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Theorem fourierdlem102 32194
Description: For a piecewise smooth function, the left and the right limits exist at any point. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
fourierdlem102.f  |-  ( ph  ->  F : RR --> RR )
fourierdlem102.t  |-  T  =  ( 2  x.  pi )
fourierdlem102.per  |-  ( (
ph  /\  x  e.  RR )  ->  ( F `
 ( x  +  T ) )  =  ( F `  x
) )
fourierdlem102.g  |-  G  =  ( ( RR  _D  F )  |`  ( -u pi (,) pi ) )
fourierdlem102.dmdv  |-  ( ph  ->  ( ( -u pi (,) pi )  \  dom  G )  e.  Fin )
fourierdlem102.gcn  |-  ( ph  ->  G  e.  ( dom 
G -cn-> CC ) )
fourierdlem102.rlim  |-  ( (
ph  /\  x  e.  ( ( -u pi [,) pi )  \  dom  G ) )  ->  (
( G  |`  (
x (,) +oo )
) lim CC  x )  =/=  (/) )
fourierdlem102.llim  |-  ( (
ph  /\  x  e.  ( ( -u pi (,] pi )  \  dom  G ) )  ->  (
( G  |`  ( -oo (,) x ) ) lim
CC  x )  =/=  (/) )
fourierdlem102.x  |-  ( ph  ->  X  e.  RR )
fourierdlem102.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 ) ) ) } )
fourierdlem102.e  |-  E  =  ( x  e.  RR  |->  ( x  +  (
( |_ `  (
( pi  -  x
)  /  T ) )  x.  T ) ) )
fourierdlem102.h  |-  H  =  ( { -u pi ,  pi ,  ( E `
 X ) }  u.  ( ( -u pi [,] pi )  \  dom  G ) )
fourierdlem102.m  |-  M  =  ( ( # `  H
)  -  1 )
fourierdlem102.q  |-  Q  =  ( iota g g 
Isom  <  ,  <  (
( 0 ... M
) ,  H ) )
Assertion
Ref Expression
fourierdlem102  |-  ( ph  ->  ( ( ( F  |`  ( -oo (,) X
) ) lim CC  X
)  =/=  (/)  /\  (
( F  |`  ( X (,) +oo ) ) lim
CC  X )  =/=  (/) ) )
Distinct variable groups:    x, E    i, F, n, x    i, G, x    g, H    g, M    i, M, n, p   
x, M    Q, g    Q, i, n, p    x, Q    T, i, n, p   
x, T    i, X, n, p    x, X    ph, g    ph, i, n, x
Allowed substitution hints:    ph( p)    P( x, g, i, n, p)    T( g)    E( g, i, n, p)    F( g, p)    G( g, n, p)    H( x, i, n, p)    X( g)

Proof of Theorem fourierdlem102
StepHypRef Expression
1 fourierdlem102.f . 2  |-  ( ph  ->  F : RR --> RR )
2 fourierdlem102.t . 2  |-  T  =  ( 2  x.  pi )
3 fourierdlem102.per . 2  |-  ( (
ph  /\  x  e.  RR )  ->  ( F `
 ( x  +  T ) )  =  ( F `  x
) )
4 fourierdlem102.x . 2  |-  ( ph  ->  X  e.  RR )
5 fourierdlem102.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 ) ) ) } )
6 fourierdlem102.m . . 3  |-  M  =  ( ( # `  H
)  -  1 )
7 2z 10917 . . . . . 6  |-  2  e.  ZZ
87a1i 11 . . . . 5  |-  ( ph  ->  2  e.  ZZ )
9 fourierdlem102.h . . . . . . . 8  |-  H  =  ( { -u pi ,  pi ,  ( E `
 X ) }  u.  ( ( -u pi [,] pi )  \  dom  G ) )
10 tpfi 7814 . . . . . . . . . 10  |-  { -u pi ,  pi , 
( E `  X
) }  e.  Fin
1110a1i 11 . . . . . . . . 9  |-  ( ph  ->  { -u pi ,  pi ,  ( E `  X ) }  e.  Fin )
12 pire 22977 . . . . . . . . . . . . . . 15  |-  pi  e.  RR
1312renegcli 9899 . . . . . . . . . . . . . 14  |-  -u pi  e.  RR
1413rexri 9663 . . . . . . . . . . . . 13  |-  -u pi  e.  RR*
1512rexri 9663 . . . . . . . . . . . . 13  |-  pi  e.  RR*
16 negpilt0 31665 . . . . . . . . . . . . . . 15  |-  -u pi  <  0
17 pipos 22979 . . . . . . . . . . . . . . 15  |-  0  <  pi
18 0re 9613 . . . . . . . . . . . . . . . 16  |-  0  e.  RR
