Theorem fouriercn 40449

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 40439 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

Step	Hyp	Ref	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 ) )
5	4	dmeqi 5325	. . . . . 6 |- dom G = dom ( ( RR _D F ) |` ( -u pi (,) pi ) )
6		ioossre 12235	. . . . . . . 8 |- ( -u pi (,) pi ) C_ RR
7		fouriercn.dv	. . . . . . . . 9 |- ( ph -> ( RR _D F ) e. ( RR -cn-> CC ) )
8		cncff 22696	. . . . . . . . 9 |- ( ( RR _D F ) e. ( RR -cn-> CC ) -> ( RR _D F ) : RR --> CC )
9		fdm 6051	. . . . . . . . 9 |- ( ( RR _D F ) : RR --> CC -> dom ( RR _D F ) = RR )
10	7, 8, 9	3syl 18	. . . . . . . 8 |- ( ph -> dom ( RR _D F ) = RR )
11	6, 10	syl5sseqr 3654	. . . . . . 7 |- ( ph -> ( -u pi (,) pi ) C_ dom ( RR _D F ) )
12		ssdmres 5420	. . . . . . 7 |- ( ( -u pi (,) pi ) C_ dom ( RR _D F ) <-> dom ( ( RR _D F ) |` ( -u pi (,) pi ) ) = ( -u pi (,) pi ) )
13	11, 12	sylib 208	. . . . . 6 |- ( ph -> dom ( ( RR _D F ) |` ( -u pi (,) pi ) ) = ( -u pi (,) pi ) )
14	5, 13	syl5eq 2668	. . . . 5 |- ( ph -> dom G = ( -u pi (,) pi ) )
15	14	difeq2d 3728	. . . 4 |- ( ph -> ( ( -u pi (,) pi ) \ dom G ) = ( ( -u pi (,) pi ) \ ( -u pi (,) pi ) ) )
16		difid 3948	. . . 4 |- ( ( -u pi (,) pi ) \ ( -u pi (,) pi ) ) = (/)
17	15, 16	syl6eq 2672	. . 3 |- ( ph -> ( ( -u pi (,) pi ) \ dom G ) = (/) )
18		0fin 8188	. . 3 |- (/) e. Fin
19	17, 18	syl6eqel 2709	. 2 |- ( ph -> ( ( -u pi (,) pi ) \ dom G ) e. Fin )
20		rescncf 22700	. . . 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 ) ) )
21	6, 7, 20	mpsyl 68	. . 3 |- ( ph -> ( ( RR _D F ) |` ( -u pi (,) pi ) ) e. ( ( -u pi (,) pi ) -cn-> CC ) )
22	4	a1i 11	. . 3 |- ( ph -> G = ( ( RR _D F ) |` ( -u pi (,) pi ) ) )
23	14	oveq1d 6665	. . 3 |- ( ph -> ( dom G -cn-> CC ) = ( ( -u pi (,) pi ) -cn-> CC ) )
24	21, 22, 23	3eltr4d 2716	. 2 |- ( ph -> G e. ( dom G -cn-> CC ) )
25		pire 24210	. . . . . 6 |- pi e. RR
26	25	renegcli 10342	. . . . 5 |- -u pi e. RR
27	25	rexri 10097	. . . . 5 |- pi e. RR*
28		icossre 12254	. . . . 5 |- ( ( -u pi e. RR /\ pi e. RR* ) -> ( -u pi [,) pi ) C_ RR )
29	26, 27, 28	mp2an 708	. . . 4 |- ( -u pi [,) pi ) C_ RR
30		eldifi 3732	. . . 4 |- ( x e. ( ( -u pi [,) pi ) \ dom G ) -> x e. ( -u pi [,) pi ) )
31	29, 30	sseldi 3601	. . 3 |- ( x e. ( ( -u pi [,) pi ) \ dom G ) -> x e. RR )
32		limcresi 23649	. . . . . 6 |- ( ( RR _D F ) lim CC x ) C_ ( ( ( RR _D F ) |` ( ( -u pi (,) pi ) i^i ( x (,) +oo ) ) ) lim CC x )
33	4	reseq1i 5392	. . . . . . . 8 |- ( G |` ( x (,) +oo ) ) = ( ( ( RR _D F ) |` ( -u pi (,) pi ) ) |` ( x (,) +oo ) )
34		resres 5409	. . . . . . . 8 |- ( ( ( RR _D F ) |` ( -u pi (,) pi ) ) |` ( x (,) +oo ) ) = ( ( RR _D F ) |` ( ( -u pi (,) pi ) i^i ( x (,) +oo ) ) )
