Mathbox for Glauco Siliprandi |
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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 23371 | . . . . . 6 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
11 | fouriercnp.j | . . . . . . 7 ⊢ 𝐽 = (topGen‘ran (,)) | |
12 | 11 | unieqi 4851 | . . . . . 6 ⊢ ∪ 𝐽 = ∪ (topGen‘ran (,)) |
13 | 10, 12 | eqtr4i 2847 | . . . . 5 ⊢ ℝ = ∪ 𝐽 |
14 | 13 | cnprcl 21853 | . . . 4 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐽)‘𝑋) → 𝑋 ∈ ℝ) |
15 | 9, 14 | syl 17 | . . 3 ⊢ (𝜑 → 𝑋 ∈ ℝ) |
16 | limcresi 24483 | . . . 4 ⊢ (𝐹 limℂ 𝑋) ⊆ ((𝐹 ↾ (-∞(,)𝑋)) limℂ 𝑋) | |
17 | eqid 2821 | . . . . . . . . . . . 12 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
18 | 17 | tgioo2 23411 | . . . . . . . . . . 11 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ) |
19 | 11, 18 | eqtri 2844 | . . . . . . . . . 10 ⊢ 𝐽 = ((TopOpen‘ℂfld) ↾t ℝ) |
20 | 19 | oveq2i 7167 | . . . . . . . . 9 ⊢ (𝐽 CnP 𝐽) = (𝐽 CnP ((TopOpen‘ℂfld) ↾t ℝ)) |
21 | 20 | fveq1i 6671 | . . . . . . . 8 ⊢ ((𝐽 CnP 𝐽)‘𝑋) = ((𝐽 CnP ((TopOpen‘ℂfld) ↾t ℝ))‘𝑋) |
22 | 9, 21 | eleqtrdi 2923 | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ ((𝐽 CnP ((TopOpen‘ℂfld) ↾t ℝ))‘𝑋)) |
23 | 17 | cnfldtop 23392 | . . . . . . . . 9 ⊢ (TopOpen‘ℂfld) ∈ Top |
24 | 23 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → (TopOpen‘ℂfld) ∈ Top) |
25 | ax-resscn 10594 | . . . . . . . . 9 ⊢ ℝ ⊆ ℂ | |
26 | 25 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → ℝ ⊆ ℂ) |
27 | unicntop 23394 | . . . . . . . . 9 ⊢ ℂ = ∪ (TopOpen‘ℂfld) | |
28 | 13, 27 | cnprest2 21898 | . . . . . . . 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 1367 | . . . . . . 7 ⊢ (𝜑 → (𝐹 ∈ ((𝐽 CnP (TopOpen‘ℂfld))‘𝑋) ↔ 𝐹 ∈ ((𝐽 CnP ((TopOpen‘ℂfld) ↾t ℝ))‘𝑋))) |
30 | 22, 29 | mpbird 259 | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ ((𝐽 CnP (TopOpen‘ℂfld))‘𝑋)) |
31 | 17, 19 | cnplimc 24485 | . . . . . . 7 ⊢ ((ℝ ⊆ ℂ ∧ 𝑋 ∈ ℝ) → (𝐹 ∈ ((𝐽 CnP (TopOpen‘ℂfld))‘𝑋) ↔ (𝐹:ℝ⟶ℂ ∧ (𝐹‘𝑋) ∈ (𝐹 limℂ 𝑋)))) |
32 | 25, 15, 31 | sylancr 589 | . . . . . 6 ⊢ (𝜑 → (𝐹 ∈ ((𝐽 CnP (TopOpen‘ℂfld))‘𝑋) ↔ (𝐹:ℝ⟶ℂ ∧ (𝐹‘𝑋) ∈ (𝐹 limℂ 𝑋)))) |
33 | 30, 32 | mpbid 234 | . . . . 5 ⊢ (𝜑 → (𝐹:ℝ⟶ℂ ∧ (𝐹‘𝑋) ∈ (𝐹 limℂ 𝑋))) |
34 | 33 | simprd 498 | . . . 4 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (𝐹 limℂ 𝑋)) |
35 | 16, 34 | sseldi 3965 | . . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ ((𝐹 ↾ (-∞(,)𝑋)) limℂ 𝑋)) |
36 | limcresi 24483 | . . . 4 ⊢ (𝐹 limℂ 𝑋) ⊆ ((𝐹 ↾ (𝑋(,)+∞)) limℂ 𝑋) | |
37 | 36, 34 | sseldi 3965 | . . 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 42527 | . 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 | ffvelrnd 6852 | . . . . . 6 ⊢ (𝜑 → (𝐹‘𝑋) ∈ ℝ) |
