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Theorem fourierdlem102 37359
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 10937 . . . . . 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 7830 . . . . . . . . . 10  |-  { -u pi ,  pi , 
( E `  X
) }  e.  Fin
1110a1i 11 . . . . . . . . 9  |-  ( ph  ->  { -u pi ,  pi ,  ( E `  X ) }  e.  Fin )
12 pire 23143 . . . . . . . . . . . . . . 15  |-  pi  e.  RR
1312renegcli 9916 . . . . . . . . . . . . . 14  |-  -u pi  e.  RR
1413rexri 9676 . . . . . . . . . . . . 13  |-  -u pi  e.  RR*
1512rexri 9676 . . . . . . . . . . . . 13  |-  pi  e.  RR*
16 negpilt0 36836 . . . . . . . . . . . . . . 15  |-  -u pi  <  0
17 pipos 23145 . . . . . . . . . . . . . . 15  |-  0  <  pi
18 0re 9626 . . . . . . . . . . . . . . . 16  |-  0  e.  RR
1913, 18, 12lttri 9742 . . . . . . . . . . . . . . 15  |-  ( (
-u pi  <  0  /\  0  <  pi )  ->  -u pi  <  pi )
2016, 17, 19mp2an 670 . . . . . . . . . . . . . 14  |-  -u pi  <  pi
2113, 12, 20ltleii 9739 . . . . . . . . . . . . 13  |-  -u pi  <_  pi
22 prunioo 11703 . . . . . . . . . . . . 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 1326 . . . . . . . . . . . 12  |-  ( (
-u pi (,) pi )  u.  { -u pi ,  pi } )  =  ( -u pi [,] pi )
2423difeq1i 3557 . . . . . . . . . . 11  |-  ( ( ( -u pi (,) pi )  u.  { -u pi ,  pi }
)  \  dom  G )  =  ( ( -u pi [,] pi )  \  dom  G )
25 difundir 3703 . . . . . . . . . . 11  |-  ( ( ( -u pi (,) pi )  u.  { -u pi ,  pi }
)  \  dom  G )  =  ( ( (
-u pi (,) pi )  \  dom  G )  u.  ( { -u pi ,  pi }  \  dom  G ) )
2624, 25eqtr3i 2433 . . . . . . . . . 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 7829 . . . . . . . . . . . 12  |-  { -u pi ,  pi }  e.  Fin
29 diffi 7786 . . . . . . . . . . . 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 7821 . . . . . . . . . . 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 659 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( -u pi (,) pi )  \  dom  G )  u.  ( { -u pi ,  pi }  \  dom  G ) )  e.  Fin )
3326, 32syl5eqel 2494 . . . . . . . . 9  |-  ( ph  ->  ( ( -u pi [,] pi )  \  dom  G )  e.  Fin )
34 unfi 7821 . . . . . . . . 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 659 . . . . . . . 8  |-  ( ph  ->  ( { -u pi ,  pi ,  ( E `
 X ) }  u.  ( ( -u pi [,] pi )  \  dom  G ) )  e. 
Fin )
369, 35syl5eqel 2494 . . . . . . 7  |-  ( ph  ->  H  e.  Fin )
37 hashcl 12475 . . . . . . 7  |-  ( H  e.  Fin  ->  ( # `
 H )  e. 
NN0 )
3836, 37syl 17 . . . . . 6  |-  ( ph  ->  ( # `  H
)  e.  NN0 )
3938nn0zd 11006 . . . . 5  |-  ( ph  ->  ( # `  H
)  e.  ZZ )
4013, 20ltneii 9729 . . . . . . 7  |-  -u pi  =/=  pi
41 hashprg 12509 . . . . . . . 8  |-  ( (
-u pi  e.  RR  /\  pi  e.  RR )  ->  ( -u pi  =/=  pi  <->  ( # `  { -u pi ,  pi }
)  =  2 ) )
4213, 12, 41mp2an 670 . . . . . . 7  |-  ( -u pi  =/=  pi  <->  ( # `  { -u pi ,  pi }
)  =  2 )
4340, 42mpbi 208 . . . . . 6  |-  ( # `  { -u pi ,  pi } )  =  2
4410elexi 3069 . . . . . . . . . 10  |-  { -u pi ,  pi , 
( E `  X
) }  e.  _V
45 ovex 6306 . . . . . . . . . . 11  |-  ( -u pi [,] pi )  e. 
