oocmnf 9691    < clt 9693    - cmin 9880   -ucneg 9881    / cdiv 10291   NNcn 10631   2c2 10681   NN0cn0 10893   (,)cioo 11660   (,]cioc 11661   [,)cico 11662   [,]cicc 11663   ...cfz 11810  ..^cfzo 11942   |_cfl 12059    seqcseq 12251   #chash 12553    ~~> cli 13625   sum_csu 13829   sincsin 14193   cosccos 14194   picpi 14196   -cn->ccncf 21986   S.citg 22655   lim CC climc 22896    _D cdv 22897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cc 8883  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-disj 4367  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-ofr 6551  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-omul 7205  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-acn 8394  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ioc 11665  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526  df-hash 12554  df-shft 13207  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-limsup 13603  df-clim 13629  df-rlim 13630  df-sum 13830  df-ef 14198  df-sin 14200  df-cos 14201  df-pi 14203  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-t1 20407  df-haus 20408  df-cmp 20479  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-ovol 22494  df-vol 22496  df-mbf 22656  df-itg1 22657  df-itg2 22658  df-ibl 22659  df-itg 22660  df-0p 22707  df-ditg 22881  df-limc 22900  df-dv 22901
This theorem is referenced by:  fourierd  38198  fourierclimd  38199
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