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Theorem fourierdlem102 38184
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 10993 . . . . . 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 7865 . . . . . . . . . 10  |-  { -u pi ,  pi , 
( E `  X
) }  e.  Fin
1110a1i 11 . . . . . . . . 9  |-  ( ph  ->  { -u pi ,  pi ,  ( E `  X ) }  e.  Fin )
12 pire 23492 . . . . . . . . . . . . . . 15  |-  pi  e.  RR
1312renegcli 9955 . . . . . . . . . . . . . 14  |-  -u pi  e.  RR
1413rexri 9711 . . . . . . . . . . . . 13  |-  -u pi  e.  RR*
1512rexri 9711 . . . . . . . . . . . . 13  |-  pi  e.  RR*
16 negpilt0 37580 . . . . . . . . . . . . . . 15  |-  -u pi  <  0
17 pipos 23494 . . . . . . . . . . . . . . 15  |-  0  <  pi
18 0re 9661 . . . . . . . . . . . . . . . 16  |-  0  e.  RR
1913, 18, 12lttri 9778 . . . . . . . . . . . . . . 15  |-  ( (
-u pi  <  0  /\  0  <  pi )  ->  -u pi  <  pi )
2016, 17, 19mp2an 686 . . . . . . . . . . . . . 14  |-  -u pi  <  pi
2113, 12, 20ltleii 9775 . . . . . . . . . . . . 13  |-  -u pi  <_  pi
22 prunioo 11787 . . . . . . . . . . . . 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 1390 . . . . . . . . . . . 12  |-  ( (
-u pi (,) pi )  u.  { -u pi ,  pi } )  =  ( -u pi [,] pi )
2423difeq1i 3536 . . . . . . . . . . 11  |-  ( ( ( -u pi (,) pi )  u.  { -u pi ,  pi }
)  \  dom  G )  =  ( ( -u pi [,] pi )  \  dom  G )
25 difundir 3687 . . . . . . . . . . 11  |-  ( ( ( -u pi (,) pi )  u.  { -u pi ,  pi }
)  \  dom  G )  =  ( ( (
-u pi (,) pi )  \  dom  G )  u.  ( { -u pi ,  pi }  \  dom  G ) )
2624, 25eqtr3i 2495 . . . . . . . . . 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 7864 . . . . . . . . . . . 12  |-  { -u pi ,  pi }  e.  Fin
29 diffi 7821 . . . . . . . . . . . 12  |-  ( {
-u pi ,  pi }  e.  Fin  ->  ( { -u pi ,  pi }  \  dom  G )  e.  Fin )
3028, 29mp1i 13 . . . . . . . . . . 11  |-  ( ph  ->  ( { -u pi ,  pi }  \  dom  G )  e.  Fin )
31 unfi 7856 . . . . . . . . . . 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 673 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( -u pi (,) pi )  \  dom  G )  u.  ( { -u pi ,  pi }  \  dom  G ) )  e.  Fin )
3326, 32syl5eqel 2553 . . . . . . . . 9  |-  ( ph  ->  ( ( -u pi [,] pi )  \  dom  G )  e.  Fin )
34 unfi 7856 . . . . . . . . 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 673 . . . . . . . 8  |-  ( ph  ->  ( { -u pi ,  pi ,  ( E `
 X ) }  u.  ( ( -u pi [,] pi )  \  dom  G ) )  e. 
Fin )
369, 35syl5eqel 2553 . . . . . . 7  |-  ( ph  ->  H  e.  Fin )
37 hashcl 12576 . . . . . . 7  |-  ( H  e.  Fin  ->  ( # `
 H )  e. 
NN0 )
3836, 37syl 17 . . . . . 6  |-  ( ph  ->  ( # `  H
)  e.  NN0 )
3938nn0zd 11061 . . . . 5  |-  ( ph  ->  ( # `  H
)  e.  ZZ )
4013, 20ltneii 9765 . . . . . . 7  |-  -u pi  =/=  pi
41 hashprg 12610 . . . . . . . 8  |-  ( (
-u pi  e.  RR  /\  pi  e.  RR )  ->  ( -u pi  =/=  pi  <->  ( # `  { -u pi ,  pi }
)  =  2 ) )
4213, 12, 41mp2an 686 . . . . . . 7  |-  ( -u pi  =/=  pi  <->  ( # `  { -u pi ,  pi }
)  =  2 )
4340, 42mpbi 213 . . . . . 6  |-  ( # `  { -u pi ,  pi } )  =  2
4410elexi 3041 . . . . . . . . . 10  |-  { -u pi ,  pi , 
( E `  X
) }  e.  _V
45 ovex 6336 . . . . . . . . . . 11  |-  ( -u pi [,] pi )  e. 
