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

Theorem fouriercn 38208
Description: If the derivative of  F is continuous, then the Fourier series for  F converges to  F everywhere and the hypothesis are simpler than those for the more general case of a piecewise smooth function ( see fourierd 38198 for a comparison). (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
fouriercn.f  |-  ( ph  ->  F : RR --> RR )
fouriercn.t  |-  T  =  ( 2  x.  pi )
fouriercn.per  |-  ( (
ph  /\  x  e.  RR )  ->  ( F `
 ( x  +  T ) )  =  ( F `  x
) )
fouriercn.dv  |-  ( ph  ->  ( RR  _D  F
)  e.  ( RR
-cn-> CC ) )
fouriercn.g  |-  G  =  ( ( RR  _D  F )  |`  ( -u pi (,) pi ) )
fouriercn.x  |-  ( ph  ->  X  e.  RR )
fouriercn.a  |-  A  =  ( n  e.  NN0  |->  ( S. ( -u pi (,) pi ) ( ( F `  x )  x.  ( cos `  (
n  x.  x ) ) )  _d x  /  pi ) )
fouriercn.b  |-  B  =  ( n  e.  NN  |->  ( S. ( -u pi (,) pi ) ( ( F `  x )  x.  ( sin `  (
n  x.  x ) ) )  _d x  /  pi ) )
Assertion
Ref Expression
fouriercn  |-  ( ph  ->  ( ( ( A `
 0 )  / 
2 )  +  sum_ n  e.  NN  ( ( ( A `  n
)  x.  ( cos `  ( n  x.  X
) ) )  +  ( ( B `  n )  x.  ( sin `  ( n  x.  X ) ) ) ) )  =  ( F `  X ) )
Distinct variable groups:    n, F, x    x, G    x, T    n, X, x    ph, x
Allowed substitution hints:    ph( n)    A( x, n)    B( x, n)    T( n)    G( n)

Proof of Theorem fouriercn
StepHypRef Expression
1 fouriercn.f . 2  |-  ( ph  ->  F : RR --> RR )
2 fouriercn.t . 2  |-  T  =  ( 2  x.  pi )
3 fouriercn.per . 2  |-  ( (
ph  /\  x  e.  RR )  ->  ( F `
 ( x  +  T ) )  =  ( F `  x
) )
4 fouriercn.g . 2  |-  G  =  ( ( RR  _D  F )  |`  ( -u pi (,) pi ) )
54dmeqi 5041 . . . . . 6  |-  dom  G  =  dom  ( ( RR 
_D  F )  |`  ( -u pi (,) pi ) )
6 ioossre 11721 . . . . . . . 8  |-  ( -u pi (,) pi )  C_  RR
7 fouriercn.dv . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  F
)  e.  ( RR
-cn-> CC ) )
8 cncff 22003 . . . . . . . . 9  |-  ( ( RR  _D  F )  e.  ( RR -cn-> CC )  ->  ( RR  _D  F ) : RR --> CC )
9 fdm 5745 . . . . . . . . 9  |-  ( ( RR  _D  F ) : RR --> CC  ->  dom  ( RR  _D  F
)  =  RR )
107, 8, 93syl 18 . . . . . . . 8  |-  ( ph  ->  dom  ( RR  _D  F )  =  RR )
116, 10syl5sseqr 3467 . . . . . . 7  |-  ( ph  ->  ( -u pi (,) pi )  C_  dom  ( RR  _D  F ) )
12 ssdmres 5132 . . . . . . 7  |-  ( (
-u pi (,) pi )  C_  dom  ( RR 
_D  F )  <->  dom  ( ( RR  _D  F )  |`  ( -u pi (,) pi ) )  =  (
-u pi (,) pi ) )
1311, 12sylib 201 . . . . . 6  |-  ( ph  ->  dom  ( ( RR 
_D  F )  |`  ( -u pi (,) pi ) )  =  (
-u pi (,) pi ) )
145, 13syl5eq 2517 . . . . 5  |-  ( ph  ->  dom  G  =  (
-u pi (,) pi ) )
1514difeq2d 3540 . . . 4  |-  ( ph  ->  ( ( -u pi (,) pi )  \  dom  G )  =  ( (
-u pi (,) pi )  \  ( -u pi (,) pi ) ) )
16 difid 3747 . . . 4  |-  ( (
