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Theorem fouriercn 37883
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 37873 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 5048 . . . . . 6  |-  dom  G  =  dom  ( ( RR 
_D  F )  |`  ( -u pi (,) pi ) )
6 ioossre 11692 . . . . . . . 8  |-  ( -u pi (,) pi )  C_  RR
7 fouriercn.dv . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  F
)  e.  ( RR
-cn-> CC ) )
8 cncff 21902 . . . . . . . . 9  |-  ( ( RR  _D  F )  e.  ( RR -cn-> CC )  ->  ( RR  _D  F ) : RR --> CC )
9 fdm 5742 . . . . . . . . 9  |-  ( ( RR  _D  F ) : RR --> CC  ->  dom  ( RR  _D  F
)  =  RR )
107, 8, 93syl 18 . . . . . . . 8  |-  ( ph  ->  dom  ( RR  _D  F )  =  RR )
116, 10syl5sseqr 3510 . . . . . . 7  |-  ( ph  ->  ( -u pi (,) pi )  C_  dom  ( RR  _D  F ) )
12 ssdmres 5138 . . . . . . 7  |-  ( (
-u pi (,) pi )  C_  dom  ( RR 
_D  F )  <->  dom  ( ( RR  _D  F )  |`  ( -u pi (,) pi ) )  =  (
-u pi (,) pi ) )
1311, 12sylib 199 . . . . . 6  |-  ( ph  ->  dom  ( ( RR 
_D  F )  |`  ( -u pi (,) pi ) )  =  (
-u pi (,) pi ) )
145, 13syl5eq 2473 . . . . 5  |-  ( ph  ->  dom  G  =  (
-u pi (,) pi ) )
1514difeq2d 3580 . . . 4  |-  ( ph  ->  ( ( -u pi (,) pi )  \  dom  G )  =  ( (
-u pi (,) pi )  \  ( -u pi (,) pi ) ) )
16 difid 3860 . . . 4  |-  ( (
-u pi (,) pi )  \  ( -u pi (,) pi ) )  =  (/)
1715, 16syl6eq 2477 . . 3  |-  ( ph  ->  ( ( -u pi (,) pi )  \  dom  G )  =  (/) )
18 0fin 7797 . . 3  |-  (/)  e.  Fin
1917, 18syl6eqel 2516 . 2  |-  ( ph  ->  ( ( -u pi (,) pi )  \  dom  G )  e.  Fin )
20 rescncf 21906 . . . 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 65 . . 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 6312 . . 3  |-  ( ph  ->  ( dom  G -cn-> CC )  =  ( (
-u pi (,) pi ) -cn-> CC ) )
2421, 22, 233eltr4d 2523 . 2  |-  ( ph  ->  G  e.  ( dom 
G -cn-> CC ) )
25 pire 23390 . . . . . 6  |-  pi  e.  RR
2625renegcli 9931 . . . . 5  |-  -u pi  e.  RR
2725rexri 9689 . . . . 5  |-  pi  e.  RR*
28 icossre 11711 . . . . 5  |-  ( (
-u pi  e.  RR  /\  pi  e.  RR* )  ->  ( -u pi [,) pi )  C_  RR )
2926, 27, 28mp2an 676 . . . 4  |-  ( -u pi [,) pi )  C_  RR
30 eldifi 3584 . . . 4  |-  ( x  e.  ( ( -u pi [,) pi )  \  dom  G )  ->  x  e.  ( -u pi [,) pi ) )
3129, 30sseldi 3459 . . 3  |-  ( x  e.  ( ( -u pi [,) pi )  \  dom  G )  ->  x  e.  RR )
32 limcresi 22817 . . . . . 6  |-  ( ( RR  _D  F ) lim
CC  x )  C_  ( ( ( RR 
_D  F )  |`  ( ( -u pi (,) pi )  i^i  (
x (,) +oo )
) ) lim CC  x
)
334reseq1i 5113 . . . . . . . 8  |-  ( G  |`  ( x (,) +oo ) )  =  ( ( ( RR  _D  F )  |`  ( -u pi (,) pi ) )  |`  ( x (,) +oo ) )
34 resres 5129 . . . . . . . 8  |-  ( ( ( RR  _D  F
)  |`  ( -u pi (,) pi ) )  |`  ( x (,) +oo ) )  =  ( ( RR  _D  F
)  |`  ( ( -u pi (,) pi )  i^i  ( x (,) +oo ) ) )