_V
46 difexg 4545 . . . . . . . . . . 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 6608 . . . . . . . . 9  |-  ( {
-u pi ,  pi ,  ( E `  X ) }  u.  ( ( -u pi [,] pi )  \  dom  G ) )  e.  _V
499, 48eqeltri 2545 . . . . . . . 8  |-  H  e. 
_V
50 negex 9893 . . . . . . . . . . 11  |-  -u pi  e.  _V
5150tpid1 4076 . . . . . . . . . 10  |-  -u pi  e.  { -u pi ,  pi ,  ( E `  X ) }
5212elexi 3041 . . . . . . . . . . 11  |-  pi  e.  _V
5352tpid2 4077 . . . . . . . . . 10  |-  pi  e.  {
-u pi ,  pi ,  ( E `  X ) }
54 prssi 4119 . . . . . . . . . 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 686 . . . . . . . . 9  |-  { -u pi ,  pi }  C_ 
{ -u pi ,  pi ,  ( E `  X ) }
56 ssun1 3588 . . . . . . . . . 10  |-  { -u pi ,  pi , 
( E `  X
) }  C_  ( { -u pi ,  pi ,  ( E `  X ) }  u.  ( ( -u pi [,] pi )  \  dom  G ) )
5756, 9sseqtr4i 3451 . . . . . . . . 9  |-  { -u pi ,  pi , 
( E `  X
) }  C_  H
5855, 57sstri 3427 . . . . . . . 8  |-  { -u pi ,  pi }  C_  H
59 hashss 12624 . . . . . . . 8  |-  ( ( H  e.  _V  /\  {
-u pi ,  pi }  C_  H )  -> 
( # `  { -u pi ,  pi }
)  <_  ( # `  H
) )
6049, 58, 59mp2an 686 . . . . . . 7  |-  ( # `  { -u pi ,  pi } )  <_  ( # `
 H )
6160a1i 11 . . . . . 6  |-  ( ph  ->  ( # `  { -u pi ,  pi }
)  <_  ( # `  H
) )
6243, 61syl5eqbrr 4430 . . . . 5  |-  ( ph  ->  2  <_  ( # `  H
) )
63 eluz2 11188 . . . . 5  |-  ( (
# `  H )  e.  ( ZZ>= `  2 )  <->  ( 2  e.  ZZ  /\  ( # `  H )  e.  ZZ  /\  2  <_  ( # `  H
) ) )
648, 39, 62, 63syl3anbrc 1214 . . . 4  |-  ( ph  ->  ( # `  H
)  e.  ( ZZ>= ` 
2 ) )
65 uz2m1nn 11256 . . . 4  |-  ( (
# `  H )  e.  ( ZZ>= `  2 )  ->  ( ( # `  H
)  -  1 )  e.  NN )
6664, 65syl 17 . . 3  |-  ( ph  ->  ( ( # `  H
)  -  1 )  e.  NN )
676, 66syl5eqel 2553 . 2  |-  ( ph  ->  M  e.  NN )
6813a1i 11 . . . . . . . . . . 11  |-  ( ph  -> 
-u pi  e.  RR )
6912a1i 11 . . . . . . . . . . 11  |-  ( ph  ->  pi  e.  RR )
70 negpitopissre 23568 . . . . . . . . . . . 12  |-  ( -u pi (,] pi )  C_  RR
7120a1i 11 . . . . . . . . . . . . . 14  |-  ( ph  -> 
-u pi  <  pi )
72 picn 23493 . . . . . . . . . . . . . . . 16  |-  pi  e.  CC
73722timesi 10753 . . . . . . . . . . . . . . 15  |-  ( 2  x.  pi )  =  ( pi  +  pi )
7472, 72subnegi 9973 . . . . . . . . . . . . . . 15  |-  ( pi 
-  -u pi )  =  ( pi  +  pi )
7573, 2, 743eqtr4i 2503 . . . . . . . . . . . . . 14  |-  T  =  ( pi  -  -u pi )
76 fourierdlem102.e . . . . . . . . . . . . . 14  |-  E  =  ( x  e.  RR  |->  ( x  +  (
( |_ `  (
( pi  -  x
)  /  T ) )  x.  T ) ) )
7768, 69, 71, 75, 76fourierdlem4 38085 . . . . . . . . . . . . 13  |-  ( ph  ->  E : RR --> ( -u pi (,] pi ) )
7877, 4ffvelrnd 6038 . . . . . . . . . . . 12  |-  ( ph  ->  ( E `  X
)  e.  ( -u pi (,] pi ) )
7970, 78sseldi 3416 . . . . . . . . . . 11  |-  ( ph  ->  ( E `  X
)  e.  RR )
8068, 69, 793jca 1210 . . . . . . . . . 10  |-  ( ph  ->  ( -u pi  e.  RR  /\  pi  e.  RR  /\  ( E `  X
)  e.  RR ) )
81 fvex 5889 . . . . . . . . . . 11  |-  ( E `
 X )  e. 
