Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fourierdlem114 Structured version   Unicode version

Theorem fourierdlem114 37353
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 10857 . . . . . 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 7750 . . . . . . . . . 10  |-  { -u pi ,  pi , 
( E `  X
) }  e.  Fin
1312a1i 11 . . . . . . . . 9  |-  ( ph  ->  { -u pi ,  pi ,  ( E `  X ) }  e.  Fin )
14 pire 23035 . . . . . . . . . . . . . . 15  |-  pi  e.  RR
1514renegcli 9836 . . . . . . . . . . . . . 14  |-  -u pi  e.  RR
1615rexri 9596 . . . . . . . . . . . . 13  |-  -u pi  e.  RR*
1714rexri 9596 . . . . . . . . . . . . 13  |-  pi  e.  RR*
18 negpilt0 36817 . . . . . . . . . . . . . . 15  |-  -u pi  <  0
19 pipos 23037 . . . . . . . . . . . . . . 15  |-  0  <  pi
20 0re 9546 . . . . . . . . . . . . . . . 16  |-  0  e.  RR
2115, 20, 14lttri 9662 . . . . . . . . . . . . . . 15  |-  ( (
-u pi  <  0  /\  0  <  pi )  ->  -u pi  <  pi )
2218, 19, 21mp2an 670 . . . . . . . . . . . . . 14  |-  -u pi  <  pi
2315, 14, 22ltleii 9659 . . . . . . . . . . . . 13  |-  -u pi  <_  pi
24 prunioo 11620 . . . . . . . . . . . . 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 1326 . . . . . . . . . . . 12  |-  ( (
-u pi (,) pi )  u.  { -u pi ,  pi } )  =  ( -u pi [,] pi )
2625difeq1i 3556 . . . . . . . . . . 11  |-  ( ( ( -u pi (,) pi )  u.  { -u pi ,  pi }
)  \  dom  G )  =  ( ( -u pi [,] pi )  \  dom  G )
27 difundir 3702 . . . . . . . . . . 11  |-  ( ( ( -u pi (,) pi )  u.  { -u pi ,  pi }
)  \  dom  G )  =  ( ( (
-u pi (,) pi )  \  dom  G )  u.  ( { -u pi ,  pi }  \  dom  G ) )
2826, 27eqtr3i 2433 . . . . . . . . . 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 7749 . . . . . . . . . . . 12  |-  { -u pi ,  pi }  e.  Fin
31 diffi 7706 . . . . . . . . . . . 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 7741 . . . . . . . . . . 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 659 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( -u pi (,) pi )  \  dom  G )  u.  ( { -u pi ,  pi }  \  dom  G ) )  e.  Fin )
3528, 34syl5eqel 2494 . . . . . . . . 9  |-  ( ph  ->  ( ( -u pi [,] pi )  \  dom  G )  e.  Fin )
36 unfi 7741 . . . . . . . . 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 659 . . . . . . . 8  |-  ( ph  ->  ( { -u pi ,  pi ,  ( E `
 X ) }  u.  ( ( -u pi [,] pi )  \  dom  G ) )  e. 
Fin )
3811, 37syl5eqel 2494 . . . . . . 7  |-  ( ph  ->  H  e.  Fin )
39 hashcl 12382 . . . . . . 7  |-  ( H  e.  Fin  ->  ( # `
 H )  e. 
NN0 )
4038, 39syl 17 . . . . . 6  |-  ( ph  ->  ( # `  H
)  e.  NN0 )
4140nn0zd 10926 . . . . 5  |-  ( ph  ->  ( # `  H
)  e.  ZZ )
4215, 22ltneii 9649 . . . . . . 7  |-  -u pi  =/=  pi
43 hashprg 12416 . . . . . . . 8  |-  ( (
-u pi  e.  RR  /\  pi  e.  RR )  ->  ( -u pi  =/=  pi  <->  ( # `  { -u pi ,  pi }
)  =  2 ) )
4415, 14, 43mp2an 670 . . . . . . 7  |-  ( -u pi  =/=  pi  <->  ( # `  { -u pi ,  pi }
)  =  2 )
4542, 44mpbi 208 . . . . . 6  |-  ( # `  { -u pi ,  pi } )  =  2
4612elexi 3068 . . . . . . . . . 10  |-  { -u pi ,  pi , 
( E `  X
) }  e.  _V
47 ovex 6262 . . . . . . . . . . 11  |-  ( -u pi [,] pi )  e. 