-u pi (,) pi )  \  ( -u pi (,) pi ) )  =  (/)
1715, 16syl6eq 2521 . . 3  |-  ( ph  ->  ( ( -u pi (,) pi )  \  dom  G )  =  (/) )
18 0fin 7817 . . 3  |-  (/)  e.  Fin
1917, 18syl6eqel 2557 . 2  |-  ( ph  ->  ( ( -u pi (,) pi )  \  dom  G )  e.  Fin )
20 rescncf 22007 . . . 4  |-  ( (
-u pi (,) pi )  C_  RR  ->  (
( RR  _D  F
)  e.  ( RR
-cn-> CC )  ->  (
( RR  _D  F
)  |`  ( -u pi (,) pi ) )  e.  ( ( -u pi (,) pi ) -cn-> CC ) ) )
216, 7, 20mpsyl 64 . . 3  |-  ( ph  ->  ( ( RR  _D  F )  |`  ( -u pi (,) pi ) )  e.  ( (
-u pi (,) pi ) -cn-> CC ) )
224a1i 11 . . 3  |-  ( ph  ->  G  =  ( ( RR  _D  F )  |`  ( -u pi (,) pi ) ) )
2314oveq1d 6323 . . 3  |-  ( ph  ->  ( dom  G -cn-> CC )  =  ( (
-u pi (,) pi ) -cn-> CC ) )
2421, 22, 233eltr4d 2564 . 2  |-  ( ph  ->  G  e.  ( dom 
G -cn-> CC ) )
25 pire 23492 . . . . . 6  |-  pi  e.  RR
2625renegcli 9955 . . . . 5  |-  -u pi  e.  RR
2725rexri 9711 . . . . 5  |-  pi  e.  RR*
28 icossre 11740 . . . . 5  |-  ( (
-u pi  e.  RR  /\  pi  e.  RR* )  ->  ( -u pi [,) pi )  C_  RR )
2926, 27, 28mp2an 686 . . . 4  |-  ( -u pi [,) pi )  C_  RR
30 eldifi 3544 . . . 4  |-  ( x  e.  ( ( -u pi [,) pi )  \  dom  G )  ->  x  e.  ( -u pi [,) pi ) )
3129, 30sseldi 3416 . . 3  |-  ( x  e.  ( ( -u pi [,) pi )  \  dom  G )  ->  x  e.  RR )
32 limcresi 22919 . . . . . 6  |-  ( ( RR  _D  F ) lim
CC  x )  C_  ( ( ( RR 
_D  F )  |`  ( ( -u pi (,) pi )  i^i  (
x (,) +oo )
) ) lim CC  x
)
334reseq1i 5107 . . . . . . . 8  |-  ( G  |`  ( x (,) +oo ) )  =  ( ( ( RR  _D  F )  |`  ( -u pi (,) pi ) )  |`  ( x (,) +oo ) )
34 resres 5123 . . . . . . . 8  |-  ( ( ( RR  _D  F
)  |`  ( -u pi (,) pi ) )  |`  ( x (,) +oo ) )  =  ( ( RR  _D  F
)  |`  ( ( -u pi (,) pi )  i^i  ( x (,) +oo ) ) )
3533, 34eqtr2i 2494 . . . . . . 7  |-  ( ( RR  _D  F )  |`  ( ( -u pi (,) pi )  i^i  (
x (,) +oo )
) )  =  ( G  |`  ( x (,) +oo ) )
3635oveq1i 6318 . . . . . 6  |-  ( ( ( RR  _D  F
)  |`  ( ( -u pi (,) pi )  i^i  ( x (,) +oo ) ) ) lim CC  x )  =  ( ( G  |`  (
x (,) +oo )
) lim CC  x )
3732, 36sseqtri 3450 . . . . 5  |-  ( ( RR  _D  F ) lim
CC  x )  C_  ( ( G  |`  ( x (,) +oo ) ) lim CC  x )
387adantr 472 . . . . . 6  |-  ( (
ph  /\  x  e.  RR )  ->  ( RR 
_D  F )  e.  ( RR -cn-> CC ) )
39 simpr 468 . . . . . 6  |-  ( (
ph  /\  x  e.  RR )  ->  x  e.  RR )
4038, 39cnlimci 22923 . . . . 5  |-  ( (
ph  /\  x  e.  RR )  ->  ( ( RR  _D  F ) `
 x )  e.  ( ( RR  _D  F ) lim CC  x ) )
4137, 40sseldi 3416 . . . 4  |-  ( (
ph  /\  x  e.  RR )  ->  ( ( RR  _D  F ) `
 x )  e.  ( ( G  |`  ( x (,) +oo ) ) lim CC  x ) )
42 ne0i 3728 . . . 4  |-  ( ( ( RR  _D  F
) `  x )  e.  ( ( G  |`  ( x (,) +oo ) ) lim CC  x )  ->  ( ( G  |`  ( x (,) +oo ) ) lim CC  x )  =/=  (/) )
4341, 42syl 17 . . 3  |-  ( (
ph  /\  x  e.  RR )  ->  ( ( G  |`  ( x (,) +oo ) ) lim CC  x )  =/=  (/) )
