![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > fouriercnp | Structured version Visualization version GIF version |
Description: If 𝐹 is continuous at the point 𝑋, then its Fourier series at 𝑋, converges to (𝐹‘𝑋). (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
fouriercnp.f | ⊢ (𝜑 → 𝐹:ℝ⟶ℝ) |
fouriercnp.t | ⊢ 𝑇 = (2 · π) |
fouriercnp.per | ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) |
fouriercnp.g | ⊢ 𝐺 = ((ℝ D 𝐹) ↾ (-π(,)π)) |
fouriercnp.dmdv | ⊢ (𝜑 → ((-π(,)π) ∖ dom 𝐺) ∈ Fin) |
fouriercnp.dvcn | ⊢ (𝜑 → 𝐺 ∈ (dom 𝐺–cn→ℂ)) |
fouriercnp.rlim | ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π[,)π) ∖ dom 𝐺)) → ((𝐺 ↾ (𝑥(,)+∞)) limℂ 𝑥) ≠ ∅) |
fouriercnp.llim | ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π(,]π) ∖ dom 𝐺)) → ((𝐺 ↾ (-∞(,)𝑥)) limℂ 𝑥) ≠ ∅) |
fouriercnp.j | ⊢ 𝐽 = (topGen‘ran (,)) |
fouriercnp.cnp | ⊢ (𝜑 → 𝐹 ∈ ((𝐽 CnP 𝐽)‘𝑋)) |
fouriercnp.a | ⊢ 𝐴 = (𝑛 ∈ ℕ0 ↦ (∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π)) |
fouriercnp.b | ⊢ 𝐵 = (𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π)) |
Ref | Expression |
---|---|
fouriercnp | ⊢ (𝜑 → (((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) = (𝐹‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fouriercnp.f | . . 3 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ) | |
2 | fouriercnp.t | . . 3 ⊢ 𝑇 = (2 · π) | |
3 | fouriercnp.per | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) | |
4 | fouriercnp.g | . . 3 ⊢ 𝐺 = ((ℝ D 𝐹) ↾ (-π(,)π)) | |
5 | fouriercnp.dmdv | . . 3 ⊢ (𝜑 → ((-π(,)π) ∖ dom 𝐺) ∈ Fin) | |
6 | fouriercnp.dvcn | . . 3 ⊢ (𝜑 → 𝐺 ∈ (dom 𝐺–cn→ℂ)) | |
7 | fouriercnp.rlim | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π[,)π) ∖ dom 𝐺)) → ((𝐺 ↾ (𝑥(,)+∞)) limℂ 𝑥) ≠ ∅) | |
8 | fouriercnp.llim | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π(,]π) ∖ dom 𝐺)) → ((𝐺 ↾ (-∞(,)𝑥)) limℂ 𝑥) ≠ ∅) | |
9 | fouriercnp.cnp | . . . 4 ⊢ (𝜑 → 𝐹 ∈ ((𝐽 CnP 𝐽)‘𝑋)) | |
10 | uniretop 24261 | . . . . . 6 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
11 | fouriercnp.j | . . . . . . 7 ⊢ 𝐽 = (topGen‘ran (,)) | |
12 | 11 | unieqi 4920 | . . . . . 6 ⊢ ∪ 𝐽 = ∪ (topGen‘ran (,)) |
13 | 10, 12 | eqtr4i 2764 | . . . . 5 ⊢ ℝ = ∪ 𝐽 |
14 | 13 | cnprcl 22731 | . . . 4 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐽)‘𝑋) → 𝑋 ∈ ℝ) |
15 | 9, 14 | syl 17 | . . 3 ⊢ (𝜑 → 𝑋 ∈ ℝ) |
16 | limcresi 25384 | . . . 4 ⊢ (𝐹 limℂ 𝑋) ⊆ ((𝐹 ↾ (-∞(,)𝑋)) limℂ 𝑋) | |
17 | eqid 2733 | . . . . . . . . . . . 12 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
18 | 17 | tgioo2 24301 | . . . . . . . . . . 11 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ) |
