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Theorem fourierdlem114 38121
Description: Fourier series convergence for periodic, piecewise smooth functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
fourierdlem114.f  |-  ( ph  ->  F : RR --> RR )
fourierdlem114.t  |-  T  =  ( 2  x.  pi )
fourierdlem114.per  |-  ( (
ph  /\  x  e.  RR )  ->  ( F `
 ( x  +  T ) )  =  ( F `  x
) )
fourierdlem114.g  |-  G  =  ( ( RR  _D  F )  |`  ( -u pi (,) pi ) )
fourierdlem114.dmdv  |-  ( ph  ->  ( ( -u pi (,) pi )  \  dom  G )  e.  Fin )
fourierdlem114.gcn  |-  ( ph  ->  G  e.  ( dom 
G -cn-> CC ) )
fourierdlem114.rlim  |-  ( (
ph  /\  x  e.  ( ( -u pi [,) pi )  \  dom  G ) )  ->  (
( G  |`  (
x (,) +oo )
) lim CC  x )  =/=  (/) )
fourierdlem114.llim  |-  ( (
ph  /\  x  e.  ( ( -u pi (,] pi )  \  dom  G ) )  ->  (
( G  |`  ( -oo (,) x ) ) lim
CC  x )  =/=  (/) )
fourierdlem114.x  |-  ( ph  ->  X  e.  RR )
fourierdlem114.l  |-  ( ph  ->  L  e.  ( ( F  |`  ( -oo (,) X ) ) lim CC  X ) )
fourierdlem114.r  |-  ( ph  ->  R  e.  ( ( F  |`  ( X (,) +oo ) ) lim CC  X ) )
fourierdlem114.a  |-  A  =  ( n  e.  NN0  |->  ( S. ( -u pi (,) pi ) ( ( F `  x )  x.  ( cos `  (
n  x.  x ) ) )  _d x  /  pi ) )
fourierdlem114.b  |-  B  =  ( n  e.  NN  |->  ( S. ( -u pi (,) pi ) ( ( F `  x )  x.  ( sin `  (
n  x.  x ) ) )  _d x  /  pi ) )
fourierdlem114.s  |-  S  =  ( n  e.  NN  |->  ( ( ( A `
 n )  x.  ( cos `  (
n  x.  X ) ) )  +  ( ( B `  n
)  x.  ( sin `  ( n  x.  X
) ) ) ) )
fourierdlem114.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 ) ) ) } )
fourierdlem114.e  |-  E  =  ( x  e.  RR  |->  ( x  +  (
( |_ `  (
( pi  -  x
)  /  T ) )  x.  T ) ) )
fourierdlem114.h  |-  H  =  ( { -u pi ,  pi ,  ( E `
 X ) }  u.  ( ( -u pi [,] pi )  \  dom  G ) )
fourierdlem114.m  |-  M  =  ( ( # `  H
)  -  1 )
fourierdlem114.q  |-  Q  =  ( iota g g 
Isom  <  ,  <  (
( 0 ... M
) ,  H ) )
Assertion
Ref Expression
fourierdlem114  |-  ( 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, n    B, n    x, E    i, F, n, x    i, G, x    g, H    i, L, n    g, M    i, M, n, p    x, M    Q, g    Q, i, n, p    x, Q    R, i, n    T, i, n, p    x, T    i, X, n, p    x, X    ph, g    ph, i, n, x
Allowed substitution hints:    ph( p)    A( x, g, i, p)    B( x, g, i, p)    P( x, g, i, n, p)    R( x, g, p)    S( x, g, i, n, p)    T( g)    E( g, i, n, p)    F( g, p)    G( g, n, p)    H( x, i, n, p)    L( x, g, p)    X( g)

Proof of Theorem fourierdlem114
StepHypRef Expression
1 fourierdlem114.f . 2  |-  ( ph  ->  F : RR --> RR )
2 fourierdlem114.t . 2  |-  T  =  ( 2  x.  pi )
3 fourierdlem114.per . 2  |-  ( (
ph  /\  x  e.  RR )  ->  ( F `
 ( x  +  T ) )  =  ( F `  x
) )
4 fourierdlem114.x . 2  |-  ( ph  ->  X  e.  RR )
5 fourierdlem114.l . 2  |-  ( ph  ->  L  e.  ( ( F  |`  ( -oo (,) X ) ) lim CC  X ) )
6 fourierdlem114.r . 2  |-  ( ph  ->  R  e.  ( ( F  |`  ( X (,) +oo ) ) lim CC  X ) )
7 fourierdlem114.p . 2  |-  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 ) ) ) } )
8 fourierdlem114.m . . 3  |-  M  =  ( ( # `  H
)  -  1 )
9 2z 10997 . . . . . 6  |-  2  e.  ZZ
109a1i 11 . . . . 5  |-  ( ph  ->  2  e.  ZZ )
11 fourierdlem114.h . . . . . . . 8  |-  H  =  ( { -u pi ,  pi ,  ( E `
 X ) }  u.  ( ( -u pi [,] pi )  \  dom  G ) )
12 tpfi 7872 . . . . . . . . . 10  |-  { -u pi ,  pi , 
( E `  X
) }  e.  Fin
1312a1i 11 . . . . . . . . 9  |-  ( ph  ->  { -u pi ,  pi ,  ( E `  X ) }  e.  Fin )
14 pire 23461 . . . . . . . . . . . . . . 15  |-  pi  e.  RR
1514renegcli 9960 . . . . . . . . . . . . . 14  |-  -u pi  e.  RR
1615rexri 9718 . . . . . . . . . . . . 13  |-  -u pi  e.  RR*
1714rexri 9718 . . . . . . . . . . . . 13  |-  pi  e.  RR*
18 negpilt0 37527 . . . . . . . . . . . . . . 15  |-  -u pi  <  0
19 pipos 23463 . . . . . . . . . . . . . . 15  |-  0  <  pi
20 0re 9668 . . . . . . . . . . . . . . . 16  |-  0  e.  RR
2115, 20, 14lttri 9785 . . . . . . . . . . . . . . 15  |-  ( (
-u pi  <  0  /\  0  <  pi )  ->  -u pi  <  pi )
2218, 19, 21mp2an 683 . . . . . . . . . . . . . 14  |-  -u pi  <  pi
2315, 14, 22ltleii 9782 . . . . . . . . . . . . 13  |-  -u pi  <_  pi
24 prunioo 11789 . . . . . . . . . . . . 13  |-  ( (
-u pi  e.  RR*  /\  pi  e.  RR*  /\  -u pi  <_  pi )  ->  (
( -u pi (,) pi )  u.  { -u pi ,  pi } )  =  ( -u pi [,] pi ) )
2516, 17, 23, 24mp3an 1373 . . . . . . . . . . . 12  |-  ( (
-u pi (,) pi )  u.  { -u pi ,  pi } )  =  ( -u pi [,] pi )
2625difeq1i 3558 . . . . . . . . . . 11  |-  ( ( ( -u pi (,) pi )  u.  { -u pi ,  pi }
)  \  dom  G )  =  ( ( -u pi [,] pi )  \  dom  G )
27 difundir 3707 . . . . . . . . . . 11  |-  ( ( ( -u pi (,) pi )  u.  { -u pi ,  pi }
)  \  dom  G )  =  ( ( (
-u pi (,) pi )  \  dom  G )  u.  ( { -u pi ,  pi }  \  dom  G ) )
2826, 27eqtr3i 2485 . . . . . . . . . 10  |-  ( (
-u pi [,] pi )  \  dom  G )  =  ( ( (
-u pi (,) pi )  \  dom  G )  u.  ( { -u pi ,  pi }  \  dom  G ) )
29 fourierdlem114.dmdv . . . . . . . . . . 11  |-  ( ph  ->  ( ( -u pi (,) pi )  \  dom  G )  e.  Fin )
30 prfi 7871 . . . . . . . . . . . 12  |-  { -u pi ,  pi }  e.  Fin
31 diffi 7828 . . . . . . . . . . . 12  |-  ( {
-u pi ,  pi }  e.  Fin  ->  ( { -u pi ,  pi }  \  dom  G )  e.  Fin )
3230, 31mp1i 13 . . . . . . . . . . 11  |-  ( ph  ->  ( { -u pi ,  pi }  \  dom  G )  e.  Fin )
33 unfi 7863 . . . . . . . . . . 11  |-  ( ( ( ( -u pi (,) pi )  \  dom  G )  e.  Fin  /\  ( { -u pi ,  pi }  \  dom  G
)  e.  Fin )  ->  ( ( ( -u pi (,) pi )  \  dom  G )  u.  ( { -u pi ,  pi }  \  dom  G ) )  e.  Fin )
3429, 32, 33syl2anc 671 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( -u pi (,) pi )  \  dom  G )  u.  ( { -u pi ,  pi }  \  dom  G ) )  e.  Fin )
3528, 34syl5eqel 2543 . . . . . . . . 9  |-  ( ph  ->  ( ( -u pi [,] pi )  \  dom  G )  e.  Fin )
36 unfi 7863 . . . . . . . . 9  |-  ( ( { -u pi ,  pi ,  ( E `  X ) }  e.  Fin  /\  ( ( -u pi [,] pi )  \  dom  G )  e.  Fin )  ->  ( { -u pi ,  pi , 
( E `  X
) }  u.  (
( -u pi [,] pi )  \  dom  G ) )  e.  Fin )
3713, 35, 36syl2anc 671 . . . . . . . 8  |-  ( ph  ->  ( { -u pi ,  pi ,  ( E `
 X ) }  u.  ( ( -u pi [,] pi )  \  dom  G ) )  e. 
Fin )
3811, 37syl5eqel 2543 . . . . . . 7  |-  ( ph  ->  H  e.  Fin )
39 hashcl 12569 . . . . . . 7  |-  ( H  e.  Fin  ->  ( # `
 H )  e. 
NN0 )
4038, 39syl 17 . . . . . 6  |-  ( ph  ->  ( # `  H
)  e.  NN0 )
4140nn0zd 11066 . . . . 5  |-  ( ph  ->  ( # `  H
)  e.  ZZ )
4215, 22ltneii 9772 . . . . . . 7  |-  -u pi  =/=  pi
43 hashprg 12603 . . . . . . . 8  |-  ( (
-u pi  e.  RR  /\  pi  e.  RR )  ->  ( -u pi  =/=  pi  <->  ( # `  { -u pi ,  pi }
)  =  2 ) )
4415, 14, 43mp2an 683 . . . . . . 7  |-  ( -u pi  =/=  pi  <->  ( # `  { -u pi ,  pi }
)  =  2 )
4542, 44mpbi 213 . . . . . 6  |-  ( # `  { -u pi ,  pi } )  =  2
4612elexi 3066 . . . . . . . . . 10  |-  { -u pi ,  pi , 
( E `  X
) }  e.  _V
47 ovex 6342 . . . . . . . . . . 11  |-  ( -u pi [,] pi )  e. 
_V
48 difexg 4564 . . . . . . . . . . 11  |-  ( (
-u pi [,] pi )  e.  _V  ->  ( ( -u pi [,] pi )  \  dom  G
)  e.  _V )
4947, 48ax-mp 5 . . . . . . . . . 10  |-  ( (
-u pi [,] pi )  \  dom  G )  e.  _V
5046, 49unex 6615 . . . . . . . . 9  |-  ( {
-u pi ,  pi ,  ( E `  X ) }  u.  ( ( -u pi [,] pi )  \  dom  G ) )  e.  _V
5111, 50eqeltri 2535 . . . . . . . 8  |-  H  e. 
_V
52 negex 9898 . . . . . . . . . . 11  |-  -u pi  e.  _V
5352tpid1 4097 . . . . . . . . . 10  |-  -u pi  e.  { -u pi ,  pi ,  ( E `  X ) }
5414elexi 3066 . . . . . . . . . . 11  |-  pi  e.  _V
5554tpid2 4098 . . . . . . . . . 10  |-  pi  e.  {
-u pi ,  pi ,  ( E `  X ) }
56 prssi 4140 . . . . . . . . . 10  |-  ( (
-u pi  e.  { -u pi ,  pi , 
( E `  X
) }  /\  pi  e.  { -u pi ,  pi ,  ( E `  X ) } )  ->  { -u pi ,  pi }  C_  { -u pi ,  pi , 
( E `  X
) } )
5753, 55, 56mp2an 683 . . . . . . . . 9  |-  { -u pi ,  pi }  C_ 
{ -u pi ,  pi ,  ( E `  X ) }
58 ssun1 3608 . . . . . . . . . 10  |-  { -u pi ,  pi , 
( E `  X
) }  C_  ( { -u pi ,  pi ,  ( E `  X ) }  u.  ( ( -u pi [,] pi )  \  dom  G ) )
5958, 11sseqtr4i 3476 . . . . . . . . 9  |-  { -u pi ,  pi , 
( E `  X
) }  C_  H
6057, 59sstri 3452 . . . . . . . 8  |-  { -u pi ,  pi }  C_  H
61 hashss 12617 . . . . . . . 8  |-  ( ( H  e.  _V  /\  {
-u pi ,  pi }  C_  H )  -> 
( # `  { -u pi ,  pi }
)  <_  ( # `  H
) )
6251, 60, 61mp2an 683 . . . . . . 7  |-  ( # `  { -u pi ,  pi } )  <_  ( # `
 H )
6362a1i 11 . . . . . 6  |-  ( ph  ->  ( # `  { -u pi ,  pi }
)  <_  ( # `  H
) )
6445, 63syl5eqbrr 4450 . . . . 5  |-  ( ph  ->  2  <_  ( # `  H
) )
65 eluz2 11193 . . . . 5  |-  ( (
# `  H )  e.  ( ZZ>= `  2 )  <->  ( 2  e.  ZZ  /\  ( # `  H )  e.  ZZ  /\  2  <_  ( # `  H
) ) )
6610, 41, 64, 65syl3anbrc 1198 . . . 4  |-  ( ph  ->  ( # `  H
)  e.  ( ZZ>= ` 
2 ) )
67 uz2m1nn 11261 . . . 4  |-  ( (
# `  H )  e.  ( ZZ>= `  2 )  ->  ( ( # `  H
)  -  1 )  e.  NN )
6866, 67syl 17 . . 3  |-  ( ph  ->  ( ( # `  H
)  -  1 )  e.  NN )
698, 68syl5eqel 2543 . 2  |-  ( ph  ->  M  e.  NN )
7015a1i 11 . . . . . . . . . . 11  |-  ( ph  -> 
-u pi  e.  RR )
7114a1i 11 . . . . . . . . . . 11  |-  ( ph  ->  pi  e.  RR )
72 negpitopissre 23537 . . . . . . . . . . . 12  |-  ( -u pi (,] pi )  C_  RR
7322a1i 11 . . . . . . . . . . . . . 14  |-  ( ph  -> 
-u pi  <  pi )
74 picn 23462 . . . . . . . . . . . . . . . 16  |-  pi  e.  CC
75742timesi 10758 . . . . . . . . . . . . . . 15  |-  ( 2  x.  pi )  =  ( pi  +  pi )
7674, 74subnegi 9978 . . . . . . . . . . . . . . 15  |-  ( pi 
-  -u pi )  =  ( pi  +  pi )
7775, 2, 763eqtr4i 2493 . . . . . . . . . . . . . 14  |-  T  =  ( pi  -  -u pi )
78 fourierdlem114.e . . . . . . . . . . . . . 14  |-  E  =  ( x  e.  RR  |->  ( x  +  (
( |_ `  (
( pi  -  x
)  /  T ) )  x.  T ) ) )
7970, 71, 73, 77, 78fourierdlem4 38010 . . . . . . . . . . . . 13  |-  ( ph  ->  E : RR --> ( -u pi (,] pi ) )
8079, 4ffvelrnd 6045 . . . . . . . . . . . 12  |-  ( ph  ->  ( E `  X
)  e.  ( -u pi (,] pi ) )
8172, 80sseldi 3441 . . . . . . . . . . 11  |-  ( ph  ->  ( E `  X
)  e.  RR )
8270, 71, 813jca 1194 . . . . . . . . . 10  |-  ( ph  ->  ( -u pi  e.  RR  /\  pi  e.  RR  /\  ( E `  X
)  e.  RR ) )
83 fvex 5897 . . . . . . . . . . 11  |-  ( E `
 X )  e. 
_V
8452, 54, 83tpss 4149 . . . . . . . . . 10  |-  ( (
-u pi  e.  RR  /\  pi  e.  RR  /\  ( E `  X )  e.  RR )  <->  { -u pi ,  pi ,  ( E `
 X ) } 
C_  RR )
8582, 84sylib 201 . . . . . . . . 9  |-  ( ph  ->  { -u pi ,  pi ,  ( E `  X ) }  C_  RR )
86 iccssre 11744 . . . . . . . . . . 11  |-  ( (
-u pi  e.  RR  /\  pi  e.  RR )  ->  ( -u pi [,] pi )  C_  RR )
8715, 14, 86mp2an 683 . . . . . . . . . 10  |-  ( -u pi [,] pi )  C_  RR
88 ssdifss 3575 . . . . . . . . . 10  |-  ( (
-u pi [,] pi )  C_  RR  ->  (
( -u pi [,] pi )  \  dom  G ) 
C_  RR )
8987, 88mp1i 13 . . . . . . . . 9  |-  ( ph  ->  ( ( -u pi [,] pi )  \  dom  G )  C_  RR )
9085, 89unssd 3621 . . . . . . . 8  |-  ( ph  ->  ( { -u pi ,  pi ,  ( E `
 X ) }  u.  ( ( -u pi [,] pi )  \  dom  G ) )  C_  RR )
9111, 90syl5eqss 3487 . . . . . . 7  |-  ( ph  ->  H  C_  RR )
92 fourierdlem114.q . . . . . . 7  |-  Q  =  ( iota g g 
Isom  <  ,  <  (
( 0 ... M
) ,  H ) )
9338, 91, 92, 8fourierdlem36 38043 . . . . . 6  |-  ( ph  ->  Q  Isom  <  ,  <  ( ( 0 ... M
) ,  H ) )
94 isof1o 6240 . . . . . 6  |-  ( Q 
Isom  <  ,  <  (
( 0 ... M
) ,  H )  ->  Q : ( 0 ... M ) -1-1-onto-> H )
95 f1of 5836 . . . . . 6  |-  ( Q : ( 0 ... M ) -1-1-onto-> H  ->  Q :
( 0 ... M
) --> H )
9693, 94, 953syl 18 . . . . 5  |-  ( ph  ->  Q : ( 0 ... M ) --> H )
9796, 91fssd 5760 . . . 4  |-  ( ph  ->  Q : ( 0 ... M ) --> RR )
98 reex 9655 . . . . 5  |-  RR  e.  _V
99 ovex 6342 . . . . 5  |-  ( 0 ... M )  e. 