4431, 43sylan2 482 . 2  |-  ( (
ph  /\  x  e.  ( ( -u pi [,) pi )  \  dom  G ) )  ->  (
( G  |`  (
x (,) +oo )
) lim CC  x )  =/=  (/) )
45 negpitopissre 23568 . . . 4  |-  ( -u pi (,] pi )  C_  RR
46 eldifi 3544 . . . 4  |-  ( x  e.  ( ( -u pi (,] pi )  \  dom  G )  ->  x  e.  ( -u pi (,] pi ) )
4745, 46sseldi 3416 . . 3  |-  ( x  e.  ( ( -u pi (,] pi )  \  dom  G )  ->  x  e.  RR )
48 limcresi 22919 . . . . . 6  |-  ( ( RR  _D  F ) lim
CC  x )  C_  ( ( ( RR 
_D  F )  |`  ( ( -u pi (,) pi )  i^i  ( -oo (,) x ) ) ) lim CC  x )
494reseq1i 5107 . . . . . . . 8  |-  ( G  |`  ( -oo (,) x
) )  =  ( ( ( RR  _D  F )  |`  ( -u pi (,) pi ) )  |`  ( -oo (,) x ) )
50 resres 5123 . . . . . . . 8  |-  ( ( ( RR  _D  F
)  |`  ( -u pi (,) pi ) )  |`  ( -oo (,) x ) )  =  ( ( RR  _D  F )  |`  ( ( -u pi (,) pi )  i^i  ( -oo (,) x ) ) )
5149, 50eqtr2i 2494 . . . . . . 7  |-  ( ( RR  _D  F )  |`  ( ( -u pi (,) pi )  i^i  ( -oo (,) x ) ) )  =  ( G  |`  ( -oo (,) x
) )
5251oveq1i 6318 . . . . . 6  |-  ( ( ( RR  _D  F
)  |`  ( ( -u pi (,) pi )  i^i  ( -oo (,) x
) ) ) lim CC  x )  =  ( ( G  |`  ( -oo (,) x ) ) lim
CC  x )
5348, 52sseqtri 3450 . . . . 5  |-  ( ( RR  _D  F ) lim
CC  x )  C_  ( ( G  |`  ( -oo (,) x ) ) lim CC  x )
5453, 40sseldi 3416 . . . 4  |-  ( (
ph  /\  x  e.  RR )  ->  ( ( RR  _D  F ) `
 x )  e.  ( ( G  |`  ( -oo (,) x ) ) lim CC  x ) )
55 ne0i 3728 . . . 4  |-  ( ( ( RR  _D  F
) `  x )  e.  ( ( G  |`  ( -oo (,) x ) ) lim CC  x )  ->  ( ( G  |`  ( -oo (,) x
) ) lim CC  x
)  =/=  (/) )
5654, 55syl 17 . . 3  |-  ( (
ph  /\  x  e.  RR )  ->  ( ( G  |`  ( -oo (,) x ) ) lim CC  x )  =/=  (/) )
5747, 56sylan2 482 . 2  |-  ( (
ph  /\  x  e.  ( ( -u pi (,] pi )  \  dom  G ) )  ->  (
( G  |`  ( -oo (,) x ) ) lim
CC  x )  =/=  (/) )
58 eqid 2471 . 2  |-  ( topGen ` 
ran  (,) )  =  (
topGen `  ran  (,) )
59 ax-resscn 9614 . . . . . . 7  |-  RR  C_  CC
6059a1i 11 . . . . . 6  |-  ( ph  ->  RR  C_  CC )
611, 60fssd 5750 . . . . . . 7  |-  ( ph  ->  F : RR --> CC )
62 ssid 3437 . . . . . . . 8  |-  RR  C_  RR
6362a1i 11 . . . . . . 7  |-  ( ph  ->  RR  C_  RR )
64 dvcn 22954 . . . . . . 7  |-  ( ( ( RR  C_  CC  /\  F : RR --> CC  /\  RR  C_  RR )  /\  dom  ( RR  _D  F
)  =  RR )  ->  F  e.  ( RR -cn-> CC ) )
6560, 61, 63, 10, 64syl31anc 1295 . . . . . 6  |-  ( ph  ->  F  e.  ( RR
-cn-> CC ) )
66 cncffvrn 22008 . . . . . 6  |-  ( ( RR  C_  CC  /\  F  e.  ( RR -cn-> CC ) )  ->  ( F  e.  ( RR -cn-> RR )  <-> 
F : RR --> RR ) )
6760, 65, 66syl2anc 673 . . . . 5  |-  ( ph  ->  ( F  e.  ( RR -cn-> RR )  <->  F : RR
--> RR ) )
681, 67mpbird 240 . . . 4  |-  ( ph  ->  F  e.  ( RR
-cn-> RR ) )
69 eqid 2471 . . . . . 6  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
