fourierclimd - Mathbox for Glauco Siliprandi

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Theorem fourierclimd 46863

Description: Fourier series convergence, for piecewise smooth functions. See fourierd 46862 for the analogous Σ equation. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
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(,)+∞)) lim_ℂ 𝑥) ≠ ∅)
fourierclimd.llim	⊢ ((𝜑 ∧ 𝑥 ∈ ((-π(,]π) ∖ dom 𝐺)) → ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥) ≠ ∅)
fourierclimd.x	⊢ (𝜑 → 𝑋 ∈ ℝ)
fourierclimd.l	⊢ (𝜑 → 𝐿 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋))
fourierclimd.r	⊢ (𝜑 → 𝑅 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 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)))))

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

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

Proof of Theorem fourierclimd Dummy variable 𝑘 is distinct from all other variables.

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(,)+∞)) lim_ℂ 𝑥) ≠ ∅)
8		fourierclimd.llim	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π(,]π) ∖ dom 𝐺)) → ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥) ≠ ∅)
9		fourierclimd.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ ℝ)
10		fourierclimd.l	. . 3 ⊢ (𝜑 → 𝐿 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋))
11		fourierclimd.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 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 2931	. . . . 5 ⊢ Ⅎ𝑘(((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))
16		nfmpt1 5214	. . . . . . . . 9 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ₀ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π))
17	12, 16	nfcxfr 2929	. . . . . . . 8 ⊢ Ⅎ𝑛𝐴
18		nfcv 2931	. . . . . . . 8 ⊢ Ⅎ𝑛𝑘
19	17, 18	nffv 6892	. . . . . . 7 ⊢ Ⅎ𝑛(𝐴‘𝑘)
20		nfcv 2931	. . . . . . 7 ⊢ Ⅎ𝑛 ·
21		nfcv 2931	. . . . . . 7 ⊢ Ⅎ𝑛(cos‘(𝑘 · 𝑋))
22	19, 20, 21	nfov 7441	. . . . . 6 ⊢ Ⅎ𝑛((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋)))
23		nfcv 2931	. . . . . 6 ⊢ Ⅎ𝑛 +
24		nfmpt1 5214	. . . . . . . . 9 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π))
25	13, 24	nfcxfr 2929	. . . . . . . 8 ⊢ Ⅎ𝑛𝐵
26	25, 18	nffv 6892	. . . . . . 7 ⊢ Ⅎ𝑛(𝐵‘𝑘)
27		nfcv 2931	. . . . . . 7 ⊢ Ⅎ𝑛(sin‘(𝑘 · 𝑋))
28	26, 20, 27	nfov 7441	. . . . . 6 ⊢ Ⅎ𝑛((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))
29	22, 23, 28	nfov 7441	. . . . 5 ⊢ Ⅎ𝑛(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋))))
30		fveq2 6882	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝐴‘𝑛) = (𝐴‘𝑘))
31		oveq1 7418	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝑛 · 𝑋) = (𝑘 · 𝑋))
32	31	fveq2d 6886	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (cos‘(𝑛 · 𝑋)) = (cos‘(𝑘 · 𝑋)))
33	30, 32	oveq12d 7429	. . . . . 6 ⊢ (𝑛 = 𝑘 → ((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) = ((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))))
34		fveq2 6882	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝐵‘𝑛) = (𝐵‘𝑘))
35	31	fveq2d 6886	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (sin‘(𝑛 · 𝑋)) = (sin‘(𝑘 · 𝑋)))
36	34, 35	oveq12d 7429	. . . . . 6 ⊢ (𝑛 = 𝑘 → ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))) = ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋))))
37	33, 36	oveq12d 7429	. . . . 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 5217	. . . 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 2792	. . 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 46861	. 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 499	1 ⊢ (𝜑 → seq1( + , 𝑆) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 ∖ cdif 3910 ∅c0 4294 class class class wbr 5113 ↦ cmpt 5196 dom cdm 5662 ↾ cres 5664 ⟶wf 6533 ‘cfv 6537 (class class class)co 7411 Fincfn 8943 ℂcc 11098 ℝcr 11099 0cc0 11100 1c1 11101 + caddc 11103 · cmul 11105 +∞cpnf 11240 -∞cmnf 11241 − cmin 11441 -cneg 11442 / cdiv 11871 ℕcn 12233 2c2 12295 ℕ₀cn0 12504 (,)cioo 13372 (,]cioc 13373 [,)cico 13374 seqcseq 14037 ⇝ cli 15535 Σcsu 15737 sincsin 16117 cosccos 16118 πcpi 16120 –cn→ccncf 25004 ∫citg 25746 lim_ℂ climc 25990 D cdv 25991

