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Mirrors > Home > MPE Home > Th. List > Mathboxes > fourierclimd | Structured version Visualization version GIF version |
Description: Fourier series convergence, for piecewise smooth functions. See fourierd 40439 for the analogous Σ equation. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
fourierclimd.f | ⊢ (𝜑 → 𝐹:ℝ⟶ℝ) |
fourierclimd.t | ⊢ 𝑇 = (2 · π) |
fourierclimd.per | ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) |
fourierclimd.g | ⊢ 𝐺 = ((ℝ D 𝐹) ↾ (-π(,)π)) |
fourierclimd.dmdv | ⊢ (𝜑 → ((-π(,)π) ∖ dom 𝐺) ∈ Fin) |
fourierclimd.dvcn | ⊢ (𝜑 → 𝐺 ∈ (dom 𝐺–cn→ℂ)) |
fourierclimd.rlim | ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π[,)π) ∖ dom 𝐺)) → ((𝐺 ↾ (𝑥(,)+∞)) limℂ 𝑥) ≠ ∅) |
fourierclimd.llim | ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π(,]π) ∖ dom 𝐺)) → ((𝐺 ↾ (-∞(,)𝑥)) limℂ 𝑥) ≠ ∅) |
fourierclimd.x | ⊢ (𝜑 → 𝑋 ∈ ℝ) |
fourierclimd.l | ⊢ (𝜑 → 𝐿 ∈ ((𝐹 ↾ (-∞(,)𝑋)) limℂ 𝑋)) |
fourierclimd.r | ⊢ (𝜑 → 𝑅 ∈ ((𝐹 ↾ (𝑋(,)+∞)) limℂ 𝑋)) |
fourierclimd.a | ⊢ 𝐴 = (𝑛 ∈ ℕ0 ↦ (∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π)) |
fourierclimd.b | ⊢ 𝐵 = (𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π)) |
fourierclimd.s | ⊢ 𝑆 = (𝑛 ∈ ℕ ↦ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) |
Ref | Expression |
---|---|
fourierclimd | ⊢ (𝜑 → seq1( + , 𝑆) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fourierclimd.f | . . 3 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ) | |
2 | fourierclimd.t | . . 3 ⊢ 𝑇 = (2 · π) | |
3 | fourierclimd.per | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) | |
4 | fourierclimd.g | . . 3 ⊢ 𝐺 = ((ℝ D 𝐹) ↾ (-π(,)π)) | |
5 | fourierclimd.dmdv | . . 3 ⊢ (𝜑 → ((-π(,)π) ∖ dom 𝐺) ∈ Fin) | |
6 | fourierclimd.dvcn | . . 3 ⊢ (𝜑 → 𝐺 ∈ (dom 𝐺–cn→ℂ)) | |
7 | fourierclimd.rlim | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π[,)π) ∖ dom 𝐺)) → ((𝐺 ↾ (𝑥(,)+∞)) limℂ 𝑥) ≠ ∅) | |
8 | fourierclimd.llim | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π(,]π) ∖ dom 𝐺)) → ((𝐺 ↾ (-∞(,)𝑥)) limℂ 𝑥) ≠ ∅) | |
9 | fourierclimd.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ ℝ) | |
10 | fourierclimd.l | . . 3 ⊢ (𝜑 → 𝐿 ∈ ((𝐹 ↾ (-∞(,)𝑋)) limℂ 𝑋)) | |
11 | fourierclimd.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ ((𝐹 ↾ (𝑋(,)+∞)) limℂ 𝑋)) | |
12 | fourierclimd.a | . . 3 ⊢ 𝐴 = (𝑛 ∈ ℕ0 ↦ (∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π)) | |
13 | fourierclimd.b | . . 3 ⊢ 𝐵 = (𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π)) | |
14 | fourierclimd.s | . . . 4 ⊢ 𝑆 = (𝑛 ∈ ℕ ↦ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) | |
15 | nfcv 2764 | . . . . 5 ⊢ Ⅎ𝑘(((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋)))) | |
16 | nfmpt1 4747 | . . . . . . . . 9 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ0 ↦ (∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π)) | |
17 | 12, 16 | nfcxfr 2762 | . . . . . . . 8 ⊢ Ⅎ𝑛𝐴 |