_V
46 difexg 4542 . . . . . . . . . . 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 6580 . . . . . . . . 9  |-  ( {
-u pi ,  pi ,  ( E `  X ) }  u.  ( ( -u pi [,] pi )  \  dom  G ) )  e.  _V
499, 48eqeltri 2486 . . . . . . . 8  |-  H  e. 
_V
50 negex 9854 . . . . . . . . . . 11  |-  -u pi  e.  _V
5150tpid1 4085 . . . . . . . . . 10  |-  -u pi  e.  { -u pi ,  pi ,  ( E `  X ) }
5212elexi 3069 . . . . . . . . . . 11  |-  pi  e.  _V
5352tpid2 4086 . . . . . . . . . 10  |-  pi  e.  {
-u pi ,  pi ,  ( E `  X ) }
54 prssi 4128 . . . . . . . . . 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 670 . . . . . . . . 9  |-  { -u pi ,  pi }  C_ 
{ -u pi ,  pi ,  ( E `  X ) }
56 ssun1 3606 . . . . . . . . . 10  |-  { -u pi ,  pi , 
( E `  X
) }  C_  ( { -u pi ,  pi ,  ( E `  X ) }  u.  ( ( -u pi [,] pi )  \  dom  G ) )
5756, 9sseqtr4i 3475 . . . . . . . . 9  |-  { -u pi ,  pi , 
( E `  X
) }  C_  H
5855, 57sstri 3451 . . . . . . . 8  |-  { -u pi ,  pi }  C_  H
59 hashss 12523 . . . . . . . 8  |-  ( ( H  e.  _V  /\  {
-u pi ,  pi }  C_  H )  -> 
( # `  { -u pi ,  pi }
)  <_  ( # `  H
) )
6049, 58, 59mp2an 670 . . . . . . 7  |-  ( # `  { -u pi ,  pi } )  <_  ( # `
 H )
6160a1i 11 . . . . . 6  |-  ( ph  ->  ( # `  { -u pi ,  pi }
)  <_  ( # `  H
) )
6243, 61syl5eqbrr 4429 . . . . 5  |-  ( ph  ->  2  <_  ( # `  H
) )
63 eluz2 11133 . . . . 5  |-  ( (
# `  H )  e.  ( ZZ>= `  2 )  <->  ( 2  e.  ZZ  /\  ( # `  H )  e.  ZZ  /\  2  <_  ( # `  H
) ) )
648, 39, 62, 63syl3anbrc 1181 . . . 4  |-  ( ph  ->  ( # `  H
)  e.  ( ZZ>= ` 
2 ) )
65 uz2m1nn 11201 . . . 4  |-  ( (
# `  H )  e.  ( ZZ>= `  2 )  ->  ( ( # `  H
)  -  1 )  e.  NN )
6664, 65syl 17 . . 3  |-  ( ph  ->  ( ( # `  H
)  -  1 )  e.  NN )
676, 66syl5eqel 2494 . 2  |-  ( ph  ->  M  e.  NN )
6813a1i 11 . . . . . . . . . . 11  |-  ( ph  -> 
-u pi  e.  RR )
6912a1i 11 . . . . . . . . . . 11  |-  ( ph  ->  pi  e.  RR )
70 negpitopissre 23219 . . . . . . . . . . . 12  |-  ( -u pi (,] pi )  C_  RR
7120a1i 11 . . . . . . . . . . . . . 14  |-  ( ph  -> 
-u pi  <  pi )
72 picn 23144 . . . . . . . . . . . . . . . 16  |-  pi  e.  CC
73722timesi 10697 . . . . . . . . . . . . . . 15  |-  ( 2  x.  pi )  =  ( pi  +  pi )
7472, 72subnegi 9934 . . . . . . . . . . . . . . 15  |-  ( pi 
-  -u pi )  =  ( pi  +  pi )
7573, 2, 743eqtr4i 2441 . . . . . . . . . . . . . 14  |-  T  =  ( pi  -  -u pi )
76 fourierdlem102.e . . . . . . . . . . . . . 14  |-  E  =  ( x  e.  RR  |->  ( x  +  (
( |_ `  (
( pi  -  x
)  /  T ) )  x.  T ) ) )
7768, 69, 71, 75, 76fourierdlem4 37261 . . . . . . . . . . . . 13  |-  ( ph  ->  E : RR --> ( -u pi (,] pi ) )
7877, 4ffvelrnd 6010 . . . . . . . . . . . 12  |-  ( ph  ->  ( E `  X
)  e.  ( -u pi (,] pi ) )
7970, 78sseldi 3440 . . . . . . . . . . 11  |-  ( ph  ->  ( E `  X
)  e.  RR )
8068, 69, 793jca 1177 . . . . . . . . . 10  |-  ( ph  ->  ( -u pi  e.  RR  /\  pi  e.  RR  /\  ( E `  X
)  e.  RR ) )
81 fvex 5859 . . . . . . . . . . 11  |-  ( E `
 X )  e. 