_V
48 difexg 4541 . . . . . . . . . . 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 6536 . . . . . . . . 9  |-  ( {
-u pi ,  pi ,  ( E `  X ) }  u.  ( ( -u pi [,] pi )  \  dom  G ) )  e.  _V
5111, 50eqeltri 2486 . . . . . . . 8  |-  H  e. 
_V
52 negex 9774 . . . . . . . . . . 11  |-  -u pi  e.  _V
5352tpid1 4084 . . . . . . . . . 10  |-  -u pi  e.  { -u pi ,  pi ,  ( E `  X ) }
5414elexi 3068 . . . . . . . . . . 11  |-  pi  e.  _V
5554tpid2 4085 . . . . . . . . . 10  |-  pi  e.  {
-u pi ,  pi ,  ( E `  X ) }
56 prssi 4127 . . . . . . . . . 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 670 . . . . . . . . 9  |-  { -u pi ,  pi }  C_ 
{ -u pi ,  pi ,  ( E `  X ) }
58 ssun1 3605 . . . . . . . . . 10  |-  { -u pi ,  pi , 
( E `  X
) }  C_  ( { -u pi ,  pi ,  ( E `  X ) }  u.  ( ( -u pi [,] pi )  \  dom  G ) )
5958, 11sseqtr4i 3474 . . . . . . . . 9  |-  { -u pi ,  pi , 
( E `  X
) }  C_  H
6057, 59sstri 3450 . . . . . . . 8  |-  { -u pi ,  pi }  C_  H
61 hashss 12430 . . . . . . . 8  |-  ( ( H  e.  _V  /\  {
-u pi ,  pi }  C_  H )  -> 
( # `  { -u pi ,  pi }
)  <_  ( # `  H
) )
6251, 60, 61mp2an 670 . . . . . . 7  |-  ( # `  { -u pi ,  pi } )  <_  ( # `
 H )
6362a1i 11 . . . . . 6  |-  ( ph  ->  ( # `  { -u pi ,  pi }
)  <_  ( # `  H
) )
6445, 63syl5eqbrr 4428 . . . . 5  |-  ( ph  ->  2  <_  ( # `  H
) )
65 eluz2 11051 . . . . 5  |-  ( (
# `  H )  e.  ( ZZ>= `  2 )  <->  ( 2  e.  ZZ  /\  ( # `  H )  e.  ZZ  /\  2  <_  ( # `  H
) ) )
6610, 41, 64, 65syl3anbrc 1181 . . . 4  |-  ( ph  ->  ( # `  H
)  e.  ( ZZ>= ` 
2 ) )
67 uz2m1nn 11119 . . . 4  |-  ( (
# `  H )  e.  ( ZZ>= `  2 )  ->  ( ( # `  H
)  -  1 )  e.  NN )
6866, 67syl 17 . . 3  |-  ( ph  ->  ( ( # `  H
)  -  1 )  e.  NN )
698, 68syl5eqel 2494 . 2  |-  ( ph  ->  M  e.  NN )
7015a1i 11 . . . . . . . . . . 11  |-  ( ph  -> 
-u pi  e.  RR )
7114a1i 11 . . . . . . . . . . 11  |-  ( ph  ->  pi  e.  RR )
72 negpitopissre 23111 . . . . . . . . . . . 12  |-  ( -u pi (,] pi )  C_  RR
7322a1i 11 . . . . . . . . . . . . . 14  |-  ( ph  -> 
-u pi  <  pi )
74 picn 23036 . . . . . . . . . . . . . . . 16  |-  pi  e.  CC
75742timesi 10617 . . . . . . . . . . . . . . 15  |-  ( 2  x.  pi )  =  ( pi  +  pi )
7674, 74subnegi 9854 . . . . . . . . . . . . . . 15  |-  ( pi 
-  -u pi )  =  ( pi  +  pi )
7775, 2, 763eqtr4i 2441 . . . . . . . . . . . . . 14  |-  T  =  ( pi  -  -u pi )
78 fourierdlem114.e . . . . . . . . . . . . . 14  |-  E  =  ( x  e.  RR  |->  ( x  +  (
( |_ `  (
( pi  -  x
)  /  T ) )  x.  T ) ) )
7970, 71, 73, 77, 78fourierdlem4 37243 . . . . . . . . . . . . 13  |-  ( ph  ->  E : RR --> ( -u pi (,] pi ) )
8079, 4ffvelrnd 5966 . . . . . . . . . . . 12  |-  ( ph  ->  ( E `  X
)  e.  ( -u pi (,] pi ) )
8172, 80sseldi 3439 . . . . . . . . . . 11  |-  ( ph  ->  ( E `  X
)  e.  RR )
8270, 71, 813jca 1177 . . . . . . . . . 10  |-  ( ph  ->  ( -u pi  e.  RR  /\  pi  e.  RR  /\  ( E `  X
)  e.  RR ) )
83 fvex 5815 . . . . . . . . . . 11  |-  ( E `
 X )  e. 