3533, 34eqtr2i 2450 . . . . . . 7  |-  ( ( RR  _D  F )  |`  ( ( -u pi (,) pi )  i^i  (
x (,) +oo )
) )  =  ( G  |`  ( x (,) +oo ) )
3635oveq1i 6307 . . . . . 6  |-  ( ( ( RR  _D  F
)  |`  ( ( -u pi (,) pi )  i^i  ( x (,) +oo ) ) ) lim CC  x )  =  ( ( G  |`  (
x (,) +oo )
) lim CC  x )
3732, 36sseqtri 3493 . . . . 5  |-  ( ( RR  _D  F ) lim
CC  x )  C_  ( ( G  |`  ( x (,) +oo ) ) lim CC  x )
387adantr 466 . . . . . 6  |-  ( (
ph  /\  x  e.  RR )  ->  ( RR 
_D  F )  e.  ( RR -cn-> CC ) )
39 simpr 462 . . . . . 6  |-  ( (
ph  /\  x  e.  RR )  ->  x  e.  RR )
4038, 39cnlimci 22821 . . . . 5  |-  ( (
ph  /\  x  e.  RR )  ->  ( ( RR  _D  F ) `
 x )  e.  ( ( RR  _D  F ) lim CC  x ) )
4137, 40sseldi 3459 . . . 4  |-  ( (
ph  /\  x  e.  RR )  ->  ( ( RR  _D  F ) `
 x )  e.  ( ( G  |`  ( x (,) +oo ) ) lim CC  x ) )
42 ne0i 3764 . . . 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 476 . 2  |-  ( (
ph  /\  x  e.  ( ( -u pi [,) pi )  \  dom  G ) )  ->  (
( G  |`  (
x (,) +oo )
) lim CC  x )  =/=  (/) )
45 negpitopissre 23466 . . . 4  |-  ( -u pi (,] pi )  C_  RR
46 eldifi 3584 . . . 4  |-  ( x  e.  ( ( -u pi (,] pi )  \  dom  G )  ->  x  e.  ( -u pi (,] pi ) )
4745, 46sseldi 3459 . . 3  |-  ( x  e.  ( ( -u pi (,] pi )  \  dom  G )  ->  x  e.  RR )
48 limcresi 22817 . . . . . 6  |-  ( ( RR  _D  F ) lim
CC  x )  C_  ( ( ( RR 
_D  F )  |`  ( ( -u pi (,) pi )  i^i  ( -oo (,) x ) ) ) lim CC  x )
494reseq1i 5113 . . . . . . . 8  |-  ( G  |`  ( -oo (,) x
) )  =  ( ( ( RR  _D  F )  |`  ( -u pi (,) pi ) )  |`  ( -oo (,) x ) )
50 resres 5129 . . . . . . . 8  |-  ( ( ( RR  _D  F
)  |`  ( -u pi (,) pi ) )  |`  ( -oo (,) x ) )  =  ( ( RR  _D  F )  |`  ( ( -u pi (,) pi )  i^i  ( -oo (,) x ) ) )
5149, 50eqtr2i 2450 . . . . . . 7  |-  ( ( RR  _D  F )  |`  ( ( -u pi (,) pi )  i^i  ( -oo (,) x ) ) )  =  ( G  |`  ( -oo (,) x
) )
5251oveq1i 6307 . . . . . 6  |-  ( ( ( RR  _D  F
)  |`  ( ( -u pi (,) pi )  i^i  ( -oo (,) x
) ) ) lim CC  x )  =  ( ( G  |`  ( -oo (,) x ) ) lim
CC  x )
5348, 52sseqtri 3493 . . . . 5  |-  ( ( RR  _D  F ) lim
CC  x )  C_  ( ( G  |`  ( -oo (,) x ) ) lim CC  x )
5453, 40sseldi 3459 . . . 4  |-  ( (
ph  /\  x  e.  RR )  ->  ( ( RR  _D  F ) `
 x )  e.  ( ( G  |`  ( -oo (,) x ) ) lim CC  x ) )
55 ne0i 3764 . . . 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 476 . 2  |-  ( (
ph  /\  x  e.  ( ( -u pi (,] pi )  \  dom  G ) )  ->  (
( G  |`  ( -oo (,) x ) ) lim
CC  x )  =/=  (/) )
58 eqid 2420 . 2  |-  ( topGen ` 
ran  (,) )  =  (
topGen `  ran  (,) )
59 ax-resscn 9592 . . . . . . 7  |-  RR  C_  CC
6059a1i 11 . . . . . 6  |-  ( ph  ->  RR  C_  CC )
611, 60fssd 5747 . . . . . . 7  |-  ( ph  ->  F : RR --> CC )
62 ssid 3480 . . . . . . . 8  |-  RR  C_  RR
6362a1i 11 . . . . . . 7  |-  ( ph  ->  RR  C_  RR )