1913, 18, 12lttri 9727 . . . . . . . . . . . . . . 15  |-  ( (
-u pi  <  0  /\  0  <  pi )  ->  -u pi  <  pi )
2016, 17, 19mp2an 672 . . . . . . . . . . . . . 14  |-  -u pi  <  pi
2113, 12, 20ltleii 9724 . . . . . . . . . . . . 13  |-  -u pi  <_  pi
22 prunioo 11674 . . . . . . . . . . . . 13  |-  ( (
-u pi  e.  RR*  /\  pi  e.  RR*  /\  -u pi  <_  pi )  ->  (
( -u pi (,) pi )  u.  { -u pi ,  pi } )  =  ( -u pi [,] pi ) )
2314, 15, 21, 22mp3an 1324 . . . . . . . . . . . 12  |-  ( (
-u pi (,) pi )  u.  { -u pi ,  pi } )  =  ( -u pi [,] pi )
2423difeq1i 3614 . . . . . . . . . . 11  |-  ( ( ( -u pi (,) pi )  u.  { -u pi ,  pi }
)  \  dom  G )  =  ( ( -u pi [,] pi )  \  dom  G )
25 difundir 3758 . . . . . . . . . . 11  |-  ( ( ( -u pi (,) pi )  u.  { -u pi ,  pi }
)  \  dom  G )  =  ( ( (
-u pi (,) pi )  \  dom  G )  u.  ( { -u pi ,  pi }  \  dom  G ) )
2624, 25eqtr3i 2488 . . . . . . . . . 10  |-  ( (
-u pi [,] pi )  \  dom  G )  =  ( ( (
-u pi (,) pi )  \  dom  G )  u.  ( { -u pi ,  pi }  \  dom  G ) )
27 fourierdlem102.dmdv . . . . . . . . . . 11  |-  ( ph  ->  ( ( -u pi (,) pi )  \  dom  G )  e.  Fin )
28 prfi 7813 . . . . . . . . . . . 12  |-  { -u pi ,  pi }  e.  Fin
29 diffi 7770 . . . . . . . . . . . 12  |-  ( {
-u pi ,  pi }  e.  Fin  ->  ( { -u pi ,  pi }  \  dom  G )  e.  Fin )
3028, 29mp1i 12 . . . . . . . . . . 11  |-  ( ph  ->  ( { -u pi ,  pi }  \  dom  G )  e.  Fin )
31 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 )
3227, 30, 31syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( -u pi (,) pi )  \  dom  G )  u.  ( { -u pi ,  pi }  \  dom  G ) )  e.  Fin )
3326, 32syl5eqel 2549 . . . . . . . . 9  |-  ( ph  ->  ( ( -u pi [,] pi )  \  dom  G )  e.  Fin )
34 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 )
3511, 33, 34syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( { -u pi ,  pi ,  ( E `
 X ) }  u.  ( ( -u pi [,] pi )  \  dom  G ) )  e. 
Fin )
369, 35syl5eqel 2549 . . . . . . 7  |-  ( ph  ->  H  e.  Fin )
37 hashcl 12431 . . . . . . 7  |-  ( H  e.  Fin  ->  ( # `
 H )  e. 
NN0 )
3836, 37syl 16 . . . . . 6  |-  ( ph  ->  ( # `  H
)  e.  NN0 )
3938nn0zd 10988 . . . . 5  |-  ( ph  ->  ( # `  H
)  e.  ZZ )
4013, 20ltneii 9714 . . . . . . 7  |-  -u pi  =/=  pi
41 hashprg 12464 . . . . . . . 8  |-  ( (
-u pi  e.  RR  /\  pi  e.  RR )  ->  ( -u pi  =/=  pi  <->  ( # `  { -u pi ,  pi }
)  =  2 ) )
4213, 12, 41mp2an 672 . . . . . . 7  |-  ( -u pi  =/=  pi  <->  ( # `  { -u pi ,  pi }
)  =  2 )
4340, 42mpbi 208 . . . . . 6  |-  ( # `  { -u pi ,  pi } )  =  2
4410elexi 3119 . . . . . . . . . 10  |-  { -u pi ,  pi , 
( E `  X
) }  e.  _V
45 ovex 6324 . . . . . . . . . . 11  |-  ( -u pi [,] pi )  e. 
_V
46 difexg 4604 . . . . . . . . . . 11  |-  ( (
-u pi [,] pi )  e.  _V  ->  ( ( -u pi [,] pi )  \  dom  G
)  e.  _V )
4745, 46ax-mp 5 . . . . . . . . . 10  |-  ( (
-u pi [,] pi )  \  dom  G )  e.  _V
4844, 47unex 6597 . . . . . . . . 9  |-  ( {
-u pi ,  pi ,  ( E `  X ) }  u.  ( ( -u pi [,] pi )  \  dom  G ) )  e.  _V
499, 48eqeltri 2541 . . . . . . . 8  |-  H  e. 
