fourierclim - Mathbox for Glauco Siliprandi

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Theorem fourierclim 46147

Description: Fourier series convergence, for piecewise smooth functions. See fourier 46148 for the analogous Σ equation. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fourierclim.f	⊢ 𝐹:ℝ⟶ℝ
fourierclim.t	⊢ 𝑇 = (2 · π)
fourierclim.per	⊢ (𝑥 ∈ ℝ → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
fourierclim.g	⊢ 𝐺 = ((ℝ D 𝐹) ↾ (-π(,)π))
fourierclim.dmdv	⊢ ((-π(,)π) ∖ dom 𝐺) ∈ Fin
fourierclim.dvcn	⊢ 𝐺 ∈ (dom 𝐺–cn→ℂ)
fourierclim.rlim	⊢ (𝑥 ∈ ((-π[,)π) ∖ dom 𝐺) → ((𝐺 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥) ≠ ∅)
fourierclim.llim	⊢ (𝑥 ∈ ((-π(,]π) ∖ dom 𝐺) → ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥) ≠ ∅)
fourierclim.x	⊢ 𝑋 ∈ ℝ
fourierclim.l	⊢ 𝐿 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)
fourierclim.r	⊢ 𝑅 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)
fourierclim.a	⊢ 𝐴 = (𝑛 ∈ ℕ₀ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π))
fourierclim.b	⊢ 𝐵 = (𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π))
fourierclim.s	⊢ 𝑆 = (𝑛 ∈ ℕ ↦ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋)))))

**Assertion**
Ref	Expression
fourierclim	⊢ seq1( + , 𝑆) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2))

Distinct variable groups: 𝑛,𝐹,𝑥 𝑥,𝐺 𝑥,𝑇 𝑛,𝑋,𝑥 Allowed substitution hints: 𝐴(𝑥,𝑛) 𝐵(𝑥,𝑛) 𝑅(𝑥,𝑛) 𝑆(𝑥,𝑛) 𝑇(𝑛) 𝐺(𝑛) 𝐿(𝑥,𝑛)

**Proof of Theorem fourierclim**
Step	Hyp	Ref	Expression
1		fourierclim.f	. . . 4 ⊢ 𝐹:ℝ⟶ℝ
2	1	a1i 11	. . 3 ⊢ (⊤ → 𝐹:ℝ⟶ℝ)
3		fourierclim.t	. . 3 ⊢ 𝑇 = (2 · π)
4		fourierclim.per	. . . 4 ⊢ (𝑥 ∈ ℝ → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
5	4	adantl 481	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
6		fourierclim.g	. . 3 ⊢ 𝐺 = ((ℝ D 𝐹) ↾ (-π(,)π))
7		fourierclim.dmdv	. . . 4 ⊢ ((-π(,)π) ∖ dom 𝐺) ∈ Fin
8	7	a1i 11	. . 3 ⊢ (⊤ → ((-π(,)π) ∖ dom 𝐺) ∈ Fin)
9		fourierclim.dvcn	. . . 4 ⊢ 𝐺 ∈ (dom 𝐺–cn→ℂ)
10	9	a1i 11	. . 3 ⊢ (⊤ → 𝐺 ∈ (dom 𝐺–cn→ℂ))
11		fourierclim.rlim	. . . 4 ⊢ (𝑥 ∈ ((-π[,)π) ∖ dom 𝐺) → ((𝐺 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥) ≠ ∅)
12	11	adantl 481	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ((-π[,)π) ∖ dom 𝐺)) → ((𝐺 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥) ≠ ∅)
13		fourierclim.llim	. . . 4 ⊢ (𝑥 ∈ ((-π(,]π) ∖ dom 𝐺) → ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥) ≠ ∅)
14	13	adantl 481	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ((-π(,]π) ∖ dom 𝐺)) → ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥) ≠ ∅)
15		fourierclim.x	. . . 4 ⊢ 𝑋 ∈ ℝ
16	15	a1i 11	. . 3 ⊢ (⊤ → 𝑋 ∈ ℝ)
17		fourierclim.l	. . . 4 ⊢ 𝐿 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)
18	17	a1i 11	. . 3 ⊢ (⊤ → 𝐿 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋))
19		fourierclim.r	. . . 4 ⊢ 𝑅 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)
20	19	a1i 11	. . 3 ⊢ (⊤ → 𝑅 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋))
21		fourierclim.a	. . 3 ⊢ 𝐴 = (𝑛 ∈ ℕ₀ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π))
22		fourierclim.b	. . 3 ⊢ 𝐵 = (𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π))
23		fourierclim.s	. . 3 ⊢ 𝑆 = (𝑛 ∈ ℕ ↦ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋)))))
24	2, 3, 5, 6, 8, 10, 12, 14, 16, 18, 20, 21, 22, 23	fourierclimd 46146	. 2 ⊢ (⊤ → seq1( + , 𝑆) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2)))
25	24	mptru 1544	1 ⊢ seq1( + , 𝑆) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1537 ⊤wtru 1538 ∈ wcel 2108 ≠ wne 2946 ∖ cdif 3973 ∅c0 4352 class class class wbr 5166 ↦ cmpt 5249 dom cdm 5700 ↾ cres 5702 ⟶wf 6571 ‘cfv 6575 (class class class)co 7450 Fincfn 9005 ℂcc 11184 ℝcr 11185 0cc0 11186 1c1 11187 + caddc 11189 · cmul 11191 +∞cpnf 11323 -∞cmnf 11324 − cmin 11522 -cneg 11523 / cdiv 11949 ℕcn 12295 2c2 12350 ℕ₀cn0 12555 (,)cioo 13409 (,]cioc 13410 [,)cico 13411 seqcseq 14054 ⇝ cli 15532 sincsin 16113 cosccos 16114 πcpi 16116 –cn→ccncf 24923 ∫citg 25674 lim_ℂ climc 25919 D cdv 25920