19 | 11, 18 | eqtri 2761 | . . . . . . . . . 10 ⊢ 𝐽 = ((TopOpen‘ℂfld) ↾t ℝ) |
20 | 19 | oveq2i 7415 | . . . . . . . . 9 ⊢ (𝐽 CnP 𝐽) = (𝐽 CnP ((TopOpen‘ℂfld) ↾t ℝ)) |
21 | 20 | fveq1i 6889 | . . . . . . . 8 ⊢ ((𝐽 CnP 𝐽)‘𝑋) = ((𝐽 CnP ((TopOpen‘ℂfld) ↾t ℝ))‘𝑋) |
22 | 9, 21 | eleqtrdi 2844 | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ ((𝐽 CnP ((TopOpen‘ℂfld) ↾t ℝ))‘𝑋)) |
23 | 17 | cnfldtop 24282 | . . . . . . . . 9 ⊢ (TopOpen‘ℂfld) ∈ Top |
24 | 23 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → (TopOpen‘ℂfld) ∈ Top) |
25 | ax-resscn 11163 | . . . . . . . . 9 ⊢ ℝ ⊆ ℂ | |
26 | 25 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → ℝ ⊆ ℂ) |
27 | unicntop 24284 | . . . . . . . . 9 ⊢ ℂ = ∪ (TopOpen‘ℂfld) | |
28 | 13, 27 | cnprest2 22776 | . . . . . . . 8 ⊢ (((TopOpen‘ℂfld) ∈ Top ∧ 𝐹:ℝ⟶ℝ ∧ ℝ ⊆ ℂ) → (𝐹 ∈ ((𝐽 CnP (TopOpen‘ℂfld))‘𝑋) ↔ 𝐹 ∈ ((𝐽 CnP ((TopOpen‘ℂfld) ↾t ℝ))‘𝑋))) |
29 | 24, 1, 26, 28 | syl3anc 1372 | . . . . . . 7 ⊢ (𝜑 → (𝐹 ∈ ((𝐽 CnP (TopOpen‘ℂfld))‘𝑋) ↔ 𝐹 ∈ ((𝐽 CnP ((TopOpen‘ℂfld) ↾t ℝ))‘𝑋))) |
30 | 22, 29 | mpbird 257 | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ ((𝐽 CnP (TopOpen‘ℂfld))‘𝑋)) |
31 | 17, 19 | cnplimc 25386 | . . . . . . 7 ⊢ ((ℝ ⊆ ℂ ∧ 𝑋 ∈ ℝ) → (𝐹 ∈ ((𝐽 CnP (TopOpen‘ℂfld))‘𝑋) ↔ (𝐹:ℝ⟶ℂ ∧ (𝐹‘𝑋) ∈ (𝐹 limℂ 𝑋)))) |
32 | 25, 15, 31 | sylancr 588 | . . . . . 6 ⊢ (𝜑 → (𝐹 ∈ ((𝐽 CnP (TopOpen‘ℂfld))‘𝑋) ↔ (𝐹:ℝ⟶ℂ ∧ (𝐹‘𝑋) ∈ (𝐹 limℂ 𝑋)))) |
33 | 30, 32 | mpbid 231 | . . . . 5 ⊢ (𝜑 → (𝐹:ℝ⟶ℂ ∧ (𝐹‘𝑋) ∈ (𝐹 limℂ 𝑋))) |
34 | 33 | simprd 497 | . . . 4 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (𝐹 limℂ 𝑋)) |
35 | 16, 34 | sselid 3979 | . . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ ((𝐹 ↾ (-∞(,)𝑋)) limℂ 𝑋)) |
36 | limcresi 25384 | . . . 4 ⊢ (𝐹 limℂ 𝑋) ⊆ ((𝐹 ↾ (𝑋(,)+∞)) limℂ 𝑋) | |
37 | 36, 34 | sselid 3979 | . . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ ((𝐹 ↾ (𝑋(,)+∞)) limℂ 𝑋)) |
38 | fouriercnp.a | . . 3 ⊢ 𝐴 = (𝑛 ∈ ℕ0 ↦ (∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π)) | |
39 | fouriercnp.b | . . 3 ⊢ 𝐵 = (𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π)) | |
40 | 1, 2, 3, 4, 5, 6, 7, 8, 15, 35, 37, 38, 39 | fourierd 44873 | . 2 ⊢ (𝜑 → (((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) = (((𝐹‘𝑋) + (𝐹‘𝑋)) / 2)) |
41 | 1, 15 | ffvelcdmd 7083 | . . . . . 6 ⊢ (𝜑 → (𝐹‘𝑋) ∈ ℝ) |
42 | 41 | recnd 11238 | . . . . 5 ⊢ (𝜑 → (𝐹‘𝑋) ∈ ℂ) |
43 | 42 | 2timesd 12451 | . . . 4 ⊢ (𝜑 → (2 · (𝐹‘𝑋)) = ((𝐹‘𝑋) + (𝐹‘𝑋))) |
44 | 43 | eqcomd 2739 | . . 3 ⊢ (𝜑 → ((𝐹‘𝑋) + (𝐹‘𝑋)) = (2 · (𝐹‘𝑋))) |
45 | 44 | oveq1d 7419 | . 2 ⊢ (𝜑 → (((𝐹‘𝑋) + (𝐹‘𝑋)) / 2) = ((2 · (𝐹‘𝑋)) / 2)) |
46 | 2cnd 12286 | . . 3 ⊢ (𝜑 → 2 ∈ ℂ) | |