64 dvcn 22852 . . . . . . 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 1267 . . . . . 6  |-  ( ph  ->  F  e.  ( RR
-cn-> CC ) )
66 cncffvrn 21907 . . . . . 6  |-  ( ( RR  C_  CC  /\  F  e.  ( RR -cn-> CC ) )  ->  ( F  e.  ( RR -cn-> RR )  <-> 
F : RR --> RR ) )
6760, 65, 66syl2anc 665 . . . . 5  |-  ( ph  ->  ( F  e.  ( RR -cn-> RR )  <->  F : RR
--> RR ) )
681, 67mpbird 235 . . . 4  |-  ( ph  ->  F  e.  ( RR
-cn-> RR ) )
69 eqid 2420 . . . . . 6  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
7069tgioo2 21798 . . . . . 6  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
7169, 70, 70cncfcn 21918 . . . . 5  |-  ( ( RR  C_  CC  /\  RR  C_  CC )  ->  ( RR -cn-> RR )  =  ( ( topGen `  ran  (,) )  Cn  ( topGen `  ran  (,) )
) )
7260, 60, 71syl2anc 665 . . . 4  |-  ( ph  ->  ( RR -cn-> RR )  =  ( ( topGen ` 
ran  (,) )  Cn  ( topGen `
 ran  (,) )
) )
7368, 72eleqtrd 2510 . . 3  |-  ( ph  ->  F  e.  ( (
topGen `  ran  (,) )  Cn  ( topGen `  ran  (,) )
) )
74 fouriercn.x . . 3  |-  ( ph  ->  X  e.  RR )
75 uniretop 21760 . . . 4  |-  RR  =  U. ( topGen `  ran  (,) )
7675cncnpi 20271 . . 3  |-  ( ( F  e.  ( (
topGen `  ran  (,) )  Cn  ( topGen `  ran  (,) )
)  /\  X  e.  RR )  ->  F  e.  ( ( ( topGen ` 
ran  (,) )  CnP  ( topGen `
 ran  (,) )
) `  X )
)
7773, 74, 76syl2anc 665 . 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 37877 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 187    /\ wa 370    = wceq 1437    e. wcel 1867    =/= wne 2616    \ cdif 3430    i^i cin 3432    C_ wss 3433   (/)c0 3758    |-> cmpt 4476   dom cdm 4846   ran crn 4847    |` cres 4848   -->wf 5589   ` cfv 5593  (class class class)co 6297   Fincfn 7569   CCcc 9533   RRcr 9534   0cc0 9535    + caddc 9538    x. cmul 9540   +oocpnf 9668   -oocmnf 9669   RR*cxr 9670   -ucneg 9857    / cdiv 10265   NNcn 10605   2c2 10655   NN0cn0 10865   (,)cioo 11631   (,]cioc 11632   [,)cico 11633   sum_csu 13730   sincsin 14094   cosccos 14095   picpi 14097   TopOpenctopn 15298   topGenctg 15314  ℂfldccnfld 18948    Cn ccn 20217    CnP ccnp 20218   -cn->ccncf 21885   S.citg 22553   lim CC climc 22794    _D cdv 22795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4530  ax-sep 4540  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6589  ax-inf2 8144  ax-cc 8861  ax-cnex 9591  ax-resscn 9592  ax-1cn 9593  ax-icn 9594  ax-addcl 9595  ax-addrcl 9596  ax-mulcl 9597  ax-mulrcl 9598  ax-mulcom 9599  ax-addass 9600  ax-mulass 9601  ax-distr 9602  ax-i2m1 9603  ax-1ne0 9604  ax-1rid 9605  ax-rnegex 9606  ax-rrecex 9607  ax-cnre 9608  ax-pre-lttri 9609  ax-pre-lttrn 9610  ax-pre-ltadd 9611  ax-pre-mulgt0 9612  ax-pre-sup 9613  ax-addf 9614  ax-mulf 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-iin 4296  df-disj 4389  df-br 4418  df-opab 