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Theorem fourierdlem94 38176
Description: For a piecewise smooth function, the left and the right limits exist at any point. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
fourierdlem94.f  |-  ( ph  ->  F : RR --> RR )
fourierdlem94.t  |-  T  =  ( 2  x.  pi )
fourierdlem94.per  |-  ( (
ph  /\  x  e.  RR )  ->  ( F `
 ( x  +  T ) )  =  ( F `  x
) )
fourierdlem94.x  |-  ( ph  ->  X  e.  RR )
fourierdlem94.p  |-  P  =  ( n  e.  NN  |->  { p  e.  ( RR  ^m  ( 0 ... n ) )  |  ( ( ( p `
 0 )  = 
-u pi  /\  (
p `  n )  =  pi )  /\  A. i  e.  ( 0..^ n ) ( p `
 i )  < 
( p `  (
i  +  1 ) ) ) } )
fourierdlem94.m  |-  ( ph  ->  M  e.  NN )
fourierdlem94.q  |-  ( ph  ->  Q  e.  ( P `
 M ) )
fourierdlem94.dvcn  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( RR 
_D  F )  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) )  e.  ( ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) -cn-> CC ) )
fourierdlem94.dvlb  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( ( RR  _D  F )  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) lim CC  ( Q `  i )
)  =/=  (/) )
fourierdlem94.dvub  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( ( RR  _D  F )  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) lim CC  ( Q `  ( i  +  1 ) ) )  =/=  (/) )
Assertion
Ref Expression
fourierdlem94  |-  ( ph  ->  ( ( ( F  |`  ( -oo (,) X
) ) lim CC  X
)  =/=  (/)  /\  (
( F  |`  ( X (,) +oo ) ) lim
CC  X )  =/=  (/) ) )
Distinct variable groups:    i, F, n, x    i, M, x, n    M, p, i, n    Q, i, x, n    Q, p    T, i, x, n    T, p    i, X, x, n    X, p    ph, i, x, n
Allowed substitution hints:    ph( p)    P( x, i, n, p)    F( p)

Proof of Theorem fourierdlem94
Dummy variables  j 
k  w  y  t  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pire 23492 . . . . 5  |-  pi  e.  RR
21renegcli 9955 . . . 4  |-  -u pi  e.  RR
32a1i 11 . . 3  |-  ( ph  -> 
-u pi  e.  RR )
41a1i 11 . . 3  |-  ( ph  ->  pi  e.  RR )
5 negpilt0 37580 . . . . 5  |-  -u pi  <  0
6 pipos 23494 . . . . 5  |-  0  <  pi
7 0re 9661 . . . . . 6  |-  0  e.  RR
82, 7, 1lttri 9778 . . . . 5  |-  ( (
-u pi  <  0  /\  0  <  pi )  ->  -u pi  <  pi )
95, 6, 8mp2an 686 . . . 4  |-  -u pi  <  pi
109a1i 11 . . 3  |-  ( ph  -> 
-u pi  <  pi )
11 fourierdlem94.p . . 3  |-  P  =  ( n  e.  NN  |->  { p  e.  ( RR  ^m  ( 0 ... n ) )  |  ( ( ( p `
 0 )  = 
-u pi  /\  (
p `  n )  =  pi )  /\  A. i  e.  ( 0..^ n ) ( p `
 i )  < 
( p `  (
i  +  1 ) ) ) } )
12 picn 23493 . . . . 5  |-  pi  e.  CC
13122timesi 10753 . . . 4  |-  ( 2  x.  pi )  =  ( pi  +  pi )
14 fourierdlem94.t . . . 4  |-  T  =  ( 2  x.  pi )
1512, 12subnegi 9973 . . . 4  |-  ( pi 
-  -u pi )  =  ( pi  +  pi )
1613, 14, 153eqtr4i 2503 . . 3  |-  T  =  ( pi  -  -u pi )
17 fourierdlem94.m . . 3  |-  ( ph  ->  M  e.  NN )
18 fourierdlem94.q . . 3  |-  ( ph  ->  Q  e.  ( P `
 M ) )
19 ssid 3437 . . . 4  |-  RR  C_  RR
2019a1i 11 . . 3  |-  ( ph  ->  RR  C_  RR )
21 fourierdlem94.f . . 3  |-  ( ph  ->  F : RR --> RR )
22 simp2 1031 . . . 4  |-  ( (
ph  /\  x  e.  RR  /\  k  e.  ZZ )  ->  x  e.  RR )
23 zre 10965 . . . . . 6  |-  ( k  e.  ZZ  ->  k  e.  RR )
24233ad2ant3 1053 . . . . 5  |-  ( (
ph  /\  x  e.  RR  /\  k  e.  ZZ )  ->  k  e.  RR )
25 2re 10701 . . . . . . . . . 10  |-  2  e.  RR
2625, 1remulcli 9675 . . . . . . . . 9  |-  ( 2  x.  pi )  e.  RR
2726a1i 11 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  pi )  e.  RR )
2814, 27syl5eqel 2553 . . . . . . 7  |-  ( ph  ->  T  e.  RR )
2928adantr 472 . . . . . 6  |-  ( (
ph  /\  k  e.  ZZ )  ->  T  e.  RR )
