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Theorem fourierdlem114 32164
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 10917 . . . . . 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 7814 . . . . . . . . . 10  |-  { -u pi ,  pi , 
( E `  X
) }  e.  Fin
1312a1i 11 . . . . . . . . 9  |-  ( ph  ->  { -u pi ,  pi ,  ( E `  X ) }  e.  Fin )
14 pire 22976 . . . . . . . . . . . . . . 15  |-  pi  e.  RR
1514renegcli 9899 . . . . . . . . . . . . . 14  |-  -u pi  e.  RR
1615rexri 9663 . . . . . . . . . . . . 13  |-  -u pi  e.  RR*
1714rexri 9663 . . . . . . . . . . . . 13  |-  pi  e.  RR*
18 negpilt0 31623 . . . . . . . . . . . . . . 15  |-  -u pi  <  0
19 pipos 22978 . . . . . . . . . . . . . . 15  |-  0  <  pi
20 0re 9613 . . . . . . . . . . . . . . . 16  |-  0  e.  RR
2115, 20, 14lttri 9727 . . . . . . . . . . . . . . 15  |-  ( (
-u pi  <  0  /\  0  <  pi )  ->  -u pi  <  pi )
2218, 19, 21mp2an 672 . . . . . . . . . . . . . 14  |-  -u pi  <  pi
2315, 14, 22ltleii 9724 . . . . . . . . . . . . 13  |-  -u pi  <_  pi
24 prunioo 11674 . . . . . . . . . . . . 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 1324 . . . . . . . . . . . 12  |-  ( (
-u pi (,) pi )  u.  { -u pi ,  pi } )  =  ( -u pi [,] pi )
2625difeq1i 3614 . . . . . . . . . . 11  |-  ( ( ( -u pi (,) pi )  u.  { -u pi ,  pi }
)  \  dom  G )  =  ( ( -u pi [,] pi )  \  dom  G )
27 difundir 3758 . . . . . . . . . . 11  |-  ( ( ( -u pi (,) pi )  u.  { -u pi ,  pi }
)  \  dom  G )  =  ( ( (
-u pi (,) pi )  \  dom  G )  u.  ( { -u pi ,  pi }  \  dom  G ) )
2826, 27eqtr3i 2488 . . . . . . . . . 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 7813 . . . . . . . . . . . 12  |-  { -u pi ,  pi }  e.  Fin
31 diffi 7770 . . . . . . . . . . . 12  |-  ( {
-u pi ,  pi }  e.  Fin  ->  ( { -u pi ,  pi }  \  dom  G )  e.  Fin )
3230, 31mp1i 12 . . . . . . . . . . 11  |-  ( ph  ->  ( { -u pi ,  pi }  \  dom  G )  e.  Fin )
33 unfi 7805 . . . . . . . . . . 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 661 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( -u pi (,) pi )  \  dom  G )  u.  ( { -u pi ,  pi }  \  dom  G ) )  e.  Fin )
3528, 34syl5eqel 2549 . . . . . . . . 9  |-  ( ph  ->  ( ( -u pi [,] pi )  \  dom  G )  e.  Fin )
36 unfi 7805 . . . . . . . . 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 661 . . . . . . . 8  |-  ( ph  ->  ( { -u pi ,  pi ,  ( E `
 X ) }  u.  ( ( -u pi [,] pi )  \  dom  G ) )  e. 
Fin )
3811, 37syl5eqel 2549 . . . . . . 7  |-  ( ph  ->  H  e.  Fin )
39 hashcl 12430 . . . . . . 7  |-  ( H  e.  Fin  ->  ( # `
 H )  e. 
NN0 )
4038, 39syl 16 . . . . . 6  |-  ( ph  ->  ( # `  H
)  e.  NN0 )
4140nn0zd 10988 . . . . 5  |-  ( ph  ->  ( # `  H
)  e.  ZZ )
4215, 22ltneii 9714 . . . . . . 7  |-  -u pi  =/=  pi
43 hashprg 12463 . . . . . . . 8  |-  ( (
-u pi  e.  RR  /\  pi  e.  RR )  ->  ( -u pi  =/=  pi  <->  ( # `  { -u pi ,  pi }
)  =  2 ) )
4415, 14, 43mp2an 672 . . . . . . 7  |-  ( -u pi  =/=  pi  <->  ( # `  { -u pi ,  pi }
)  =  2 )
4542, 44mpbi 208 . . . . . 6  |-  ( # `  { -u pi ,  pi } )  =  2
4612elexi 3119 . . . . . . . . . 10  |-  { -u pi ,  pi , 
( E `  X
) }  e.  _V
47 ovex 6324 . . . . . . . . . . 11  |-  ( -u pi [,] pi )  e. 