_V
8250, 52, 81tpss 4129 . . . . . . . . . 10  |-  ( (
-u pi  e.  RR  /\  pi  e.  RR  /\  ( E `  X )  e.  RR )  <->  { -u pi ,  pi ,  ( E `
 X ) } 
C_  RR )
8380, 82sylib 201 . . . . . . . . 9  |-  ( ph  ->  { -u pi ,  pi ,  ( E `  X ) }  C_  RR )
84 iccssre 11741 . . . . . . . . . . 11  |-  ( (
-u pi  e.  RR  /\  pi  e.  RR )  ->  ( -u pi [,] pi )  C_  RR )
8513, 12, 84mp2an 686 . . . . . . . . . 10  |-  ( -u pi [,] pi )  C_  RR
86 ssdifss 3553 . . . . . . . . . 10  |-  ( (
-u pi [,] pi )  C_  RR  ->  (
( -u pi [,] pi )  \  dom  G ) 
C_  RR )
8785, 86mp1i 13 . . . . . . . . 9  |-  ( ph  ->  ( ( -u pi [,] pi )  \  dom  G )  C_  RR )
8883, 87unssd 3601 . . . . . . . 8  |-  ( ph  ->  ( { -u pi ,  pi ,  ( E `
 X ) }  u.  ( ( -u pi [,] pi )  \  dom  G ) )  C_  RR )
899, 88syl5eqss 3462 . . . . . . 7  |-  ( ph  ->  H  C_  RR )
90 fourierdlem102.q . . . . . . 7  |-  Q  =  ( iota g g 
Isom  <  ,  <  (
( 0 ... M
) ,  H ) )
9136, 89, 90, 6fourierdlem36 38118 . . . . . 6  |-  ( ph  ->  Q  Isom  <  ,  <  ( ( 0 ... M
) ,  H ) )
92 isof1o 6234 . . . . . 6  |-  ( Q 
Isom  <  ,  <  (
( 0 ... M
) ,  H )  ->  Q : ( 0 ... M ) -1-1-onto-> H )
93 f1of 5828 . . . . . 6  |-  ( Q : ( 0 ... M ) -1-1-onto-> H  ->  Q :
( 0 ... M
) --> H )
9491, 92, 933syl 18 . . . . 5  |-  ( ph  ->  Q : ( 0 ... M ) --> H )
9594, 89fssd 5750 . . . 4  |-  ( ph  ->  Q : ( 0 ... M ) --> RR )
96 reex 9648 . . . . 5  |-  RR  e.  _V
97 ovex 6336 . . . . 5  |-  ( 0 ... M )  e. 