_V
8452, 54, 83tpss 4136 . . . . . . . . . 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 11577 . . . . . . . . . . 11  |-  ( (
-u pi  e.  RR  /\  pi  e.  RR )  ->  ( -u pi [,] pi )  C_  RR )
8715, 14, 86mp2an 670 . . . . . . . . . 10  |-  ( -u pi [,] pi )  C_  RR
88 ssdifss 3573 . . . . . . . . . 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 3618 . . . . . . . 8  |-  ( ph  ->  ( { -u pi ,  pi ,  ( E `
 X ) }  u.  ( ( -u pi [,] pi )  \  dom  G ) )  C_  RR )
9111, 90syl5eqss 3485 . . . . . . 7  |-  ( ph  ->  H  C_  RR )
92 fourierdlem114.q . . . . . . 7  |-  Q  =  ( iota g g 
Isom  <  ,  <  (
( 0 ... M
) ,  H ) )
9338, 91, 92, 8fourierdlem36 37275 . . . . . 6  |-  ( ph  ->  Q  Isom  <  ,  <  ( ( 0 ... M
) ,  H ) )
94 isof1o 6160 . . . . . 6  |-  ( Q 
Isom  <  ,  <  (
( 0 ... M
) ,  H )  ->  Q : ( 0 ... M ) -1-1-onto-> H )
95 f1of 5755 . . . . . 6  |-  ( Q : ( 0 ... M ) -1-1-onto-> H  ->  Q :
( 0 ... M
) --> H )
9693, 94, 953syl 20 . . . . 5  |-  ( ph  ->  Q : ( 0 ... M ) --> H )
9796, 91fssd 5679 . . . 4  |-  ( ph  ->  Q : ( 0 ... M ) --> RR )
98 reex 9533 . . . . 5  |-  RR  e.  _V
99 ovex 6262 . . . . 5  |-  ( 0 ... M )  e. 
_V
10098, 99elmap 7405 . . . 4  |-  ( Q  e.  ( RR  ^m  ( 0 ... M
) )  <->  Q :
( 0 ... M
) --> RR )
10197, 100sylibr 212 . . 3  |-  ( ph  ->  Q  e.  ( RR 
^m  ( 0 ... M ) ) )
102 fveq2 5805 . . . . . . . . . . 11  |-  ( 0  =  i  ->  ( Q `  0 )  =  ( Q `  i ) )
103102adantl 464 . . . . . . . . . 10  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  =  i )  -> 
( Q `  0
)  =  ( Q `
 i ) )
10497ffvelrnda 5965 . . . . . . . . . . . 12  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  ( Q `  i )  e.  RR )
105104leidd 10079 . . . . . . . . . . 11  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  ( Q `  i )  <_  ( Q `  i
) )
106105adantr 463 . . . . . . . . . 10  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  =  i )  -> 
( Q `  i
)  <_  ( Q `  i ) )
107103, 106eqbrtrd 4414 . . . . . . . . 9  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  =  i )  -> 
( Q `  0
)  <_  ( Q `  i ) )
108 elfzelz 11659 . . . . . . . . . . . . 13  |-  ( i  e.  ( 0 ... M )  ->  i  e.  ZZ )
109108zred 10928 . . . . . . . . . . . 12  |-  ( i  e.  ( 0 ... M )  ->  i  e.  RR )
110109ad2antlr 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  0  =  i )  ->  i  e.  RR )
111 elfzle1 11660 . . . . . . . . . . . 12  |-  ( i  e.  ( 0 ... M )  ->  0  <_  i )
112111ad2antlr 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  0  =  i )  ->  0  <_  i )
113 neqne 36791 . . . . . . . . . . . . 13  |-  ( -.  0  =  i  -> 
0  =/=  i )
114113necomd 2674 . . . . . . . . . . . 12  |-  ( -.  0  =  i  -> 
i  =/=  0 )
115114adantl 464 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  0  =  i )  ->  i  =/=  0 )
116110, 112, 115ne0gt0d 9674 . . . . . . . . . 10  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  0  =  i )  ->  0  <  i )
117 nnssnn0 10759 . . . . . . . . . . . . . . . . 17  |-  NN  C_  NN0
118 nn0uz 11079 . . . . . . . . . . . . . . . . 17  |-  NN0  =  ( ZZ>= `  0 )
119117, 118sseqtri 3473 . . . . . . . . . . . . . . . 16  |-  NN  C_  ( ZZ>= `  0 )
120119, 69sseldi 3439 . . . . . . . . . . . . . . 15  |-  ( ph  ->  M  e.  ( ZZ>= ` 
0 ) )
121 eluzfz1 11664 . . . . . . . . . . . . . . 15  |-  ( M  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... M