_V
48 difexg 4604 . . . . . . . . . . 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 6597 . . . . . . . . 9  |-  ( {
-u pi ,  pi ,  ( E `  X ) }  u.  ( ( -u pi [,] pi )  \  dom  G ) )  e.  _V
5111, 50eqeltri 2541 . . . . . . . 8  |-  H  e. 
_V
52 negex 9837 . . . . . . . . . . 11  |-  -u pi  e.  _V
5352tpid1 4145 . . . . . . . . . 10  |-  -u pi  e.  { -u pi ,  pi ,  ( E `  X ) }
5414elexi 3119 . . . . . . . . . . 11  |-  pi  e.  _V
5554tpid2 4146 . . . . . . . . . 10  |-  pi  e.  {
-u pi ,  pi ,  ( E `  X ) }
56 prssi 4188 . . . . . . . . . 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 672 . . . . . . . . 9  |-  { -u pi ,  pi }  C_ 
{ -u pi ,  pi ,  ( E `  X ) }
58 ssun1 3663 . . . . . . . . . 10  |-  { -u pi ,  pi , 
( E `  X
) }  C_  ( { -u pi ,  pi ,  ( E `  X ) }  u.  ( ( -u pi [,] pi )  \  dom  G ) )
5958, 11sseqtr4i 3532 . . . . . . . . 9  |-  { -u pi ,  pi , 
( E `  X
) }  C_  H
6057, 59sstri 3508 . . . . . . . 8  |-  { -u pi ,  pi }  C_  H
61 hashss 12477 . . . . . . . 8  |-  ( ( H  e.  _V  /\  {
-u pi ,  pi }  C_  H )  -> 
( # `  { -u pi ,  pi }
)  <_  ( # `  H
) )
6251, 60, 61mp2an 672 . . . . . . 7  |-  ( # `  { -u pi ,  pi } )  <_  ( # `
 H )
6362a1i 11 . . . . . 6  |-  ( ph  ->  ( # `  { -u pi ,  pi }
)  <_  ( # `  H
) )
6445, 63syl5eqbrr 4490 . . . . 5  |-  ( ph  ->  2  <_  ( # `  H
) )
65 eluz2 11112 . . . . 5  |-  ( (
# `  H )  e.  ( ZZ>= `  2 )  <->  ( 2  e.  ZZ  /\  ( # `  H )  e.  ZZ  /\  2  <_  ( # `  H
) ) )
6610, 41, 64, 65syl3anbrc 1180 . . . 4  |-  ( ph  ->  ( # `  H
)  e.  ( ZZ>= ` 
2 ) )
67 uz2m1nn 11181 . . . 4  |-  ( (
# `  H )  e.  ( ZZ>= `  2 )  ->  ( ( # `  H
)  -  1 )  e.  NN )
6866, 67syl 16 . . 3  |-  ( ph  ->  ( ( # `  H
)  -  1 )  e.  NN )
698, 68syl5eqel 2549 . 2  |-  ( ph  ->  M  e.  NN )
7015a1i 11 . . . . . . . . . . 11  |-  ( ph  -> 
-u pi  e.  RR )
7114a1i 11 . . . . . . . . . . 11  |-  ( ph  ->  pi  e.  RR )
72 negpitopissre 23052 . . . . . . . . . . . 12  |-  ( -u pi (,] pi )  C_  RR
7322a1i 11 . . . . . . . . . . . . . 14  |-  ( ph  -> 
-u pi  <  pi )
74 picn 22977 . . . . . . . . . . . . . . . 16  |-  pi  e.  CC
75742timesi 10677 . . . . . . . . . . . . . . 15  |-  ( 2  x.  pi )  =  ( pi  +  pi )
7674, 74subnegi 9917 . . . . . . . . . . . . . . 15  |-  ( pi 
-  -u pi )  =  ( pi  +  pi )
7775, 2, 763eqtr4i 2496 . . . . . . . . . . . . . 14  |-  T  =  ( pi  -  -u pi )
78 fourierdlem114.e . . . . . . . . . . . . . 14  |-  E  =  ( x  e.  RR  |->  ( x  +  (
( |_ `  (
( pi  -  x
)  /  T ) )  x.  T ) ) )
7970, 71, 73, 77, 78fourierdlem4 32054 . . . . . . . . . . . . 13  |-  ( ph  ->  E : RR --> ( -u pi (,] pi ) )
8079, 4ffvelrnd 6033 . . . . . . . . . . . 12  |-  ( ph  ->  ( E `  X
)  e.  ( -u pi (,] pi ) )
8172, 80sseldi 3497 . . . . . . . . . . 11  |-  ( ph  ->  ( E `  X
)  e.  RR )
8270, 71, 813jca 1176 . . . . . . . . . 10  |-  ( ph  ->  ( -u pi  e.  RR  /\  pi  e.  RR  /\  ( E `  X
)  e.  RR ) )
83 fvex 5882 . . . . . . . . . . 11  |-  ( E `
 X )  e. 