30293adant2 1049 . . . . 5  |-  ( (
ph  /\  x  e.  RR  /\  k  e.  ZZ )  ->  T  e.  RR )
3124, 30remulcld 9689 . . . 4  |-  ( (
ph  /\  x  e.  RR  /\  k  e.  ZZ )  ->  ( k  x.  T )  e.  RR )
3222, 31readdcld 9688 . . 3  |-  ( (
ph  /\  x  e.  RR  /\  k  e.  ZZ )  ->  ( x  +  ( k  x.  T
) )  e.  RR )
33 simp1 1030 . . . 4  |-  ( (
ph  /\  x  e.  RR  /\  k  e.  ZZ )  ->  ph )
34 simp3 1032 . . . 4  |-  ( (
ph  /\  x  e.  RR  /\  k  e.  ZZ )  ->  k  e.  ZZ )
35 ax-resscn 9614 . . . . . . . . 9  |-  RR  C_  CC
3635a1i 11 . . . . . . . 8  |-  ( ph  ->  RR  C_  CC )
3721, 36fssd 5750 . . . . . . 7  |-  ( ph  ->  F : RR --> CC )
3837adantr 472 . . . . . 6  |-  ( (
ph  /\  k  e.  ZZ )  ->  F : RR
--> CC )
3938adantr 472 . . . . 5  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  x  e.  RR )  ->  F : RR --> CC )
4029adantr 472 . . . . 5  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  x  e.  RR )  ->  T  e.  RR )
41 simplr 770 . . . . 5  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  x  e.  RR )  ->  k  e.  ZZ )
42 simpr 468 . . . . 5  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  x  e.  RR )  ->  x  e.  RR )
43 eleq1 2537 . . . . . . . . 9  |-  ( x  =  y  ->  (
x  e.  RR  <->  y  e.  RR ) )
4443anbi2d 718 . . . . . . . 8  |-  ( x  =  y  ->  (
( ph  /\  x  e.  RR )  <->  ( ph  /\  y  e.  RR ) ) )
45 oveq1 6315 . . . . . . . . . 10  |-  ( x  =  y  ->  (
x  +  T )  =  ( y  +  T ) )
4645fveq2d 5883 . . . . . . . . 9  |-  ( x  =  y  ->  ( F `  ( x  +  T ) )  =  ( F `  (
y  +  T ) ) )
47 fveq2 5879 . . . . . . . . 9  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
4846, 47eqeq12d 2486 . . . . . . . 8  |-  ( x  =  y  ->  (
( F `  (
x  +  T ) )  =  ( F `
 x )  <->  ( F `  ( y  +  T
) )  =  ( F `  y ) ) )
4944, 48imbi12d 327 . . . . . . 7  |-  ( x  =  y  ->  (
( ( ph  /\  x  e.  RR )  ->  ( F `  (
x  +  T ) )  =  ( F `
 x ) )  <-> 
( ( ph  /\  y  e.  RR )  ->  ( F `  (
y  +  T ) )  =  ( F `
 y ) ) ) )
50 fourierdlem94.per . . . . . . 7  |-  ( (
ph  /\  x  e.  RR )  ->  ( F `
 ( x  +  T ) )  =  ( F `  x
) )
5149, 50chvarv 2120 . . . . . 6  |-  ( (
ph  /\  y  e.  RR )  ->  ( F `
 ( y  +  T ) )  =  ( F `  y
) )
5251adant423 37430 . . . . 5  |-  ( ( ( ( ph  /\  k  e.  ZZ )  /\  x  e.  RR )  /\  y  e.  RR )  ->  ( F `  ( y  +  T
) )  =  ( F `  y ) )
5339, 40, 41, 42, 52fperiodmul 37610 . . . 4  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  x  e.  RR )  ->  ( F `  ( x  +  ( k  x.  T ) ) )  =  ( F `  x ) )
5433, 34, 22, 53syl21anc 1291 . . 3  |-  ( (
ph  /\  x  e.  RR  /\  k  e.  ZZ )  ->  ( F `  ( x  +  (
k  x.  T ) ) )  =  ( F `  x ) )
5535a1i 11 . . . 4  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  RR  C_  CC )
56 ioossre 11721 . . . . . . . 8  |-  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) )  C_  RR
5756a1i 11 . . . . . . 7  |-  ( ph  ->  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) )  C_  RR )
5821, 57fssresd 5762 . . . . . 6  |-  ( ph  ->  ( F  |`  (
( Q `  i
) (,) ( Q `
 ( i  +  1 ) ) ) ) : ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) --> RR )
5958, 36fssd 5750 . . . . 5  |-  ( ph  ->  ( F  |`  (
( Q `  i
) (,) ( Q `
 ( i  +  1 ) ) ) ) : ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) --> CC )
6059adantr 472 . . . 4  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( F  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) : ( ( Q `  i
) (,) ( Q `
 ( i  +  1 ) ) ) --> CC )
6156a1i 11 . . . 4  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( Q `
 i ) (,) ( Q `  (
i  +  1 ) ) )  C_  RR )
6237adantr 472 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  F : RR --> CC )