18 | nfcv 2764 | . . . . . . . 8 ⊢ Ⅎ𝑛𝑘 | |
19 | 17, 18 | nffv 6198 | . . . . . . 7 ⊢ Ⅎ𝑛(𝐴‘𝑘) |
20 | nfcv 2764 | . . . . . . 7 ⊢ Ⅎ𝑛 · | |
21 | nfcv 2764 | . . . . . . 7 ⊢ Ⅎ𝑛(cos‘(𝑘 · 𝑋)) | |
22 | 19, 20, 21 | nfov 6676 | . . . . . 6 ⊢ Ⅎ𝑛((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) |
23 | nfcv 2764 | . . . . . 6 ⊢ Ⅎ𝑛 + | |
24 | nfmpt1 4747 | . . . . . . . . 9 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π)) | |
25 | 13, 24 | nfcxfr 2762 | . . . . . . . 8 ⊢ Ⅎ𝑛𝐵 |
26 | 25, 18 | nffv 6198 | . . . . . . 7 ⊢ Ⅎ𝑛(𝐵‘𝑘) |
27 | nfcv 2764 | . . . . . . 7 ⊢ Ⅎ𝑛(sin‘(𝑘 · 𝑋)) | |
28 | 26, 20, 27 | nfov 6676 | . . . . . 6 ⊢ Ⅎ𝑛((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋))) |
29 | 22, 23, 28 | nfov 6676 | . . . . 5 ⊢ Ⅎ𝑛(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))) |
30 | fveq2 6191 | . . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝐴‘𝑛) = (𝐴‘𝑘)) | |
31 | oveq1 6657 | . . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝑛 · 𝑋) = (𝑘 · 𝑋)) | |
32 | 31 | fveq2d 6195 | . . . . . . 7 ⊢ (𝑛 = 𝑘 → (cos‘(𝑛 · 𝑋)) = (cos‘(𝑘 · 𝑋))) |
33 | 30, 32 | oveq12d 6668 | . . . . . 6 ⊢ (𝑛 = 𝑘 → ((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) = ((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋)))) |
34 | fveq2 6191 | . . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝐵‘𝑛) = (𝐵‘𝑘)) | |
35 | 31 | fveq2d 6195 | . . . . . . 7 ⊢ (𝑛 = 𝑘 → (sin‘(𝑛 · 𝑋)) = (sin‘(𝑘 · 𝑋))) |
36 | 34, 35 | oveq12d 6668 | . . . . . 6 ⊢ (𝑛 = 𝑘 → ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))) = ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))) |
37 | 33, 36 | oveq12d 6668 | . . . . 5 ⊢ (𝑛 = 𝑘 → (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋)))) = (((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋))))) |
38 | 15, 29, 37 | cbvmpt 4749 | . . . 4 ⊢ (𝑛 ∈ ℕ ↦ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) = (𝑘 ∈ ℕ ↦ (((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋))))) |
39 | 14, 38 | eqtri 2644 | . . 3 ⊢ 𝑆 = (𝑘 ∈ ℕ ↦ (((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋))))) |
40 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 39 | fourierdlem115 40438 | . 2 ⊢ (𝜑 → (seq1( + , 𝑆) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2)) ∧ (((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) = ((𝐿 + 𝑅) / 2))) |
41 | 40 | simpld 475 | 1 ⊢ (𝜑 → seq1( + , 𝑆) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∖ cdif 3571 ∅c0 3915 class class class wbr 4653 ↦ cmpt 4729 dom cdm 5114 ↾ cres 5116 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 +∞cpnf 10071 -∞cmnf 10072 − cmin 10266 -cneg 10267 / cdiv 10684 ℕcn 11020 2c2 11070 ℕ0cn0 11292 (,)cioo 12175 (,]cioc 12176 [,)cico 12177 seqcseq 12801 ⇝ cli 14215 Σcsu 14416 sincsin 14794 cosccos 14795 πcpi 14797 –cn→ccncf 22679 ∫citg 23387 limℂ 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: fourierclim 40441 |
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