_V
8452, 54, 83tpss 4197 . . . . . . . . . 10  |-  ( (
-u pi  e.  RR  /\  pi  e.  RR  /\  ( E `  X )  e.  RR )  <->  { -u pi ,  pi ,  ( E `
 X ) } 
C_  RR )
8582, 84sylib 196 . . . . . . . . 9  |-  ( ph  ->  { -u pi ,  pi ,  ( E `  X ) }  C_  RR )
86 iccssre 11631 . . . . . . . . . . 11  |-  ( (
-u pi  e.  RR  /\  pi  e.  RR )  ->  ( -u pi [,] pi )  C_  RR )
8715, 14, 86mp2an 672 . . . . . . . . . 10  |-  ( -u pi [,] pi )  C_  RR
88 ssdifss 3631 . . . . . . . . . 10  |-  ( (
-u pi [,] pi )  C_  RR  ->  (
( -u pi [,] pi )  \  dom  G ) 
C_  RR )
8987, 88mp1i 12 . . . . . . . . 9  |-  ( ph  ->  ( ( -u pi [,] pi )  \  dom  G )  C_  RR )
9085, 89unssd 3676 . . . . . . . 8  |-  ( ph  ->  ( { -u pi ,  pi ,  ( E `
 X ) }  u.  ( ( -u pi [,] pi )  \  dom  G ) )  C_  RR )
9111, 90syl5eqss 3543 . . . . . . 7  |-  ( ph  ->  H  C_  RR )
92 fourierdlem114.q . . . . . . 7  |-  Q  =  ( iota g g 
Isom  <  ,  <  (
( 0 ... M
) ,  H ) )
9338, 91, 92, 8fourierdlem36 32086 . . . . . 6  |-  ( ph  ->  Q  Isom  <  ,  <  ( ( 0 ... M
) ,  H ) )
94 isof1o 6222 . . . . . 6  |-  ( Q 
Isom  <  ,  <  (
( 0 ... M
) ,  H )  ->  Q : ( 0 ... M ) -1-1-onto-> H )
95 f1of 5822 . . . . . 6  |-  ( Q : ( 0 ... M ) -1-1-onto-> H  ->  Q :
( 0 ... M
) --> H )
9693, 94, 953syl 20 . . . . 5  |-  ( ph  ->  Q : ( 0 ... M ) --> H )
9796, 91fssd 5746 . . . 4  |-  ( ph  ->  Q : ( 0 ... M ) --> RR )
98 reex 9600 . . . . 5  |-  RR  e.  _V
99 ovex 6324 . . . . 5  |-  ( 0 ... M )  e. 
_V
10098, 99elmap 7466 . . . 4  |-  ( Q  e.  ( RR  ^m  ( 0 ... M
) )  <->  Q :
( 0 ... M
) --> RR )
10197, 100sylibr 212 . . 3  |-  ( ph  ->  Q  e.  ( RR 
^m  ( 0 ... M ) ) )
102 fveq2 5872 . . . . . . . . . . 11  |-  ( 0  =  i  ->  ( Q `  0 )  =  ( Q `  i ) )
103102adantl 466 . . . . . . . . . 10  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  =  i )  -> 
( Q `  0
)  =  ( Q `
 i ) )
10497ffvelrnda 6032 . . . . . . . . . . . 12  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  ( Q `  i )  e.  RR )
105104leidd 10140 . . . . . . . . . . 11  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  ( Q `  i )  <_  ( Q `  i
) )
106105adantr 465 . . . . . . . . . 10  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  =  i )  -> 
( Q `  i
)  <_  ( Q `  i ) )
107103, 106eqbrtrd 4476 . . . . . . . . 9  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  =  i )  -> 
( Q `  0
)  <_  ( Q `  i ) )
108 elfzelz 11713 . . . . . . . . . . . . 13  |-  ( i  e.  ( 0 ... M )  ->  i  e.  ZZ )
109108zred 10990 . . . . . . . . . . . 12  |-  ( i  e.  ( 0 ... M )  ->  i  e.  RR )
110109ad2antlr 726 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  0  =  i )  ->  i  e.  RR )
111 elfzle1 11714 . . . . . . . . . . . 12  |-  ( i  e.  ( 0 ... M )  ->  0  <_  i )
112111ad2antlr 726 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  0  =  i )  ->  0  <_  i )
113 neqne 31595 . . . . . . . . . . . . 13  |-  ( -.  0  =  i  -> 
0  =/=  i )
114113necomd 2728 . . . . . . . . . . . 12  |-  ( -.  0  =  i  -> 
i  =/=  0 )
115114adantl 466 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  0  =  i )  ->  i  =/=  0 )
116110, 112, 115ne0gt0d 9739 . . . . . . . . . 10  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  0  =  i )  ->  0  <  i )
117 nnssnn0 10819 . . . . . . . . . . . . . . . . 17  |-  NN  C_  NN0
118 nn0uz 11140 . . . . . . . . . . . . . . . . 17  |-  NN0  =  ( ZZ>= `  0 )
119117, 118sseqtri 3531 . . . . . . . . . . . . . . . 16  |-  NN  C_  ( ZZ>= `  0 )
120119, 69sseldi 3497 . . . . . . . . . . . . . . 15  |-  ( ph  ->  M  e.  ( ZZ>= ` 
0 ) )
121 eluzfz1 11718 . . . . . . . . . . . . . . 15  |-  ( M  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... M
) )
122120, 121syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  0  e.  ( 0 ... M ) )