) )
122120, 121syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  0  e.  ( 0 ... M ) )
12396, 122ffvelrnd 5966 . . . . . . . . . . . . 13  |-  ( ph  ->  ( Q `  0
)  e.  H )
12491, 123sseldd 3442 . . . . . . . . . . . 12  |-  ( ph  ->  ( Q `  0
)  e.  RR )
125124ad2antrr 724 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  <  i )  ->  ( Q `  0 )  e.  RR )
126104adantr 463 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  <  i )  ->  ( Q `  i )  e.  RR )
127 simpr 459 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  <  i )  ->  0  <  i )
12893ad2antrr 724 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  <  i )  ->  Q  Isom  <  ,  <  (
( 0 ... M
) ,  H ) )
129122anim1i 566 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  (
0  e.  ( 0 ... M )  /\  i  e.  ( 0 ... M ) ) )
130129adantr 463 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  <  i )  ->  (
0  e.  ( 0 ... M )  /\  i  e.  ( 0 ... M ) ) )
131 isorel 6161 . . . . . . . . . . . . 13  |-  ( ( Q  Isom  <  ,  <  ( ( 0 ... M
) ,  H )  /\  ( 0  e.  ( 0 ... M
)  /\  i  e.  ( 0 ... M
) ) )  -> 
( 0  <  i  <->  ( Q `  0 )  <  ( Q `  i ) ) )
132128, 130, 131syl2anc 659 . . . . . . . . . . . 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 9685 . . . . . . . . . 10  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  <  i )  ->  ( Q `  0 )  <_  ( Q `  i
) )
135116, 134syldan 468 . . . . . . . . 9  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  0  =  i )  ->  ( Q `  0
)  <_  ( Q `  i ) )
136107, 135pm2.61dan 792 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  ( Q `  0 )  <_  ( Q `  i
) )
137136adantr 463 . . . . . . 7  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  ( Q `  i )  =  -u pi )  -> 
( Q `  0
)  <_  ( Q `  i ) )
138 simpr 459 . . . . . . 7  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  ( Q `  i )  =  -u pi )  -> 
( Q `  i
)  =  -u pi )
139137, 138breqtrd 4418 . . . . . 6  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  ( Q `  i )  =  -u pi )  -> 
( Q `  0
)  <_  -u pi )
14070rexrd 9593 . . . . . . . 8  |-  ( ph  -> 
-u pi  e.  RR* )
14171rexrd 9593 . . . . . . . 8  |-  ( ph  ->  pi  e.  RR* )
142 lbicc2 11607 . . . . . . . . . . . . . 14  |-  ( (
-u pi  e.  RR*  /\  pi  e.  RR*  /\  -u pi  <_  pi )  ->  -u pi  e.  ( -u pi [,] pi ) )
14316, 17, 23, 142mp3an 1326 . . . . . . . . . . . . 13  |-  -u pi  e.  ( -u pi [,] pi )
144143a1i 11 . . . . . . . . . . . 12  |-  ( ph  -> 
-u pi  e.  (
-u pi [,] pi ) )
145 ubicc2 11608 . . . . . . . . . . . . . 14  |-  ( (
-u pi  e.  RR*  /\  pi  e.  RR*  /\  -u pi  <_  pi )  ->  pi  e.  ( -u pi [,] pi ) )
14616, 17, 23, 145mp3an 1326 . . . . . . . . . . . . 13  |-  pi  e.  ( -u pi [,] pi )
147146a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  pi  e.  ( -u pi [,] pi ) )
148 iocssicc 11583 . . . . . . . . . . . . 13  |-  ( -u pi (,] pi )  C_  ( -u pi [,] pi )
149148, 80sseldi 3439 . . . . . . . . . . . 12  |-  ( ph  ->  ( E `  X
)  e.  ( -u pi [,] pi ) )
150 tpssi 4137 . . . . . . . . . . . 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 1230 . . . . . . . . . . 11  |-  ( ph  ->  { -u pi ,  pi ,  ( E `  X ) }  C_  ( -u pi [,] pi ) )
152 difssd 3570 . . . . . . . . . . 11  |-  ( ph  ->  ( ( -u pi [,] pi )  \  dom  G )  C_  ( -u pi [,] pi ) )
153151, 152unssd 3618 . . . . . . . . . 10  |-  ( ph  ->  ( { -u pi ,  pi ,  ( E `