35	33, 34	eqtr2i 2645	. . . . . . 7 |- ( ( RR _D F ) |` ( ( -u pi (,) pi ) i^i ( x (,) +oo ) ) ) = ( G |` ( x (,) +oo ) )
36	35	oveq1i 6660	. . . . . 6 |- ( ( ( RR _D F ) |` ( ( -u pi (,) pi ) i^i ( x (,) +oo ) ) ) lim CC x ) = ( ( G |` ( x (,) +oo ) ) lim CC x )
37	32, 36	sseqtri 3637	. . . . 5 |- ( ( RR _D F ) lim CC x ) C_ ( ( G |` ( x (,) +oo ) ) lim CC x )
38	7	adantr 481	. . . . . 6 |- ( ( ph /\ x e. RR ) -> ( RR _D F ) e. ( RR -cn-> CC ) )
39		simpr 477	. . . . . 6 |- ( ( ph /\ x e. RR ) -> x e. RR )
40	38, 39	cnlimci 23653	. . . . 5 |- ( ( ph /\ x e. RR ) -> ( ( RR _D F ) ` x ) e. ( ( RR _D F ) lim CC x ) )
41	37, 40	sseldi 3601	. . . 4 |- ( ( ph /\ x e. RR ) -> ( ( RR _D F ) ` x ) e. ( ( G |` ( x (,) +oo ) ) lim CC x ) )
42		ne0i 3921	. . . 4 |- ( ( ( RR _D F ) ` x ) e. ( ( G |` ( x (,) +oo ) ) lim CC x ) -> ( ( G |` ( x (,) +oo ) ) lim CC x ) =/= (/) )
43	41, 42	syl 17	. . 3 |- ( ( ph /\ x e. RR ) -> ( ( G |` ( x (,) +oo ) ) lim CC x ) =/= (/) )
44	31, 43	sylan2 491	. 2 |- ( ( ph /\ x e. ( ( -u pi [,) pi ) \ dom G ) ) -> ( ( G |` ( x (,) +oo ) ) lim CC x ) =/= (/) )
45		negpitopissre 24286	. . . 4 |- ( -u pi (,] pi ) C_ RR
46		eldifi 3732	. . . 4 |- ( x e. ( ( -u pi (,] pi ) \ dom G ) -> x e. ( -u pi (,] pi ) )
47	45, 46	sseldi 3601	. . 3 |- ( x e. ( ( -u pi (,] pi ) \ dom G ) -> x e. RR )
48		limcresi 23649	. . . . . 6 |- ( ( RR _D F ) lim CC x ) C_ ( ( ( RR _D F ) |` ( ( -u pi (,) pi ) i^i ( -oo (,) x ) ) ) lim CC x )
49	4	reseq1i 5392	. . . . . . . 8 |- ( G |` ( -oo (,) x ) ) = ( ( ( RR _D F ) |` ( -u pi (,) pi ) ) |` ( -oo (,) x ) )
50		resres 5409	. . . . . . . 8 |- ( ( ( RR _D F ) |` ( -u pi (,) pi ) ) |` ( -oo (,) x ) ) = ( ( RR _D F ) |` ( ( -u pi (,) pi ) i^i ( -oo (,) x ) ) )
51	49, 50	eqtr2i 2645	. . . . . . 7 |- ( ( RR _D F ) |` ( ( -u pi (,) pi ) i^i ( -oo (,) x ) ) ) = ( G |` ( -oo (,) x ) )
52	51	oveq1i 6660	. . . . . 6 |- ( ( ( RR _D F ) |` ( ( -u pi (,) pi ) i^i ( -oo (,) x ) ) ) lim CC x ) = ( ( G |` ( -oo (,) x ) ) lim CC x )
53	48, 52	sseqtri 3637	. . . . 5 |- ( ( RR _D F ) lim CC x ) C_ ( ( G |` ( -oo (,) x ) ) lim CC x )
54	53, 40	sseldi 3601	. . . 4 |- ( ( ph /\ x e. RR ) -> ( ( RR _D F ) ` x ) e. ( ( G |` ( -oo (,) x ) ) lim CC x ) )
55		ne0i 3921	. . . 4 |- ( ( ( RR _D F ) ` x ) e. ( ( G |` ( -oo (,) x ) ) lim CC x ) -> ( ( G |` ( -oo (,) x ) ) lim CC x ) =/= (/) )
56	54, 55	syl 17	. . 3 |- ( ( ph /\ x e. RR ) -> ( ( G |` ( -oo (,) x ) ) lim CC x ) =/= (/) )
57	47, 56	sylan2 491	. 2 |- ( ( ph /\ x e. ( ( -u pi (,] pi ) \ dom G ) ) -> ( ( G |` ( -oo (,) x ) ) lim CC x ) =/= (/) )
58		eqid 2622	. 2 |- ( topGen ` ran (,) ) = ( topGen ` ran (,) )
59		ax-resscn 9993	. . . . . . 7 |- RR C_ CC
60	59	a1i 11	. . . . . 6 |- ( ph -> RR C_ CC )
61	1, 60	fssd 6057	. . . . . . 7 |- ( ph -> F : RR --> CC )