12396, 122ffvelrnd 6033 . . . . . . . . . . . . 13  |-  ( ph  ->  ( Q `  0
)  e.  H )
12491, 123sseldd 3500 . . . . . . . . . . . 12  |-  ( ph  ->  ( Q `  0
)  e.  RR )
125124ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  <  i )  ->  ( Q `  0 )  e.  RR )
126104adantr 465 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  <  i )  ->  ( Q `  i )  e.  RR )
127 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  <  i )  ->  0  <  i )
12893ad2antrr 725 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  <  i )  ->  Q  Isom  <  ,  <  (
( 0 ... M
) ,  H ) )
129122anim1i 568 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  (
0  e.  ( 0 ... M )  /\  i  e.  ( 0 ... M ) ) )
130129adantr 465 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  <  i )  ->  (
0  e.  ( 0 ... M )  /\  i  e.  ( 0 ... M ) ) )
131 isorel 6223 . . . . . . . . . . . . 13  |-  ( ( Q  Isom  <  ,  <  ( ( 0 ... M
) ,  H )  /\  ( 0  e.  ( 0 ... M
)  /\  i  e.  ( 0 ... M
) ) )  -> 
( 0  <  i  <->  ( Q `  0 )  <  ( Q `  i ) ) )
132128, 130, 131syl2anc 661 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  <  i )  ->  (
0  <  i  <->  ( Q `  0 )  < 
( Q `  i
) ) )
133127, 132mpbid 210 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  <  i )  ->  ( Q `  0 )  <  ( Q `  i
) )
134125, 126, 133ltled 9750 . . . . . . . . . 10  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  <  i )  ->  ( Q `  0 )  <_  ( Q `  i
) )
135116, 134syldan 470 . . . . . . . . 9  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  0  =  i )  ->  ( Q `  0
)  <_  ( Q `  i ) )
136107, 135pm2.61dan 791 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  ( Q `  0 )  <_  ( Q `  i
) )
137136adantr 465 . . . . . . 7  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  ( Q `  i )  =  -u pi )  -> 
( Q `  0
)  <_  ( Q `  i ) )
138 simpr 461 . . . . . . 7  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  ( Q `  i )  =  -u pi )  -> 
( Q `  i
)  =  -u pi )
139137, 138breqtrd 4480 . . . . . 6  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  ( Q `  i )  =  -u pi )  -> 
( Q `  0
)  <_  -u pi )
14070rexrd 9660 . . . . . . . 8  |-  ( ph  -> 
-u pi  e.  RR* )
14171rexrd 9660 . . . . . . . 8  |-  ( ph  ->  pi  e.  RR* )
142 lbicc2 11661 . . . . . . . . . . . . . 14  |-  ( (
-u pi  e.  RR*  /\  pi  e.  RR*  /\  -u pi  <_  pi )  ->  -u pi  e.  ( -u pi [,] pi ) )
14316, 17, 23, 142mp3an 1324 . . . . . . . . . . . . 13  |-  -u pi  e.  ( -u pi [,] pi )
144143a1i 11 . . . . . . . . . . . 12  |-  ( ph  -> 
-u pi  e.  (
-u pi [,] pi ) )
145 ubicc2 11662 . . . . . . . . . . . . . 14  |-  ( (
-u pi  e.  RR*  /\  pi  e.  RR*  /\  -u pi  <_  pi )  ->  pi  e.  ( -u pi [,] pi ) )
14616, 17, 23, 145mp3an 1324 . . . . . . . . . . . . 13  |-  pi  e.  ( -u pi [,] pi )
147146a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  pi  e.  ( -u pi [,] pi ) )
148 iocssicc 11637 . . . . . . . . . . . . 13  |-  ( -u pi (,] pi )  C_  ( -u pi [,] pi )
149148, 80sseldi 3497 . . . . . . . . . . . 12  |-  ( ph  ->  ( E `  X
)  e.  ( -u pi [,] pi ) )
150 tpssi 4198 . . . . . . . . . . . 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 1228 . . . . . . . . . . 11  |-  ( ph  ->  { -u pi ,  pi ,  ( E `  X ) }  C_  ( -u pi [,] pi ) )
152 difssd 3628 . . . . . . . . . . 11  |-  ( ph  ->  ( ( -u pi [,] pi )  \  dom  G )  C_  ( -u pi [,] pi ) )
153151, 152unssd 3676 . . . . . . . . . 10  |-  ( ph  ->  ( { -u pi ,  pi ,  ( E `
 X ) }  u.  ( ( -u pi [,] pi )  \  dom  G ) )  C_  ( -u pi [,] pi ) )
15411, 153syl5eqss 3543 . . . . . . . . 9  |-  ( ph  ->  H  C_  ( -u pi [,] pi ) )
155154, 123sseldd 3500 . . . . . . . 8  |-  ( ph  ->  ( Q `  0
)  e.  ( -u pi [,] pi ) )
156 iccgelb 11606 . . . . . . . 8  |-  ( (
-u pi  e.  RR*  /\  pi  e.  RR*  /\  ( Q `  0 )  e.  ( -u pi [,] pi ) )  ->  -u pi  <_  ( Q `  0
) )
157140, 141, 155, 156syl3anc 1228 . . . . . . 7  |-  ( ph  -> 
-u pi  <_  ( Q `  0 )
)
158157ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  ( Q `  i )  =  -u pi )  ->  -u pi  <_  ( Q `  0 ) )
159124ad2antrr 725 . . . . . . 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 9744 . . . . . 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 922 . . . . 5  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  ( Q `  i )  =  -u pi )  -> 
( Q `  0
)  =  -u pi )
16359, 53sselii 3496 . . . . . . 7  |-  -u pi  e.  H
164 f1ofo 5829 . . . . . . . . 9  |-  ( Q : ( 0 ... M ) -1-1-onto-> H  ->  Q :
( 0 ... M
) -onto-> H )
16594, 164syl 16 . . . . . . . 8  |-  ( Q 
Isom  <  ,  <  (
( 0 ... M
) ,  H )  ->  Q : ( 0 ... M )
-onto-> H )
166 forn 5804 . . . . . . . 8  |-  ( Q : ( 0 ... M ) -onto-> H  ->  ran  Q  =  H )
16793, 165, 1663syl 20 . . . . . . 7  |-  ( ph  ->  ran  Q  =  H )
168163, 167syl5eleqr 2552 . . . . . 6  |-  ( ph  -> 
-u pi  e.  ran  Q )
169 ffn 5737 . . . . . . 7  |-  ( Q : ( 0 ... M ) --> H  ->  Q  Fn  ( 0 ... M ) )
170 fvelrnb 5920 . . . . . . 7  |-  ( Q  Fn  ( 0 ... M )  ->  ( -u pi  e.  ran  Q  <->  E. i  e.  ( 0 ... M ) ( Q `  i )  =  -u pi ) )
17196, 169, 1703syl 20 . . . . . 6  |-  ( ph  ->  ( -u pi  e.  ran  Q  <->  E. i  e.  ( 0 ... M ) ( Q `  i
)  =  -u pi ) )
172168, 171mpbid 210 . . . . 5  |-  ( ph  ->  E. i  e.  ( 0 ... M ) ( Q `  i
)  =  -u pi )
173162, 172r19.29a 2999 . . . 4  |-  ( ph  ->  ( Q `  0
)  =  -u pi )
17459, 55sselii 3496 . . . . . . 7  |-  pi  e.  H
175174, 167syl5eleqr 2552 . . . . . 6  |-  ( ph  ->  pi  e.  ran  Q
)
176 fvelrnb 5920 . . . . . . 7  |-  ( Q  Fn  ( 0 ... M )  ->  (
pi  e.  ran  Q  <->  E. i  e.  ( 0 ... M ) ( Q `  i )  =  pi ) )
17796, 169, 1763syl 20 . . . . . 6  |-  ( ph  ->  ( pi  e.  ran  Q  <->  E. i  e.  (
0 ... M ) ( Q `  i )  =  pi ) )
178175, 177mpbid 210 . . . . 5  |-  ( ph  ->  E. i  e.  ( 0 ... M ) ( Q `  i
)  =  pi )
17996, 154fssd 5746 . . . . . . . . . 10  |-  ( ph  ->  Q : ( 0 ... M ) --> (
-u pi [,] pi ) )
180 eluzfz2 11719 . . . . . . . . . . 11  |-  ( M  e.  ( ZZ>= `  0
)  ->  M  e.  ( 0 ... M
) )
181120, 180syl 16 . . . . . . . . . 10  |-  ( ph  ->  M  e.  ( 0 ... M ) )
182179, 181ffvelrnd 6033 . . . . . . . . 9  |-  ( ph  ->  ( Q `  M
)  e.  ( -u pi [,] pi ) )
183 iccleub 11605 . . . . . . . . 9  |-  ( (
-u pi  e.  RR*  /\  pi  e.  RR*  /\  ( Q `  M )  e.  ( -u pi [,] pi ) )  ->  ( Q `  M )  <_  pi )
184140, 141, 182, 183syl3anc 1228 . . . . . . . 8  |-  ( ph  ->  ( Q `  M
)  <_  pi )
1851843ad2ant1 1017 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 0 ... M
)  /\  ( Q `  i )  =  pi )  ->  ( Q `  M )  <_  pi )
186 id 22 . . . . . . . . . 10  |-  ( ( Q `  i )  =  pi  ->  ( Q `  i )  =  pi )
187186eqcomd 2465 . . . . . . . . 9  |-  ( ( Q `  i )  =  pi  ->  pi  =  ( Q `  i ) )
1881873ad2ant3 1019 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 0 ... M
)  /\  ( Q `  i )  =  pi )  ->  pi  =  ( Q `  i ) )
189105adantr 465 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  =  M )  ->  ( Q `  i )  <_  ( Q `  i
) )
190 fveq2 5872 . . . . . . . . . . . 12  |-  ( i  =  M  ->  ( Q `  i )  =  ( Q `  M ) )
191190adantl 466 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  =  M )  ->  ( Q `  i )  =  ( Q `  M ) )
192189, 191breqtrd 4480 . . . . . . . . . 10  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  =  M )  ->  ( Q `  i )  <_  ( Q `  M
) )
193109ad2antlr 726 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  i  =  M )  ->  i  e.  RR )
194 elfzel2 11711 . . . . . . . . . . . . . 14  |-  ( i  e.  ( 0 ... M )  ->  M  e.  ZZ )