 X ) }  u.  ( ( -u pi [,] pi )  \  dom  G ) )  C_  ( -u pi [,] pi ) )
15411, 153syl5eqss 3485 . . . . . . . . 9  |-  ( ph  ->  H  C_  ( -u pi [,] pi ) )
155154, 123sseldd 3442 . . . . . . . 8  |-  ( ph  ->  ( Q `  0
)  e.  ( -u pi [,] pi ) )
156 iccgelb 11552 . . . . . . . 8  |-  ( (
-u pi  e.  RR*  /\  pi  e.  RR*  /\  ( Q `  0 )  e.  ( -u pi [,] pi ) )  ->  -u pi  <_  ( Q `  0
) )
157140, 141, 155, 156syl3anc 1230 . . . . . . 7  |-  ( ph  -> 
-u pi  <_  ( Q `  0 )
)
158157ad2antrr 724 . . . . . 6  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  ( Q `  i )  =  -u pi )  ->  -u pi  <_  ( Q `  0 ) )
159124ad2antrr 724 . . . . . . 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 9679 . . . . . 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 923 . . . . 5  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  ( Q `  i )  =  -u pi )  -> 
( Q `  0
)  =  -u pi )
16359, 53sselii 3438 . . . . . . 7  |-  -u pi  e.  H
164 f1ofo 5762 . . . . . . . . 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 5737 . . . . . . . 8  |-  ( Q : ( 0 ... M ) -onto-> H  ->  ran  Q  =  H )
16793, 165, 1663syl 20 . . . . . . 7  |-  ( ph  ->  ran  Q  =  H )
168163, 167syl5eleqr 2497 . . . . . 6  |-  ( ph  -> 
-u pi  e.  ran  Q )
169 ffn 5670 . . . . . . 7  |-  ( Q : ( 0 ... M ) --> H  ->  Q  Fn  ( 0 ... M ) )
170 fvelrnb 5852 . . . . . . 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 2948 . . . 4  |-  ( ph  ->  ( Q `  0
)  =  -u pi )
17459, 55sselii 3438 . . . . . . 7  |-  pi  e.  H
175174, 167syl5eleqr 2497 . . . . . 6  |-  ( ph  ->  pi  e.  ran  Q
)
176 fvelrnb 5852 . . . . . . 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 5679 . . . . . . . . . 10  |-  ( ph  ->  Q : ( 0 ... M ) --> (
-u pi [,] pi ) )
180 eluzfz2 11665 . . . . . . . . . . 11  |-  ( M  e.  ( ZZ>= `  0
)  ->  M  e.  ( 0 ... M
) )
181120, 180syl 17 . . . . . . . . . 10  |-  ( ph  ->  M  e.  ( 0 ... M ) )
182179, 181ffvelrnd 5966 . . . . . . . . 9  |-  ( ph  ->  ( Q `  M
)  e.  ( -u pi [,] pi ) )
183 iccleub 11551 . . . . . . . . 9  |-  ( (
-u pi  e.  RR*  /\  pi  e.  RR*  /\  ( Q `  M )  e.  ( -u pi [,] pi ) )  ->  ( Q `  M )  <_  pi )
184140, 141, 182, 183syl3anc 1230 . . . . . . . 8  |-  ( ph  ->  ( Q `  M
)  <_  pi )
1851843ad2ant1 1018 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 0 ... M
)  /\  ( Q `  i )  =  pi )  ->  ( Q `  M )  <_  pi )
186 id 22 . . . . . . . . . 10  |-  ( ( Q `  i )  =  pi  ->  ( Q `  i )  =  pi )
187186eqcomd 2410 . . . . . . . . 9  |-  ( ( Q `  i )  =  pi  ->  pi  =  ( Q `  i ) )
1881873ad2ant3 1020 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 0 ... M
)  /\  ( Q `  i )  =  pi )  ->  pi  =  ( Q `  i ) )
189105adantr 463 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  =  M )  ->  ( Q `  i )  <_  ( Q `  i
) )
190 fveq2 5805 . . . . . . . . . . . 12  |-  ( i  =  M  ->  ( Q `  i )  =  ( Q `  M ) )
191190adantl 464 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  =  M )  ->  ( Q `  i )  =  ( Q `  M ) )
192189, 191breqtrd 4418 . . . . . . . . . 10  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  =  M )  ->  ( Q `  i )  <_  ( Q `  M
) )
193109ad2antlr 725 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  i  =  M )  ->  i  e.  RR )
194 elfzel2 11657 . . . . . . . . . . . . . 14  |-  ( i  e.  ( 0 ... M )  ->  M  e.  ZZ )