_V
50 negex 9837 . . . . . . . . . . 11  |-  -u pi  e.  _V
5150tpid1 4145 . . . . . . . . . 10  |-  -u pi  e.  { -u pi ,  pi ,  ( E `  X ) }
5212elexi 3119 . . . . . . . . . . 11  |-  pi  e.  _V
5352tpid2 4146 . . . . . . . . . 10  |-  pi  e.  {
-u pi ,  pi ,  ( E `  X ) }
54 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
) } )
5551, 53, 54mp2an 672 . . . . . . . . 9  |-  { -u pi ,  pi }  C_ 
{ -u pi ,  pi ,  ( E `  X ) }
56 ssun1 3663 . . . . . . . . . 10  |-  { -u pi ,  pi , 
( E `  X
) }  C_  ( { -u pi ,  pi ,  ( E `  X ) }  u.  ( ( -u pi [,] pi )  \  dom  G ) )
5756, 9sseqtr4i 3532 . . . . . . . . 9  |-  { -u pi ,  pi , 
( E `  X
) }  C_  H
5855, 57sstri 3508 . . . . . . . 8  |-  { -u pi ,  pi }  C_  H
59 hashss 12478 . . . . . . . 8  |-  ( ( H  e.  _V  /\  {
-u pi ,  pi }  C_  H )  -> 
( # `  { -u pi ,  pi }
)  <_  ( # `  H
) )
6049, 58, 59mp2an 672 . . . . . . 7  |-  ( # `  { -u pi ,  pi } )  <_  ( # `
 H )
6160a1i 11 . . . . . 6  |-  ( ph  ->  ( # `  { -u pi ,  pi }
)  <_  ( # `  H
) )
6243, 61syl5eqbrr 4490 . . . . 5  |-  ( ph  ->  2  <_  ( # `  H
) )
63 eluz2 11112 . . . . 5  |-  ( (
# `  H )  e.  ( ZZ>= `  2 )  <->  ( 2  e.  ZZ  /\  ( # `  H )  e.  ZZ  /\  2  <_  ( # `  H
) ) )
648, 39, 62, 63syl3anbrc 1180 . . . 4  |-  ( ph  ->  ( # `  H
)  e.  ( ZZ>= ` 
2 ) )
65 uz2m1nn 11181 . . . 4  |-  ( (
# `  H )  e.  ( ZZ>= `  2 )  ->  ( ( # `  H
)  -  1 )  e.  NN )
6664, 65syl 16 . . 3  |-  ( ph  ->  ( ( # `  H
)  -  1 )  e.  NN )
676, 66syl5eqel 2549 . 2  |-  ( ph  ->  M  e.  NN )
6813a1i 11 . . . . . . . . . . 11  |-  ( ph  -> 
-u pi  e.  RR )
6912a1i 11 . . . . . . . . . . 11  |-  ( ph  ->  pi  e.  RR )
70 negpitopissre 23053 . . . . . . . . . . . 12  |-  ( -u pi (,] pi )  C_  RR
7120a1i 11 . . . . . . . . . . . . . 14  |-  ( ph  -> 
-u pi  <  pi )
72 picn 22978 . . . . . . . . . . . . . . . 16  |-  pi  e.  CC
73722timesi 10677 . . . . . . . . . . . . . . 15  |-  ( 2  x.  pi )  =  ( pi  +  pi )
7472, 72subnegi 9917 . . . . . . . . . . . . . . 15  |-  ( pi 
-  -u pi )  =  ( pi  +  pi )
7573, 2, 743eqtr4i 2496 . . . . . . . . . . . . . 14  |-  T  =  ( pi  -  -u pi )
76 fourierdlem102.e . . . . . . . . . . . . . 14  |-  E  =  ( x  e.  RR  |->  ( x  +  (
( |_ `  (
( pi  -  x
)  /  T ) )  x.  T ) ) )
7768, 69, 71, 75, 76fourierdlem4 32096 . . . . . . . . . . . . 13  |-  ( ph  ->  E : RR --> ( -u pi (,] pi ) )
7877, 4ffvelrnd 6033 . . . . . . . . . . . 12  |-  ( ph  ->  ( E `  X
)  e.  ( -u pi (,] pi ) )
7970, 78sseldi 3497 . . . . . . . . . . 11  |-  ( ph  ->  ( E `  X
)  e.  RR )
8068, 69, 793jca 1176 . . . . . . . . . 10  |-  ( ph  ->  ( -u pi  e.  RR  /\  pi  e.  RR  /\  ( E `  X
)  e.  RR ) )
81 fvex 5882 . . . . . . . . . . 11  |-  ( E `
 X )  e. 
_V
8250, 52, 81tpss 4197 . . . . . . . . . 10  |-  ( (
-u pi  e.  RR  /\  pi  e.  RR  /\  ( E `  X )  e.  RR )  <->  { -u pi ,  pi ,  ( E `
 X ) } 
C_  RR )
8380, 82sylib 196 . . . . . . . . 9  |-  ( ph  ->  { -u pi ,  pi ,  ( E `  X ) }  C_  RR )
84 iccssre 11631 . . . . . . . . . . 11  |-  ( (
-u pi  e.  RR  /\  pi  e.  RR )  ->  ( -u pi [,] pi )  C_  RR )
8513, 12, 84mp2an 672 . . . . . . . . . 10  |-  ( -u pi [,] pi )  C_  RR
86 ssdifss 3631 . . . . . . . . . 10  |-  ( (
-u pi [,] pi )  C_  RR  ->  (
( -u pi [,] pi )  \  dom  G ) 
C_  RR )
8785, 86mp1i 12 . . . . . . . . 9  |-  ( ph  ->  ( ( -u pi [,] pi )  \  dom  G )  C_  RR )
8883, 87unssd 3676 . . . . . . . 8  |-  ( ph  ->  ( { -u pi ,  pi ,  ( E `
 X ) }  u.  ( ( -u pi [,] pi )  \  dom  G ) )  C_  RR )
899, 88syl5eqss 3543 . . . . . . 7  |-  ( ph  ->  H  C_  RR )
90 fourierdlem102.q . . . . . . 7  |-  Q  =  ( iota g g 
Isom  <  ,  <  (
( 0 ... M
) ,  H ) )
9136, 89, 90, 6fourierdlem36 32128 . . . . . 6  |-  ( ph  ->  Q  Isom  <  ,  <  ( ( 0 ... M
) ,  H ) )
92 isof1o 6222 . . . . . 6  |-  ( Q 
Isom  <  ,  <  (
( 0 ... M
) ,  H )  ->  Q : ( 0 ... M ) -1-1-onto-> H )
93 f1of 5822 . . . . . 6  |-  ( Q : ( 0 ... M ) -1-1-onto-> H  ->  Q :
( 0 ... M
) --> H )
9491, 92, 933syl 20 . . . . 5  |-  ( ph  ->  Q : ( 0 ... M ) --> H )
9594, 89fssd 5746 . . . 4  |-  ( ph  ->  Q : ( 0 ... M ) --> RR )
96 reex 9600 . . . . 5  |-  RR  e.  _V
97 ovex 6324 . . . . 5  |-  ( 0 ... M )  e. 