47 | 2ne0 12312 | . . . 4 ⊢ 2 ≠ 0 | |
48 | 47 | a1i 11 | . . 3 ⊢ (𝜑 → 2 ≠ 0) |
49 | 42, 46, 48 | divcan3d 11991 | . 2 ⊢ (𝜑 → ((2 · (𝐹‘𝑋)) / 2) = (𝐹‘𝑋)) |
50 | 40, 45, 49 | 3eqtrd 2777 | 1 ⊢ (𝜑 → (((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) = (𝐹‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∖ cdif 3944 ⊆ wss 3947 ∅c0 4321 ∪ cuni 4907 ↦ cmpt 5230 dom cdm 5675 ran crn 5676 ↾ cres 5677 ⟶wf 6536 ‘cfv 6540 (class class class)co 7404 Fincfn 8935 ℂcc 11104 ℝcr 11105 0cc0 11106 + caddc 11109 · cmul 11111 +∞cpnf 11241 -∞cmnf 11242 -cneg 11441 / cdiv 11867 ℕcn 12208 2c2 12263 ℕ0cn0 12468 (,)cioo 13320 (,]cioc 13321 [,)cico 13322 Σcsu 15628 sincsin 16003 cosccos 16004 πcpi 16006 ↾t crest 17362 TopOpenctopn 17363 topGenctg 17379 ℂfldccnfld 20929 Topctop 22377 CnP ccnp 22711 –cn→ccncf 24374 ∫citg 25117 limℂ climc 25361 D cdv 25362 |
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 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-inf2 9632 ax-cc 10426 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186 |
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 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-symdif 4241 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-disj 5113 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7665 df-ofr 7666 df-om 7851 df-1st 7970 df-2nd 7971 df-supp 8142 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-oadd 8465 df-omul 8466 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-dju 9892 df-card 9930 df-acn 9933 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-xnn0 12541 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ioc 13325 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15010 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15411 df-clim 15428 df-rlim 15429 df-sum 15629 df-ef 16007 df-sin 16009 df-cos 16010 df-pi 16012 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-mulg 18945 df-cntz 19175 df-cmn 19643 df-psmet 20921 df-xmet 20922 df-met 20923 df-bl 20924 df-mopn 20925 df-fbas 20926 df-fg 20927 df-cnfld 20930 df-top 22378 df-topon 22395 df-topsp 22417 df-bases 22431 df-cld 22505 df-ntr 22506 df-cls 22507 df-nei 22584 df-lp 22622 df-perf 22623 df-cn 22713 df-cnp 22714 df-t1 22800 df-haus 22801 df-cmp 22873 df-tx 23048 df-hmeo 23241 df-fil 23332 df-fm 23424 df-flim 23425 df-flf 23426 df-xms 23808 df-ms 23809 df-tms 23810 df-cncf 24376 df-ovol 24963 df-vol 24964 df-mbf 25118 df-itg1 25119 df-itg2 25120 df-ibl 25121 df-itg 25122 df-0p 25169 df-ditg 25346 df-limc 25365 df-dv 25366 |
This theorem is referenced by: fouriercn 44883 |
Copyright terms: Public domain | W3C validator |