195194zred 10990 . . . . . . . . . . . . 13  |-  ( i  e.  ( 0 ... M )  ->  M  e.  RR )
196195ad2antlr 726 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  i  =  M )  ->  M  e.  RR )
197 elfzle2 11715 . . . . . . . . . . . . 13  |-  ( i  e.  ( 0 ... M )  ->  i  <_  M )
198197ad2antlr 726 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  i  =  M )  ->  i  <_  M )
199 neqne 31595 . . . . . . . . . . . . . 14  |-  ( -.  i  =  M  -> 
i  =/=  M )
200199necomd 2728 . . . . . . . . . . . . 13  |-  ( -.  i  =  M  ->  M  =/=  i )
201200adantl 466 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  i  =  M )  ->  M  =/=  i )
202193, 196, 198, 201leneltd 31655 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  i  =  M )  ->  i  <  M )
203104adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  <  M )  ->  ( Q `  i )  e.  RR )
20487, 182sseldi 3497 . . . . . . . . . . . . 13  |-  ( ph  ->  ( Q `  M
)  e.  RR )
205204ad2antrr 725 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  <  M )  ->  ( Q `  M )  e.  RR )
206 simpr 461 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  <  M )  ->  i  <  M )
20793ad2antrr 725 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  <  M )  ->  Q  Isom  <  ,  <  (
( 0 ... M
) ,  H ) )
208 simpr 461 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  i  e.  ( 0 ... M
) )
209181adantr 465 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  M  e.  ( 0 ... M
) )
210208, 209jca 532 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  (
i  e.  ( 0 ... M )  /\  M  e.  ( 0 ... M ) ) )
211210adantr 465 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  <  M )  ->  (
i  e.  ( 0 ... M )  /\  M  e.  ( 0 ... M ) ) )
212 isorel 6223 . . . . . . . . . . . . . 14  |-  ( ( Q  Isom  <  ,  <  ( ( 0 ... M
) ,  H )  /\  ( i  e.  ( 0 ... M
)  /\  M  e.  ( 0 ... M
) ) )  -> 
( i  <  M  <->  ( Q `  i )  <  ( Q `  M ) ) )
213207, 211, 212syl2anc 661 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  <  M )  ->  (
i  <  M  <->  ( Q `  i )  <  ( Q `  M )
) )
214206, 213mpbid 210 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  <  M )  ->  ( Q `  i )  <  ( Q `  M
) )
215203, 205, 214ltled 9750 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  <  M )  ->  ( Q `  i )  <_  ( Q `  M
) )
216202, 215syldan 470 . . . . . . . . . 10  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  i  =  M )  ->  ( Q `  i
)  <_  ( Q `  M ) )
217192, 216pm2.61dan 791 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  ( Q `  i )  <_  ( Q `  M
) )
2182173adant3 1016 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 0 ... M
)  /\  ( Q `  i )  =  pi )  ->  ( Q `  i )  <_  ( Q `  M )
)
219188, 218eqbrtrd 4476 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 0 ... M
)  /\  ( Q `  i )  =  pi )  ->  pi  <_  ( Q `  M ) )
2202043ad2ant1 1017 . . . . . . . 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 9744 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 0 ... M
)  /\  ( Q `  i )  =  pi )  ->  ( ( Q `  M )  =  pi  <->  ( ( Q `
 M )  <_  pi  /\  pi  <_  ( Q `  M )
) ) )
223185, 219, 222mpbir2and 922 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0 ... M
)  /\  ( Q `  i )  =  pi )  ->  ( Q `  M )  =  pi )
224223rexlimdv3a 2951 . . . . 5  |-  ( ph  ->  ( E. i  e.  ( 0 ... M
) ( Q `  i )  =  pi 
->  ( Q `  M
)  =  pi ) )
225178, 224mpd 15 . . . 4  |-  ( ph  ->  ( Q `  M
)  =  pi )
226 elfzoelz 11825 . . . . . . . . 9  |-  ( i  e.  ( 0..^ M )  ->  i  e.  ZZ )
227226zred 10990 . . . . . . . 8  |-  ( i  e.  ( 0..^ M )  ->  i  e.  RR )
228227ltp1d 10496 . . . . . . 7  |-  ( i  e.  ( 0..^ M )  ->  i  <  ( i  +  1 ) )
229228adantl 466 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  i  <  (
i  +  1 ) )