4477  df-mpt 4478  df-tr 4513  df-eprel 4757  df-id 4761  df-po 4767  df-so 4768  df-fr 4805  df-se 4806  df-we 4807  df-xp 4852  df-rel 4853  df-cnv 4854  df-co 4855  df-dm 4856  df-rn 4857  df-res 4858  df-ima 4859  df-pred 5391  df-ord 5437  df-on 5438  df-lim 5439  df-suc 5440  df-iota 5557  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-isom 5602  df-riota 6259  df-ov 6300  df-oprab 6301  df-mpt2 6302  df-of 6537  df-ofr 6538  df-om 6699  df-1st 6799  df-2nd 6800  df-supp 6918  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-omul 7187  df-er 7363  df-map 7474  df-pm 7475  df-ixp 7523  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7882  df-fi 7923  df-sup 7954  df-inf 7955  df-oi 8023  df-card 8370  df-acn 8373  df-cda 8594  df-pnf 9673  df-mnf 9674  df-xr 9675  df-ltxr 9676  df-le 9677  df-sub 9858  df-neg 9859  df-div 10266  df-nn 10606  df-2 10664  df-3 10665  df-4 10666  df-5 10667  df-6 10668  df-7 10669  df-8 10670  df-9 10671  df-10 10672  df-n0 10866  df-z 10934  df-dec 11048  df-uz 11156  df-q 11261  df-rp 11299  df-xneg 11405  df-xadd 11406  df-xmul 11407  df-ioo 11635  df-ioc 11636  df-ico 11637  df-icc 11638  df-fz 11779  df-fzo 11910  df-fl 12021  df-mod 12090  df-seq 12207  df-exp 12266  df-fac 12453  df-bc 12481  df-hash 12509  df-shft 13109  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-limsup 13504  df-clim 13530  df-rlim 13531  df-sum 13731  df-ef 14099  df-sin 14101  df-cos 14102  df-pi 14104  df-struct 15101  df-ndx 15102  df-slot 15103  df-base 15104  df-sets 15105  df-ress 15106  df-plusg 15181  df-mulr 15182  df-starv 15183  df-sca 15184  df-vsca 15185  df-ip 15186  df-tset 15187  df-ple 15188  df-ds 15190  df-unif 15191  df-hom 15192  df-cco 15193  df-rest 15299  df-topn 15300  df-0g 15318  df-gsum 15319  df-topgen 15320  df-pt 15321  df-prds 15324  df-xrs 15378  df-qtop 15384  df-imas 15385  df-xps 15388  df-mre 15470  df-mrc 15471  df-acs 15473  df-mgm 16466  df-sgrp 16505  df-mnd 16515  df-submnd 16561  df-mulg 16654  df-cntz 16949  df-cmn 17410  df-psmet 18940  df-xmet 18941  df-met 18942  df-bl 18943  df-mopn 18944  df-fbas 18945  df-fg 18946  df-cnfld 18949  df-top 19898  df-bases 19899  df-topon 19900  df-topsp 19901  df-cld 20011  df-ntr 20012  df-cls 20013  df-nei 20091  df-lp 20129  df-perf 20130  df-cn 20220  df-cnp 20221  df-t1 20307  df-haus 20308  df-cmp 20379  df-tx 20554  df-hmeo 20747  df-fil 20838  df-fm 20930  df-flim 20931  df-flf 20932  df-xms 21312  df-ms 21313  df-tms 21314  df-cncf 21887  df-ovol 22393  df-vol 22395  df-mbf 22554  df-itg1 22555  df-itg2 22556  df-ibl 22557  df-itg 22558  df-0p 22605  df-ditg 22779  df-limc 22798  df-dv 22799
This theorem is referenced by: (None)
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