_V
10098, 99elmap 7525 . . . 4  |-  ( Q  e.  ( RR  ^m  ( 0 ... M
) )  <->  Q :
( 0 ... M
) --> RR )
10197, 100sylibr 217 . . 3  |-  ( ph  ->  Q  e.  ( RR 
^m  ( 0 ... M ) ) )
102 fveq2 5887 . . . . . . . . . . 11  |-  ( 0  =  i  ->  ( Q `  0 )  =  ( Q `  i ) )
103102adantl 472 . . . . . . . . . 10  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  =  i )  -> 
( Q `  0
)  =  ( Q `
 i ) )
10497ffvelrnda 6044 . . . . . . . . . . . 12  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  ( Q `  i )  e.  RR )
105104leidd 10207 . . . . . . . . . . 11  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  ( Q `  i )  <_  ( Q `  i
) )
106105adantr 471 . . . . . . . . . 10  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  =  i )  -> 
( Q `  i
)  <_  ( Q `  i ) )
107103, 106eqbrtrd 4436 . . . . . . . . 9  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  =  i )  -> 
( Q `  0
)  <_  ( Q `  i ) )
108 elfzelz 11828 . . . . . . . . . . . . 13  |-  ( i  e.  ( 0 ... M )  ->  i  e.  ZZ )
109108zred 11068 . . . . . . . . . . . 12  |-  ( i  e.  ( 0 ... M )  ->  i  e.  RR )
110109ad2antlr 738 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  0  =  i )  ->  i  e.  RR )
111 elfzle1 11830 . . . . . . . . . . . 12  |-  ( i  e.  ( 0 ... M )  ->  0  <_  i )
112111ad2antlr 738 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  0  =  i )  ->  0  <_  i )
113 neqne 2642 . . . . . . . . . . . . 13  |-  ( -.  0  =  i  -> 
0  =/=  i )
114113necomd 2690 . . . . . . . . . . . 12  |-  ( -.  0  =  i  -> 
i  =/=  0 )
115114adantl 472 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  0  =  i )  ->  i  =/=  0 )
116110, 112, 115ne0gt0d 9797 . . . . . . . . . 10  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  0  =  i )  ->  0  <  i )
117 nnssnn0 10900 . . . . . . . . . . . . . . . . 17  |-  NN  C_  NN0
118 nn0uz 11221 . . . . . . . . . . . . . . . . 17  |-  NN0  =  ( ZZ>= `  0 )
119117, 118sseqtri 3475 . . . . . . . . . . . . . . . 16  |-  NN  C_  ( ZZ>= `  0 )
120119, 69sseldi 3441 . . . . . . . . . . . . . . 15  |-  ( ph  ->  M  e.  ( ZZ>= ` 
0 ) )
121 eluzfz1 11834 . . . . . . . . . . . . . . 15  |-  ( M  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... M
) )
122120, 121syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  0  e.  ( 0 ... M ) )
12396, 122ffvelrnd 6045 . . . . . . . . . . . . 13  |-  ( ph  ->  ( Q `  0
)  e.  H )
12491, 123sseldd 3444 . . . . . . . . . . . 12  |-  ( ph  ->  ( Q `  0
)  e.  RR )
125124ad2antrr 737 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  <  i )  ->  ( Q `  0 )  e.  RR )
126104adantr 471 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  <  i )  ->  ( Q `  i )  e.  RR )
127 simpr 467 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  <  i )  ->  0  <  i )
12893ad2antrr 737 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  <  i )  ->  Q  Isom  <  ,  <  (
( 0 ... M
) ,  H ) )
129122anim1i 576 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  (
0  e.  ( 0 ... M )  /\  i  e.  ( 0 ... M ) ) )
130129adantr 471 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  <  i )  ->  (
0  e.  ( 0 ... M )  /\  i  e.  ( 0 ... M ) ) )
131 isorel 6241 . . . . . . . . . . . . 13  |-  ( ( Q  Isom  <  ,  <  ( ( 0 ... M
) ,  H )  /\  ( 0  e.  ( 0 ... M
)  /\  i  e.  ( 0 ... M
) ) )  -> 
( 0  <  i  <->  ( Q `  0 )  <  ( Q `  i ) ) )
132128, 130, 131syl2anc 671 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  <  i )  ->  (
0  <  i  <->  ( Q `  0 )  < 
( Q `  i
) ) )
133127, 132mpbid 215 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  <  i )  ->  ( Q `  0 )  <  ( Q `  i
) )
134125, 126, 133ltled 9808 . . . . . . . . . 10  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  <  i )  ->  ( Q `  0 )  <_  ( Q `  i
) )
135116, 134syldan 477 . . . . . . . . 9  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  0  =  i )  ->  ( Q `  0
)  <_  ( Q `  i ) )
136107, 135pm2.61dan 805 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  ( Q `  0 )  <_  ( Q `  i
) )
137136adantr 471 . . . . . . 7  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  ( Q `  i )  =  -u pi )  -> 
( Q `  0
)  <_  ( Q `  i ) )
138 simpr 467 . . . . . . 7  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  ( Q `  i )  =  -u pi )  -> 
( Q `  i
)  =  -u pi )
139137, 138breqtrd 4440 . . . . . 6  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  ( Q `  i )  =  -u pi )  -> 
( Q `  0
)  <_  -u pi )
14070rexrd 9715 . . . . . . . 8  |-  ( ph  -> 
-u pi  e.  RR* )
14171rexrd 9715 . . . . . . . 8  |-  ( ph  ->  pi  e.  RR* )
142 lbicc2 11776 . . . . . . . . . . . . . 14  |-  ( (
-u pi  e.  RR*  /\  pi  e.  RR*  /\  -u pi  <_  pi )  ->  -u pi  e.  ( -u pi [,] pi ) )
14316, 17, 23, 142mp3an 1373 . . . . . . . . . . . . 13  |-  -u pi  e.  ( -u pi [,] pi )
144143a1i 11 . . . . . . . . . . . 12  |-  ( ph  -> 
-u pi  e.  (
-u pi [,] pi ) )
145 ubicc2 11777 . . . . . . . . . . . . . 14  |-  ( (
-u pi  e.  RR*  /\  pi  e.  RR*  /\  -u pi  <_  pi )  ->  pi  e.  ( -u pi [,] pi ) )
14616, 17, 23, 145mp3an 1373 . . . . . . . . . . . . 13  |-  pi  e.  ( -u pi [,] pi )