230 elfzofz 11840 . . . . . . . 8  |-  ( i  e.  ( 0..^ M )  ->  i  e.  ( 0 ... M
) )
231 fzofzp1 11911 . . . . . . . 8  |-  ( i  e.  ( 0..^ M )  ->  ( i  +  1 )  e.  ( 0 ... M
) )
232230, 231jca 532 . . . . . . 7  |-  ( i  e.  ( 0..^ M )  ->  ( i  e.  ( 0 ... M
)  /\  ( i  +  1 )  e.  ( 0 ... M
) ) )
233 isorel 6223 . . . . . . 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 477 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( i  < 
( i  +  1 )  <->  ( Q `  i )  <  ( Q `  ( i  +  1 ) ) ) )
235229, 234mpbid 210 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( Q `  i )  <  ( Q `  ( i  +  1 ) ) )
236235ralrimiva 2871 . . . 4  |-  ( ph  ->  A. i  e.  ( 0..^ M ) ( Q `  i )  <  ( Q `  ( i  +  1 ) ) )
237173, 225, 236jca31 534 . . 3  |-  ( ph  ->  ( ( ( Q `
 0 )  = 
-u pi  /\  ( Q `  M )  =  pi )  /\  A. i  e.  ( 0..^ M ) ( Q `
 i )  < 
( Q `  (
i  +  1 ) ) ) )
2387fourierdlem2 32052 . . . 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 16 . . 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 922 . 2  |-  ( ph  ->  Q  e.  ( P `
 M ) )
241 fourierdlem114.g . . . . 5  |-  G  =  ( ( RR  _D  F )  |`  ( -u pi (,) pi ) )
242241reseq1i 5279 . . . 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 465 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  Q : ( 0 ... M ) --> ( -u pi [,] pi ) )
246 simpr 461 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  i  e.  ( 0..^ M ) )
247243, 244, 245, 246fourierdlem27 32077 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( Q `
 i ) (,) ( Q `  (
i  +  1 ) ) )  C_  ( -u pi (,) pi ) )
248247resabs1d 5313 . . . 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 2511 . . 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 32088 . . 3  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( G  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) )  e.  ( ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) -cn-> CC ) )
252249, 251eqeltrd 2545 . 2  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( RR 
_D  F )  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) )  e.  ( ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) -cn-> CC ) )
253249oveq1d 6311 . . 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 465 . . . . 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 714 . . . . 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 714 . . . . 5  |-  ( ( ( ph  /\  i  e.  ( 0..^ M ) )  /\  x  e.  ( ( -u pi (,] pi )  \  dom  G ) )  ->  (
( G  |`  ( -oo (,) x ) ) lim
CC  x )  =/=  (/) )
25993adantr 465 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  Q  Isom  <  ,  <  ( ( 0 ... M ) ,  H ) )
260259, 94, 953syl 20 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  Q : ( 0 ... M ) --> H )
26181adantr 465 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( E `  X )  e.  RR )
262259, 165, 1663syl 20 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ran  Q  =  H )
263254, 256, 258, 259, 260, 246, 235, 247, 261, 11, 262fourierdlem46 32096 . . . 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 459 . . 3  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( G  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) lim CC  ( Q `  i )
)  =/=  (/) )
265253, 264eqnetrd 2750 . 2  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( ( RR  _D  F )  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) lim CC  ( Q `  i )
)  =/=  (/) )
266249oveq1d 6311 . . 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 463 . . 3  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( G  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) lim CC  ( Q `  ( i  +  1 ) ) )  =/=  (/) )
268266, 267eqnetrd 2750 . 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 4148 . . . . 5  |-  ( E `
 X )  e. 
{ -u pi ,  pi ,  ( E `  X ) }
273 elun1 3667 . . . . 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 12 . . . 4  |-  ( ph  ->  ( E `  X
)  e.  ( {
-u pi ,  pi ,  ( E `  X ) }  u.  ( ( -u pi [,] pi )  \  dom  G ) ) )
275274, 11syl6eleqr 2556 . . 3  |-  ( ph  ->  ( E `  X
)  e.  H )
276275, 167eleqtrrd 2548 . 2  |-  ( ph  ->  ( E `  X
)  e.  ran  Q
)
2771, 2, 3, 4, 5, 6, 7, 69, 240, 252, 265, 268, 269, 270, 271, 78, 276fourierdlem113 32163 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 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   E.wrex 2808   {crab 2811   _Vcvv 3109    \ cdif 3468    u. cun 3469    C_ wss 3471   (/)c0 3793   {cpr 4034   {ctp 4036   class class class wbr 4456    |-> cmpt 4515   dom cdm 5008   ran crn 5009    |` cres 5010   iotacio 5555    Fn wfn 5589   -->wf 5590   -onto->wfo 5592   -1-1-onto->wf1o 5593   ` cfv 5594    Isom wiso 5595  (class class class)co 6296    ^m cmap 7438   Fincfn 7535   CCcc 9507   RRcr 9508   0cc0 9509   1c1 9510    + caddc 9512    x. cmul 9514   +oocpnf 9642   -oocmnf 9643   RR*cxr 9644    < clt 9645    <_ cle 9646    - cmin 9824   -ucneg 9825    / cdiv 10227   NNcn 10556   2c2 10606   NN0cn0 10816   ZZcz 10885   ZZ>=cuz 11106   (,)cioo 11554   (,]cioc 11555   [,)cico 11556   [,]cicc 11557   ...cfz 11697  ..^cfzo 11820   |_cfl 11929    seqcseq 12109   #chash 12407    ~~> cli 13318   sum_csu 13519   sincsin 13810   cosccos 13811   picpi 13813   -cn->ccncf 21505   S.citg 22152   lim CC climc 22391    _D cdv 22392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cc 8832  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588  ax-mulf 9589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-disj 4428  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-ofr 6540  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-omul 7153  df-er 7329  df-map 7440  df-pm 7441  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-fi 7889  df-sup 7919  df-oi 7953  df-card 8337  df-acn 8340  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ioo 11558  df-ioc 11559  df-ico 11560  df-icc 11561  df-fz 11698  df-fzo 11821  df-fl 11931  df-mod 11999  df-seq 12110  df-exp 12169  df-fac 12356  df-bc 12383  df-hash 12408  df-shft 12911  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080  df-limsup 13305  df-clim 13322  df-rlim 13323  df-sum 13520  df-ef 13814  df-sin 13816  df-cos 13817  df-pi 13819  df-struct 14645  df-ndx 14646  df-slot 14647  df-base 14648  df-sets 14649  df-ress 14650  df-plusg 14724  df-mulr 14725  df-starv 14726  df-sca 14727  df-vsca 14728  df-ip 14729  df-tset 14730  df-ple 14731  df-ds 14733  df-unif 14734  df-hom 14735  df-cco 14736  df-rest 14839  df-topn 14840  df-0g 14858  df-gsum 14859  df-topgen 14860  df-pt 14861  df-prds 14864  df-xrs 14918  df-qtop 14923  df-imas 14924  df-xps 14926  df-mre 15002  df-mrc 15003  df-acs 15005  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-submnd 16093  df-mulg 16186  df-cntz 16481  df-cmn 16926  df-psmet 18537  df-xmet 18538  df-met 18539  df-bl 18540  df-mopn 18541  df-fbas 18542  df-fg 18543  df-cnfld 18547  df-top 19525  df-bases 19527  df-topon 19528  df-topsp 19529  df-cld 19646  df-ntr 19647  df-cls 19648  df-nei 19725  df-lp 19763  df-perf 19764  df-cn 19854  df-cnp 19855  df-t1 19941  df-haus 19942  df-cmp 20013  df-tx 20188  df-hmeo 20381  df-fil 20472  df-fm 20564  df-flim 20565  df-flf 20566  df-xms 20948  df-ms 20949  df-tms 20950  df-cncf 21507  df-ovol 22001  df-vol 22002  df-mbf 22153  df-itg1 22154  df-itg2 22155  df-ibl 22156  df-itg 22157  df-0p 22202  df-ditg 22376  df-limc 22395  df-dv 22396
This theorem is referenced by:  fourierdlem115  32165
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