42 | 41 | recnd 10669 | . . . . 5 ⊢ (𝜑 → (𝐹‘𝑋) ∈ ℂ) |
43 | 42 | 2timesd 11881 | . . . 4 ⊢ (𝜑 → (2 · (𝐹‘𝑋)) = ((𝐹‘𝑋) + (𝐹‘𝑋))) |
44 | 43 | eqcomd 2827 | . . 3 ⊢ (𝜑 → ((𝐹‘𝑋) + (𝐹‘𝑋)) = (2 · (𝐹‘𝑋))) |
45 | 44 | oveq1d 7171 | . 2 ⊢ (𝜑 → (((𝐹‘𝑋) + (𝐹‘𝑋)) / 2) = ((2 · (𝐹‘𝑋)) / 2)) |
46 | 2cnd 11716 | . . 3 ⊢ (𝜑 → 2 ∈ ℂ) | |
47 | 2ne0 11742 | . . . 4 ⊢ 2 ≠ 0 | |
48 | 47 | a1i 11 | . . 3 ⊢ (𝜑 → 2 ≠ 0) |
49 | 42, 46, 48 | divcan3d 11421 | . 2 ⊢ (𝜑 → ((2 · (𝐹‘𝑋)) / 2) = (𝐹‘𝑋)) |
50 | 40, 45, 49 | 3eqtrd 2860 | 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 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∖ cdif 3933 ⊆ wss 3936 ∅c0 4291 ∪ cuni 4838 ↦ cmpt 5146 dom cdm 5555 ran crn 5556 ↾ cres 5557 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 Fincfn 8509 ℂcc 10535 ℝcr 10536 0cc0 10537 + caddc 10540 · cmul 10542 +∞cpnf 10672 -∞cmnf 10673 -cneg 10871 / cdiv 11297 ℕcn 11638 2c2 11693 ℕ0cn0 11898 (,)cioo 12739 (,]cioc 12740 [,)cico 12741 Σcsu 15042 sincsin 15417 cosccos 15418 πcpi 15420 ↾t crest 16694 TopOpenctopn 16695 topGenctg 16711 ℂfldccnfld 20545 Topctop 21501 CnP ccnp 21833 –cn→ccncf 23484 ∫citg 24219 limℂ climc 24460 D cdv 24461 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cc 9857 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-symdif 4219 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-disj 5032 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-ofr 7410 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-omul 8107 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-dju 9330 df-card 9368 df-acn 9371 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-xnn0 11969 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ioo 12743 df-ioc 12744 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-fac 13635 df-bc 13664 df-hash 13692 df-shft 14426 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-limsup 14828 df-clim 14845 df-rlim 14846 df-sum 15043 df-ef 15421 df-sin 15423 df-cos 15424 df-pi 15426 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-fbas 20542 df-fg 20543 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-ntr 21628 df-cls 21629 df-nei 21706 df-lp 21744 df-perf 21745 df-cn 21835 df-cnp 21836 df-t1 21922 df-haus 21923 df-cmp 21995 df-tx 22170 df-hmeo 22363 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-xms 22930 df-ms 22931 df-tms 22932 df-cncf 23486 df-ovol 24065 df-vol 24066 df-mbf 24220 df-itg1 24221 df-itg2 24222 df-ibl 24223 df-itg 24224 df-0p 24271 df-ditg 24445 df-limc 24464 df-dv 24465 |
This theorem is referenced by: fouriercn 42537 |
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