_V
9896, 97elmap 7518 . . . 4  |-  ( Q  e.  ( RR  ^m  ( 0 ... M
) )  <->  Q :
( 0 ... M
) --> RR )
9995, 98sylibr 217 . . 3  |-  ( ph  ->  Q  e.  ( RR 
^m  ( 0 ... M ) ) )
100 fveq2 5879 . . . . . . . . . . 11  |-  ( 0  =  i  ->  ( Q `  0 )  =  ( Q `  i ) )
101100adantl 473 . . . . . . . . . 10  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  =  i )  -> 
( Q `  0
)  =  ( Q `
 i ) )
10295ffvelrnda 6037 . . . . . . . . . . . 12  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  ( Q `  i )  e.  RR )
103102leidd 10201 . . . . . . . . . . 11  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  ( Q `  i )  <_  ( Q `  i
) )
104103adantr 472 . . . . . . . . . 10  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  =  i )  -> 
( Q `  i
)  <_  ( Q `  i ) )
105101, 104eqbrtrd 4416 . . . . . . . . 9  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  =  i )  -> 
( Q `  0
)  <_  ( Q `  i ) )
106 elfzelz 11826 . . . . . . . . . . . . 13  |-  ( i  e.  ( 0 ... M )  ->  i  e.  ZZ )
107106zred 11063 . . . . . . . . . . . 12  |-  ( i  e.  ( 0 ... M )  ->  i  e.  RR )
108107ad2antlr 741 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  0  =  i )  ->  i  e.  RR )
109 elfzle1 11828 . . . . . . . . . . . 12  |-  ( i  e.  ( 0 ... M )  ->  0  <_  i )
110109ad2antlr 741 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  0  =  i )  ->  0  <_  i )
111 neqne 2651 . . . . . . . . . . . . 13  |-  ( -.  0  =  i  -> 
0  =/=  i )
112111necomd 2698 . . . . . . . . . . . 12  |-  ( -.  0  =  i  -> 
i  =/=  0 )
113112adantl 473 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  0  =  i )  ->  i  =/=  0 )
114108, 110, 113ne0gt0d 9789 . . . . . . . . . 10  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  0  =  i )  ->  0  <  i )
115 nnssnn0 10896 . . . . . . . . . . . . . . . . 17  |-  NN  C_  NN0
116 nn0uz 11217 . . . . . . . . . . . . . . . . 17  |-  NN0  =  ( ZZ>= `  0 )
117115, 116sseqtri 3450 . . . . . . . . . . . . . . . 16  |-  NN  C_  ( ZZ>= `  0 )
118117, 67sseldi 3416 . . . . . . . . . . . . . . 15  |-  ( ph  ->  M  e.  ( ZZ>= ` 
0 ) )
119 eluzfz1 11832 . . . . . . . . . . . . . . 15  |-  ( M  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... M
) )
120118, 119syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  0  e.  ( 0 ... M ) )
12194, 120ffvelrnd 6038 . . . . . . . . . . . . 13  |-  ( ph  ->  ( Q `  0
)  e.  H )
12289, 121sseldd 3419 . . . . . . . . . . . 12  |-  ( ph  ->  ( Q `  0
)  e.  RR )
123122ad2antrr 740 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  <  i )  ->  ( Q `  0 )  e.  RR )
124102adantr 472 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  <  i )  ->  ( Q `  i )  e.  RR )
125 simpr 468 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  <  i )  ->  0  <  i )
12691ad2antrr 740 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  <  i )  ->  Q  Isom  <  ,  <  (
( 0 ... M
) ,  H ) )
127120anim1i 578 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  (
0  e.  ( 0 ... M )  /\  i  e.  ( 0 ... M ) ) )
128127adantr 472 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  <  i )  ->  (
0  e.  ( 0 ... M )  /\  i  e.  ( 0 ... M ) ) )
129 isorel 6235 . . . . . . . . . . . . 13  |-  ( ( Q  Isom  <  ,  <  ( ( 0 ... M
) ,  H )  /\  ( 0  e.  ( 0 ... M
)  /\  i  e.  ( 0 ... M
) ) )  -> 
( 0  <  i  <->  ( Q `  0 )  <  ( Q `  i ) ) )
130126, 128, 129syl2anc 673 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  <  i )  ->  (
0  <  i  <->  ( Q `  0 )  < 
( Q `  i
) ) )
131125, 130mpbid 215 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  <  i )  ->  ( Q `  0 )  <  ( Q `  i
) )
132123, 124, 131ltled 9800 . . . . . . . . . 10  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  <  i )  ->  ( Q `  0 )  <_  ( Q `  i
) )