_V
8250, 52, 81tpss 4137 . . . . . . . . . 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 11660 . . . . . . . . . . 11  |-  ( (
-u pi  e.  RR  /\  pi  e.  RR )  ->  ( -u pi [,] pi )  C_  RR )
8513, 12, 84mp2an 670 . . . . . . . . . 10  |-  ( -u pi [,] pi )  C_  RR
86 ssdifss 3574 . . . . . . . . . 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 3619 . . . . . . . 8  |-  ( ph  ->  ( { -u pi ,  pi ,  ( E `
 X ) }  u.  ( ( -u pi [,] pi )  \  dom  G ) )  C_  RR )
899, 88syl5eqss 3486 . . . . . . 7  |-  ( ph  ->  H  C_  RR )
90 fourierdlem102.q . . . . . . 7  |-  Q  =  ( iota g g 
Isom  <  ,  <  (
( 0 ... M
) ,  H ) )
9136, 89, 90, 6fourierdlem36 37293 . . . . . 6  |-  ( ph  ->  Q  Isom  <  ,  <  ( ( 0 ... M
) ,  H ) )
92 isof1o 6204 . . . . . 6  |-  ( Q 
Isom  <  ,  <  (
( 0 ... M
) ,  H )  ->  Q : ( 0 ... M ) -1-1-onto-> H )
93 f1of 5799 . . . . . 6  |-  ( Q : ( 0 ... M ) -1-1-onto-> H  ->  Q :
( 0 ... M
) --> H )
9491, 92, 933syl 18 . . . . 5  |-  ( ph  ->  Q : ( 0 ... M ) --> H )
9594, 89fssd 5723 . . . 4  |-  ( ph  ->  Q : ( 0 ... M ) --> RR )
96 reex 9613 . . . . 5  |-  RR  e.  _V
97 ovex 6306 . . . . 5  |-  ( 0 ... M )  e. 
_V
9896, 97elmap 7485 . . . 4  |-  ( Q  e.  ( RR  ^m  ( 0 ... M
) )  <->  Q :
( 0 ... M
) --> RR )
9995, 98sylibr 212 . . 3  |-  ( ph  ->  Q  e.  ( RR 
^m  ( 0 ... M ) ) )
100 fveq2 5849 . . . . . . . . . . 11  |-  ( 0  =  i  ->  ( Q `  0 )  =  ( Q `  i ) )
101100adantl 464 . . . . . . . . . 10  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  =  i )  -> 
( Q `  0
)  =  ( Q `
 i ) )
10295ffvelrnda 6009 . . . . . . . . . . . 12  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  ( Q `  i )  e.  RR )
103102leidd 10159 . . . . . . . . . . 11  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  ( Q `  i )  <_  ( Q `  i
) )
104103adantr 463 . . . . . . . . . 10  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  =  i )  -> 
( Q `  i
)  <_  ( Q `  i ) )
105101, 104eqbrtrd 4415 . . . . . . . . 9  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  =  i )  -> 
( Q `  0
)  <_  ( Q `  i ) )
106 elfzelz 11742 . . . . . . . . . . . . 13  |-  ( i  e.  ( 0 ... M )  ->  i  e.  ZZ )
107106zred 11008 . . . . . . . . . . . 12  |-  ( i  e.  ( 0 ... M )  ->  i  e.  RR )
108107ad2antlr 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  0  =  i )  ->  i  e.  RR )
109 elfzle1 11743 . . . . . . . . . . . 12  |-  ( i  e.  ( 0 ... M )  ->  0  <_  i )
110109ad2antlr 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  0  =  i )  ->  0  <_  i )
111 neqne 36810 . . . . . . . . . . . . 13  |-  ( -.  0  =  i  -> 
0  =/=  i )
112111necomd 2674 . . . . . . . . . . . 12  |-  ( -.  0  =  i  -> 
i  =/=  0 )
113112adantl 464 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  0  =  i )  ->  i  =/=  0 )
114108, 110, 113ne0gt0d 9754 . . . . . . . . . 10  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  0  =  i )  ->  0  <  i )
115 nnssnn0 10839 . . . . . . . . . . . . . . . . 17  |-  NN  C_  NN0
116 nn0uz 11161 . . . . . . . . . . . . . . . . 17  |-  NN0  =  ( ZZ>= `  0 )
117115, 116sseqtri 3474 . . . . . . . . . . . . . . . 16  |-  NN  C_  ( ZZ>= `  0 )
118117, 67sseldi 3440 . . . . . . . . . . . . . . 15  |-  ( ph  ->  M  e.  ( ZZ>= ` 
0 ) )
119 eluzfz1 11747 . . . . . . . . . . . . . . 15  |-  ( M  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... M
) )
120118, 119syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  0  e.  ( 0 ... M ) )
12194, 120ffvelrnd 6010 . . . . . . . . . . . . 13  |-  ( ph  ->  ( Q `  0
)  e.  H )
12289, 121sseldd 3443 . . . . . . . . . . . 12  |-  ( ph  ->  ( Q `  0
)  e.  RR )
123122ad2antrr 724 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  <  i )  ->  ( Q `  0 )  e.  RR )
124102adantr 463 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  <  i )  ->  ( Q `  i )  e.  RR )
125 simpr 459 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  <  i )  ->  0  <  i )
12691ad2antrr 724 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  <  i )  ->  Q  Isom  <  ,  <  (
( 0 ... M
) ,  H ) )
127120anim1i 566 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  (
0  e.  ( 0 ... M )  /\  i  e.  ( 0 ... M ) ) )
128127adantr 463 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  <  i )  ->  (
0  e.  ( 0 ... M )  /\  i  e.  ( 0 ... M ) ) )
129 isorel 6205 . . . . . . . . . . . . 13  |-  ( ( Q  Isom  <  ,  <  ( ( 0 ... M
) ,  H )  /\  ( 0  e.  ( 0 ... M
)  /\  i  e.  ( 0 ... M
) ) )  -> 
( 0  <  i  <->  ( Q `  0 )  <  ( Q `  i ) ) )
130126, 128, 129syl2anc 659 . . . . . . . . . . . 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 9765 . . . . . . . . . 10  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  0  <  i )  ->  ( Q `  0 )  <_  ( Q `  i
) )
133114, 132syldan 468 . . . . . . . . 9  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  0  =  i )  ->  ( Q `  0
)  <_  ( Q `  i ) )
134105, 133pm2.61dan 792 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  ( Q `  0 )  <_  ( Q `  i
) )
135134adantr 463 . . . . . . 7  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  ( Q `  i )  =  -u pi )  -> 
( Q `  0
)  <_  ( Q `  i ) )
136 simpr 459 . . . . . . 7  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  ( Q `  i )  =  -u pi )  -> 
( Q `  i
)  =  -u pi )
137135, 136breqtrd 4419 . . . . . 6  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  ( Q `  i )  =  -u pi )  -> 
( Q `  0
)  <_  -u pi )
13868rexrd 9673 . . . . . . . 8  |-  ( ph  -> 
-u pi  e.  RR* )
13969rexrd 9673 . . . . . . . 8  |-  ( ph  ->  pi  e.  RR* )
140 lbicc2 11690 . . . . . . . . . . . . . 14  |-  ( (
-u pi  e.  RR*  /\  pi  e.  RR*  /\  -u pi  <_  pi )  ->  -u pi  e.  ( -u pi [,] pi ) )
14114, 15, 21, 140mp3an 1326 . . . . . . . . . . . . 13  |-  -u pi  e.  ( -u pi [,] pi )
142141a1i 11 . . . . . . . . . . . 12  |-  ( ph  -> 
-u pi  e.  (
-u pi [,] pi ) )
143 ubicc2 11691 . . . . . . . . . . . . . 14  |-  ( (
-u pi  e.  RR*  /\  pi  e.  RR*  /\  -u pi  <_  pi )  ->  pi  e.  ( -u pi [,] pi ) )
14414, 15, 21, 143mp3an 1326 . . . . . . . . . . . . 13  |-  pi  e.  ( -u pi [,] pi )
145144a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  pi  e.  ( -u pi [,] pi ) )
146 iocssicc 11666 . . . . . . . . . . . . 13  |-  ( -u pi (,] pi )  C_  ( -u pi [,] pi )
147146, 78sseldi 3440 . . . . . . . . . . . 12  |-  ( ph  ->  ( E `  X
)  e.  ( -u pi [,] pi ) )
148 tpssi 4138 . . . . . . . . . . . 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 1230 . . . . . . . . . . 11  |-  ( ph  ->  { -u pi ,  pi ,  ( E `  X ) }  C_  ( -u pi [,] pi ) )
150 difssd 3571 . . . . . . . . . . 11  |-  ( ph  ->  ( ( -u pi [,] pi )  \  dom  G )  C_  ( -u pi [,] pi ) )
151149, 150unssd 3619 . . . . . . . . . 10  |-  ( ph  ->  ( { -u pi ,  pi ,  ( E `
 X ) }  u.  ( ( -u pi [,] pi )  \  dom  G ) )  C_  ( -u pi [,] pi ) )
1529, 151syl5eqss 3486 . . . . . . . . 9  |-  ( ph  ->  H  C_  ( -u pi [,] pi ) )
153152, 121sseldd 3443 . . . . . . . 8  |-  ( ph  ->  ( Q `  0
)  e.  ( -u pi [,] pi ) )
154 iccgelb 11635 . . . . . . . 8  |-  ( (
-u pi  e.  RR*  /\  pi  e.  RR*  /\  ( Q `  0 )  e.  ( -u pi [,] pi ) )  ->  -u pi  <_  ( Q `  0
) )
155138, 139, 153, 154syl3anc 1230 . . . . . . 7  |-  ( ph  -> 
-u pi  <_  ( Q `  0 )
)
156155ad2antrr 724 . . . . . 6  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  ( Q `  i )  =  -u pi )  ->  -u pi  <_  ( Q `  0 ) )
157122ad2antrr 724 . . . . . . 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 9759 . . . . . 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 923 . . . . 5  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  ( Q `  i )  =  -u pi )  -> 
( Q `  0
)  =  -u pi )
16157, 51sselii 3439 . . . . . . 7  |-  -u pi  e.  H
162 f1ofo 5806 . . . . . . . . 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 5781 . . . . . . . 8  |-  ( Q : ( 0 ... M ) -onto-> H  ->  ran  Q  =  H )
16591, 163, 1643syl 18 . . . . . . 7  |-  ( ph  ->  ran  Q  =  H )
166161, 165syl5eleqr 2497 . . . . . 6  |-  ( ph  -> 
-u pi  e.  ran  Q )
167 ffn 5714 . . . . . . 7  |-  ( Q : ( 0 ... M ) --> H  ->  Q  Fn  ( 0 ... M ) )
168 fvelrnb 5896 . . . . . . 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 210 . . . . 5  |-  ( ph  ->  E. i  e.  ( 0 ... M ) ( Q `  i
)  =  -u pi )
171160, 170r19.29a 2949 . . . 4  |-  ( ph  ->  ( Q `  0
)  =  -u pi )
17257, 53sselii 3439 . . . . . . 7  |-  pi  e.  H
173172, 165syl5eleqr 2497 . . . . . 6  |-  ( ph  ->  pi  e.  ran  Q
)
174 fvelrnb 5896 . . . . . . 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 210 . . . . 5  |-  ( ph  ->  E. i  e.  ( 0 ... M ) ( Q `  i
)  =  pi )
17794, 152fssd 5723 . . . . . . . . . 10  |-  ( ph  ->  Q : ( 0 ... M ) --> (
-u pi [,] pi ) )
178 eluzfz2 11748 . . . . . . . . . . 11  |-  ( M  e.  ( ZZ>= `  0
)  ->  M  e.  ( 0 ... M
) )
179118, 178syl 17 . . . . . . . . . 10  |-  ( ph  ->  M  e.  ( 0 ... M ) )
180177, 179ffvelrnd 6010 . . . . . . . . 9  |-  ( ph  ->  ( Q `  M
)  e.  ( -u pi [,] pi ) )
181 iccleub 11634 . . . . . . . . 9  |-  ( (
-u pi  e.  RR*  /\  pi  e.  RR*  /\  ( Q `  M )  e.  ( -u pi [,] pi ) )  ->  ( Q `  M )  <_  pi )
182138, 139, 180, 181syl3anc 1230 . . . . . . . 8  |-  ( ph  ->  ( Q `  M
)  <_  pi )
1831823ad2ant1 1018 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 0 ... M
)  /\  ( Q `  i )  =  pi )  ->  ( Q `  M )  <_  pi )
184 id 22 . . . . . . . . . 10  |-  ( ( Q `  i )  =  pi  ->  ( Q `  i )  =  pi )
185184eqcomd 2410 . . . . . . . . 9  |-  ( ( Q `  i )  =  pi  ->  pi  =  ( Q `  i ) )
1861853ad2ant3 1020 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 0 ... M
)  /\  ( Q `  i )  =  pi )  ->  pi  =  ( Q `  i ) )
187103adantr 463 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  =  M )  ->  ( Q `  i )  <_  ( Q `  i
) )
188 fveq2 5849 . . . . . . . . . . . 12  |-  ( i  =  M  ->  ( Q `  i )  =  ( Q `  M ) )
189188adantl 464 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  =  M )  ->  ( Q `  i )  =  ( Q `  M ) )
190187, 189breqtrd 4419 . . . . . . . . . 10  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  =  M )  ->  ( Q `  i )  <_  ( Q `  M
) )
