fouriercnp - Mathbox for Glauco Siliprandi

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Theorem fouriercnp 42504

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.)

**Hypotheses**
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(,)+∞)) lim_ℂ 𝑥) ≠ ∅)
fouriercnp.llim	⊢ ((𝜑 ∧ 𝑥 ∈ ((-π(,]π) ∖ dom 𝐺)) → ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 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))

**Assertion**
Ref	Expression
fouriercnp	⊢ (𝜑 → (((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) = (𝐹‘𝑋))

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) 𝑇(𝑛) 𝐺(𝑛) 𝐽(𝑥,𝑛)

**Proof of Theorem fouriercnp**
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(,)+∞)) lim_ℂ 𝑥) ≠ ∅)
8		fouriercnp.llim	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π(,]π) ∖ dom 𝐺)) → ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥) ≠ ∅)
9		fouriercnp.cnp	. . . 4 ⊢ (𝜑 → 𝐹 ∈ ((𝐽 CnP 𝐽)‘𝑋))
10		uniretop 23365	. . . . . 6 ⊢ ℝ = ∪ (topGen‘ran (,))
11		fouriercnp.j	. . . . . . 7 ⊢ 𝐽 = (topGen‘ran (,))
12	11	unieqi 4841	. . . . . 6 ⊢ ∪ 𝐽 = ∪ (topGen‘ran (,))
13	10, 12	eqtr4i 2847	. . . . 5 ⊢ ℝ = ∪ 𝐽
14	13	cnprcl 21847	. . . 4 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐽)‘𝑋) → 𝑋 ∈ ℝ)
15	9, 14	syl 17	. . 3 ⊢ (𝜑 → 𝑋 ∈ ℝ)
16		limcresi 24477	. . . 4 ⊢ (𝐹 lim_ℂ 𝑋) ⊆ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)
17		eqid 2821	. . . . . . . . . . . 12 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
18	17	tgioo2 23405	. . . . . . . . . . 11 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
19	11, 18	eqtri 2844	. . . . . . . . . 10 ⊢ 𝐽 = ((TopOpen‘ℂ_fld) ↾_t ℝ)
20	19	oveq2i 7161	. . . . . . . . 9 ⊢ (𝐽 CnP 𝐽) = (𝐽 CnP ((TopOpen‘ℂ_fld) ↾_t ℝ))
21	20	fveq1i 6666	. . . . . . . 8 ⊢ ((𝐽 CnP 𝐽)‘𝑋) = ((𝐽 CnP ((TopOpen‘ℂ_fld) ↾_t ℝ))‘𝑋)
22	9, 21	eleqtrdi 2923	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ ((𝐽 CnP ((TopOpen‘ℂ_fld) ↾_t ℝ))‘𝑋))
23	17	cnfldtop 23386	. . . . . . . . 9 ⊢ (TopOpen‘ℂ_fld) ∈ Top
24	23	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (TopOpen‘ℂ_fld) ∈ Top)
25		ax-resscn 10588	. . . . . . . . 9 ⊢ ℝ ⊆ ℂ
26	25	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ ⊆ ℂ)
27		unicntop 23388	. . . . . . . . 9 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
28	13, 27	cnprest2 21892	. . . . . . . 8 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ 𝐹:ℝ⟶ℝ ∧ ℝ ⊆ ℂ) → (𝐹 ∈ ((𝐽 CnP (TopOpen‘ℂ_fld))‘𝑋) ↔ 𝐹 ∈ ((𝐽 CnP ((TopOpen‘ℂ_fld) ↾_t ℝ))‘𝑋)))
29	24, 1, 26, 28	syl3anc 1367	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∈ ((𝐽 CnP (TopOpen‘ℂ_fld))‘𝑋) ↔ 𝐹 ∈ ((𝐽 CnP ((TopOpen‘ℂ_fld) ↾_t ℝ))‘𝑋)))
30	22, 29	mpbird 259	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ ((𝐽 CnP (TopOpen‘ℂ_fld))‘𝑋))
31	17, 19	cnplimc 24479	. . . . . . 7 ⊢ ((ℝ ⊆ ℂ ∧ 𝑋 ∈ ℝ) → (𝐹 ∈ ((𝐽 CnP (TopOpen‘ℂ_fld))‘𝑋) ↔ (𝐹:ℝ⟶ℂ ∧ (𝐹‘𝑋) ∈ (𝐹 lim_ℂ 𝑋))))
32	25, 15, 31	sylancr 589	. . . . . 6 ⊢ (𝜑 → (𝐹 ∈ ((𝐽 CnP (TopOpen‘ℂ_fld))‘𝑋) ↔ (𝐹:ℝ⟶ℂ ∧ (𝐹‘𝑋) ∈ (𝐹 lim_ℂ 𝑋))))
33	30, 32	mpbid 234	. . . . 5 ⊢ (𝜑 → (𝐹:ℝ⟶ℂ ∧ (𝐹‘𝑋) ∈ (𝐹 lim_ℂ 𝑋)))
34	33	simprd 498	. . . 4 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (𝐹 lim_ℂ 𝑋))
35	16, 34	sseldi 3965	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋))
36		limcresi 24477	. . . 4 ⊢ (𝐹 lim_ℂ 𝑋) ⊆ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)
37	36, 34	sseldi 3965	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 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 42500	. 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 6847	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑋) ∈ ℝ)
42	41	recnd 10663	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑋) ∈ ℂ)
43	42	2timesd 11874	. . . 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 7165	. 2 ⊢ (𝜑 → (((𝐹‘𝑋) + (𝐹‘𝑋)) / 2) = ((2 · (𝐹‘𝑋)) / 2))
46		2cnd 11709	. . 3 ⊢ (𝜑 → 2 ∈ ℂ)
47		2ne0 11735	. . . 4 ⊢ 2 ≠ 0
48	47	a1i 11	. . 3 ⊢ (𝜑 → 2 ≠ 0)
49	42, 46, 48	divcan3d 11415	. 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 1533 ∈ wcel 2110 ≠ wne 3016 ∖ cdif 3933 ⊆ wss 3936 ∅c0 4291 ∪ cuni 4832 ↦ cmpt 5139 dom cdm 5550 ran crn 5551 ↾ cres 5552 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 Fincfn 8503 ℂcc 10529 ℝcr 10530 0cc0 10531 + caddc 10534 · cmul 10536 +∞cpnf 10666 -∞cmnf 10667 -cneg 10865 / cdiv 11291 ℕcn 11632 2c2 11686 ℕ₀cn0 11891 (,)cioo 12732 (,]cioc 12733 [,)cico 12734 Σcsu 15036 sincsin 15411 cosccos 15412 πcpi 15414 ↾_t crest 16688 TopOpenctopn 16689 topGenctg 16705 ℂ_fldccnfld 20539 Topctop 21495 CnP ccnp 21827 –cn→ccncf 23478 ∫citg 24213 lim_ℂ climc 24454 D cdv 24455