195194zred 10928 . . . . . . . . . . . . 13  |-  ( i  e.  ( 0 ... M )  ->  M  e.  RR )
196195ad2antlr 725 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  i  =  M )  ->  M  e.  RR )
197 elfzle2 11661 . . . . . . . . . . . . 13  |-  ( i  e.  ( 0 ... M )  ->  i  <_  M )
198197ad2antlr 725 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  i  =  M )  ->  i  <_  M )
199 neqne 36791 . . . . . . . . . . . . . 14  |-  ( -.  i  =  M  -> 
i  =/=  M )
200199necomd 2674 . . . . . . . . . . . . 13  |-  ( -.  i  =  M  ->  M  =/=  i )
201200adantl 464 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  i  =  M )  ->  M  =/=  i )
202193, 196, 198, 201leneltd 36845 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  i  =  M )  ->  i  <  M )
203104adantr 463 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  <  M )  ->  ( Q `  i )  e.  RR )
20487, 182sseldi 3439 . . . . . . . . . . . . 13  |-  ( ph  ->  ( Q `  M
)  e.  RR )
205204ad2antrr 724 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  <  M )  ->  ( Q `  M )  e.  RR )
206 simpr 459 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  <  M )  ->  i  <  M )
20793ad2antrr 724 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  <  M )  ->  Q  Isom  <  ,  <  (
( 0 ... M
) ,  H ) )
208 simpr 459 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  i  e.  ( 0 ... M
) )
209181adantr 463 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  M  e.  ( 0 ... M
) )
210208, 209jca 530 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  (
i  e.  ( 0 ... M )  /\  M  e.  ( 0 ... M ) ) )
211210adantr 463 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  <  M )  ->  (
i  e.  ( 0 ... M )  /\  M  e.  ( 0 ... M ) ) )
212 isorel 6161 . . . . . . . . . . . . . 14  |-  ( ( Q  Isom  <  ,  <  ( ( 0 ... M
) ,  H )  /\  ( i  e.  ( 0 ... M
)  /\  M  e.  ( 0 ... M
) ) )  -> 
( i  <  M  <->  ( Q `  i )  <  ( Q `  M ) ) )
213207, 211, 212syl2anc 659 . . . . . . . . . . . . 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 9685 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  <  M )  ->  ( Q `  i )  <_  ( Q `  M
) )
216202, 215syldan 468 . . . . . . . . . 10  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  i  =  M )  ->  ( Q `  i
)  <_  ( Q `  M ) )
217192, 216pm2.61dan 792 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  ( Q `  i )  <_  ( Q `  M
) )
2182173adant3 1017 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 0 ... M
)  /\  ( Q `  i )  =  pi )  ->  ( Q `  i )  <_  ( Q `  M )
)
219188, 218eqbrtrd 4414 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 0 ... M
)  /\  ( Q `  i )  =  pi )  ->  pi  <_  ( Q `  M ) )
2202043ad2ant1 1018 . . . . . . . 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 9679 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 0 ... M
)  /\  ( Q `  i )  =  pi )  ->  ( ( Q `  M )  =  pi  <->  ( ( Q `
 M )  <_  pi  /\  pi  <_  ( Q `  M )
) ) )
223185, 219, 222mpbir2and 923 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0 ... M
)  /\  ( Q `  i )  =  pi )  ->  ( Q `  M )  =  pi )
224223rexlimdv3a 2897 . . . . 5  |-  ( ph  ->  ( E. i  e.  ( 0 ... M
) ( Q `  i )  =  pi 
->  ( Q `  M
)  =  pi ) )
225178, 224mpd 15 . . . 4  |-  ( ph  ->  ( Q `  M
)  =  pi )
226 elfzoelz 11772 . . . . . . . . 9  |-  ( i  e.  ( 0..^ M )  ->  i  e.  ZZ )
227226zred 10928 . . . . . . . 8  |-  ( i  e.  ( 0..^ M )  ->  i  e.  RR )