62		ssid 3624	. . . . . . . 8 |- RR C_ RR
63	62	a1i 11	. . . . . . 7 |- ( ph -> RR C_ RR )
64		dvcn 23684	. . . . . . 7 |- ( ( ( RR C_ CC /\ F : RR --> CC /\ RR C_ RR ) /\ dom ( RR _D F ) = RR ) -> F e. ( RR -cn-> CC ) )
65	60, 61, 63, 10, 64	syl31anc 1329	. . . . . 6 |- ( ph -> F e. ( RR -cn-> CC ) )
66		cncffvrn 22701	. . . . . 6 |- ( ( RR C_ CC /\ F e. ( RR -cn-> CC ) ) -> ( F e. ( RR -cn-> RR ) <-> F : RR --> RR ) )
67	60, 65, 66	syl2anc 693	. . . . 5 |- ( ph -> ( F e. ( RR -cn-> RR ) <-> F : RR --> RR ) )
68	1, 67	mpbird 247	. . . 4 |- ( ph -> F e. ( RR -cn-> RR ) )
69		eqid 2622	. . . . . 6 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
70	69	tgioo2 22606	. . . . . 6 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
71	69, 70, 70	cncfcn 22712	. . . . 5 |- ( ( RR C_ CC /\ RR C_ CC ) -> ( RR -cn-> RR ) = ( ( topGen ` ran (,) ) Cn ( topGen ` ran (,) ) ) )
72	60, 60, 71	syl2anc 693	. . . 4 |- ( ph -> ( RR -cn-> RR ) = ( ( topGen ` ran (,) ) Cn ( topGen ` ran (,) ) ) )
73	68, 72	eleqtrd 2703	. . 3 |- ( ph -> F e. ( ( topGen ` ran (,) ) Cn ( topGen ` ran (,) ) ) )
74		fouriercn.x	. . 3 |- ( ph -> X e. RR )
75		uniretop 22566	. . . 4 |- RR = U. ( topGen ` ran (,) )
76	75	cncnpi 21082	. . 3 |- ( ( F e. ( ( topGen ` ran (,) ) Cn ( topGen ` ran (,) ) ) /\ X e. RR ) -> F e. ( ( ( topGen ` ran (,) ) CnP ( topGen ` ran (,) ) ) ` X ) )
77	73, 74, 76	syl2anc 693	. 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 ) )
80	1, 2, 3, 4, 19, 24, 44, 57, 58, 77, 78, 79	fouriercnp 40443	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 ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   |-> cmpt 4729   dom cdm 5114   ran crn 5115   |` cres 5116   -->wf 5884   ` cfv 5888  (class class class)co 6650   Fincfn 7955   CCcc 9934   RRcr 9935   0cc0 9936   + caddc 9939   x. cmul 9941   +oocpnf 10071   -oocmnf 10072   RR*cxr 10073   -ucneg 10267   / cdiv 10684   NNcn 11020   2c2 11070   NN0cn0 11292   (,)cioo 12175   (,]cioc 12176   [,)cico 12177   sum_csu 14416   sincsin 14794   cosccos 14795   picpi 14797   TopOpenctopn 16082   topGenctg 16098  ℂ_fldccnfld 19746   Cn ccn 21028   CnP ccnp 21029   -cn->ccncf 22679   S.citg 23387   lim CC climc 23626   _D cdv 23627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cc 9257  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-ofr 6898  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-xnn0 11364  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-t1 21118  df-haus 21119  df-cmp 21190  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-ovol 23233  df-vol 23234  df-mbf 23388  df-itg1 23389  df-itg2 23390  df-ibl 23391  df-itg 23392  df-0p 23437  df-ditg 23611  df-limc 23630  df-dv 23631

This theorem is referenced by: (None)