147146a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  pi  e.  ( -u pi [,] pi ) )
148 iocssicc 11750 . . . . . . . . . . . . 13  |-  ( -u pi (,] pi )  C_  ( -u pi [,] pi )
149148, 80sseldi 3441 . . . . . . . . . . . 12  |-  ( ph  ->  ( E `  X
)  e.  ( -u pi [,] pi ) )
150 tpssi 4150 . . . . . . . . . . . 12  |-  ( (
-u pi  e.  (
-u pi [,] pi )  /\  pi  e.  (
-u pi [,] pi )  /\  ( E `  X )  e.  (
-u pi [,] pi ) )  ->  { -u pi ,  pi , 
( E `  X
) }  C_  ( -u pi [,] pi ) )
151144, 147, 149, 150syl3anc 1276 . . . . . . . . . . 11  |-  ( ph  ->  { -u pi ,  pi ,  ( E `  X ) }  C_  ( -u pi [,] pi ) )
152 difssd 3572 . . . . . . . . . . 11  |-  ( ph  ->  ( ( -u pi [,] pi )  \  dom  G )  C_  ( -u pi [,] pi ) )
153151, 152unssd 3621 . . . . . . . . . 10  |-  ( ph  ->  ( { -u pi ,  pi ,  ( E `
 X ) }  u.  ( ( -u pi [,] pi )  \  dom  G ) )  C_  ( -u pi [,] pi ) )
15411, 153syl5eqss 3487 . . . . . . . . 9  |-  ( ph  ->  H  C_  ( -u pi [,] pi ) )
155154, 123sseldd 3444 . . . . . . . 8  |-  ( ph  ->  ( Q `  0
)  e.  ( -u pi [,] pi ) )
156 iccgelb 11719 . . . . . . . 8  |-  ( (
-u pi  e.  RR*  /\  pi  e.  RR*  /\  ( Q `  0 )  e.  ( -u pi [,] pi ) )  ->  -u pi  <_  ( Q `  0
) )
157140, 141, 155, 156syl3anc 1276 . . . . . . 7  |-  ( ph  -> 
-u pi  <_  ( Q `  0 )
)
158157ad2antrr 737 . . . . . 6  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  ( Q `  i )  =  -u pi )  ->  -u pi  <_  ( Q `  0 ) )
159124ad2antrr 737 . . . . . . 7  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  ( Q `  i )  =  -u pi )  -> 
( Q `  0
)  e.  RR )
16015a1i 11 . . . . . . 7  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  ( Q `  i )  =  -u pi )  ->  -u pi  e.  RR )
161159, 160letri3d 9802 . . . . . 6  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  ( Q `  i )  =  -u pi )  -> 
( ( Q ` 
0 )  =  -u pi 
<->  ( ( Q ` 
0 )  <_  -u pi  /\  -u pi  <_  ( Q `
 0 ) ) ) )
162139, 158, 161mpbir2and 938 . . . . 5  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  ( Q `  i )  =  -u pi )  -> 
( Q `  0
)  =  -u pi )
16359, 53sselii 3440 . . . . . . 7  |-  -u pi  e.  H
164 f1ofo 5843 . . . . . . . . 9  |-  ( Q : ( 0 ... M ) -1-1-onto-> H  ->  Q :
( 0 ... M
) -onto-> H )
16594, 164syl 17 . . . . . . . 8  |-  ( Q 
Isom  <  ,  <  (
( 0 ... M
) ,  H )  ->  Q : ( 0 ... M )
-onto-> H )
166 forn 5818 . . . . . . . 8  |-  ( Q : ( 0 ... M ) -onto-> H  ->  ran  Q  =  H )
16793, 165, 1663syl 18 . . . . . . 7  |-  ( ph  ->  ran  Q  =  H )
168163, 167syl5eleqr 2546 . . . . . 6  |-  ( ph  -> 
-u pi  e.  ran  Q )
169 ffn 5750 . . . . . . 7  |-  ( Q : ( 0 ... M ) --> H  ->  Q  Fn  ( 0 ... M ) )
170 fvelrnb 5934 . . . . . . 7  |-  ( Q  Fn  ( 0 ... M )  ->  ( -u pi  e.  ran  Q  <->  E. i  e.  ( 0 ... M ) ( Q `  i )  =  -u pi ) )
17196, 169, 1703syl 18 . . . . . 6  |-  ( ph  ->  ( -u pi  e.  ran  Q  <->  E. i  e.  ( 0 ... M ) ( Q `  i
)  =  -u pi ) )
172168, 171mpbid 215 . . . . 5  |-  ( ph  ->  E. i  e.  ( 0 ... M ) ( Q `  i
)  =  -u pi )
173162, 172r19.29a 2943 . . . 4  |-  ( ph  ->  ( Q `  0
)  =  -u pi )
17459, 55sselii 3440 . . . . . . 7  |-  pi  e.  H
175174, 167syl5eleqr 2546 . . . . . 6  |-  ( ph  ->  pi  e.  ran  Q
)
176 fvelrnb 5934 . . . . . . 7  |-  ( Q  Fn  ( 0 ... M )  ->  (
pi  e.  ran  Q  <->  E. i  e.  ( 0 ... M ) ( Q `  i )  =  pi ) )
17796, 169, 1763syl 18 . . . . . 6  |-  ( ph  ->  ( pi  e.  ran  Q  <->  E. i  e.  (
0 ... M ) ( Q `  i )  =  pi ) )
178175, 177mpbid 215 . . . . 5  |-  ( ph  ->  E. i  e.  ( 0 ... M ) ( Q `  i
)  =  pi )
17996, 154fssd 5760 . . . . . . . . . 10  |-  ( ph  ->  Q : ( 0 ... M ) --> (
-u pi [,] pi ) )
180 eluzfz2 11835 . . . . . . . . . . 11  |-  ( M  e.  ( ZZ>= `  0
)  ->  M  e.  ( 0 ... M
) )
181120, 180syl 17 . . . . . . . . . 10  |-  ( ph  ->  M  e.  ( 0 ... M ) )
182179, 181ffvelrnd 6045 . . . . . . . . 9  |-  ( ph  ->  ( Q `  M
)  e.  ( -u pi [,] pi ) )
183 iccleub 11718 . . . . . . . . 9  |-  ( (
-u pi  e.  RR*  /\  pi  e.  RR*  /\  ( Q `  M )  e.  ( -u pi [,] pi ) )  ->  ( Q `  M )  <_  pi )
184140, 141, 182, 183syl3anc 1276 . . . . . . . 8  |-  ( ph  ->  ( Q `  M
)  <_  pi )
1851843ad2ant1 1035 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 0 ... M
)  /\  ( Q `  i )  =  pi )  ->  ( Q `  M )  <_  pi )
186 id 22 . . . . . . . . . 10  |-  ( ( Q `  i )  =  pi  ->  ( Q `  i )  =  pi )
187186eqcomd 2467 . . . . . . . . 9  |-  ( ( Q `  i )  =  pi  ->  pi  =  ( Q `  i ) )
1881873ad2ant3 1037 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 0 ... M
)  /\  ( Q `  i )  =  pi )  ->  pi  =  ( Q `  i ) )
189105adantr 471 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  =  M )  ->  ( Q `  i )  <_  ( Q `  i
) )
190 fveq2 5887 . . . . . . . . . . . 12  |-  ( i  =  M  ->  ( Q `  i )  =  ( Q `  M ) )
191190adantl 472 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  =  M )  ->  ( Q `  i )  =  ( Q `  M ) )
192189, 191breqtrd 4440 . . . . . . . . . 10  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  =  M )  ->  ( Q `  i )  <_  ( Q `  M
) )
193109ad2antlr 738 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  i  =  M )  ->  i  e.  RR )
194 elfzel2 11826 . . . . . . . . . . . . . 14  |-  ( i  e.  ( 0 ... M )  ->  M  e.  ZZ )