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-inf2 9098 ax-cc 9851 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 ax-mulf 10611

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3497 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 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-disj 5025 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-ofr 7404 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-omul 8101 df-er 8283 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-fi 8869 df-sup 8900 df-inf 8901 df-oi 8968 df-dju 9324 df-card 9362 df-acn 9365 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-xnn0 11962 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ioo 12736 df-ioc 12737 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-fl 13156 df-mod 13232 df-seq 13364 df-exp 13424 df-fac 13628 df-bc 13657 df-hash 13685 df-shft 14420 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-limsup 14822 df-clim 14839 df-rlim 14840 df-sum 15037 df-ef 15415 df-sin 15417 df-cos 15418 df-pi 15420 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-hom 16583 df-cco 16584 df-rest 16690 df-topn 16691 df-0g 16709 df-gsum 16710 df-topgen 16711 df-pt 16712 df-prds 16715 df-xrs 16769 df-qtop 16774 df-imas 16775 df-xps 16777 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-mulg 18219 df-cntz 18441 df-cmn 18902 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-fbas 20536 df-fg 20537 df-cnfld 20540 df-top 21496 df-topon 21513 df-topsp 21535 df-bases 21548 df-cld 21621 df-ntr 21622 df-cls 21623 df-nei 21700 df-lp 21738 df-perf 21739 df-cn 21829 df-cnp 21830 df-t1 21916 df-haus 21917 df-cmp 21989 df-tx 22164 df-hmeo 22357 df-fil 22448 df-fm 22540 df-flim 22541 df-flf 22542 df-xms 22924 df-ms 22925 df-tms 22926 df-cncf 23480 df-ovol 24059 df-vol 24060 df-mbf 24214 df-itg1 24215 df-itg2 24216 df-ibl 24217 df-itg 24218 df-0p 24265 df-ditg 24439 df-limc 24458 df-dv 24459

This theorem is referenced by: fouriercn 42510

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