_V
9896, 97elmap 7466 . . . 4  |-  ( Q  e.  ( RR  ^m  ( 0 ... M
) )  <->  Q :
( 0 ... M
) --> RR )
9995, 98sylibr 212 . . 3  |-  ( ph  ->  Q  e.  ( RR 
^m  ( 0 ... M ) ) )
100 fveq2 5872 . . . . . . . . . . 11  |-  ( 0  =  i  ->  ( Q `  0 )  =  ( Q `  i ) )
101100adantl 466 . . . . . . . . . 10  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  =  i )  -> 
( Q `  0
)  =  ( Q `
 i ) )
10295ffvelrnda 6032 . . . . . . . . . . . 12  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  ( Q `  i )  e.  RR )
103102leidd 10140 . . . . . . . . . . 11  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  ( Q `  i )  <_  ( Q `  i
) )
104103adantr 465 . . . . . . . . . 10  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  =  i )  -> 
( Q `  i
)  <_  ( Q `  i ) )
105101, 104eqbrtrd 4476 . . . . . . . . 9  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  =  i )  -> 
( Q `  0
)  <_  ( Q `  i ) )
106 elfzelz 11713 . . . . . . . . . . . . 13  |-  ( i  e.  ( 0 ... M )  ->  i  e.  ZZ )
107106zred 10990 . . . . . . . . . . . 12  |-  ( i  e.  ( 0 ... M )  ->  i  e.  RR )
108107ad2antlr 726 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  0  =  i )  ->  i  e.  RR )
109 elfzle1 11714 . . . . . . . . . . . 12  |-  ( i  e.  ( 0 ... M )  ->  0  <_  i )
110109ad2antlr 726 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  0  =  i )  ->  0  <_  i )
111 neqne 31637 . . . . . . . . . . . . 13  |-  ( -.  0  =  i  -> 
0  =/=  i )
112111necomd 2728 . . . . . . . . . . . 12  |-  ( -.  0  =  i  -> 
i  =/=  0 )
113112adantl 466 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  0  =  i )  ->  i  =/=  0 )
114108, 110, 113ne0gt0d 9739 . . . . . . . . . 10  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  0  =  i )  ->  0  <  i )
115 nnssnn0 10819 . . . . . . . . . . . . . . . . 17  |-  NN  C_  NN0
116 nn0uz 11140 . . . . . . . . . . . . . . . . 17  |-  NN0  =  ( ZZ>= `  0 )
117115, 116sseqtri 3531 . . . . . . . . . . . . . . . 16  |-  NN  C_  ( ZZ>= `  0 )
118117, 67sseldi 3497 . . . . . . . . . . . . . . 15  |-  ( ph  ->  M  e.  ( ZZ>= ` 
0 ) )
119 eluzfz1 11718 . . . . . . . . . . . . . . 15  |-  ( M  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... M
) )
120118, 119syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  0  e.  ( 0 ... M ) )
12194, 120ffvelrnd 6033 . . . . . . . . . . . . 13  |-  ( ph  ->  ( Q `  0
)  e.  H )
12289, 121sseldd 3500 . . . . . . . . . . . 12  |-  ( ph  ->  ( Q `  0
)  e.  RR )
123122ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  <  i )  ->  ( Q `  0 )  e.  RR )
124102adantr 465 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  <  i )  ->  ( Q `  i )  e.  RR )
125 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  <  i )  ->  0  <  i )
12691ad2antrr 725 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  <  i )  ->  Q  Isom  <  ,  <  (
( 0 ... M
) ,  H ) )
127120anim1i 568 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  (
0  e.  ( 0 ... M )  /\  i  e.  ( 0 ... M ) ) )
128127adantr 465 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  <  i )  ->  (
0  e.  ( 0 ... M )  /\  i  e.  ( 0 ... M ) ) )
129 isorel 6223 . . . . . . . . . . . . 13  |-  ( ( Q  Isom  <  ,  <  ( ( 0 ... M
) ,  H )  /\  ( 0  e.  ( 0 ... M
)  /\  i  e.  ( 0 ... M
) ) )  -> 
( 0  <  i  <->  ( Q `  0 )  <  ( Q `  i ) ) )
130126, 128, 129syl2anc 661 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  <  i )  ->  (
0  <  i  <->  ( Q `  0 )  < 
( Q `  i
) ) )
131125, 130mpbid 210 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  <  i )  ->  ( Q `  0 )  <  ( Q `  i
) )
132123, 124, 131ltled 9750 . . . . . . . . . 10  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  <  i )  ->  ( Q `  0 )  <_  ( Q `  i
) )
133114, 132syldan 470 . . . . . . . . 9  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  0  =  i )  ->  ( Q `  0
)  <_  ( Q `  i ) )
134105, 133pm2.61dan 791 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  ( Q `  0 )  <_  ( Q `  i
) )
135134adantr 465 . . . . . . 7  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  ( Q `  i )  =  -u pi )  -> 
( Q `  0
)  <_  ( Q `  i ) )
136 simpr 461 . . . . . . 7  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  ( Q `  i )  =  -u pi )  -> 