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-inf2 9712 ax-cc 10506 ax-cnex 11242 ax-resscn 11243 ax-1cn 11244 ax-icn 11245 ax-addcl 11246 ax-addrcl 11247 ax-mulcl 11248 ax-mulrcl 11249 ax-mulcom 11250 ax-addass 11251 ax-mulass 11252 ax-distr 11253 ax-i2m1 11254 ax-1ne0 11255 ax-1rid 11256 ax-rnegex 11257 ax-rrecex 11258 ax-cnre 11259 ax-pre-lttri 11260 ax-pre-lttrn 11261 ax-pre-ltadd 11262 ax-pre-mulgt0 11263 ax-pre-sup 11264 ax-addf 11265

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-symdif 4272 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-disj 5134 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6334 df-ord 6400 df-on 6401 df-lim 6402 df-suc 6403 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-isom 6584 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-of 7716 df-ofr 7717 df-om 7906 df-1st 8032 df-2nd 8033 df-supp 8204 df-frecs 8324 df-wrecs 8355 df-recs 8429 df-rdg 8468 df-1o 8524 df-2o 8525 df-oadd 8528 df-omul 8529 df-er 8765 df-map 8888 df-pm 8889 df-ixp 8958 df-en 9006 df-dom 9007 df-sdom 9008 df-fin 9009 df-fsupp 9434 df-fi 9482 df-sup 9513 df-inf 9514 df-oi 9581 df-dju 9972 df-card 10010 df-acn 10013 df-pnf 11328 df-mnf 11329 df-xr 11330 df-ltxr 11331 df-le 11332 df-sub 11524 df-neg 11525 df-div 11950 df-nn 12296 df-2 12358 df-3 12359 df-4 12360 df-5 12361 df-6 12362 df-7 12363 df-8 12364 df-9 12365 df-n0 12556 df-xnn0 12628 df-z 12642 df-dec 12761 df-uz 12906 df-q 13016 df-rp 13060 df-xneg 13177 df-xadd 13178 df-xmul 13179 df-ioo 13413 df-ioc 13414 df-ico 13415 df-icc 13416 df-fz 13570 df-fzo 13714 df-fl 13845 df-mod 13923 df-seq 14055 df-exp 14115 df-fac 14325 df-bc 14354 df-hash 14382 df-shft 15118 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-limsup 15519 df-clim 15536 df-rlim 15537 df-sum 15737 df-ef 16117 df-sin 16119 df-cos 16120 df-pi 16122 df-struct 17196 df-sets 17213 df-slot 17231 df-ndx 17243 df-base 17261 df-ress 17290 df-plusg 17326 df-mulr 17327 df-starv 17328 df-sca 17329 df-vsca 17330 df-ip 17331 df-tset 17332 df-ple 17333 df-ds 17335 df-unif 17336 df-hom 17337 df-cco 17338 df-rest 17484 df-topn 17485 df-0g 17503 df-gsum 17504 df-topgen 17505 df-pt 17506 df-prds 17509 df-xrs 17564 df-qtop 17569 df-imas 17570 df-xps 17572 df-mre 17646 df-mrc 17647 df-acs 17649 df-mgm 18680 df-sgrp 18759 df-mnd 18775 df-submnd 18821 df-mulg 19110 df-cntz 19359 df-cmn 19826 df-psmet 21381 df-xmet 21382 df-met 21383 df-bl 21384 df-mopn 21385 df-fbas 21386 df-fg 21387 df-cnfld 21390 df-top 22923 df-topon 22940 df-topsp 22962 df-bases 22976 df-cld 23050 df-ntr 23051 df-cls 23052 df-nei 23129 df-lp 23167 df-perf 23168 df-cn 23258 df-cnp 23259 df-t1 23345 df-haus 23346 df-cmp 23418 df-tx 23593 df-hmeo 23786 df-fil 23877 df-fm 23969 df-flim 23970 df-flf 23971 df-xms 24353 df-ms 24354 df-tms 24355 df-cncf 24925 df-ovol 25520 df-vol 25521 df-mbf 25675 df-itg1 25676 df-itg2 25677 df-ibl 25678 df-itg 25679 df-0p 25726 df-ditg 25904 df-limc 25923 df-dv 25924

This theorem is referenced by: fouriersw 46154

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