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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 24286 | . . . . . 6 β’ β = βͺ (topGenβran (,)) | |
11 | fouriercnp.j | . . . . . . 7 β’ π½ = (topGenβran (,)) | |
12 | 11 | unieqi 4921 | . . . . . 6 β’ βͺ π½ = βͺ (topGenβran (,)) |
13 | 10, 12 | eqtr4i 2763 | . . . . 5 β’ β = βͺ π½ |
14 | 13 | cnprcl 22756 | . . . 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 25409 | . . . 4 β’ (πΉ limβ π) β ((πΉ βΎ (-β(,)π)) limβ π) | |
17 | eqid 2732 | . . . . . . . . . . . 12 β’ (TopOpenββfld) = (TopOpenββfld) | |
18 | 17 | tgioo2 24326 | . . . . . . . . . . 11 β’ (topGenβran (,)) = ((TopOpenββfld) βΎt β) |
19 | 11, 18 | eqtri 2760 | . . . . . . . . . 10 β’ π½ = ((TopOpenββfld) βΎt β) |
20 | 19 | oveq2i 7422 | . . . . . . . . 9 β’ (π½ CnP π½) = (π½ CnP ((TopOpenββfld) βΎt β)) |
21 | 20 | fveq1i 6892 | . . . . . . . 8 β’ ((π½ CnP π½)βπ) = ((π½ CnP ((TopOpenββfld) βΎt β))βπ) |
22 | 9, 21 | eleqtrdi 2843 | . . . . . . 7 β’ (π β πΉ β ((π½ CnP ((TopOpenββfld) βΎt β))βπ)) |
23 | 17 | cnfldtop 24307 | . . . . . . . . 9 β’ (TopOpenββfld) β Top |
24 | 23 | a1i 11 | . . . . . . . 8 β’ (π β (TopOpenββfld) β Top) |
25 | ax-resscn 11169 | . . . . . . . . 9 β’ β β β | |
26 | 25 | a1i 11 | . . . . . . . 8 β’ (π β β β β) |
27 | unicntop 24309 | . . . . . . . . 9 β’ β = βͺ (TopOpenββfld) | |
28 | 13, 27 | cnprest2 22801 | . . . . . . . 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 1371 | . . . . . . 7 β’ (π β (πΉ β ((π½ CnP (TopOpenββfld))βπ) β πΉ β ((π½ CnP ((TopOpenββfld) βΎt β))βπ))) |
30 | 22, 29 | mpbird 256 | . . . . . 6 β’ (π β πΉ β ((π½ CnP (TopOpenββfld))βπ)) |
31 | 17, 19 | cnplimc 25411 | . . . . . . 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 587 | . . . . . 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 496 | . . . 4 β’ (π β (πΉβπ) β (πΉ limβ π)) |
35 | 16, 34 | sselid 3980 | . . 3 β’ (π β (πΉβπ) β ((πΉ βΎ (-β(,)π)) limβ π)) |
36 | limcresi 25409 | . . . 4 β’ (πΉ limβ π) β ((πΉ βΎ (π(,)+β)) limβ π) | |
37 | 36, 34 | sselid 3980 | . . 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 45017 | . 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 7087 | . . . . . 6 β’ (π β (πΉβπ) β β) |
42 | 41 | recnd 11244 | . . . . 5 β’ (π β (πΉβπ) β β) |
43 | 42 | 2timesd 12457 | . . . 4 β’ (π β (2 Β· (πΉβπ)) = ((πΉβπ) + (πΉβπ))) |
44 | 43 | eqcomd 2738 | . . 3 β’ (π β ((πΉβπ) + (πΉβπ)) = (2 Β· (πΉβπ))) |
45 | 44 | oveq1d 7426 | . 2 β’ (π β (((πΉβπ) + (πΉβπ)) / 2) = ((2 Β· (πΉβπ)) / 2)) |
46 | 2cnd 12292 | . . 3 β’ (π β 2 β β) | |
47 | 2ne0 12318 | . . . 4 β’ 2 β 0 | |
48 | 47 | a1i 11 | . . 3 β’ (π β 2 β 0) |
49 | 42, 46, 48 | divcan3d 11997 | . 2 β’ (π β ((2 Β· (πΉβπ)) / 2) = (πΉβπ)) |