( Q `  i
)  =  -u pi )
137135, 136breqtrd 4480 . . . . . 6  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  ( Q `  i )  =  -u pi )  -> 
( Q `  0
)  <_  -u pi )
13868rexrd 9660 . . . . . . . 8  |-  ( ph  -> 
-u pi  e.  RR* )
13969rexrd 9660 . . . . . . . 8  |-  ( ph  ->  pi  e.  RR* )
140 lbicc2 11661 . . . . . . . . . . . . . 14  |-  ( (
-u pi  e.  RR*  /\  pi  e.  RR*  /\  -u pi  <_  pi )  ->  -u pi  e.  ( -u pi [,] pi ) )
14114, 15, 21, 140mp3an 1324 . . . . . . . . . . . . 13  |-  -u pi  e.  ( -u pi [,] pi )
142141a1i 11 . . . . . . . . . . . 12  |-  ( ph  -> 
-u pi  e.  (
-u pi [,] pi ) )
143 ubicc2 11662 . . . . . . . . . . . . . 14  |-  ( (
-u pi  e.  RR*  /\  pi  e.  RR*  /\  -u pi  <_  pi )  ->  pi  e.  ( -u pi [,] pi ) )
14414, 15, 21, 143mp3an 1324 . . . . . . . . . . . . 13  |-  pi  e.  ( -u pi [,] pi )
145144a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  pi  e.  ( -u pi [,] pi ) )
146 iocssicc 11637 . . . . . . . . . . . . 13  |-  ( -u pi (,] pi )  C_  ( -u pi [,] pi )
147146, 78sseldi 3497 . . . . . . . . . . . 12  |-  ( ph  ->  ( E `  X
)  e.  ( -u pi [,] pi ) )
148 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 ) )
149142, 145, 147, 148syl3anc 1228 . . . . . . . . . . 11  |-  ( ph  ->  { -u pi ,  pi ,  ( E `  X ) }  C_  ( -u pi [,] pi ) )
150 difssd 3628 . . . . . . . . . . 11  |-  ( ph  ->  ( ( -u pi [,] pi )  \  dom  G )  C_  ( -u pi [,] pi ) )
151149, 150unssd 3676 . . . . . . . . . 10  |-  ( ph  ->  ( { -u pi ,  pi ,  ( E `
 X ) }  u.  ( ( -u pi [,] pi )  \  dom  G ) )  C_  ( -u pi [,] pi ) )
1529, 151syl5eqss 3543 . . . . . . . . 9  |-  ( ph  ->  H  C_  ( -u pi [,] pi ) )
153152, 121sseldd 3500 . . . . . . . 8  |-  ( ph  ->  ( Q `  0
)  e.  ( -u pi [,] pi ) )
154 iccgelb 11606 . . . . . . . 8  |-  ( (
-u pi  e.  RR*  /\  pi  e.  RR*  /\  ( Q `  0 )  e.  ( -u pi [,] pi ) )  ->  -u pi  <_  ( Q `  0
) )
155138, 139, 153, 154syl3anc 1228 . . . . . . 7  |-  ( ph  -> 
-u pi  <_  ( Q `  0 )
)
156155ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  ( Q `  i )  =  -u pi )  ->  -u pi  <_  ( Q `  0 ) )
157122ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  ( Q `  i )  =  -u pi )  -> 
( Q `  0
)  e.  RR )
15813a1i 11 . . . . . . 7  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  ( Q `  i )  =  -u pi )  ->  -u pi  e.  RR )
159157, 158letri3d 9744 . . . . . 6  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  ( Q `  i )  =  -u pi )  -> 
( ( Q ` 
0 )  =  -u pi 
<->  ( ( Q ` 
0 )  <_  -u pi  /\  -u pi  <_  ( Q `
 0 ) ) ) )
160137, 156, 159mpbir2and 922 . . . . 5  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  ( Q `  i )  =  -u pi )  -> 
( Q `  0
)  =  -u pi )
16157, 51sselii 3496 . . . . . . 7  |-  -u pi  e.  H
162 f1ofo 5829 . . . . . . . . 9  |-  ( Q : ( 0 ... M ) -1-1-onto-> H  ->  Q :
( 0 ... M
) -onto-> H )
16392, 162syl 16 . . . . . . . 8  |-  ( Q 
Isom  <  ,  <  (
( 0 ... M
) ,  H )  ->  Q : ( 0 ... M )
-onto-> H )
164 forn 5804 . . . . . . . 8  |-  ( Q : ( 0 ... M ) -onto-> H  ->  ran  Q  =  H )
16591, 163, 1643syl 20 . . . . . . 7  |-  ( ph  ->  ran  Q  =  H )
166161, 165syl5eleqr 2552 . . . . . 6  |-  ( ph  -> 
-u pi  e.  ran  Q )
167 ffn 5737 . . . . . . 7  |-  ( Q : ( 0 ... M ) --> H  ->  Q  Fn  ( 0 ... M ) )
168 fvelrnb 5920 . . . . . . 7  |-  ( Q  Fn  ( 0 ... M )  ->  ( -u pi  e.  ran  Q  <->  E. i  e.  ( 0 ... M ) ( Q `  i )  =  -u pi ) )
16994, 167, 1683syl 20 . . . . . 6  |-  ( ph  ->  ( -u pi  e.  ran  Q  <->  E. i  e.  ( 0 ... M ) ( Q `  i
)  =  -u pi ) )
170166, 169mpbid 210 . . . . 5  |-  ( ph  ->  E. i  e.  ( 0 ... M ) ( Q `  i
)  =  -u pi )
171160, 170r19.29a 2999 . . . 4  |-  ( ph  ->  ( Q `  0
)  =  -u pi )
17257, 53sselii 3496 . . . . . . 7  |-  pi  e.  H
173172, 165syl5eleqr 2552 . . . . . 6  |-  ( ph  ->  pi  e.  ran  Q
)
174 fvelrnb 5920 . . . . . . 7  |-  ( Q  Fn  ( 0 ... M )  ->  (
pi  e.  ran  Q  <->  E. i  e.  ( 0 ... M ) ( Q `  i )  =  pi ) )
17594, 167, 1743syl 20 . . . . . 6  |-  ( ph  ->  ( pi  e.  ran  Q  <->  E. i  e.  (
0 ... M ) ( Q `  i )  =  pi ) )
176173, 175mpbid 210 . . . . 5  |-  ( ph  ->  E. i  e.  ( 0 ... M ) ( Q `  i
)  =  pi )