195194zred 11068 . . . . . . . . . . . . 13  |-  ( i  e.  ( 0 ... M )  ->  M  e.  RR )
196195ad2antlr 738 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  i  =  M )  ->  M  e.  RR )
197 elfzle2 11831 . . . . . . . . . . . . 13  |-  ( i  e.  ( 0 ... M )  ->  i  <_  M )
198197ad2antlr 738 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  i  =  M )  ->  i  <_  M )
199 neqne 2642 . . . . . . . . . . . . . 14  |-  ( -.  i  =  M  -> 
i  =/=  M )
200199necomd 2690 . . . . . . . . . . . . 13  |-  ( -.  i  =  M  ->  M  =/=  i )
201200adantl 472 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  i  =  M )  ->  M  =/=  i )
202193, 196, 198, 201leneltd 9814 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  i  =  M )  ->  i  <  M )
203104adantr 471 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  <  M )  ->  ( Q `  i )  e.  RR )
20487, 182sseldi 3441 . . . . . . . . . . . . 13  |-  ( ph  ->  ( Q `  M
)  e.  RR )
205204ad2antrr 737 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  <  M )  ->  ( Q `  M )  e.  RR )
206 simpr 467 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  <  M )  ->  i  <  M )
20793ad2antrr 737 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  <  M )  ->  Q  Isom  <  ,  <  (
( 0 ... M
) ,  H ) )
208 simpr 467 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  i  e.  ( 0 ... M
) )
209181adantr 471 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  M  e.  ( 0 ... M
) )
210208, 209jca 539 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  (
i  e.  ( 0 ... M )  /\  M  e.  ( 0 ... M ) ) )
211210adantr 471 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  <  M )  ->  (
i  e.  ( 0 ... M )  /\  M  e.  ( 0 ... M ) ) )
212 isorel 6241 . . . . . . . . . . . . . 14  |-  ( ( Q  Isom  <  ,  <  ( ( 0 ... M
) ,  H )  /\  ( i  e.  ( 0 ... M
)  /\  M  e.  ( 0 ... M
) ) )  -> 
( i  <  M  <->  ( Q `  i )  <  ( Q `  M ) ) )
213207, 211, 212syl2anc 671 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  <  M )  ->  (
i  <  M  <->  ( Q `  i )  <  ( Q `  M )
) )
214206, 213mpbid 215 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  <  M )  ->  ( Q `  i )  <  ( Q `  M
) )
215203, 205, 214ltled 9808 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  <  M )  ->  ( Q `  i )  <_  ( Q `  M
) )
216202, 215syldan 477 . . . . . . . . . 10  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  i  =  M )  ->  ( Q `  i
)  <_  ( Q `  M ) )
217192, 216pm2.61dan 805 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  ( Q `  i )  <_  ( Q `  M
) )
2182173adant3 1034 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 0 ... M
)  /\  ( Q `  i )  =  pi )  ->  ( Q `  i )  <_  ( Q `  M )
)
219188, 218eqbrtrd 4436 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 0 ... M
)  /\  ( Q `  i )  =  pi )  ->  pi  <_  ( Q `  M ) )
2202043ad2ant1 1035 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 0 ... M
)  /\  ( Q `  i )  =  pi )  ->  ( Q `  M )  e.  RR )
22114a1i 11 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 0 ... M
)  /\  ( Q `  i )  =  pi )  ->  pi  e.  RR )
222220, 221letri3d 9802 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 0 ... M
)  /\  ( Q `  i )  =  pi )  ->  ( ( Q `  M )  =  pi  <->  ( ( Q `
 M )  <_  pi  /\  pi  <_  ( Q `  M )
) ) )
223185, 219, 222mpbir2and 938 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0 ... M
)  /\  ( Q `  i )  =  pi )  ->  ( Q `  M )  =  pi )
224223rexlimdv3a 2892 . . . . 5  |-  ( ph  ->  ( E. i  e.  ( 0 ... M
) ( Q `  i )  =  pi 
->  ( Q `  M
)  =  pi ) )
225178, 224mpd 15 . . . 4  |-  ( ph  ->  ( Q `  M
)  =  pi )
226 elfzoelz 11950 . . . . . . . . 9  |-  ( i  e.  ( 0..^ M )  ->  i  e.  ZZ )
227226zred 11068 . . . . . . . 8  |-  ( i  e.  ( 0..^ M )  ->  i  e.  RR )
228227ltp1d 10564 . . . . . . 7  |-  ( i  e.  ( 0..^ M )  ->  i  <  ( i  +  1 ) )
229228adantl 472 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  i  <  (
i  +  1 ) )
230 elfzofz 11965 . . . . . . . 8  |-  ( i  e.  ( 0..^ M )  ->  i  e.  ( 0 ... M
) )
231 fzofzp1 12038 . . . . . . . 8  |-  ( i  e.  ( 0..^ M )  ->  ( i  +  1 )  e.  ( 0 ... M
) )
232230, 231jca 539 . . . . . . 7  |-  ( i  e.  ( 0..^ M )  ->  ( i  e.  ( 0 ... M
)  /\  ( i  +  1 )  e.  ( 0 ... M
) ) )
233 isorel 6241 . . . . . . 7  |-  ( ( Q  Isom  <  ,  <  ( ( 0 ... M
) ,  H )  /\  ( i  e.  ( 0 ... M
)  /\  ( i  +  1 )  e.  ( 0 ... M
) ) )  -> 
( i  <  (
i  +  1 )  <-> 
( Q `  i
)  <  ( Q `  ( i  +  1 ) ) ) )
23493, 232, 233syl2an 484 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( i  < 
( i  +  1 )  <->  ( Q `  i )  <  ( Q `  ( i  +  1 ) ) ) )
235229, 234mpbid 215 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( Q `  i )  <  ( Q `  ( i  +  1 ) ) )
236235ralrimiva 2813 . . . 4  |-  ( ph  ->  A. i  e.  ( 0..^ M ) ( Q `  i )  <  ( Q `  ( i  +  1 ) ) )
237173, 225, 236jca31 541 . . 3  |-  ( ph  ->  ( ( ( Q `
 0 )  = 
-u pi  /\  ( Q `  M )  =  pi )  /\  A. i  e.  ( 0..^ M ) ( Q `
 i )  < 
( Q `  (
i  +  1 ) ) ) )
2387fourierdlem2 38008 . . . 4  |-  ( 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 ) ) ) ) ) )
23969, 238syl 17 . . 3  |-  ( 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 ) ) ) ) ) )
240101, 237, 239mpbir2and 938 . 2  |-  ( ph  ->  Q  e.  ( P `
 M ) )
241 fourierdlem114.g . . . . 5  |-  G  =  ( ( RR  _D  F )  |`  ( -u pi (,) pi ) )
242241reseq1i 5119 . . . 4  |-  ( G  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) )  =  ( ( ( RR  _D  F )  |`  ( -u pi (,) pi ) )  |`  ( ( Q `  i ) (,) ( Q `  (
i  +  1 ) ) ) )
24316a1i 11 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  -u pi  e.  RR* )
24417a1i 11 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  pi  e.  RR* )
245179adantr 471 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  Q : ( 0 ... M ) --> ( -u pi [,] pi ) )
246 simpr 467 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  i  e.  ( 0..^ M ) )
247243, 244, 245, 246fourierdlem27 38033 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( Q `
 i ) (,) ( Q `  (
i  +  1 ) ) )  C_  ( -u pi (,) pi ) )
248247resabs1d 5152 . . . 4  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( ( RR  _D  F )  |`  ( -u pi (,) pi ) )  |`  (
( Q `  i
) (,) ( Q `
 ( i  +  1 ) ) ) )  =  ( ( RR  _D  F )  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) )
249242, 248syl5req 2508 . . 3  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( RR 
_D  F )  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) )  =  ( G  |`  ( ( Q `  i ) (,) ( Q `  (
i  +  1 ) ) ) ) )
250 fourierdlem114.gcn . . . 4  |-  ( ph  ->  G  e.  ( dom 
G -cn-> CC ) )
251250, 7, 69, 240, 11, 167fourierdlem38 38045 . . 3  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( G  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) )  e.  ( ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) -cn-> CC ) )
252249, 251eqeltrd 2539 . 2  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( RR 
_D  F )  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) )  e.  ( ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) -cn-> CC ) )
253249oveq1d 6329 . . 3  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( ( RR  _D  F )  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) lim CC  ( Q `  i )
)  =  ( ( G  |`  ( ( Q `  i ) (,) ( Q `  (
i  +  1 ) ) ) ) lim CC  ( Q `  i ) ) )
254250adantr 471 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  G  e.  ( dom  G -cn-> CC ) )
255 fourierdlem114.rlim . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( -u pi [,) pi )  \  dom  G ) )  ->  (
( G  |`  (
x (,) +oo )
) lim CC  x )  =/=  (/) )
256255adantlr 726 . . . . 5  |-  ( ( ( ph  /\  i  e.  ( 0..^ M ) )  /\  x  e.  ( ( -u pi [,) pi )  \  dom  G ) )  ->  (
( G  |`  (
x (,) +oo )
) lim CC  x )  =/=  (/) )
257 fourierdlem114.llim . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( -u pi (,] pi )  \  dom  G ) )  ->  (
( G  |`  ( -oo (,) x ) ) lim
CC  x )  =/=  (/) )
258257adantlr 726 . . . . 5  |-  ( ( ( ph  /\  i  e.  ( 0..^ M ) )  /\  x  e.  ( ( -u pi (,] pi )  \  dom  G ) )  ->  (
( G  |`  ( -oo (,) x ) ) lim
CC  x )  =/=  (/) )
25993adantr 471 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  Q  Isom  <  ,  <  ( ( 0 ... M ) ,  H ) )
260259, 94, 953syl 18 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  Q : ( 0 ... M ) --> H )
26181adantr 471 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( E `  X )  e.  RR )
262259, 165, 1663syl 18 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ran  Q  =  H )
263254, 256, 258, 259, 260, 246, 235, 247, 261, 11, 262fourierdlem46 38053 . . . 4  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( ( G  |`  ( ( Q `  i ) (,) ( Q `  (
i  +  1 ) ) ) ) lim CC  ( Q `  i ) )  =/=  (/)  /\  (
( G  |`  (
( Q `  i
) (,) ( Q `
 ( i  +  1 ) ) ) ) lim CC  ( Q `
 ( i  +  1 ) ) )  =/=  (/) ) )
264263simpld 465 . . 3  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( G  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) lim CC  ( Q `  i )
)  =/=  (/) )
265253, 264eqnetrd 2702 . 2  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( ( RR  _D  F )  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) lim CC  ( Q `  i )
)  =/=  (/) )
266249oveq1d 6329 . . 3  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( ( RR  _D  F )  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) lim CC  ( Q `  ( i  +  1 ) ) )  =  ( ( G  |`  ( ( Q `  i ) (,) ( Q `  (
i  +  1 ) ) ) ) lim CC  ( Q `  ( i  +  1 ) ) ) )
267263simprd 469 . . 3  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( G  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) lim CC  ( Q `  ( i  +  1 ) ) )  =/=  (/) )
268266, 267eqnetrd 2702 . 2  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( ( RR  _D  F )  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) lim CC  ( Q `  ( i  +  1 ) ) )  =/=  (/) )
269 fourierdlem114.a . 2  |-  A  =  ( n  e.  NN0  |->  ( S. ( -u pi (,) pi ) ( ( F `  x )  x.  ( cos `  (
n  x.  x ) ) )  _d x  /  pi ) )
270 fourierdlem114.b . 2  |-  B  =  ( n  e.  NN  |->  ( S. ( -u pi (,) pi ) ( ( F `  x )  x.  ( sin `  (
n  x.  x ) ) )  _d x  /  pi ) )
271 fourierdlem114.s . 2  |-  S  =  ( n  e.  NN  |->  ( ( ( A `
 n )  x.  ( cos `  (
n  x.  X ) ) )  +  ( ( B `  n
)  x.  ( sin `  ( n  x.  X
) ) ) ) )
27283tpid3 4100 . . . . 5  |-  ( E `
 X )  e. 
{ -u pi ,  pi ,  ( E `  X ) }
273 elun1 3612 . . . . 5  |-  ( ( E `  X )  e.  { -u pi ,  pi ,  ( E `
 X ) }  ->  ( E `  X )  e.  ( { -u pi ,  pi ,  ( E `  X ) }  u.  ( ( -u pi [,] pi )  \  dom  G ) ) )
274272, 273mp1i 13 . . . 4  |-  ( ph  ->  ( E `  X
)  e.  ( {
-u pi ,  pi ,  ( E `  X ) }  u.  ( ( -u pi [,] pi )  \  dom  G ) ) )
275274, 11syl6eleqr 2550 . . 3  |-  ( ph  ->  ( E `  X
)  e.  H )
276275, 167eleqtrrd 2542 . 2  |-  ( ph  ->  ( E `  X
)  e.  ran  Q
)
2771, 2, 3, 4, 5, 6, 7, 69, 240, 252, 265, 268, 269, 270, 271, 78, 276fourierdlem113 38120 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:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 375    /\ w3a 991    = wceq 1454    e. wcel 1897    =/= wne 2632   A.wral 2748   E.wrex 2749   {crab 2752   _Vcvv 3056    \ cdif 3412    u. cun 3413    C_ wss 3415   (/)c0 3742   {cpr 3981   {ctp 3983   class class class wbr 4415    |-> cmpt 4474   dom cdm 4852   ran crn 4853    |` cres 4854   iotacio 5562    Fn wfn 5595   -->wf 5596   -onto->wfo 5598   -1-1-onto->wf1o 5599   ` cfv 5600    Isom wiso 5601  (class class class)co 6314    ^m cmap 7497   Fincfn 7594   CCcc 9562   RRcr 9563   0cc0 9564   1c1 9565    + caddc 9567    x. cmul 9569   +oocpnf 9697   -oocmnf 9698   RR*cxr 9699    < clt 9700    <_ cle 9701    - cmin 9885   -ucneg 9886    / cdiv 10296   NNcn 10636   2c2 10686   NN0cn0 10897   ZZcz 10965   ZZ>=cuz 11187   (,)cioo 11663   (,]cioc 11664   [,)cico 11665   [,]cicc 11666   ...cfz 11812  ..^cfzo 11945   |_cfl 12057    seqcseq 12244   #chash 12546    ~~> cli 13596   sum_csu 13800   sincsin 14164   cosccos 14165   picpi 14167   -cn->ccncf 21956   S.citg 22624   lim CC climc 22865    _D cdv 22866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-inf2 8171  ax-cc 8890  ax-cnex 9620  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-mulcom 9628  ax-addass 9629  ax-mulass 9630  ax-distr 9631  ax-i2m1 9632  ax-1ne0 9633  ax-1rid 9634  ax-rnegex 9635  ax-rrecex 9636  ax-cnre 9637  ax-pre-lttri 9638  ax-pre-lttrn 9639  ax-pre-ltadd 9640  ax-pre-mulgt0 9641  ax-pre-sup 9642  ax-addf 9643  ax-mulf 9644
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-fal 1460  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-iun 4293  df-iin 4294  df-disj 4387  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-se 4812  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-isom 5609  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-of 6557  df-ofr 6558  df-om 6719  df-1st 6819  df-2nd 6820  df-supp 6941  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-1o 7207  df-2o 7208  df-oadd 7211  df-omul 7212  df-er 7388  df-map 7499  df-pm 7500  df-ixp 7548  df-en 7595  df-dom 7596  df-sdom 7597  df-fin 7598  df-fsupp 7909  df-fi 7950  df-sup 7981  df-inf 7982  df-oi 8050  df-card 8398  df-acn 8401  df-cda 8623  df-pnf 9702  df-mnf 9703  df-xr 9704  df-ltxr 9705  df-le 9706  df-sub 9887  df-neg 9888  df-div 10297  df-nn 10637  df-2 10695  df-3 10696  df-4 10697  df-5 10698  df-6 10699  df-7 10700  df-8 10701  df-9 10702  df-10 10703  df-n0 10898  df-z 10966  df-dec 11080  df-uz 11188  df-q 11293  df-rp 11331  df-xneg 11437  df-xadd 11438  df-xmul 11439  df-ioo 11667  df-ioc 11668  df-ico 11669  df-icc 11670  df-fz 11813  df-fzo 11946  df-fl 12059  df-mod 12128  df-seq 12245  df-exp 12304  df-fac 12491  df-bc 12519  df-hash 12547  df-shft 13178  df-cj 13210  df-re 13211  df-im 13212  df-sqrt 13346  df-abs 13347  df-limsup 13574  df-clim 13600  df-rlim 13601  df-sum 13801  df-ef 14169  df-sin 14171  df-cos 14172  df-pi 14174  df-struct 15171  df-ndx 15172  df-slot 15173  df-base 15174  df-sets 15175  df-ress 15176  df-plusg 15251  df-mulr 15252  df-starv 15253  df-sca 15254  df-vsca 15255  df-ip 15256  df-tset 15257  df-ple 15258  df-ds 15260  df-unif 15261  df-hom 15262  df-cco 15263  df-rest 15369  df-topn 15370  df-0g 15388  df-gsum 15389  df-topgen 15390  df-pt 15391  df-prds 15394  df-xrs 15448  df-qtop 15454  df-imas 15455  df-xps 15458  df-mre 15540  df-mrc 15541  df-acs 15543  df-mgm 16536  df-sgrp 16575  df-mnd 16585  df-submnd 16631  df-mulg 16724  df-cntz 17019  df-cmn 17480  df-psmet 19010  df-xmet 19011  df-met 19012  df-bl 19013  df-mopn 19014  df-fbas 19015  df-fg 19016  df-cnfld 19019  df-top 19969  df-bases 19970  df-topon 19971  df-topsp 19972  df-cld 20082  df-ntr 20083  df-cls 20084  df-nei 20162  df-lp 20200  df-perf 20201  df-cn 20291  df-cnp 20292  df-t1 20378  df-haus 20379  df-cmp 20450  df-tx 20625  df-hmeo 20818  df-fil 20909  df-fm 21001  df-flim 21002  df-flf 21003  df-xms 21383  df-ms 21384  df-tms 21385  df-cncf 21958  df-ovol 22464  df-vol 22466  df-mbf 22625  df-itg1 22626  df-itg2 22627  df-ibl 22628  df-itg 22629  df-0p 22676  df-ditg 22850  df-limc 22869  df-dv 22870
This theorem is referenced by:  fourierdlem115  38122
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