50 | 40, 45, 49 | 3eqtrd 2776 | 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 396 = wceq 1541 β wcel 2106 β wne 2940 β cdif 3945 β wss 3948 β c0 4322 βͺ cuni 4908 β¦ cmpt 5231 dom cdm 5676 ran crn 5677 βΎ cres 5678 βΆwf 6539 βcfv 6543 (class class class)co 7411 Fincfn 8941 βcc 11110 βcr 11111 0cc0 11112 + caddc 11115 Β· cmul 11117 +βcpnf 11247 -βcmnf 11248 -cneg 11447 / cdiv 11873 βcn 12214 2c2 12269 β0cn0 12474 (,)cioo 13326 (,]cioc 13327 [,)cico 13328 Ξ£csu 15634 sincsin 16009 cosccos 16010 Οcpi 16012 βΎt crest 17368 TopOpenctopn 17369 topGenctg 17385 βfldccnfld 20950 Topctop 22402 CnP ccnp 22736 βcnβccncf 24399 β«citg 25142 limβ climc 25386 D cdv 25387 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-inf2 9638 ax-cc 10432 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-symdif 4242 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-disj 5114 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-ofr 7673 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-oadd 8472 df-omul 8473 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-dju 9898 df-card 9936 df-acn 9939 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12475 df-xnn0 12547 df-z 12561 df-dec 12680 df-uz 12825 df-q 12935 df-rp 12977 df-xneg 13094 df-xadd 13095 df-xmul 13096 df-ioo 13330 df-ioc 13331 df-ico 13332 df-icc 13333 df-fz 13487 df-fzo 13630 df-fl 13759 df-mod 13837 df-seq 13969 df-exp 14030 df-fac 14236 df-bc 14265 df-hash 14293 df-shft 15016 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-limsup 15417 df-clim 15434 df-rlim 15435 df-sum 15635 df-ef 16013 df-sin 16015 df-cos 16016 df-pi 16018 df-struct 17082 df-sets 17099 df-slot 17117 df-ndx 17129 df-base 17147 df-ress 17176 df-plusg 17212 df-mulr 17213 df-starv 17214 df-sca 17215 df-vsca 17216 df-ip 17217 df-tset 17218 df-ple 17219 df-ds 17221 df-unif 17222 df-hom 17223 df-cco 17224 df-rest 17370 df-topn 17371 df-0g 17389 df-gsum 17390 df-topgen 17391 df-pt 17392 df-prds 17395 df-xrs 17450 df-qtop 17455 df-imas 17456 df-xps 17458 df-mre 17532 df-mrc 17533 df-acs 17535 df-mgm 18563 df-sgrp 18612 df-mnd 18628 df-submnd 18674 df-mulg 18953 df-cntz 19183 df-cmn 19652 df-psmet 20942 df-xmet 20943 df-met 20944 df-bl 20945 df-mopn 20946 df-fbas 20947 df-fg 20948 df-cnfld 20951 df-top 22403 df-topon 22420 df-topsp 22442 df-bases 22456 df-cld 22530 df-ntr 22531 df-cls 22532 df-nei 22609 df-lp 22647 df-perf 22648 df-cn 22738 df-cnp 22739 df-t1 22825 df-haus 22826 df-cmp 22898 df-tx 23073 df-hmeo 23266 df-fil 23357 df-fm 23449 df-flim 23450 df-flf 23451 df-xms 23833 df-ms 23834 df-tms 23835 df-cncf 24401 df-ovol 24988 df-vol 24989 df-mbf 25143 df-itg1 25144 df-itg2 25145 df-ibl 25146 df-itg 25147 df-0p 25194 df-ditg 25371 df-limc 25390 df-dv 25391 |
This theorem is referenced by: fouriercn 45027 |
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