17794, 152fssd 5746 . . . . . . . . . 10  |-  ( ph  ->  Q : ( 0 ... M ) --> (
-u pi [,] pi ) )
178 eluzfz2 11719 . . . . . . . . . . 11  |-  ( M  e.  ( ZZ>= `  0
)  ->  M  e.  ( 0 ... M
) )
179118, 178syl 16 . . . . . . . . . 10  |-  ( ph  ->  M  e.  ( 0 ... M ) )
180177, 179ffvelrnd 6033 . . . . . . . . 9  |-  ( ph  ->  ( Q `  M
)  e.  ( -u pi [,] pi ) )
181 iccleub 11605 . . . . . . . . 9  |-  ( (
-u pi  e.  RR*  /\  pi  e.  RR*  /\  ( Q `  M )  e.  ( -u pi [,] pi ) )  ->  ( Q `  M )  <_  pi )
182138, 139, 180, 181syl3anc 1228 . . . . . . . 8  |-  ( ph  ->  ( Q `  M
)  <_  pi )
1831823ad2ant1 1017 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 0 ... M
)  /\  ( Q `  i )  =  pi )  ->  ( Q `  M )  <_  pi )
184 id 22 . . . . . . . . . 10  |-  ( ( Q `  i )  =  pi  ->  ( Q `  i )  =  pi )
185184eqcomd 2465 . . . . . . . . 9  |-  ( ( Q `  i )  =  pi  ->  pi  =  ( Q `  i ) )
1861853ad2ant3 1019 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 0 ... M
)  /\  ( Q `  i )  =  pi )  ->  pi  =  ( Q `  i ) )
187103adantr 465 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  =  M )  ->  ( Q `  i )  <_  ( Q `  i
) )
188 fveq2 5872 . . . . . . . . . . . 12  |-  ( i  =  M  ->  ( Q `  i )  =  ( Q `  M ) )
189188adantl 466 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  =  M )  ->  ( Q `  i )  =  ( Q `  M ) )
190187, 189breqtrd 4480 . . . . . . . . . 10  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  =  M )  ->  ( Q `  i )  <_  ( Q `  M
) )
191107ad2antlr 726 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  i  =  M )  ->  i  e.  RR )
192 elfzel2 11711 . . . . . . . . . . . . . 14  |-  ( i  e.  ( 0 ... M )  ->  M  e.  ZZ )
193192zred 10990 . . . . . . . . . . . . 13  |-  ( i  e.  ( 0 ... M )  ->  M  e.  RR )
194193ad2antlr 726 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  i  =  M )  ->  M  e.  RR )
195 elfzle2 11715 . . . . . . . . . . . . 13  |-  ( i  e.  ( 0 ... M )  ->  i  <_  M )
196195ad2antlr 726 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  i  =  M )  ->  i  <_  M )
197 neqne 31637 . . . . . . . . . . . . . 14  |-  ( -.  i  =  M  -> 
i  =/=  M )
198197necomd 2728 . . . . . . . . . . . . 13  |-  ( -.  i  =  M  ->  M  =/=  i )
199198adantl 466 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  i  =  M )  ->  M  =/=  i )
200191, 194, 196, 199leneltd 31697 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  i  =  M )  ->  i  <  M )
201102adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  <  M )  ->  ( Q `  i )  e.  RR )
20285, 180sseldi 3497 . . . . . . . . . . . . 13  |-  ( ph  ->  ( Q `  M
)  e.  RR )
203202ad2antrr 725 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  <  M )  ->  ( Q `  M )  e.  RR )
204 simpr 461 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  <  M )  ->  i  <  M )
20591ad2antrr 725 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  <  M )  ->  Q  Isom  <  ,  <  (
( 0 ... M
) ,  H ) )
206 simpr 461 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  i  e.  ( 0 ... M
) )
207179adantr 465 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  M  e.  ( 0 ... M
) )
208206, 207jca 532 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  (
i  e.  ( 0 ... M )  /\  M  e.  ( 0 ... M ) ) )
209208adantr 465 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  <  M )  ->  (
i  e.  ( 0 ... M )  /\  M  e.  ( 0 ... M ) ) )
210 isorel 6223 . . . . . . . . . . . . . 14  |-  ( ( Q  Isom  <  ,  <  ( ( 0 ... M
) ,  H )  /\  ( i  e.  ( 0 ... M
)  /\  M  e.  ( 0 ... M
) ) )  -> 
( i  <  M  <->  ( Q `  i )  <  ( Q `  M ) ) )
211205, 209, 210syl2anc 661 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  <  M )  ->  (
i  <  M  <->  ( Q `  i )  <  ( Q `  M )
) )
212204, 211mpbid 210 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  <  M )  ->  ( Q `  i )  <  ( Q `  M
) )
213201, 203, 212ltled 9750 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  <  M )  ->  ( Q `  i )  <_  ( Q `  M
) )
214200, 213syldan 470 . . . . . . . . . 10  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  i  =  M )  ->  ( Q `  i
)  <_  ( Q `  M ) )
215190, 214pm2.61dan 791 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  ( Q `  i )  <_  ( Q `  M
) )