133114, 132syldan 478 . . . . . . . . 9  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  0  =  i )  ->  ( Q `  0
)  <_  ( Q `  i ) )
134105, 133pm2.61dan 808 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  ( Q `  0 )  <_  ( Q `  i
) )
135134adantr 472 . . . . . . 7  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  ( Q `  i )  =  -u pi )  -> 
( Q `  0
)  <_  ( Q `  i ) )
136 simpr 468 . . . . . . 7  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  ( Q `  i )  =  -u pi )  -> 
( Q `  i
)  =  -u pi )
137135, 136breqtrd 4420 . . . . . 6  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  ( Q `  i )  =  -u pi )  -> 
( Q `  0
)  <_  -u pi )
13868rexrd 9708 . . . . . . . 8  |-  ( ph  -> 
-u pi  e.  RR* )
13969rexrd 9708 . . . . . . . 8  |-  ( ph  ->  pi  e.  RR* )
140 lbicc2 11774 . . . . . . . . . . . . . 14  |-  ( (
-u pi  e.  RR*  /\  pi  e.  RR*  /\  -u pi  <_  pi )  ->  -u pi  e.  ( -u pi [,] pi ) )
14114, 15, 21, 140mp3an 1390 . . . . . . . . . . . . 13  |-  -u pi  e.  ( -u pi [,] pi )
142141a1i 11 . . . . . . . . . . . 12  |-  ( ph  -> 
-u pi  e.  (
-u pi [,] pi ) )
143 ubicc2 11775 . . . . . . . . . . . . . 14  |-  ( (
-u pi  e.  RR*  /\  pi  e.  RR*  /\  -u pi  <_  pi )  ->  pi  e.  ( -u pi [,] pi ) )
14414, 15, 21, 143mp3an 1390 . . . . . . . . . . . . 13  |-  pi  e.  ( -u pi [,] pi )
145144a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  pi  e.  ( -u pi [,] pi ) )
146 iocssicc 11747 . . . . . . . . . . . . 13  |-  ( -u pi (,] pi )  C_  ( -u pi [,] pi )
147146, 78sseldi 3416 . . . . . . . . . . . 12  |-  ( ph  ->  ( E `  X
)  e.  ( -u pi [,] pi ) )
148 tpssi 4130 . . . . . . . . . . . 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 1292 . . . . . . . . . . 11  |-  ( ph  ->  { -u pi ,  pi ,  ( E `  X ) }  C_  ( -u pi [,] pi ) )
150 difssd 3550 . . . . . . . . . . 11  |-  ( ph  ->  ( ( -u pi [,] pi )  \  dom  G )  C_  ( -u pi [,] pi ) )
151149, 150unssd 3601 . . . . . . . . . 10  |-  ( ph  ->  ( { -u pi ,  pi ,  ( E `
 X ) }  u.  ( ( -u pi [,] pi )  \  dom  G ) )  C_  ( -u pi [,] pi ) )
1529, 151syl5eqss 3462 . . . . . . . . 9  |-  ( ph  ->  H  C_  ( -u pi [,] pi ) )
153152, 121sseldd 3419 . . . . . . . 8  |-  ( ph  ->  ( Q `  0
)  e.  ( -u pi [,] pi ) )
154 iccgelb 11716 . . . . . . . 8  |-  ( (
-u pi  e.  RR*  /\  pi  e.  RR*  /\  ( Q `  0 )  e.  ( -u pi [,] pi ) )  ->  -u pi  <_  ( Q `  0
) )
155138, 139, 153, 154syl3anc 1292 . . . . . . 7  |-  ( ph  -> 
-u pi  <_  ( Q `  0 )
)
156155ad2antrr 740 . . . . . 6  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  ( Q `  i )  =  -u pi )  ->  -u pi  <_  ( Q `  0 ) )
157122ad2antrr 740 . . . . . . 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 9794 . . . . . 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 936 . . . . 5  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  ( Q `  i )  =  -u pi )  -> 
( Q `  0
)  =  -u pi )
16157, 51sselii 3415 . . . . . . 7  |-  -u pi  e.  H
162 f1ofo 5835 . . . . . . . . 9  |-  ( Q : ( 0 ... M ) -1-1-onto-> H  ->  Q :
( 0 ... M
) -onto-> H )
16392, 162syl 17 . . . . . . . 8  |-  ( Q 
Isom  <  ,  <  (
( 0 ... M
) ,  H )  ->  Q : ( 0 ... M )
-onto-> H )
164 forn 5809 . . . . . . . 8  |-  ( Q : ( 0 ... M ) -onto-> H  ->  ran  Q  =  H )
16591, 163, 1643syl 18 . . . . . . 7  |-  ( ph  ->  ran  Q  =  H )
166161, 165syl5eleqr 2556 . . . . . 6  |-  ( ph  -> 
-u pi  e.  ran  Q )
167 ffn 5739 . . . . . . 7  |-  ( Q : ( 0 ... M ) --> H  ->  Q  Fn  ( 0 ... M ) )
168 fvelrnb 5926 . . . . . . 7  |-  ( Q  Fn  ( 0 ... M )  ->  ( -u pi  e.  ran  Q  <->  E. i  e.  ( 0 ... M ) ( Q `  i )  =  -u pi ) )
16994, 167, 1683syl 18 . . . . . 6  |-  ( ph  ->  ( -u pi  e.  ran  Q  <->  E. i  e.  ( 0 ... M ) ( Q `  i
)  =  -u pi ) )