191107ad2antlr 725 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  i  =  M )  ->  i  e.  RR )
192 elfzel2 11740 . . . . . . . . . . . . . 14  |-  ( i  e.  ( 0 ... M )  ->  M  e.  ZZ )
193192zred 11008 . . . . . . . . . . . . 13  |-  ( i  e.  ( 0 ... M )  ->  M  e.  RR )
194193ad2antlr 725 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  i  =  M )  ->  M  e.  RR )
195 elfzle2 11744 . . . . . . . . . . . . 13  |-  ( i  e.  ( 0 ... M )  ->  i  <_  M )
196195ad2antlr 725 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  i  =  M )  ->  i  <_  M )
197 neqne 36810 . . . . . . . . . . . . . 14  |-  ( -.  i  =  M  -> 
i  =/=  M )
198197necomd 2674 . . . . . . . . . . . . 13  |-  ( -.  i  =  M  ->  M  =/=  i )
199198adantl 464 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  i  =  M )  ->  M  =/=  i )
200191, 194, 196, 199leneltd 36864 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  i  =  M )  ->  i  <  M )
201102adantr 463 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  <  M )  ->  ( Q `  i )  e.  RR )
20285, 180sseldi 3440 . . . . . . . . . . . . 13  |-  ( ph  ->  ( Q `  M
)  e.  RR )
203202ad2antrr 724 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  <  M )  ->  ( Q `  M )  e.  RR )
204 simpr 459 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  <  M )  ->  i  <  M )
20591ad2antrr 724 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  <  M )  ->  Q  Isom  <  ,  <  (
( 0 ... M
) ,  H ) )
206 simpr 459 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  i  e.  ( 0 ... M
) )
207179adantr 463 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  M  e.  ( 0 ... M
) )
208206, 207jca 530 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  (
i  e.  ( 0 ... M )  /\  M  e.  ( 0 ... M ) ) )
209208adantr 463 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  <  M )  ->  (
i  e.  ( 0 ... M )  /\  M  e.  ( 0 ... M ) ) )
210 isorel 6205 . . . . . . . . . . . . . 14  |-  ( ( Q  Isom  <  ,  <  ( ( 0 ... M
) ,  H )  /\  ( i  e.  ( 0 ... M
)  /\  M  e.  ( 0 ... M
) ) )  -> 
( i  <  M  <->  ( Q `  i )  <  ( Q `  M ) ) )
211205, 209, 210syl2anc 659 . . . . . . . . . . . . 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 9765 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  i  <  M )  ->  ( Q `  i )  <_  ( Q `  M
) )
214200, 213syldan 468 . . . . . . . . . 10  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  -.  i  =  M )  ->  ( Q `  i
)  <_  ( Q `  M ) )
215190, 214pm2.61dan 792 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  ( Q `  i )  <_  ( Q `  M
) )
2162153adant3 1017 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 0 ... M
)  /\  ( Q `  i )  =  pi )  ->  ( Q `  i )  <_  ( Q `  M )
)
217186, 216eqbrtrd 4415 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 0 ... M
)  /\  ( Q `  i )  =  pi )  ->  pi  <_  ( Q `  M ) )
2182023ad2ant1 1018 . . . . . . . 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 9759 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 0 ... M
)  /\  ( Q `  i )  =  pi )  ->  ( ( Q `  M )  =  pi  <->  ( ( Q `
 M )  <_  pi  /\  pi  <_  ( Q `  M )
) ) )
221183, 217, 220mpbir2and 923 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0 ... M
)  /\  ( Q `  i )  =  pi )  ->  ( Q `  M )  =  pi )
222221rexlimdv3a 2898 . . . . 5  |-  ( ph  ->  ( E. i  e.  ( 0 ... M
) ( Q `  i )  =  pi 
->  ( Q `  M
)  =  pi ) )
223176, 222mpd 15 . . . 4  |-  ( ph  ->  ( Q `  M
)  =  pi )
224 elfzoelz 11859 . . . . . . . . 9  |-  ( i  e.  ( 0..^ M )  ->  i  e.  ZZ )
225224zred 11008 . . . . . . . 8  |-  ( i  e.  ( 0..^ M )  ->  i  e.  RR )
226225ltp1d 10516 . . . . . . 7  |-  ( i  e.  ( 0..^ M )  ->  i  <  ( i  +  1 ) )
227226adantl 464 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  i  <  (
i  +  1 ) )
228 elfzofz 11874 . . . . . . . 8  |-  ( i  e.  ( 0..^ M )  ->  i  e.  ( 0 ... M
) )
229 fzofzp1 11946 . . . . . . . 8  |-  ( i  e.  ( 0..^ M )  ->  ( i  +  1 )  e.  ( 0 ... M
) )
230228, 229jca 530 . . . . . . 7  |-  ( i  e.  ( 0..^ M )  ->  ( i  e.  ( 0 ... M
)  /\  ( i  +  1 )  e.  ( 0 ... M
) ) )
231 isorel 6205 . . . . . . 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 475 . . . . . 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 2818 . . . 4  |-  ( ph  ->  A. i  e.  ( 0..^ M ) ( Q `  i )  <  ( Q `  ( i  +  1 ) ) )
235171, 223, 234jca31 532 . . 3  |-  ( ph  ->  ( ( ( Q `
 0 )  = 
-u pi  /\  ( Q `  M )  =  pi )  /\  A. i  e.  ( 0..^ M ) ( Q `
 i )  < 
( Q `  (
i  +  1 ) ) ) )
2365fourierdlem2 37259 . . . 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 923 . 2  |-  ( ph  ->  Q  e.  ( P `
 M ) )
239 fourierdlem102.g . . . . 5  |-  G  =  ( ( RR  _D  F )  |`  ( -u pi (,) pi ) )
240239reseq1i 5090 . . . 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 463 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  Q : ( 0 ... M ) --> ( -u pi [,] pi ) )
244 simpr 459 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  i  e.  ( 0..^ M ) )
245241, 242, 243, 244fourierdlem27 37284 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( Q `
 i ) (,) ( Q `  (
i  +  1 ) ) )  C_  ( -u pi (,) pi ) )
246245resabs1d 5123 . . . 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 2456 . . 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 37295 . . 3  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( G  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) )  e.  ( ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) -cn-> CC ) )
250247, 249eqeltrd 2490 . 2  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( RR 
_D  F )  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) )  e.  ( ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) -cn-> CC ) )
251247oveq1d 6293 . . 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 463 . . . . 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 713 . . . . 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 713 . . . . 5  |-  ( ( ( ph  /\  i  e.  ( 0..^ M ) )  /\  x  e.  ( ( -u pi (,] pi )  \  dom  G ) )  ->  (
( G  |`  ( -oo (,) x ) ) lim
CC  x )  =/=  (/) )
25791adantr 463 . . . . 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 463 . . . . 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 37303 . . . 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 457 . . 3  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( G  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) lim CC  ( Q `  i )
)  =/=  (/) )
263251, 262eqnetrd 2696 . 2  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( ( RR  _D  F )  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) lim CC  ( Q `  i )
)  =/=  (/) )
264247oveq1d 6293 . . 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 461 . . 3  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( G  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) lim CC  ( Q `  ( i  +  1 ) ) )  =/=  (/) )