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-inf2 9610 ax-cc 10419 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177 ax-pre-sup 11178 ax-addf 11179

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-symdif 4214 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-iin 4963 df-disj 5081 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-se 5616 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7675 df-ofr 7676 df-om 7863 df-1st 7986 df-2nd 7987 df-supp 8157 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8453 df-2o 8454 df-oadd 8457 df-omul 8458 df-er 8694 df-map 8826 df-pm 8827 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fsupp 9322 df-fi 9371 df-sup 9402 df-inf 9403 df-oi 9472 df-dju 9887 df-card 9925 df-acn 9928 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-div 11872 df-nn 12234 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12505 df-xnn0 12578 df-z 12592 df-dec 12712 df-uz 12863 df-q 12973 df-rp 13017 df-xneg 13137 df-xadd 13138 df-xmul 13139 df-ioo 13376 df-ioc 13377 df-ico 13378 df-icc 13379 df-fz 13536 df-fzo 13683 df-fl 13825 df-mod 13903 df-seq 14038 df-exp 14098 df-fac 14310 df-bc 14339 df-hash 14367 df-shft 15104 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-limsup 15522 df-clim 15539 df-rlim 15540 df-sum 15738 df-ef 16121 df-sin 16123 df-cos 16124 df-pi 16126 df-struct 17207 df-sets 17224 df-slot 17242 df-ndx 17254 df-base 17270 df-ress 17291 df-plusg 17323 df-mulr 17324 df-starv 17325 df-sca 17326 df-vsca 17327 df-ip 17328 df-tset 17329 df-ple 17330 df-ds 17332 df-unif 17333 df-hom 17334 df-cco 17335 df-rest 17475 df-topn 17476 df-0g 17494 df-gsum 17495 df-topgen 17496 df-pt 17497 df-prds 17500 df-xrs 17556 df-qtop 17561 df-imas 17562 df-xps 17564 df-mre 17638 df-mrc 17639 df-acs 17641 df-mgm 18698 df-sgrp 18777 df-mnd 18793 df-submnd 18842 df-mulg 19134 df-cntz 19387 df-cmn 19852 df-psmet 21483 df-xmet 21484 df-met 21485 df-bl 21486 df-mopn 21487 df-fbas 21488 df-fg 21489 df-cnfld 21492 df-top 23020 df-topon 23037 df-topsp 23059 df-bases 23072 df-cld 23145 df-ntr 23146 df-cls 23147 df-nei 23224 df-lp 23262 df-perf 23263 df-cn 23353 df-cnp 23354 df-t1 23440 df-haus 23441 df-cmp 23513 df-tx 23688 df-hmeo 23881 df-fil 23972 df-fm 24064 df-flim 24065 df-flf 24066 df-xms 24446 df-ms 24447 df-tms 24448 df-cncf 25006 df-ovol 25592 df-vol 25593 df-mbf 25747 df-itg1 25748 df-itg2 25749 df-ibl 25750 df-itg 25751 df-0p 25798 df-ditg 25975 df-limc 25994 df-dv 25995

This theorem is referenced by: fourierclim 46864

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