170166, 169mpbid 215 . . . . 5  |-  ( ph  ->  E. i  e.  ( 0 ... M ) ( Q `  i
)  =  -u pi )
171160, 170r19.29a 2918 . . . 4  |-  ( ph  ->  ( Q `  0
)  =  -u pi )
17257, 53sselii 3415 . . . . . . 7  |-  pi  e.  H
173172, 165syl5eleqr 2556 . . . . . 6  |-  ( ph  ->  pi  e.  ran  Q
)
174 fvelrnb 5926 . . . . . . 7  |-  ( Q  Fn  ( 0 ... M )  ->  (
pi  e.  ran  Q  <->  E. i  e.  ( 0 ... M ) ( Q `  i )  =  pi ) )
17594, 167, 1743syl 18 . . . . . 6  |-  ( ph  ->  ( pi  e.  ran  Q  <->  E. i  e.  (
0 ... M ) ( Q `  i )  =  pi ) )
176173, 175mpbid 215 . . . . 5  |-  ( ph  ->  E. i  e.  ( 0 ... M ) ( Q `  i
)  =  pi )
17794, 152fssd 5750 . . . . . . . . . 10  |-  ( ph  ->  Q : ( 0 ... M ) --> (
-u pi [,] pi ) )
178 eluzfz2 11833 . . . . . . . . . . 11  |-  ( M  e.  ( ZZ>= `  0
)  ->  M  e.  ( 0 ... M
) )
179118, 178syl 17 . . . . . . . . . 10  |-  ( ph  ->  M  e.  ( 0 ... M ) )
180177, 179ffvelrnd 6038 . . . . . . . . 9  |-  ( ph  ->  ( Q `  M
)  e.  ( -u pi [,] pi ) )
181 iccleub 11715 . . . . . . . . 9  |-  ( (
-u pi  e.  RR*  /\  pi  e.  RR*  /\  ( Q `  M )  e.  ( -u pi [,] pi ) )  ->  ( Q `  M )  <_  pi )
182138, 139, 180, 181syl3anc 1292 . . . . . . . 8  |-  ( ph  ->  ( Q `  M
)  <_  pi )
1831823ad2ant1 1051 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 0 ... M
)  /\  ( Q `  i )  =  pi )  ->  ( Q `  M )  <_  pi )
184 id 22 . . . . . . . . . 10  |-  ( ( Q `  i )  =  pi  ->  ( Q `  i )  =  pi )
185184eqcomd 2477 . . . . . . . . 9  |-  ( ( Q `  i )  =  pi  ->  pi  =  ( Q `  i ) )
1861853ad2ant3 1053 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 0 ... M
)  /\  ( Q `  i )  =  pi )  ->  pi  =  ( Q `  i ) )
187103adantr 472 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  =  M )  ->  ( Q `  i )  <_  ( Q `  i
) )
188 fveq2 5879 . . . . . . . . . . . 12  |-  ( i  =  M  ->  ( Q `  i )  =  ( Q `  M ) )
189188adantl 473 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  =  M )  ->  ( Q `  i )  =  ( Q `  M ) )
190187, 189breqtrd 4420 . . . . . . . . . 10  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  =  M )  ->  ( Q `  i )  <_  ( Q `  M
) )
191107ad2antlr 741 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  i  =  M )  ->  i  e.  RR )
192 elfzel2 11824 . . . . . . . . . . . . . 14  |-  ( i  e.  ( 0 ... M )  ->  M  e.  ZZ )
193192zred 11063 . . . . . . . . . . . . 13  |-  ( i  e.  ( 0 ... M )  ->  M  e.  RR )
194193ad2antlr 741 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  i  =  M )  ->  M  e.  RR )
195 elfzle2 11829 . . . . . . . . . . . . 13  |-  ( i  e.  ( 0 ... M )  ->  i  <_  M )
196195ad2antlr 741 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  i  =  M )  ->  i  <_  M )
197 neqne 2651 . . . . . . . . . . . . . 14  |-  ( -.  i  =  M  -> 
i  =/=  M )
198197necomd 2698 . . . . . . . . . . . . 13  |-  ( -.  i  =  M  ->  M  =/=  i )
199198adantl 473 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  i  =  M )  ->  M  =/=  i )
200191, 194, 196, 199leneltd 9806 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  i  =  M )  ->  i  <  M )
201102adantr 472 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  <  M )  ->  ( Q `  i )  e.  RR )
20285, 180sseldi 3416 . . . . . . . . . . . . 13  |-  ( ph  ->  ( Q `  M
)  e.  RR )
203202ad2antrr 740 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  <  M )  ->  ( Q `  M )  e.  RR )
204 simpr 468 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  <  M )  ->  i  <  M )
20591ad2antrr 740 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  <  M )  ->  Q  Isom  <  ,  <  (
( 0 ... M
) ,  H ) )
206 simpr 468 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  i  e.  ( 0 ... M
) )
207179adantr 472 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  M  e.  ( 0 ... M
) )
208206, 207jca 541 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  (