228227ltp1d 10436 . . . . . . 7  |-  ( i  e.  ( 0..^ M )  ->  i  <  ( i  +  1 ) )
229228adantl 464 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  i  <  (
i  +  1 ) )
230 elfzofz 11787 . . . . . . . 8  |-  ( i  e.  ( 0..^ M )  ->  i  e.  ( 0 ... M
) )
231 fzofzp1 11859 . . . . . . . 8  |-  ( i  e.  ( 0..^ M )  ->  ( i  +  1 )  e.  ( 0 ... M
) )
232230, 231jca 530 . . . . . . 7  |-  ( i  e.  ( 0..^ M )  ->  ( i  e.  ( 0 ... M
)  /\  ( i  +  1 )  e.  ( 0 ... M
) ) )
233 isorel 6161 . . . . . . 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 475 . . . . . 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 2817 . . . 4  |-  ( ph  ->  A. i  e.  ( 0..^ M ) ( Q `  i )  <  ( Q `  ( i  +  1 ) ) )
237173, 225, 236jca31 532 . . 3  |-  ( ph  ->  ( ( ( Q `
 0 )  = 
-u pi  /\  ( Q `  M )  =  pi )  /\  A. i  e.  ( 0..^ M ) ( Q `
 i )  < 
( Q `  (
i  +  1 ) ) ) )
2387fourierdlem2 37241 . . . 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 923 . 2  |-  ( ph  ->  Q  e.  ( P `
 M ) )
241 fourierdlem114.g . . . . 5  |-  G  =  ( ( RR  _D  F )  |`  ( -u pi (,) pi ) )
242241reseq1i 5211 . . . 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 463 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  Q : ( 0 ... M ) --> ( -u pi [,] pi ) )
246 simpr 459 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  i  e.  ( 0..^ M ) )
247243, 244, 245, 246fourierdlem27 37266 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( Q `
 i ) (,) ( Q `  (
i  +  1 ) ) )  C_  ( -u pi (,) pi ) )
248247resabs1d 5244 . . . 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 2456 . . 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 37277 . . 3  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( G  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) )  e.  ( ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) -cn-> CC ) )
252249, 251eqeltrd 2490 . 2  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( RR 
_D  F )  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) )  e.  ( ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) -cn-> CC ) )
253249oveq1d 6249 . . 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 463 . . . . 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 713 . . . . 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 713 . . . . 5  |-  ( ( ( ph  /\  i  e.  ( 0..^ M ) )  /\  x  e.  ( ( -u pi (,] pi )  \  dom  G ) )  ->  (
( G  |`  ( -oo (,) x ) ) lim
CC  x )  =/=  (/) )
25993adantr 463 . . . . 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 463 . . . . 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 37285 . . . 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 457 . . 3  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( G  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) lim CC  ( Q `  i )
)  =/=  (/) )
265253, 264eqnetrd 2696 . 2  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( ( RR  _D  F )  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) lim CC  ( Q `  i )
)  =/=  (/) )
266249oveq1d 6249 . . 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 461 . . 3  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( G  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) lim CC  ( Q `  ( i  +  1 ) ) )  =/=  (/) )
268266, 267eqnetrd 2696 . 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 4087 . . . . 5  |-  ( E `
 X )  e. 
