fouriercnp - Mathbox for Glauco Siliprandi

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Theorem fouriercnp 43269

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 23477	. . . . . 6 ⊢ ℝ = ∪ (topGen‘ran (,))
11		fouriercnp.j	. . . . . . 7 ⊢ 𝐽 = (topGen‘ran (,))
12	11	unieqi 4814	. . . . . 6 ⊢ ∪ 𝐽 = ∪ (topGen‘ran (,))
13	10, 12	eqtr4i 2784	. . . . 5 ⊢ ℝ = ∪ 𝐽
14	13	cnprcl 21958	. . . 4 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐽)‘𝑋) → 𝑋 ∈ ℝ)
15	9, 14	syl 17	. . 3 ⊢ (𝜑 → 𝑋 ∈ ℝ)
16		limcresi 24597	. . . 4 ⊢ (𝐹 lim_ℂ 𝑋) ⊆ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)
17		eqid 2758	. . . . . . . . . . . 12 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
18	17	tgioo2 23517	. . . . . . . . . . 11 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
19	11, 18	eqtri 2781	. . . . . . . . . 10 ⊢ 𝐽 = ((TopOpen‘ℂ_fld) ↾_t ℝ)
20	19	oveq2i 7167	. . . . . . . . 9 ⊢ (𝐽 CnP 𝐽) = (𝐽 CnP ((TopOpen‘ℂ_fld) ↾_t ℝ))
21	20	fveq1i 6664	. . . . . . . 8 ⊢ ((𝐽 CnP 𝐽)‘𝑋) = ((𝐽 CnP ((TopOpen‘ℂ_fld) ↾_t ℝ))‘𝑋)
22	9, 21	eleqtrdi 2862	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ ((𝐽 CnP ((TopOpen‘ℂ_fld) ↾_t ℝ))‘𝑋))
23	17	cnfldtop 23498	. . . . . . . . 9 ⊢ (TopOpen‘ℂ_fld) ∈ Top
24	23	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (TopOpen‘ℂ_fld) ∈ Top)
25		ax-resscn 10645	. . . . . . . . 9 ⊢ ℝ ⊆ ℂ
26	25	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ ⊆ ℂ)
27		unicntop 23500	. . . . . . . . 9 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
28	13, 27	cnprest2 22003	. . . . . . . 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 1368	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∈ ((𝐽 CnP (TopOpen‘ℂ_fld))‘𝑋) ↔ 𝐹 ∈ ((𝐽 CnP ((TopOpen‘ℂ_fld) ↾_t ℝ))‘𝑋)))
30	22, 29	mpbird 260	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ ((𝐽 CnP (TopOpen‘ℂ_fld))‘𝑋))
31	17, 19	cnplimc 24599	. . . . . . 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 590	. . . . . 6 ⊢ (𝜑 → (𝐹 ∈ ((𝐽 CnP (TopOpen‘ℂ_fld))‘𝑋) ↔ (𝐹:ℝ⟶ℂ ∧ (𝐹‘𝑋) ∈ (𝐹 lim_ℂ 𝑋))))
33	30, 32	mpbid 235	. . . . 5 ⊢ (𝜑 → (𝐹:ℝ⟶ℂ ∧ (𝐹‘𝑋) ∈ (𝐹 lim_ℂ 𝑋)))
34	33	simprd 499	. . . 4 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (𝐹 lim_ℂ 𝑋))
35	16, 34	sseldi 3892	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋))
36		limcresi 24597	. . . 4 ⊢ (𝐹 lim_ℂ 𝑋) ⊆ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)
37	36, 34	sseldi 3892	. . 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 43265	. 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 6849	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑋) ∈ ℝ)
42	41	recnd 10720	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑋) ∈ ℂ)
43	42	2timesd 11930	. . . 4 ⊢ (𝜑 → (2 · (𝐹‘𝑋)) = ((𝐹‘𝑋) + (𝐹‘𝑋)))
44	43	eqcomd 2764	. . 3 ⊢ (𝜑 → ((𝐹‘𝑋) + (𝐹‘𝑋)) = (2 · (𝐹‘𝑋)))
45	44	oveq1d 7171	. 2 ⊢ (𝜑 → (((𝐹‘𝑋) + (𝐹‘𝑋)) / 2) = ((2 · (𝐹‘𝑋)) / 2))
46		2cnd 11765	. . 3 ⊢ (𝜑 → 2 ∈ ℂ)
47		2ne0 11791	. . . 4 ⊢ 2 ≠ 0
48	47	a1i 11	. . 3 ⊢ (𝜑 → 2 ≠ 0)
49	42, 46, 48	divcan3d 11472	. 2 ⊢ (𝜑 → ((2 · (𝐹‘𝑋)) / 2) = (𝐹‘𝑋))
50	40, 45, 49	3eqtrd 2797	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 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2951 ∖ cdif 3857 ⊆ wss 3860 ∅c0 4227 ∪ cuni 4801 ↦ cmpt 5116 dom cdm 5528 ran crn 5529 ↾ cres 5530 ⟶wf 6336 ‘cfv 6340 (class class class)co 7156 Fincfn 8540 ℂcc 10586 ℝcr 10587 0cc0 10588 + caddc 10591 · cmul 10593 +∞cpnf 10723 -∞cmnf 10724 -cneg 10922 / cdiv 11348 ℕcn 11687 2c2 11742 ℕ₀cn0 11947 (,)cioo 12792 (,]cioc 12793 [,)cico 12794 Σcsu 15103 sincsin 15478 cosccos 15479 πcpi 15481 ↾_t crest 16765 TopOpenctopn 16766 topGenctg 16782 ℂ_fldccnfld 20179 Topctop 21606 CnP ccnp 21938 –cn→ccncf 23590 ∫citg 24331 lim_ℂ climc 24574 D cdv 24575