i  e.  ( 0 ... M )  /\  M  e.  ( 0 ... M ) ) )
209208adantr 472 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  <  M )  ->  (
i  e.  ( 0 ... M )  /\  M  e.  ( 0 ... M ) ) )
210 isorel 6235 . . . . . . . . . . . . . 14  |-  ( ( Q  Isom  <  ,  <  ( ( 0 ... M
) ,  H )  /\  ( i  e.  ( 0 ... M
)  /\  M  e.  ( 0 ... M
) ) )  -> 
( i  <  M  <->  ( Q `  i )  <  ( Q `  M ) ) )
211205, 209, 210syl2anc 673 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  <  M )  ->  (
i  <  M  <->  ( Q `  i )  <  ( Q `  M )
) )
212204, 211mpbid 215 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  <  M )  ->  ( Q `  i )  <  ( Q `  M
) )
213201, 203, 212ltled 9800 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  <  M )  ->  ( Q `  i )  <_  ( Q `  M
) )
214200, 213syldan 478 . . . . . . . . . 10  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  i  =  M )  ->  ( Q `  i
)  <_  ( Q `  M ) )
215190, 214pm2.61dan 808 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  ( Q `  i )  <_  ( Q `  M
) )
2162153adant3 1050 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 0 ... M
)  /\  ( Q `  i )  =  pi )  ->  ( Q `  i )  <_  ( Q `  M )
)
217186, 216eqbrtrd 4416 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 0 ... M
)  /\  ( Q `  i )  =  pi )  ->  pi  <_  ( Q `  M ) )
2182023ad2ant1 1051 . . . . . . . 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 9794 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 0 ... M
)  /\  ( Q `  i )  =  pi )  ->  ( ( Q `  M )  =  pi  <->  ( ( Q `
 M )  <_  pi  /\  pi  <_  ( Q `  M )
) ) )
221183, 217, 220mpbir2and 936 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0 ... M
)  /\  ( Q `  i )  =  pi )  ->  ( Q `  M )  =  pi )
222221rexlimdv3a 2873 . . . . 5  |-  ( ph  ->  ( E. i  e.  ( 0 ... M
) ( Q `  i )  =  pi 
->  ( Q `  M
)  =  pi ) )
223176, 222mpd 15 . . . 4  |-  ( ph  ->  ( Q `  M
)  =  pi )
224 elfzoelz 11947 . . . . . . . . 9  |-  ( i  e.  ( 0..^ M )  ->  i  e.  ZZ )
225224zred 11063 . . . . . . . 8  |-  ( i  e.  ( 0..^ M )  ->  i  e.  RR )
226225ltp1d 10559 . . . . . . 7  |-  ( i  e.  ( 0..^ M )  ->  i  <  ( i  +  1 ) )
227226adantl 473 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  i  <  (
i  +  1 ) )
228 elfzofz 11962 . . . . . . . 8  |-  ( i  e.  ( 0..^ M )  ->  i  e.  ( 0 ... M
) )
229 fzofzp1 12037 . . . . . . . 8  |-  ( i  e.  ( 0..^ M )  ->  ( i  +  1 )  e.  ( 0 ... M
) )
230228, 229jca 541 . . . . . . 7  |-  ( i  e.  ( 0..^ M )  ->  ( i  e.  ( 0 ... M
)  /\  ( i  +  1 )  e.  ( 0 ... M
) ) )
231 isorel 6235 . . . . . . 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 485 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( i  < 
( i  +  1 )  <->  ( Q `  i )  <  ( Q `  ( i  +  1 ) ) ) )
233227, 232mpbid 215 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( Q `  i )  <  ( Q `  ( i  +  1 ) ) )
234233ralrimiva 2809 . . . 4  |-  ( ph  ->  A. i  e.  ( 0..^ M ) ( Q `  i )  <  ( Q `  ( i  +  1 ) ) )
235171, 223, 234jca31 543 . . 3  |-  ( ph  ->  ( ( ( Q `
 0 )  = 
-u pi  /\  ( Q `  M )  =  pi )  /\  A. i  e.  ( 0..^ M ) ( Q `
 i )  < 
( Q `  (
i  +  1 ) ) ) )
2365fourierdlem2 38083 . . . 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 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 ) ) ) ) ) )
23899, 235, 237mpbir2and 936 . 2  |-  ( ph  ->  Q  e.  ( P `
 M ) )
239 fourierdlem102.g . . . . 5  |-  G  =  ( ( RR  _D  F )  |`  ( -u pi (,) pi ) )
240239reseq1i 5107 . . . 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 472 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  Q : ( 0 ... M ) --> ( -u pi [,] pi ) )
244 simpr 468 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  i  e.  ( 0..^ M ) )
245241, 242, 243, 244fourierdlem27 38108 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( Q `