7069tgioo2 21899 . . . . . 6  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
7169, 70, 70cncfcn 22019 . . . . 5  |-  ( ( RR  C_  CC  /\  RR  C_  CC )  ->  ( RR -cn-> RR )  =  ( ( topGen `  ran  (,) )  Cn  ( topGen `  ran  (,) )
) )
7260, 60, 71syl2anc 673 . . . 4  |-  ( ph  ->  ( RR -cn-> RR )  =  ( ( topGen ` 
ran  (,) )  Cn  ( topGen `
 ran  (,) )
) )
7368, 72eleqtrd 2551 . . 3  |-  ( ph  ->  F  e.  ( (
topGen `  ran  (,) )  Cn  ( topGen `  ran  (,) )
) )
74 fouriercn.x . . 3  |-  ( ph  ->  X  e.  RR )
75 uniretop 21861 . . . 4  |-  RR  =  U. ( topGen `  ran  (,) )
7675cncnpi 20371 . . 3  |-  ( ( F  e.  ( (
topGen `  ran  (,) )  Cn  ( topGen `  ran  (,) )
)  /\  X  e.  RR )  ->  F  e.  ( ( ( topGen ` 
ran  (,) )  CnP  ( topGen `
 ran  (,) )
) `  X )
)
7773, 74, 76syl2anc 673 . 2  |-  ( ph  ->  F  e.  ( ( ( topGen `  ran  (,) )  CnP  ( topGen `  ran  (,) )
) `  X )
)
78 fouriercn.a . 2  |-  A  =  ( n  e.  NN0  |->  ( S. ( -u pi (,) pi ) ( ( F `  x )  x.  ( cos `  (
n  x.  x ) ) )  _d x  /  pi ) )
79 fouriercn.b . 2  |-  B  =  ( n  e.  NN  |->  ( S. ( -u pi (,) pi ) ( ( F `  x )  x.  ( sin `  (
n  x.  x ) ) )  _d x  /  pi ) )
801, 2, 3, 4, 19, 24, 44, 57, 58, 77, 78, 79fouriercnp 38202 1  |-  ( ph  ->  ( ( ( A `
 0 )  / 
2 )  +  sum_ n  e.  NN  ( ( ( A `  n
)  x.  ( cos `  ( n  x.  X
) ) )  +  ( ( B `  n )  x.  ( sin `  ( n  x.  X ) ) ) ) )  =  ( F `  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641    \ cdif 3387    i^i cin 3389    C_ wss 3390   (/)c0 3722    |-> cmpt 4454   dom cdm 4839   ran crn 4840    |` cres 4841   -->wf 5585   ` cfv 5589  (class class class)co 6308   Fincfn 7587   CCcc 9555   RRcr 9556   0cc0 9557    + caddc 9560    x. cmul 9562   +oocpnf 9690   -oocmnf 9691   RR*cxr 9692   -ucneg 9881    / cdiv 10291   NNcn 10631   2c2 10681   NN0cn0 10893   (,)cioo 11660   (,]cioc 11661   [,)cico 11662   sum_csu 13829   sincsin 14193   cosccos 14194   picpi 14196   TopOpenctopn 15398   topGenctg 15414  ℂfldccnfld 19047    Cn ccn 20317    CnP ccnp 20318   -cn->ccncf 21986   S.citg 22655   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-cc 8883  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-disj 4367  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-ofr 6551  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-omul 7205  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-acn 8394  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-mod 12130  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-t1 20407  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-ovol 22494  df-vol 22496  df-mbf 22656  df-itg1 22657  df-itg2 22658  df-ibl 22659  df-itg 22660  df-0p 22707  df-ditg 22881  df-limc 22900  df-dv 22901
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator