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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fouriercnp | Structured version Visualization version GIF version |
Description: If πΉ is continuous at the point π, then its Fourier series at π, converges to (πΉβπ). (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
fouriercnp.f | β’ (π β πΉ:ββΆβ) |
fouriercnp.t | β’ π = (2 Β· Ο) |
fouriercnp.per | β’ ((π β§ π₯ β β) β (πΉβ(π₯ + π)) = (πΉβπ₯)) |
fouriercnp.g | β’ πΊ = ((β D πΉ) βΎ (-Ο(,)Ο)) |
fouriercnp.dmdv | β’ (π β ((-Ο(,)Ο) β dom πΊ) β Fin) |
fouriercnp.dvcn | β’ (π β πΊ β (dom πΊβcnββ)) |
fouriercnp.rlim | β’ ((π β§ π₯ β ((-Ο[,)Ο) β dom πΊ)) β ((πΊ βΎ (π₯(,)+β)) limβ π₯) β β ) |
fouriercnp.llim | β’ ((π β§ π₯ β ((-Ο(,]Ο) β dom πΊ)) β ((πΊ βΎ (-β(,)π₯)) limβ π₯) β β ) |
fouriercnp.j | β’ π½ = (topGenβran (,)) |
fouriercnp.cnp | β’ (π β πΉ β ((π½ CnP π½)βπ)) |
fouriercnp.a | β’ π΄ = (π β β0 β¦ (β«(-Ο(,)Ο)((πΉβπ₯) Β· (cosβ(π Β· π₯))) dπ₯ / Ο)) |
fouriercnp.b | β’ π΅ = (π β β β¦ (β«(-Ο(,)Ο)((πΉβπ₯) Β· (sinβ(π Β· π₯))) dπ₯ / Ο)) |
Ref | Expression |
---|---|
fouriercnp | β’ (π β (((π΄β0) / 2) + Ξ£π β β (((π΄βπ) Β· (cosβ(π Β· π))) + ((π΅βπ) Β· (sinβ(π Β· π))))) = (πΉβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fouriercnp.f | . . 3 β’ (π β πΉ:ββΆβ) | |
2 | fouriercnp.t | . . 3 β’ π = (2 Β· Ο) | |
3 | fouriercnp.per | . . 3 β’ ((π β§ π₯ β β) β (πΉβ(π₯ + π)) = (πΉβπ₯)) | |
4 | fouriercnp.g | . . 3 β’ πΊ = ((β D πΉ) βΎ (-Ο(,)Ο)) | |
5 | fouriercnp.dmdv | . . 3 β’ (π β ((-Ο(,)Ο) β dom πΊ) β Fin) | |
6 | fouriercnp.dvcn | . . 3 β’ (π β πΊ β (dom πΊβcnββ)) | |
7 | fouriercnp.rlim | . . 3 β’ ((π β§ π₯ β ((-Ο[,)Ο) β dom πΊ)) β ((πΊ βΎ (π₯(,)+β)) limβ π₯) β β ) | |
8 | fouriercnp.llim | . . 3 β’ ((π β§ π₯ β ((-Ο(,]Ο) β dom πΊ)) β ((πΊ βΎ (-β(,)π₯)) limβ π₯) β β ) | |
9 | fouriercnp.cnp | . . . 4 β’ (π β πΉ β ((π½ CnP π½)βπ)) | |
10 | uniretop 24279 | . . . . . 6 β’ β = βͺ (topGenβran (,)) | |
11 | fouriercnp.j | . . . . . . 7 β’ π½ = (topGenβran (,)) | |
12 | 11 | unieqi 4922 | . . . . . 6 β’ βͺ π½ = βͺ (topGenβran (,)) |
13 | 10, 12 | eqtr4i 2764 | . . . . 5 β’ β = βͺ π½ |
14 | 13 | cnprcl 22749 | . . . 4 β’ (πΉ β ((π½ CnP π½)βπ) β π β β) |
15 | 9, 14 | syl 17 | . . 3 β’ (π β π β β) |
16 | limcresi 25402 | . . . 4 β’ (πΉ limβ π) β ((πΉ βΎ (-β(,)π)) limβ π) | |
17 | eqid 2733 | . . . . . . . . . . . 12 β’ (TopOpenββfld) = (TopOpenββfld) | |
18 | 17 | tgioo2 24319 | . . . . . . . . . . 11 β’ (topGenβran (,)) = ((TopOpenββfld) βΎt β) |
19 | 11, 18 | eqtri 2761 | . . . . . . . . . 10 β’ π½ = ((TopOpenββfld) βΎt β) |
20 | 19 | oveq2i 7420 | . . . . . . . . 9 β’ (π½ CnP π½) = (π½ CnP ((TopOpenββfld) βΎt β)) |
21 | 20 | fveq1i 6893 | . . . . . . . 8 β’ ((π½ CnP π½)βπ) = ((π½ CnP ((TopOpenββfld) βΎt β))βπ) |
22 | 9, 21 | eleqtrdi 2844 | . . . . . . 7 β’ (π β πΉ β ((π½ CnP ((TopOpenββfld) βΎt β))βπ)) |
23 | 17 | cnfldtop 24300 | . . . . . . . . 9 β’ (TopOpenββfld) β Top |
24 | 23 | a1i 11 | . . . . . . . 8 β’ (π β (TopOpenββfld) β Top) |
25 | ax-resscn 11167 | . . . . . . . . 9 β’ β β β | |
26 | 25 | a1i 11 | . . . . . . . 8 β’ (π β β β β) |
27 | unicntop 24302 | . . . . . . . . 9 β’ β = βͺ (TopOpenββfld) | |
28 | 13, 27 | cnprest2 22794 | . . . . . . . 8 β’ (((TopOpenββfld) β Top β§ πΉ:ββΆβ β§ β β β) β (πΉ β ((π½ CnP (TopOpenββfld))βπ) β πΉ β ((π½ CnP ((TopOpenββfld) βΎt β))βπ))) |
29 | 24, 1, 26, 28 | syl3anc 1372 | . . . . . . 7 β’ (π β (πΉ β ((π½ CnP (TopOpenββfld))βπ) β πΉ β ((π½ CnP ((TopOpenββfld) βΎt β))βπ))) |
30 | 22, 29 | mpbird 257 | . . . . . 6 β’ (π β πΉ β ((π½ CnP (TopOpenββfld))βπ)) |
31 | 17, 19 | cnplimc 25404 | . . . . . . 7 β’ ((β β β β§ π β β) β (πΉ β ((π½ CnP (TopOpenββfld))βπ) β (πΉ:ββΆβ β§ (πΉβπ) β (πΉ limβ π)))) |
32 | 25, 15, 31 | sylancr 588 | . . . . . 6 β’ (π β (πΉ β ((π½ CnP (TopOpenββfld))βπ) β (πΉ:ββΆβ β§ (πΉβπ) β (πΉ limβ π)))) |
33 | 30, 32 | mpbid 231 | . . . . 5 β’ (π β (πΉ:ββΆβ β§ (πΉβπ) β (πΉ limβ π))) |
34 | 33 | simprd 497 | . . . 4 β’ (π β (πΉβπ) β (πΉ limβ π)) |
35 | 16, 34 | sselid 3981 | . . 3 β’ (π β (πΉβπ) β ((πΉ βΎ (-β(,)π)) limβ π)) |
36 | limcresi 25402 | . . . 4 β’ (πΉ limβ π) β ((πΉ βΎ (π(,)+β)) limβ π) | |
37 | 36, 34 | sselid 3981 | . . 3 β’ (π β (πΉβπ) β ((πΉ βΎ (π(,)+β)) limβ π)) |
38 | fouriercnp.a | . . 3 β’ π΄ = (π β β0 β¦ (β«(-Ο(,)Ο)((πΉβπ₯) Β· (cosβ(π Β· π₯))) dπ₯ / Ο)) | |
39 | fouriercnp.b | . . 3 β’ π΅ = (π β β β¦ (β«(-Ο(,)Ο)((πΉβπ₯) Β· (sinβ(π Β· π₯))) dπ₯ / Ο)) | |
40 | 1, 2, 3, 4, 5, 6, 7, 8, 15, 35, 37, 38, 39 | fourierd 44938 | . 2 β’ (π β (((π΄β0) / 2) + Ξ£π β β (((π΄βπ) Β· (cosβ(π Β· π))) + ((π΅βπ) Β· (sinβ(π Β· π))))) = (((πΉβπ) + (πΉβπ)) / 2)) |
41 | 1, 15 | ffvelcdmd 7088 | . . . . . 6 β’ (π β (πΉβπ) β β) |
42 | 41 | recnd 11242 | . . . . 5 β’ (π β (πΉβπ) β β) |
43 | 42 | 2timesd 12455 | . . . 4 β’ (π β (2 Β· (πΉβπ)) = ((πΉβπ) + (πΉβπ))) |
44 | 43 | eqcomd 2739 | . . 3 β’ (π β ((πΉβπ) + (πΉβπ)) = (2 Β· (πΉβπ))) |
45 | 44 | oveq1d 7424 | . 2 β’ (π β (((πΉβπ) + (πΉβπ)) / 2) = ((2 Β· (πΉβπ)) / 2)) |
46 | 2cnd 12290 | . . 3 β’ (π β 2 β β) | |
47 | 2ne0 12316 | . . . 4 β’ 2 β 0 | |
48 | 47 | a1i 11 | . . 3 β’ (π β 2 β 0) |
49 | 42, 46, 48 | divcan3d 11995 | . 2 β’ (π β ((2 Β· (πΉβπ)) / 2) = (πΉβπ)) |
50 | 40, 45, 49 | 3eqtrd 2777 | 1 β’ (π β (((π΄β0) / 2) + Ξ£π β β (((π΄βπ) Β· (cosβ(π Β· π))) + ((π΅βπ) Β· (sinβ(π Β· π))))) = (πΉβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 β wne 2941 β cdif 3946 β wss 3949 β c0 4323 βͺ cuni 4909 β¦ cmpt 5232 dom cdm 5677 ran crn 5678 βΎ cres 5679 βΆwf 6540 βcfv 6544 (class class class)co 7409 Fincfn 8939 βcc 11108 βcr 11109 0cc0 11110 + caddc 11113 Β· cmul 11115 +βcpnf 11245 -βcmnf 11246 -cneg 11445 / cdiv 11871 βcn 12212 2c2 12267 β0cn0 12472 (,)cioo 13324 (,]cioc 13325 [,)cico 13326 Ξ£csu 15632 sincsin 16007 cosccos 16008 Οcpi 16010 βΎt crest 17366 TopOpenctopn 17367 topGenctg 17383 βfldccnfld 20944 Topctop 22395 CnP ccnp 22729 βcnβccncf 24392 β«citg 25135 limβ climc 25379 D cdv 25380 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cc 10430 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189 ax-mulf 11190 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-symdif 4243 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-disj 5115 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-ofr 7671 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-oadd 8470 df-omul 8471 df-er 8703 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9505 df-dju 9896 df-card 9934 df-acn 9937 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-xnn0 12545 df-z 12559 df-dec 12678 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ioo 13328 df-ioc 13329 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-fl 13757 df-mod 13835 df-seq 13967 df-exp 14028 df-fac 14234 df-bc 14263 df-hash 14291 df-shft 15014 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-limsup 15415 df-clim 15432 df-rlim 15433 df-sum 15633 df-ef 16011 df-sin 16013 df-cos 16014 df-pi 16016 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-hom 17221 df-cco 17222 df-rest 17368 df-topn 17369 df-0g 17387 df-gsum 17388 df-topgen 17389 df-pt 17390 df-prds 17393 df-xrs 17448 df-qtop 17453 df-imas 17454 df-xps 17456 df-mre 17530 df-mrc 17531 df-acs 17533 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-submnd 18672 df-mulg 18951 df-cntz 19181 df-cmn 19650 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-fbas 20941 df-fg 20942 df-cnfld 20945 df-top 22396 df-topon 22413 df-topsp 22435 df-bases 22449 df-cld 22523 df-ntr 22524 df-cls 22525 df-nei 22602 df-lp 22640 df-perf 22641 df-cn 22731 df-cnp 22732 df-t1 22818 df-haus 22819 df-cmp 22891 df-tx 23066 df-hmeo 23259 df-fil 23350 df-fm 23442 df-flim 23443 df-flf 23444 df-xms 23826 df-ms 23827 df-tms 23828 df-cncf 24394 df-ovol 24981 df-vol 24982 df-mbf 25136 df-itg1 25137 df-itg2 25138 df-ibl 25139 df-itg 25140 df-0p 25187 df-ditg 25364 df-limc 25383 df-dv 25384 |
This theorem is referenced by: fouriercn 44948 |
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