6319a1i 11 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  RR  C_  RR )
64 eqid 2471 . . . . . . . 8  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
6564tgioo2 21899 . . . . . . . 8  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
6664, 65dvres 22945 . . . . . . 7  |-  ( ( ( RR  C_  CC  /\  F : RR --> CC )  /\  ( RR  C_  RR  /\  ( ( Q `
 i ) (,) ( Q `  (
i  +  1 ) ) )  C_  RR ) )  ->  ( RR  _D  ( F  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) )  =  ( ( RR  _D  F )  |`  (
( int `  ( topGen `
 ran  (,) )
) `  ( ( Q `  i ) (,) ( Q `  (
i  +  1 ) ) ) ) ) )
6755, 62, 63, 61, 66syl22anc 1293 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( RR  _D  ( F  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) )  =  ( ( RR  _D  F )  |`  ( ( int `  ( topGen `
 ran  (,) )
) `  ( ( Q `  i ) (,) ( Q `  (
i  +  1 ) ) ) ) ) )
6867dmeqd 5042 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  dom  ( RR  _D  ( F  |`  (
( Q `  i
) (,) ( Q `
 ( i  +  1 ) ) ) ) )  =  dom  ( ( RR  _D  F )  |`  (
( int `  ( topGen `
 ran  (,) )
) `  ( ( Q `  i ) (,) ( Q `  (
i  +  1 ) ) ) ) ) )
69 ioontr 37707 . . . . . . . 8  |-  ( ( int `  ( topGen ` 
ran  (,) ) ) `  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) )  =  ( ( Q `  i
) (,) ( Q `
 ( i  +  1 ) ) )
7069reseq2i 5108 . . . . . . 7  |-  ( ( RR  _D  F )  |`  ( ( int `  ( topGen `
 ran  (,) )
) `  ( ( Q `  i ) (,) ( Q `  (
i  +  1 ) ) ) ) )  =  ( ( RR 
_D  F )  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) )
7170dmeqi 5041 . . . . . 6  |-  dom  (
( RR  _D  F
)  |`  ( ( int `  ( topGen `  ran  (,) )
) `  ( ( Q `  i ) (,) ( Q `  (
i  +  1 ) ) ) ) )  =  dom  ( ( RR  _D  F )  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) )
7271a1i 11 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  dom  ( ( RR  _D  F )  |`  ( ( int `  ( topGen `
 ran  (,) )
) `  ( ( Q `  i ) (,) ( Q `  (
i  +  1 ) ) ) ) )  =  dom  ( ( RR  _D  F )  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) )
73 fourierdlem94.dvcn . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( RR 
_D  F )  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) )  e.  ( ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) -cn-> CC ) )
74 cncff 22003 . . . . . 6  |-  ( ( ( RR  _D  F
)  |`  ( ( Q `
 i ) (,) ( Q `  (
i  +  1 ) ) ) )  e.  ( ( ( Q `
 i ) (,) ( Q `  (
i  +  1 ) ) ) -cn-> CC )  ->  ( ( RR 
_D  F )  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) : ( ( Q `  i
) (,) ( Q `
 ( i  +  1 ) ) ) --> CC )
75 fdm 5745 . . . . . 6  |-  ( ( ( RR  _D  F
)  |`  ( ( Q `
 i ) (,) ( Q `  (
i  +  1 ) ) ) ) : ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) --> CC  ->  dom  ( ( RR  _D  F )  |`  (
( Q `  i
) (,) ( Q `
 ( i  +  1 ) ) ) )  =  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) )
7673, 74, 753syl 18 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  dom  ( ( RR  _D  F )  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) )  =  ( ( Q `  i
) (,) ( Q `
 ( i  +  1 ) ) ) )
7768, 72, 763eqtrd 2509 . . . 4  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  dom  ( RR  _D  ( F  |`  (
( Q `  i
) (,) ( Q `
 ( i  +  1 ) ) ) ) )  =  ( ( Q `  i
) (,) ( Q `
 ( i  +  1 ) ) ) )
78 dvcn 22954 . . . 4  |-  ( ( ( RR  C_  CC  /\  ( F  |`  (
( Q `  i
) (,) ( Q `
 ( i  +  1 ) ) ) ) : ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) --> CC 
/\  ( ( Q `
 i ) (,) ( Q `  (
i  +  1 ) ) )  C_  RR )  /\  dom  ( RR 
_D  ( F  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) )  =  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) )  ->  ( F  |`  ( ( Q `
 i ) (,) ( Q `  (
i  +  1 ) ) ) )  e.  ( ( ( Q `
 i ) (,) ( Q `  (
i  +  1 ) ) ) -cn-> CC ) )
7955, 60, 61, 77, 78syl31anc 1295 . . 3  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( F  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) )  e.  ( ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) -cn-> CC ) )
8061, 35syl6ss 3430 . . . 4  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( Q `
 i ) (,) ( Q `  (
i  +  1 ) ) )  C_  CC )
8111fourierdlem2 38083 . . . . . . . . . . . 12  |-  ( M  e.  NN  ->  ( Q  e.  ( P `  M )  <->  ( Q  e.  ( RR  ^m  (
0 ... M ) )  /\  ( ( ( Q `  0 )  =  -u pi  /\  ( Q `  M )  =  pi )  /\  A. i  e.  ( 0..^ M ) ( Q `
 i )  < 
( Q `  (
i  +  1 ) ) ) ) ) )
8217, 81syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( Q  e.  ( P `  M )  <-> 
( Q  e.  ( RR  ^m  ( 0 ... M ) )  /\  ( ( ( Q `  0 )  =  -u pi  /\  ( Q `  M )  =  pi )  /\  A. i  e.  ( 0..^ M ) ( Q `
 i )  < 
( Q `  (
i  +  1 ) ) ) ) ) )
8318, 82mpbid 215 . . . . . . . . . 10  |-  ( ph  ->  ( Q  e.  ( RR  ^m  ( 0 ... M ) )  /\  ( ( ( Q `  0 )  =  -u pi  /\  ( Q `  M )  =  pi )  /\  A. i  e.  ( 0..^ M ) ( Q `
 i )  < 
( Q `  (
i  +  1 ) ) ) ) )
8483simpld 466 . . . . . . . . 9  |-  ( ph  ->  Q  e.  ( RR 
^m  ( 0 ... M ) ) )
85 elmapi 7511 . . . . . . . . 9  |-  ( Q  e.  ( RR  ^m  ( 0 ... M
) )  ->  Q : ( 0 ... M ) --> RR )
8684, 85syl 17 . . . . . . . 8  |-  ( ph  ->  Q : ( 0 ... M ) --> RR )
8786adantr 472 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  Q : ( 0 ... M ) --> RR )
88 elfzofz 11962 . . . . . . . 8  |-  ( i  e.  ( 0..^ M )  ->  i  e.  ( 0 ... M
) )
8988adantl 473 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  i  e.  ( 0 ... M ) )
9087, 89ffvelrnd 6038 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( Q `  i )  e.  RR )
9190rexrd 9708 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( Q `  i )  e.  RR* )
92 fzofzp1 12037 . . . . . . 7  |-  ( i  e.  ( 0..^ M )  ->  ( i  +  1 )  e.  ( 0 ... M
) )
9392adantl 473 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( i  +  1 )  e.  ( 0 ... M ) )
9487, 93ffvelrnd 6038 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( Q `  ( i  +  1 ) )  e.  RR )
9583simprrd 775 . . . . . 6  |-  ( ph  ->  A. i  e.  ( 0..^ M ) ( Q `  i )  <  ( Q `  ( i  +  1 ) ) )
9695r19.21bi 2776 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( Q `  i )  <  ( Q `  ( i  +  1 ) ) )
9764, 91, 94, 96lptioo2cn 37823 . . . 4  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( Q `  ( i  +  1 ) )  e.  ( ( limPt `  ( TopOpen ` fld ) ) `  (
( Q `  i
) (,) ( Q `
 ( i  +  1 ) ) ) ) )
9858adantr 472 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( F  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) : ( ( Q `  i
) (,) ( Q `
 ( i  +  1 ) ) ) --> RR )
9936, 37, 20dvbss 22935 . . . . . . . 8  |-  ( ph  ->  dom  ( RR  _D  F )  C_  RR )
100 dvfre 22984 . . . . . . . . 9  |-  ( ( F : RR --> RR  /\  RR  C_  RR )  -> 
( RR  _D  F
) : dom  ( RR  _D  F ) --> RR )
10121, 20, 100syl2anc 673 . . . . . . . 8  |-  ( ph  ->  ( RR  _D  F
) : dom  ( RR  _D  F ) --> RR )
10283simprd 470 . . . . . . . . 9  |-  ( ph  ->  ( ( ( Q `
 0 )  = 
-u pi  /\  ( Q `  M )  =  pi )  /\  A. i  e.  ( 0..^ M ) ( Q `
 i )  < 
( Q `  (
i  +  1 ) ) ) )
103102simplld 769 . . . . . . . 8  |-  ( ph  ->  ( Q `  0
)  =  -u pi )
104102simplrd 771 . . . . . . . 8  |-  ( ph  ->  ( Q `  M
)  =  pi )
10573, 74syl 17 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( RR 
_D  F )  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) : ( ( Q `  i
) (,) ( Q `
 ( i  +  1 ) ) ) --> CC )
10694rexrd 9708 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( Q `  ( i  +  1 ) )  e.  RR* )
10764, 106, 90, 96lptioo1cn 37824 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( Q `  i )  e.  ( ( limPt `  ( TopOpen ` fld ) ) `  (
( Q `  i
) (,) ( Q `
 ( i  +  1 ) ) ) ) )
108 fourierdlem94.dvlb . . . . . . . . 9  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( ( RR  _D  F )  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) lim CC  ( Q `  i )
)  =/=  (/) )
109105, 80, 107, 108, 64ellimciota 37791 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( iota x x  e.  ( (
( RR  _D  F
)  |`  ( ( Q `
 i ) (,) ( Q `  (
i  +  1 ) ) ) ) lim CC  ( Q `  i ) ) )  e.  ( ( ( RR  _D  F )  |`  (
( Q `  i
) (,) ( Q `
 ( i  +  1 ) ) ) ) lim CC  ( Q `
 i ) ) )
110 fourierdlem94.dvub . . . . . . . . 9  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( ( RR  _D  F )  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) lim CC  ( Q `  ( i  +  1 ) ) )  =/=  (/) )
111105, 80, 97, 110, 64ellimciota 37791 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( iota x x  e.  ( (
( RR  _D  F
)  |`  ( ( Q `
 i ) (,) ( Q `  (
i  +  1 ) ) ) ) lim CC  ( Q `  ( i  +  1 ) ) ) )  e.  ( ( ( RR  _D  F )  |`  (
( Q `  i
) (,) ( Q `
 ( i  +  1 ) ) ) ) lim CC  ( Q `
 ( i  +  1 ) ) ) )
11223adantl 473 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ZZ )  ->  k  e.  RR )
113112, 29remulcld 9689 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ZZ )  ->  ( k  x.  T )  e.  RR )
11438adantr 472 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  t  e.  RR )  ->  F : RR --> CC )
11529adantr 472 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  t  e.  RR )  ->  T  e.  RR )
116 simplr 770 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  t  e.  RR )  ->  k  e.  ZZ )
117 simpr 468 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  t  e.  RR )  ->  t  e.  RR )
11850adant423 37430 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  k  e.  ZZ )  /\  t  e.  RR )  /\  x  e.  RR )  ->  ( F `  ( x  +  T
) )  =  ( F `  x ) )
119114, 115, 116, 117, 118fperiodmul 37610 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  t  e.  RR )  ->  ( F `  ( t  +  ( k  x.  T ) ) )  =  ( F `  t ) )
120 eqid 2471 . . . . . . . . . . 11  |-  ( RR 
_D  F )  =  ( RR  _D  F
)
12138, 113, 119, 120fperdvper 37887 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  t  e.  dom  ( RR  _D  F ) )  -> 
( ( t  +  ( k  x.  T
) )  e.  dom  ( RR  _D  F
)  /\  ( ( RR  _D  F ) `  ( t  +  ( k  x.  T ) ) )  =  ( ( RR  _D  F
) `  t )
) )
122121an32s 821 . . . . . . . . 9  |-  ( ( ( ph  /\  t  e.  dom  ( RR  _D  F ) )  /\  k  e.  ZZ )  ->  ( ( t  +  ( k  x.  T
) )  e.  dom  ( RR  _D  F
)  /\  ( ( RR  _D  F ) `  ( t  +  ( k  x.  T ) ) )  =  ( ( RR  _D  F
) `  t )
) )
123122simpld 466 . . . . . . . 8  |-  ( ( ( ph  /\  t  e.  dom  ( RR  _D  F ) )  /\  k  e.  ZZ )  ->  ( t  +  ( k  x.  T ) )  e.  dom  ( RR  _D  F ) )
124122simprd 470 . . . . . . . 8  |-  ( ( ( ph  /\  t  e.  dom  ( RR  _D  F ) )  /\  k  e.  ZZ )  ->  ( ( RR  _D  F ) `  (
t  +  ( k  x.  T ) ) )  =  ( ( RR  _D  F ) `
 t ) )
125 fveq2 5879 . . . . . . . . . 10  |-  ( j  =  i  ->  ( Q `  j )  =  ( Q `  i ) )
126 oveq1 6315 . . . . . . . . . . 11  |-  ( j  =  i  ->  (
j  +  1 )  =  ( i  +  1 ) )
127126fveq2d 5883 . . . . . . . . . 10  |-  ( j  =  i  ->  ( Q `  ( j  +  1 ) )  =  ( Q `  ( i  +  1 ) ) )
128125, 127oveq12d 6326 . . . . . . . . 9  |-  ( j  =  i  ->  (
( Q `  j
) (,) ( Q `
 ( j  +  1 ) ) )  =  ( ( Q `
 i ) (,) ( Q `  (
i  +  1 ) ) ) )
129128cbvmptv 4488 . . . . . . . 8  |-  ( j  e.  ( 0..^ M )  |->  ( ( Q `
 j ) (,) ( Q `  (
j  +  1 ) ) ) )  =  ( i  e.  ( 0..^ M )  |->  ( ( Q `  i
) (,) ( Q `
 ( i  +  1 ) ) ) )
130 eqid 2471 . . . . . . . 8  |-  ( t  e.  RR  |->  ( t  +  ( ( |_
`  ( ( pi 
-  t )  /  T ) )  x.  T ) ) )  =  ( t  e.  RR  |->  ( t  +  ( ( |_ `  ( ( pi  -  t )  /  T
) )  x.  T
) ) )
13199, 101, 3, 4, 10, 16, 17, 86, 103, 104, 73, 109, 111, 123, 124, 129, 130fourierdlem71 38153 . . . . . . 7  |-  ( ph  ->  E. z  e.  RR  A. t  e.  dom  ( RR  _D  F ) ( abs `  ( ( RR  _D  F ) `
 t ) )  <_  z )
132131adantr 472 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  E. z  e.  RR  A. t  e.  dom  ( RR  _D  F ) ( abs `  ( ( RR  _D  F ) `
 t ) )  <_  z )
133 nfv 1769 . . . . . . . . . 10  |-  F/ t ( ph  /\  i  e.  ( 0..^ M ) )
134 nfra1 2785 . . . . . . . . . 10  |-  F/ t A. t  e.  dom  ( RR  _D  F
) ( abs `  (
( RR  _D  F
) `  t )
)  <_  z
135133, 134nfan 2031 . . . . . . . . 9  |-  F/ t ( ( ph  /\  i  e.  ( 0..^ M ) )  /\  A. t  e.  dom  ( RR  _D  F ) ( abs `  ( ( RR  _D  F ) `
 t ) )  <_  z )
13667, 70syl6eq 2521 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( RR  _D  ( F  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) )  =  ( ( RR  _D  F )  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) )
137136fveq1d 5881 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( RR 
_D  ( F  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) ) `  t )  =  ( ( ( RR  _D  F )  |`  (
( Q `  i
) (,) ( Q `
 ( i  +  1 ) ) ) ) `  t ) )
138 fvres 5893 . . . . . . . . . . . . . 14  |-  ( t  e.  ( ( Q `
 i ) (,) ( Q `  (
i  +  1 ) ) )  ->  (
( ( RR  _D  F )  |`  (
( Q `  i
) (,) ( Q `
 ( i  +  1 ) ) ) ) `  t )  =  ( ( RR 
_D  F ) `  t ) )
139137, 138sylan9eq 2525 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  i  e.  ( 0..^ M ) )  /\  t  e.  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) )  ->  (
( RR  _D  ( F  |`  ( ( Q `
 i ) (,) ( Q `  (
i  +  1 ) ) ) ) ) `
 t )  =  ( ( RR  _D  F ) `  t
) )
140139fveq2d 5883 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0..^ M ) )  /\  t  e.  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) )  ->  ( abs `  ( ( RR 
_D  ( F  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) ) `  t ) )  =  ( abs `  (
( RR  _D  F
) `  t )
) )
141140adantlr 729 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  i  e.  ( 0..^ M ) )  /\  A. t  e.  dom  ( RR  _D  F ) ( abs `  ( ( RR  _D  F ) `
 t ) )  <_  z )  /\  t  e.  ( ( Q `  i ) (,) ( Q `  (
i  +  1 ) ) ) )  -> 
( abs `  (
( RR  _D  ( F  |`  ( ( Q `
 i ) (,) ( Q `  (
i  +  1 ) ) ) ) ) `
 t ) )  =  ( abs `  (
( RR  _D  F
) `  t )
) )
142 simplr 770 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  i  e.  ( 0..^ M ) )  /\  A. t  e.  dom  ( RR  _D  F ) ( abs `  ( ( RR  _D  F ) `
 t ) )  <_  z )  /\  t  e.  ( ( Q `  i ) (,) ( Q `  (
i  +  1 ) ) ) )  ->  A. t  e.  dom  ( RR  _D  F
) ( abs `  (
( RR  _D  F
) `  t )
)  <_  z )
143 ssdmres 5132 . . . . . . . . . . . . . . 15  |-  ( ( ( Q `  i
) (,) ( Q `
 ( i  +  1 ) ) ) 
C_  dom  ( RR  _D  F )  <->  dom  ( ( RR  _D  F )  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) )  =  ( ( Q `  i
) (,) ( Q `
 ( i  +  1 ) ) ) )
14476, 143sylibr 217 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( Q `
 i ) (,) ( Q `  (
i  +  1 ) ) )  C_  dom  ( RR  _D  F
) )
145144ad2antrr 740 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  i  e.  ( 0..^ M ) )  /\  A. t  e.  dom  ( RR  _D  F ) ( abs `  ( ( RR  _D  F ) `
 t ) )  <_  z )  /\  t  e.  ( ( Q `  i ) (,) ( Q `  (
i  +  1 ) ) ) )  -> 
( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) )  C_  dom  ( RR 
_D  F ) )
146 simpr 468 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  i  e.  ( 0..^ M ) )  /\  A. t  e.  dom  ( RR  _D  F ) ( abs `  ( ( RR  _D  F ) `
 t ) )  <_  z )  /\  t  e.  ( ( Q `  i ) (,) ( Q `  (
i  +  1 ) ) ) )  -> 
t  e.  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) )
147145, 146sseldd 3419 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  i  e.  ( 0..^ M ) )  /\  A. t  e.  dom  ( RR  _D  F ) ( abs `  ( ( RR  _D  F ) `
 t ) )  <_  z )  /\  t  e.  ( ( Q `  i ) (,) ( Q `  (
i  +  1 ) ) ) )  -> 
t  e.  dom  ( RR  _D  F ) )
148 rspa 2774 . . . . . . . . . . . 12  |-  ( ( A. t  e.  dom  ( RR  _D  F
) ( abs `  (
( RR  _D  F
) `  t )
)  <_  z  /\  t  e.  dom  ( RR 
_D  F ) )  ->  ( abs `  (
( RR  _D  F
) `  t )
)  <_  z )
149142, 147, 148syl2anc 673 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  i  e.  ( 0..^ M ) )  /\  A. t  e.  dom  ( RR  _D  F ) ( abs `  ( ( RR  _D  F ) `
 t ) )  <_  z )  /\  t  e.  ( ( Q `  i ) (,) ( Q `  (
i  +  1 ) ) ) )  -> 
( abs `  (
( RR  _D  F
) `  t )
)  <_  z )
150141, 149eqbrtrd 4416 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  i  e.  ( 0..^ M ) )  /\  A. t  e.  dom  ( RR  _D  F ) ( abs `  ( ( RR  _D  F ) `
 t ) )  <_  z )  /\  t  e.  ( ( Q `  i ) (,) ( Q `  (
i  +  1 ) ) ) )  -> 
( abs `  (
( RR  _D  ( F  |`  ( ( Q `
 i ) (,) ( Q `  (
i  +  1 ) ) ) ) ) `
 t ) )  <_  z )
151150ex 441 . . . . . . . . 9  |-  ( ( ( ph  /\  i  e.  ( 0..^ M ) )  /\  A. t  e.  dom  ( RR  _D  F ) ( abs `  ( ( RR  _D  F ) `  t
) )  <_  z
)  ->  ( t  e.  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) )  ->  ( abs `  ( ( RR  _D  ( F  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) ) `  t ) )  <_  z )
)
152135, 151ralrimi 2800 . . . . . . . 8  |-  ( ( ( ph  /\  i  e.  ( 0..^ M ) )  /\  A. t  e.  dom  ( RR  _D  F ) ( abs `  ( ( RR  _D  F ) `  t
) )  <_  z
)  ->  A. t  e.  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ( abs `  (
( RR  _D  ( F  |`  ( ( Q `
 i ) (,) ( Q `  (
i  +  1 ) ) ) ) ) `
 t ) )  <_  z )
153152ex 441 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( A. t  e.  dom  ( RR  _D  F ) ( abs `  ( ( RR  _D  F ) `  t
) )  <_  z  ->  A. t  e.  ( ( Q `  i
) (,) ( Q `
 ( i  +  1 ) ) ) ( abs `  (
( RR  _D  ( F  |`  ( ( Q `
 i ) (,) ( Q `  (
i  +  1 ) ) ) ) ) `
 t ) )  <_  z ) )
154153reximdv 2857 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( E. z  e.  RR  A. t  e. 
dom  ( RR  _D  F ) ( abs `  ( ( RR  _D  F ) `  t
) )  <_  z  ->  E. z  e.  RR  A. t  e.  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ( abs `  ( ( RR  _D  ( F  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) ) `  t ) )  <_ 
z ) )
155132, 154mpd 15 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  E. z  e.  RR  A. t  e.  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ( abs `  ( ( RR  _D  ( F  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) ) `  t ) )  <_ 
z )
15690, 94, 98, 77, 155ioodvbdlimc2 37907 . . . 4  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( F  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) lim CC  ( Q `  ( i  +  1 ) ) )  =/=  (/) )
15760, 80, 97, 156, 64ellimciota 37791 . . 3  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( iota y
y  e.  ( ( F  |`  ( ( Q `  i ) (,) ( Q `  (
i  +  1 ) ) ) ) lim CC  ( Q `  ( i  +  1 ) ) ) )  e.  ( ( F  |`  (
( Q `  i
) (,) ( Q `
 ( i  +  1 ) ) ) ) lim CC  ( Q `
 ( i  +  1 ) ) ) )
158 fourierdlem94.x . . 3  |-  ( ph  ->  X  e.  RR )
159 oveq2 6316 . . . . . . 7  |-  ( y  =  x  ->  (
pi  -  y )  =  ( pi  -  x ) )
160159oveq1d 6323 . . . . . 6  |-  ( y  =  x  ->  (
( pi  -  y
)  /  T )  =  ( ( pi 
-  x )  /  T ) )
161160fveq2d 5883 . . . . 5  |-  ( y  =  x  ->  ( |_ `  ( ( pi 
