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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 46242 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 2904 | . . . . 5 ⊢ Ⅎ𝑘(((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋)))) | |
| 16 | nfmpt1 5249 | . . . . . . . . 9 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ0 ↦ (∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π)) | |
| 17 | 12, 16 | nfcxfr 2902 | . . . . . . . 8 ⊢ Ⅎ𝑛𝐴 | 
| 18 | nfcv 2904 | . . . . . . . 8 ⊢ Ⅎ𝑛𝑘 | |
| 19 | 17, 18 | nffv 6915 | . . . . . . 7 ⊢ Ⅎ𝑛(𝐴‘𝑘) | 
| 20 | nfcv 2904 | . . . . . . 7 ⊢ Ⅎ𝑛 · | |
| 21 | nfcv 2904 | . . . . . . 7 ⊢ Ⅎ𝑛(cos‘(𝑘 · 𝑋)) | |
| 22 | 19, 20, 21 | nfov 7462 | . . . . . 6 ⊢ Ⅎ𝑛((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) | 
| 23 | nfcv 2904 | . . . . . 6 ⊢ Ⅎ𝑛 + | |
| 24 | nfmpt1 5249 | . . . . . . . . 9 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π)) | |
| 25 | 13, 24 | nfcxfr 2902 | . . . . . . . 8 ⊢ Ⅎ𝑛𝐵 | 
| 26 | 25, 18 | nffv 6915 | . . . . . . 7 ⊢ Ⅎ𝑛(𝐵‘𝑘) | 
| 27 | nfcv 2904 | . . . . . . 7 ⊢ Ⅎ𝑛(sin‘(𝑘 · 𝑋)) | |
| 28 | 26, 20, 27 | nfov 7462 | . . . . . 6 ⊢ Ⅎ𝑛((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋))) | 
| 29 | 22, 23, 28 | nfov 7462 | . . . . 5 ⊢ Ⅎ𝑛(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))) | 
| 30 | fveq2 6905 | . . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝐴‘𝑛) = (𝐴‘𝑘)) | |
| 31 | oveq1 7439 | . . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝑛 · 𝑋) = (𝑘 · 𝑋)) | |
| 32 | 31 | fveq2d 6909 | . . . . . . 7 ⊢ (𝑛 = 𝑘 → (cos‘(𝑛 · 𝑋)) = (cos‘(𝑘 · 𝑋))) | 
| 33 | 30, 32 | oveq12d 7450 | . . . . . 6 ⊢ (𝑛 = 𝑘 → ((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) = ((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋)))) | 
| 34 | fveq2 6905 | . . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝐵‘𝑛) = (𝐵‘𝑘)) | |
| 35 | 31 | fveq2d 6909 | . . . . . . 7 ⊢ (𝑛 = 𝑘 → (sin‘(𝑛 · 𝑋)) = (sin‘(𝑘 · 𝑋))) | 
| 36 | 34, 35 | oveq12d 7450 | . . . . . 6 ⊢ (𝑛 = 𝑘 → ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))) = ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))) | 
| 37 | 33, 36 | oveq12d 7450 | . . . . 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 5252 | . . . 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 2764 | . . 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 46241 | . 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 494 | 1 ⊢ (𝜑 → seq1( + , 𝑆) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ∖ cdif 3947 ∅c0 4332 class class class wbr 5142 ↦ cmpt 5224 dom cdm 5684 ↾ cres 5686 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 Fincfn 8986 ℂcc 11154 ℝcr 11155 0cc0 11156 1c1 11157 + caddc 11159 · cmul 11161 +∞cpnf 11293 -∞cmnf 11294 − cmin 11493 -cneg 11494 / cdiv 11921 ℕcn 12267 2c2 12322 ℕ0cn0 12528 (,)cioo 13388 (,]cioc 13389 [,)cico 13390 seqcseq 14043 ⇝ cli 15521 Σcsu 15723 sincsin 16100 cosccos 16101 πcpi 16103 –cn→ccncf 24903 ∫citg 25654 limℂ climc 25898 D cdv 25899 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 ax-cc 10476 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 ax-addf 11235 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-symdif 4252 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-disj 5110 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-ofr 7699 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-oadd 8511 df-omul 8512 df-er 8746 df-map 8869 df-pm 8870 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-fi 9452 df-sup 9483 df-inf 9484 df-oi 9551 df-dju 9942 df-card 9980 df-acn 9983 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-xnn0 12602 df-z 12616 df-dec 12736 df-uz 12880 df-q 12992 df-rp 13036 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ioo 13392 df-ioc 13393 df-ico 13394 df-icc 13395 df-fz 13549 df-fzo 13696 df-fl 13833 df-mod 13911 df-seq 14044 df-exp 14104 df-fac 14314 df-bc 14343 df-hash 14371 df-shft 15107 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-limsup 15508 df-clim 15525 df-rlim 15526 df-sum 15724 df-ef 16104 df-sin 16106 df-cos 16107 df-pi 16109 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-submnd 18798 df-mulg 19087 df-cntz 19336 df-cmn 19801 df-psmet 21357 df-xmet 21358 df-met 21359 df-bl 21360 df-mopn 21361 df-fbas 21362 df-fg 21363 df-cnfld 21366 df-top 22901 df-topon 22918 df-topsp 22940 df-bases 22954 df-cld 23028 df-ntr 23029 df-cls 23030 df-nei 23107 df-lp 23145 df-perf 23146 df-cn 23236 df-cnp 23237 df-t1 23323 df-haus 23324 df-cmp 23396 df-tx 23571 df-hmeo 23764 df-fil 23855 df-fm 23947 df-flim 23948 df-flf 23949 df-xms 24331 df-ms 24332 df-tms 24333 df-cncf 24905 df-ovol 25500 df-vol 25501 df-mbf 25655 df-itg1 25656 df-itg2 25657 df-ibl 25658 df-itg 25659 df-0p 25706 df-ditg 25883 df-limc 25902 df-dv 25903 | 
| This theorem is referenced by: fourierclim 46244 | 
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