266264, 265eqnetrd 2696 . 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 37351 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2754   E.wrex 2755   {crab 2758   _Vcvv 3059    \ cdif 3411    u. cun 3412    C_ wss 3414   (/)c0 3738   {cpr 3974   {ctp 3976   class class class wbr 4395    |-> cmpt 4453   dom cdm 4823   ran crn 4824    |` cres 4825   iotacio 5531    Fn wfn 5564   -->wf 5565   -onto->wfo 5567   -1-1-onto->wf1o 5568   ` cfv 5569    Isom wiso 5570  (class class class)co 6278    ^m cmap 7457   Fincfn 7554   CCcc 9520   RRcr 9521   0cc0 9522   1c1 9523    + caddc 9525    x. cmul 9527   +oocpnf 9655   -oocmnf 9656   RR*cxr 9657    < clt 9658    <_ cle 9659    - cmin 9841   -ucneg 9842    / cdiv 10247   NNcn 10576   2c2 10626   NN0cn0 10836   ZZcz 10905   ZZ>=cuz 11127   (,)cioo 11582   (,]cioc 11583   [,)cico 11584   [,]cicc 11585   ...cfz 11726  ..^cfzo 11854   |_cfl 11964   #chash 12452   picpi 14011   -cn->ccncf 21672   lim CC climc 22558    _D cdv 22559
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 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600  ax-addf 9601  ax-mulf 9602
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 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6521  df-om 6684  df-1st 6784  df-2nd 6785  df-supp 6903  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-2o 7168  df-oadd 7171  df-er 7348  df-map 7459  df-pm 7460  df-ixp 7508  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-fsupp 7864  df-fi 7905  df-sup 7935  df-oi 7969  df-card 8352  df-cda 8580  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-5 10638  df-6 10639  df-7 10640  df-8 10641  df-9 10642  df-10 10643  df-n0 10837  df-z 10906  df-dec 11020  df-uz 11128  df-q 11228  df-rp 11266  df-xneg 11371  df-xadd 11372  df-xmul 11373  df-ioo 11586  df-ioc 11587  df-ico 11588  df-icc 11589  df-fz 11727  df-fzo 11855  df-fl 11966  df-seq 12152  df-exp 12211  df-fac 12398  df-bc 12425  df-hash 12453  df-shft 13049  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-limsup 13443  df-clim 13460  df-rlim 13461  df-sum 13658  df-ef 14012  df-sin 14014  df-cos 14015  df-pi 14017  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-ress 14848  df-plusg 14922  df-mulr 14923  df-starv 14924  df-sca 14925  df-vsca 14926  df-ip 14927  df-tset 14928  df-ple 14929  df-ds 14931  df-unif 14932  df-hom 14933  df-cco 14934  df-rest 15037  df-topn 15038  df-0g 15056  df-gsum 15057  df-topgen 15058  df-pt 15059  df-prds 15062  df-xrs 15116  df-qtop 15121  df-imas 15122  df-xps 15124  df-mre 15200  df-mrc 15201  df-acs 15203  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-submnd 16291  df-mulg 16384  df-cntz 16679  df-cmn 17124  df-psmet 18731  df-xmet 18732  df-met 18733  df-bl 18734  df-mopn 18735  df-fbas 18736  df-fg 18737  df-cnfld 18741  df-top 19691  df-bases 19693  df-topon 19694  df-topsp 19695  df-cld 19812  df-ntr 19813  df-cls 19814  df-nei 19892  df-lp 19930  df-perf 19931  df-cn 20021  df-cnp 20022  df-haus 20109  df-cmp 20180  df-tx 20355  df-hmeo 20548  df-fil 20639  df-fm 20731  df-flim 20732  df-flf 20733  df-xms 21115  df-ms 21116  df-tms 21117  df-cncf 21674  df-limc 22562  df-dv 22563
This theorem is referenced by:  fourierdlem106  37363
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