-  y )  /  T ) )  =  ( |_ `  (
( pi  -  x
)  /  T ) ) )
162161oveq1d 6323 . . . 4  |-  ( y  =  x  ->  (
( |_ `  (
( pi  -  y
)  /  T ) )  x.  T )  =  ( ( |_
`  ( ( pi 
-  x )  /  T ) )  x.  T ) )
163162cbvmptv 4488 . . 3  |-  ( y  e.  RR  |->  ( ( |_ `  ( ( pi  -  y )  /  T ) )  x.  T ) )  =  ( x  e.  RR  |->  ( ( |_
`  ( ( pi 
-  x )  /  T ) )  x.  T ) )
164 id 22 . . . . 5  |-  ( z  =  x  ->  z  =  x )
165 fveq2 5879 . . . . 5  |-  ( z  =  x  ->  (
( y  e.  RR  |->  ( ( |_ `  ( ( pi  -  y )  /  T
) )  x.  T
) ) `  z
)  =  ( ( y  e.  RR  |->  ( ( |_ `  (
( pi  -  y
)  /  T ) )  x.  T ) ) `  x ) )
166164, 165oveq12d 6326 . . . 4  |-  ( z  =  x  ->  (
z  +  ( ( y  e.  RR  |->  ( ( |_ `  (
( pi  -  y
)  /  T ) )  x.  T ) ) `  z ) )  =  ( x  +  ( ( y  e.  RR  |->  ( ( |_ `  ( ( pi  -  y )  /  T ) )  x.  T ) ) `
 x ) ) )
167166cbvmptv 4488 . . 3  |-  ( z  e.  RR  |->  ( z  +  ( ( y  e.  RR  |->  ( ( |_ `  ( ( pi  -  y )  /  T ) )  x.  T ) ) `
 z ) ) )  =  ( x  e.  RR  |->  ( x  +  ( ( y  e.  RR  |->  ( ( |_ `  ( ( pi  -  y )  /  T ) )  x.  T ) ) `
 x ) ) )
1683, 4, 10, 11, 16, 17, 18, 20, 21, 32, 54, 79, 157, 158, 163, 167fourierdlem49 38131 . 2  |-  ( ph  ->  ( ( F  |`  ( -oo (,) X ) ) lim CC  X )  =/=  (/) )
16990, 94, 98, 77, 155ioodvbdlimc1 37904 . . . 4  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( F  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) lim CC  ( Q `  i )
)  =/=  (/) )
17060, 80, 107, 169, 64ellimciota 37791 . . 3  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( iota y
y  e.  ( ( F  |`  ( ( Q `  i ) (,) ( Q `  (
i  +  1 ) ) ) ) lim CC  ( Q `  i ) ) )  e.  ( ( F  |`  (
( Q `  i
) (,) ( Q `
 ( i  +  1 ) ) ) ) lim CC  ( Q `
 i ) ) )
171 biid 244 . . 3  |-  ( ( ( ( ( ph  /\  i  e.  ( 0..^ M ) )  /\  w  e.  ( ( Q `  i ) [,) ( Q `  (
i  +  1 ) ) ) )  /\  k  e.  ZZ )  /\  w  =  ( X  +  ( k  x.  T ) ) )  <-> 
( ( ( (
ph  /\  i  e.  ( 0..^ M ) )  /\  w  e.  ( ( Q `  i
) [,) ( Q `
 ( i  +  1 ) ) ) )  /\  k  e.  ZZ )  /\  w  =  ( X  +  ( k  x.  T
) ) ) )
1723, 4, 10, 11, 16, 17, 18, 21, 32, 54, 79, 170, 158, 163, 167, 171fourierdlem48 38130 . 2  |-  ( ph  ->  ( ( F  |`  ( X (,) +oo )
) lim CC  X )  =/=  (/) )
173168, 172jca 541 1  |-  ( ph  ->  ( ( ( F  |`  ( -oo (,) X
) ) lim CC  X
)  =/=  (/)  /\  (
( F  |`  ( X (,) +oo ) ) lim
CC  X )  =/=  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   E.wrex 2757   {crab 2760    C_ wss 3390   (/)c0 3722   class class class wbr 4395    |-> cmpt 4454   dom cdm 4839   ran crn 4840    |` cres 4841   iotacio 5551   -->wf 5585   ` cfv 5589  (class class class)co 6308    ^m cmap 7490   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560    x. cmul 9562   +oocpnf 9690   -oocmnf 9691    < clt 9693    <_ cle 9694    - cmin 9880   -ucneg 9881    / cdiv 10291   NNcn 10631   2c2 10681   ZZcz 10961   (,)cioo 11660   [,)cico 11662   ...cfz 11810  ..^cfzo 11942   |_cfl 12059   abscabs 13374   picpi 14196   TopOpenctopn 15398   topGenctg 15414  ℂfldccnfld 19047   intcnt 20109   -cn->ccncf 21986   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-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-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-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-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-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-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-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-limc 22900  df-dv 22901
This theorem is referenced by:  fourierdlem102  38184
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