{ -u pi ,  pi ,  ( E `  X ) }
273 elun1 3609 . . . . 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 2501 . . 3  |-  ( ph  ->  ( E `  X
)  e.  H )
276275, 167eleqtrrd 2493 . 2  |-  ( ph  ->  ( E `  X
)  e.  ran  Q
)
2771, 2, 3, 4, 5, 6, 7, 69, 240, 252, 265, 268, 269, 270, 271, 78, 276fourierdlem113 37352 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2753   E.wrex 2754   {crab 2757   _Vcvv 3058    \ cdif 3410    u. cun 3411    C_ wss 3413   (/)c0 3737   {cpr 3973   {ctp 3975   class class class wbr 4394    |-> cmpt 4452   dom cdm 4942   ran crn 4943    |` cres 4944   iotacio 5487    Fn wfn 5520   -->wf 5521   -onto->wfo 5523   -1-1-onto->wf1o 5524   ` cfv 5525    Isom wiso 5526  (class class class)co 6234    ^m cmap 7377   Fincfn 7474   CCcc 9440   RRcr 9441   0cc0 9442   1c1 9443    + caddc 9445    x. cmul 9447   +oocpnf 9575   -oocmnf 9576   RR*cxr 9577    < clt 9578    <_ cle 9579    - cmin 9761   -ucneg 9762    / cdiv 10167   NNcn 10496   2c2 10546   NN0cn0 10756   ZZcz 10825   ZZ>=cuz 11045   (,)cioo 11500   (,]cioc 11501   [,)cico 11502   [,]cicc 11503   ...cfz 11643  ..^cfzo 11767   |_cfl 11877    seqcseq 12061   #chash 12359    ~~> cli 13363   sum_csu 13564   sincsin 13900   cosccos 13901   picpi 13903   -cn->ccncf 21564   S.citg 22211   lim CC climc 22450    _D cdv 22451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-inf2 8011  ax-cc 8767  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518  ax-pre-mulgt0 9519  ax-pre-sup 9520  ax-addf 9521  ax-mulf 9522
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-iin 4273  df-disj 4366  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-of 6477  df-ofr 6478  df-om 6639  df-1st 6738  df-2nd 6739  df-supp 6857  df-recs 6999  df-rdg 7033  df-1o 7087  df-2o 7088  df-oadd 7091  df-omul 7092  df-er 7268  df-map 7379  df-pm 7380  df-ixp 7428  df-en 7475  df-dom 7476  df-sdom 7477  df-fin 7478  df-fsupp 7784  df-fi 7825  df-sup 7855  df-oi 7889  df-card 8272  df-acn 8275  df-cda 8500  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-sub 9763  df-neg 9764  df-div 10168  df-nn 10497  df-2 10555  df-3 10556  df-4 10557  df-5 10558  df-6 10559  df-7 10560  df-8 10561  df-9 10562  df-10 10563  df-n0 10757  df-z 10826  df-dec 10940  df-uz 11046  df-q 11146  df-rp 11184  df-xneg 11289  df-xadd 11290  df-xmul 11291  df-ioo 11504  df-ioc 11505  df-ico 11506  df-icc 11507  df-fz 11644  df-fzo 11768  df-fl 11879  df-mod 11948  df-seq 12062  df-exp 12121  df-fac 12308  df-bc 12335  df-hash 12360  df-shft 12956  df-cj 12988  df-re 12989  df-im 12990  df-sqrt 13124  df-abs 13125  df-limsup 13350  df-clim 13367  df-rlim 13368  df-sum 13565  df-ef 13904  df-sin 13906  df-cos 13907  df-pi 13909  df-struct 14735  df-ndx 14736  df-slot 14737  df-base 14738  df-sets 14739  df-ress 14740  df-plusg 14814  df-mulr 14815  df-starv 14816  df-sca 14817  df-vsca 14818  df-ip 14819  df-tset 14820  df-ple 14821  df-ds 14823  df-unif 14824  df-hom 14825  df-cco 14826  df-rest 14929  df-topn 14930  df-0g 14948  df-gsum 14949  df-topgen 14950  df-pt 14951  df-prds 14954  df-xrs 15008  df-qtop 15013  df-imas 15014  df-xps 15016  df-mre 15092  df-mrc 15093  df-acs 15095  df-mgm 16088  df-sgrp 16127  df-mnd 16137  df-submnd 16183  df-mulg 16276  df-cntz 16571  df-cmn 17016  df-psmet 18623  df-xmet 18624  df-met 18625  df-bl 18626  df-mopn 18627  df-fbas 18628  df-fg 18629  df-cnfld 18633  df-top 19583  df-bases 19585  df-topon 19586  df-topsp 19587  df-cld 19704  df-ntr 19705  df-cls 19706  df-nei 19784  df-lp 19822  df-perf 19823  df-cn 19913  df-cnp 19914  df-t1 20000  df-haus 20001  df-cmp 20072  df-tx 20247  df-hmeo 20440  df-fil 20531  df-fm 20623  df-flim 20624  df-flf 20625  df-xms 21007  df-ms 21008  df-tms 21009  df-cncf 21566  df-ovol 22060  df-vol 22061  df-mbf 22212  df-itg1 22213  df-itg2 22214  df-ibl 22215  df-itg 22216  df-0p 22261  df-ditg 22435  df-limc 22454  df-dv 22455
This theorem is referenced by:  fourierdlem115  37354
  Copyright terms: Public domain W3C validator