2162153adant3 1016 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 0 ... M
)  /\  ( Q `  i )  =  pi )  ->  ( Q `  i )  <_  ( Q `  M )
)
217186, 216eqbrtrd 4476 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 0 ... M
)  /\  ( Q `  i )  =  pi )  ->  pi  <_  ( Q `  M ) )
2182023ad2ant1 1017 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 0 ... M
)  /\  ( Q `  i )  =  pi )  ->  ( Q `  M )  e.  RR )
21912a1i 11 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 0 ... M
)  /\  ( Q `  i )  =  pi )  ->  pi  e.  RR )
220218, 219letri3d 9744 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 0 ... M
)  /\  ( Q `  i )  =  pi )  ->  ( ( Q `  M )  =  pi  <->  ( ( Q `
 M )  <_  pi  /\  pi  <_  ( Q `  M )
) ) )
221183, 217, 220mpbir2and 922 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0 ... M
)  /\  ( Q `  i )  =  pi )  ->  ( Q `  M )  =  pi )
222221rexlimdv3a 2951 . . . . 5  |-  ( ph  ->  ( E. i  e.  ( 0 ... M
) ( Q `  i )  =  pi 
->  ( Q `  M
)  =  pi ) )
223176, 222mpd 15 . . . 4  |-  ( ph  ->  ( Q `  M
)  =  pi )
224 elfzoelz 11826 . . . . . . . . 9  |-  ( i  e.  ( 0..^ M )  ->  i  e.  ZZ )
225224zred 10990 . . . . . . . 8  |-  ( i  e.  ( 0..^ M )  ->  i  e.  RR )
226225ltp1d 10496 . . . . . . 7  |-  ( i  e.  ( 0..^ M )  ->  i  <  ( i  +  1 ) )
227226adantl 466 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  i  <  (
i  +  1 ) )
228 elfzofz 11841 . . . . . . . 8  |-  ( i  e.  ( 0..^ M )  ->  i  e.  ( 0 ... M
) )
229 fzofzp1 11912 . . . . . . . 8  |-  ( i  e.  ( 0..^ M )  ->  ( i  +  1 )  e.  ( 0 ... M
) )
230228, 229jca 532 . . . . . . 7  |-  ( i  e.  ( 0..^ M )  ->  ( i  e.  ( 0 ... M
)  /\  ( i  +  1 )  e.  ( 0 ... M
) ) )
231 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 ) ) ) )
23291, 230, 231syl2an 477 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( i  < 
( i  +  1 )  <->  ( Q `  i )  <  ( Q `  ( i  +  1 ) ) ) )
233227, 232mpbid 210 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( Q `  i )  <  ( Q `  ( i  +  1 ) ) )
234233ralrimiva 2871 . . . 4  |-  ( ph  ->  A. i  e.  ( 0..^ M ) ( Q `  i )  <  ( Q `  ( i  +  1 ) ) )
235171, 223, 234jca31 534 . . 3  |-  ( ph  ->  ( ( ( Q `
 0 )  = 
-u pi  /\  ( Q `  M )  =  pi )  /\  A. i  e.  ( 0..^ M ) ( Q `
 i )  < 
( Q `  (
i  +  1 ) ) ) )
2365fourierdlem2 32094 . . . 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 ) ) ) ) ) )
23767, 236syl 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 ) ) ) ) ) )
23899, 235, 237mpbir2and 922 . 2  |-  ( ph  ->  Q  e.  ( P `
 M ) )
239 fourierdlem102.g . . . . 5  |-  G  =  ( ( RR  _D  F )  |`  ( -u pi (,) pi ) )
240239reseq1i 5279 . . . 4  |-  ( G  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) )  =  ( ( ( RR  _D  F )  |`  ( -u pi (,) pi ) )  |`  ( ( Q `  i ) (,) ( Q `  (
i  +  1 ) ) ) )
24114a1i 11 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  -u pi  e.  RR* )
24215a1i 11 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  pi  e.  RR* )
243177adantr 465 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  Q : ( 0 ... M ) --> ( -u pi [,] pi ) )
244 simpr 461 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  i  e.  ( 0..^ M ) )
245241, 242, 243, 244fourierdlem27 32119 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( Q `
 i ) (,) ( Q `  (
i  +  1 ) ) )  C_  ( -u pi (,) pi ) )
246245resabs1d 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 ) ) ) ) )
247240, 246syl5req 2511 . . 3  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( RR 
_D  F )  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) )  =  ( G  |`  ( ( Q `  i ) (,) ( Q `  (
i  +  1 ) ) ) ) )
248 fourierdlem102.gcn . . . 4  |-  ( ph  ->  G  e.  ( dom 
G -cn-> CC ) )
249248, 5, 67, 238, 9, 165fourierdlem38 32130 . . 3  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( G  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) )  e.  ( ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) -cn-> CC ) )
250247, 249eqeltrd 2545 . 2  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( RR 
_D  F )  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) )  e.  ( ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) -cn-> CC ) )
251247oveq1d 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 ) ) )