 i ) (,) ( Q `  (
i  +  1 ) ) )  C_  ( -u pi (,) pi ) )
246245resabs1d 5140 . . . 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 2518 . . 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 38120 . . 3  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( G  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) )  e.  ( ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) -cn-> CC ) )
250247, 249eqeltrd 2549 . 2  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( RR 
_D  F )  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) )  e.  ( ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) -cn-> CC ) )
251247oveq1d 6323 . . 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 472 . . . . 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 729 . . . . 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 729 . . . . 5  |-  ( ( ( ph  /\  i  e.  ( 0..^ M ) )  /\  x  e.  ( ( -u pi (,] pi )  \  dom  G ) )  ->  (
( G  |`  ( -oo (,) x ) ) lim
CC  x )  =/=  (/) )
25791adantr 472 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  Q  Isom  <  ,  <  ( ( 0 ... M ) ,  H ) )
258257, 92, 933syl 18 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  Q : ( 0 ... M ) --> H )
25979adantr 472 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( E `  X )  e.  RR )
260257, 163, 1643syl 18 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ran  Q  =  H )
261252, 254, 256, 257, 258, 244, 233, 245, 259, 9, 260fourierdlem46 38128 . . . 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 466 . . 3  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( G  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) lim CC  ( Q `  i )
)  =/=  (/) )
263251, 262eqnetrd 2710 . 2  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( ( RR  _D  F )  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) lim CC  ( Q `  i )
)  =/=  (/) )
264247oveq1d 6323 . . 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 470 . . 3  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( G  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) lim CC  ( Q `  ( i  +  1 ) ) )  =/=  (/) )
266264, 265eqnetrd 2710 . 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 38176 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 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   E.wrex 2757   {crab 2760   _Vcvv 3031    \ cdif 3387    u. cun 3388    C_ wss 3390   (/)c0 3722   {cpr 3961   {ctp 3963   class class class wbr 4395    |-> cmpt 4454   dom cdm 4839   ran crn 4840    |` cres 4841   iotacio 5551    Fn wfn 5584   -->wf 5585   -onto->wfo 5587   -1-1-onto->wf1o 5588   ` cfv 5589    Isom wiso 5590  (class class class)co 6308    ^m cmap 7490   Fincfn 7587   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560    x. cmul 9562   +oocpnf 9690   -oocmnf 9691   RR*cxr 9692    < clt 9693    <_ cle 9694    - cmin 9880   -ucneg 9881    / cdiv 10291   NNcn 10631   2c2 10681   NN0cn0 10893   ZZcz 10961   ZZ>=cuz 11182   (,)cioo 11660   (,]cioc 11661   [,)cico 11662   [,]cicc 11663   ...cfz 11810  ..^cfzo 11942   |_cfl 12059   #chash 12553   picpi 14196   -cn->ccncf 21986   lim CC climc 22896    _D cdv 22897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ioc 11665  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526  df-hash 12554  df-shft 13207  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-limsup 13603  df-clim 13629  df-rlim 13630  df-sum 13830  df-ef 14198  df-sin 14200  df-cos 14201  df-pi 14203  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-cmp 20479  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-limc 22900  df-dv 22901
This theorem is referenced by:  fourierdlem106  38188
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