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5160 ax-sep 5173 ax-nul 5180 ax-pow 5238 ax-pr 5302 ax-un 7465 ax-inf2 9150 ax-cc 9908 ax-cnex 10644 ax-resscn 10645 ax-1cn 10646 ax-icn 10647 ax-addcl 10648 ax-addrcl 10649 ax-mulcl 10650 ax-mulrcl 10651 ax-mulcom 10652 ax-addass 10653 ax-mulass 10654 ax-distr 10655 ax-i2m1 10656 ax-1ne0 10657 ax-1rid 10658 ax-rnegex 10659 ax-rrecex 10660 ax-cnre 10661 ax-pre-lttri 10662 ax-pre-lttrn 10663 ax-pre-ltadd 10664 ax-pre-mulgt0 10665 ax-pre-sup 10666 ax-addf 10667 ax-mulf 10668

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-pss 3879 df-symdif 4149 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4802 df-int 4842 df-iun 4888 df-iin 4889 df-disj 5002 df-br 5037 df-opab 5099 df-mpt 5117 df-tr 5143 df-id 5434 df-eprel 5439 df-po 5447 df-so 5448 df-fr 5487 df-se 5488 df-we 5489 df-xp 5534 df-rel 5535 df-cnv 5536 df-co 5537 df-dm 5538 df-rn 5539 df-res 5540 df-ima 5541 df-pred 6131 df-ord 6177 df-on 6178 df-lim 6179 df-suc 6180 df-iota 6299 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-isom 6349 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7411 df-ofr 7412 df-om 7586 df-1st 7699 df-2nd 7700 df-supp 7842 df-wrecs 7963 df-recs 8024 df-rdg 8062 df-1o 8118 df-2o 8119 df-oadd 8122 df-omul 8123 df-er 8305 df-map 8424 df-pm 8425 df-ixp 8493 df-en 8541 df-dom 8542 df-sdom 8543 df-fin 8544 df-fsupp 8880 df-fi 8921 df-sup 8952 df-inf 8953 df-oi 9020 df-dju 9376 df-card 9414 df-acn 9417 df-pnf 10728 df-mnf 10729 df-xr 10730 df-ltxr 10731 df-le 10732 df-sub 10923 df-neg 10924 df-div 11349 df-nn 11688 df-2 11750 df-3 11751 df-4 11752 df-5 11753 df-6 11754 df-7 11755 df-8 11756 df-9 11757 df-n0 11948 df-xnn0 12020 df-z 12034 df-dec 12151 df-uz 12296 df-q 12402 df-rp 12444 df-xneg 12561 df-xadd 12562 df-xmul 12563 df-ioo 12796 df-ioc 12797 df-ico 12798 df-icc 12799 df-fz 12953 df-fzo 13096 df-fl 13224 df-mod 13300 df-seq 13432 df-exp 13493 df-fac 13697 df-bc 13726 df-hash 13754 df-shft 14487 df-cj 14519 df-re 14520 df-im 14521 df-sqrt 14655 df-abs 14656 df-limsup 14889 df-clim 14906 df-rlim 14907 df-sum 15104 df-ef 15482 df-sin 15484 df-cos 15485 df-pi 15487 df-struct 16556 df-ndx 16557 df-slot 16558 df-base 16560 df-sets 16561 df-ress 16562 df-plusg 16649 df-mulr 16650 df-starv 16651 df-sca 16652 df-vsca 16653 df-ip 16654 df-tset 16655 df-ple 16656 df-ds 16658 df-unif 16659 df-hom 16660 df-cco 16661 df-rest 16767 df-topn 16768 df-0g 16786 df-gsum 16787 df-topgen 16788 df-pt 16789 df-prds 16792 df-xrs 16846 df-qtop 16851 df-imas 16852 df-xps 16854 df-mre 16928 df-mrc 16929 df-acs 16931 df-mgm 17931 df-sgrp 17980 df-mnd 17991 df-submnd 18036 df-mulg 18305 df-cntz 18527 df-cmn 18988 df-psmet 20171 df-xmet 20172 df-met 20173 df-bl 20174 df-mopn 20175 df-fbas 20176 df-fg 20177 df-cnfld 20180 df-top 21607 df-topon 21624 df-topsp 21646 df-bases 21659 df-cld 21732 df-ntr 21733 df-cls 21734 df-nei 21811 df-lp 21849 df-perf 21850 df-cn 21940 df-cnp 21941 df-t1 22027 df-haus 22028 df-cmp 22100 df-tx 22275 df-hmeo 22468 df-fil 22559 df-fm 22651 df-flim 22652 df-flf 22653 df-xms 23035 df-ms 23036 df-tms 23037 df-cncf 23592 df-ovol 24177 df-vol 24178 df-mbf 24332 df-itg1 24333 df-itg2 24334 df-ibl 24335 df-itg 24336 df-0p 24383 df-ditg 24559 df-limc 24578 df-dv 24579

This theorem is referenced by: fouriercn 43275

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