252248adantr 465 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  G  e.  ( dom  G -cn-> CC ) )
253 fourierdlem102.rlim . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( -u pi [,) pi )  \  dom  G ) )  ->  (
( G  |`  (
x (,) +oo )
) lim CC  x )  =/=  (/) )
254253adantlr 714 . . . . 5  |-  ( ( ( ph  /\  i  e.  ( 0..^ M ) )  /\  x  e.  ( ( -u pi [,) pi )  \  dom  G ) )  ->  (
( G  |`  (
x (,) +oo )
) lim CC  x )  =/=  (/) )
255 fourierdlem102.llim . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( -u pi (,] pi )  \  dom  G ) )  ->  (
( G  |`  ( -oo (,) x ) ) lim
CC  x )  =/=  (/) )
256255adantlr 714 . . . . 5  |-  ( ( ( ph  /\  i  e.  ( 0..^ M ) )  /\  x  e.  ( ( -u pi (,] pi )  \  dom  G ) )  ->  (
( G  |`  ( -oo (,) x ) ) lim
CC  x )  =/=  (/) )
25791adantr 465 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  Q  Isom  <  ,  <  ( ( 0 ... M ) ,  H ) )
258257, 92, 933syl 20 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  Q : ( 0 ... M ) --> H )
25979adantr 465 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( E `  X )  e.  RR )
260257, 163, 1643syl 20 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ran  Q  =  H )
261252, 254, 256, 257, 258, 244, 233, 245, 259, 9, 260fourierdlem46 32138 . . . 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 ) ) )  =/=  (/) ) )
262261simpld 459 . . 3  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( G  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) lim CC  ( Q `  i )
)  =/=  (/) )
263251, 262eqnetrd 2750 . 2  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( ( RR  _D  F )  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) lim CC  ( Q `  i )
)  =/=  (/) )
264247oveq1d 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 ) ) ) )
265261simprd 463 . . 3  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( G  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) lim CC  ( Q `  ( i  +  1 ) ) )  =/=  (/) )
266264, 265eqnetrd 2750 . 2  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( ( RR  _D  F )  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) lim CC  ( Q `  ( i  +  1 ) ) )  =/=  (/) )
2671, 2, 3, 4, 5, 67, 238, 250, 263, 266fourierdlem94 32186 1  |-  ( ph  ->  ( ( ( F  |`  ( -oo (,) X
) ) lim CC  X
)  =/=  (/)  /\  (
( F  |`  ( X (,) +oo ) ) lim
CC  X )  =/=  (/) ) )
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 11821   |_cfl 11930   #chash 12408   picpi 13814   -cn->ccncf 21506   lim CC climc 22392    _D cdv 22393
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-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-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-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-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-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 11822  df-fl 11932  df-seq 12111  df-exp 12170  df-fac 12357  df-bc 12384  df-hash 12409  df-shft 12912  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-limsup 13306  df-clim 13323  df-rlim 13324  df-sum 13521  df-ef 13815  df-sin 13817  df-cos 13818  df-pi 13820  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-starv 14727  df-sca 14728  df-vsca 14729  df-ip 14730  df-tset 14731  df-ple 14732  df-ds 14734  df-unif 14735  df-hom 14736  df-cco 14737  df-rest 14840  df-topn 14841  df-0g 14859  df-gsum 14860  df-topgen 14861  df-pt 14862  df-prds 14865  df-xrs 14919  df-qtop 14924  df-imas 14925  df-xps 14927  df-mre 15003  df-mrc 15004  df-acs 15006  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-submnd 16094  df-mulg 16187  df-cntz 16482  df-cmn 16927  df-psmet 18538  df-xmet 18539  df-met 18540  df-bl 18541  df-mopn 18542  df-fbas 18543  df-fg 18544  df-cnfld 18548  df-top 19526  df-bases 19528  df-topon 19529  df-topsp 19530  df-cld 19647  df-ntr 19648  df-cls 19649  df-nei 19726  df-lp 19764  df-perf 19765  df-cn 19855  df-cnp 19856  df-haus 19943  df-cmp 20014  df-tx 20189  df-hmeo 20382  df-fil 20473  df-fm 20565  df-flim 20566  df-flf 20567  df-xms 20949  df-ms 20950  df-tms 20951  df-cncf 21508  df-limc 22396  df-dv 22397
This theorem is referenced by:  fourierdlem106  32198
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