fouriercnp - Mathbox for Glauco Siliprandi

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Theorem fouriercnp 47235

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 25081	. . . . . 6 ⊢ ℝ = ∪ (topGen‘ran (,))
11		fouriercnp.j	. . . . . . 7 ⊢ 𝐽 = (topGen‘ran (,))
12	11	unieqi 4879	. . . . . 6 ⊢ ∪ 𝐽 = ∪ (topGen‘ran (,))
13	10, 12	eqtr4i 2787	. . . . 5 ⊢ ℝ = ∪ 𝐽
14	13	cnprcl 23563	. . . 4 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐽)‘𝑋) → 𝑋 ∈ ℝ)
15	9, 14	syl 18	. . 3 ⊢ (𝜑 → 𝑋 ∈ ℝ)
16		limcresi 26205	. . . 4 ⊢ (𝐹 lim_ℂ 𝑋) ⊆ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)
17		tgioo4 25124	. . . . . . . . . . 11 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
18	11, 17	eqtri 2784	. . . . . . . . . 10 ⊢ 𝐽 = ((TopOpen‘ℂ_fld) ↾_t ℝ)
19	18	oveq2i 7431	. . . . . . . . 9 ⊢ (𝐽 CnP 𝐽) = (𝐽 CnP ((TopOpen‘ℂ_fld) ↾_t ℝ))
20	19	fveq1i 6886	. . . . . . . 8 ⊢ ((𝐽 CnP 𝐽)‘𝑋) = ((𝐽 CnP ((TopOpen‘ℂ_fld) ↾_t ℝ))‘𝑋)
21	9, 20	eleqtrdi 2871	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ ((𝐽 CnP ((TopOpen‘ℂ_fld) ↾_t ℝ))‘𝑋))
22		eqid 2761	. . . . . . . . . 10 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
23	22	cnfldtop 25102	. . . . . . . . 9 ⊢ (TopOpen‘ℂ_fld) ∈ Top
24	23	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (TopOpen‘ℂ_fld) ∈ Top)
25		ax-resscn 11257	. . . . . . . . 9 ⊢ ℝ ⊆ ℂ
26	25	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ ⊆ ℂ)
27		unicntop 25104	. . . . . . . . 9 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
28	13, 27	cnprest2 23608	. . . . . . . 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 1398	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∈ ((𝐽 CnP (TopOpen‘ℂ_fld))‘𝑋) ↔ 𝐹 ∈ ((𝐽 CnP ((TopOpen‘ℂ_fld) ↾_t ℝ))‘𝑋)))
30	21, 29	mpbird 260	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ ((𝐽 CnP (TopOpen‘ℂ_fld))‘𝑋))
31	22, 18	cnplimc 26207	. . . . . . 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 599	. . . . . 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 501	. . . 4 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (𝐹 lim_ℂ 𝑋))
35	16, 34	sselid 3929	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋))
36		limcresi 26205	. . . 4 ⊢ (𝐹 lim_ℂ 𝑋) ⊆ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)
37	36, 34	sselid 3929	. . 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 47231	. 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	ffvelcdmd 7085	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑋) ∈ ℝ)
42	41	recnd 11337	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑋) ∈ ℂ)
43	42	2timesd 12589	. . . 4 ⊢ (𝜑 → (2 · (𝐹‘𝑋)) = ((𝐹‘𝑋) + (𝐹‘𝑋)))
44	43	eqcomd 2767	. . 3 ⊢ (𝜑 → ((𝐹‘𝑋) + (𝐹‘𝑋)) = (2 · (𝐹‘𝑋)))
45	44	oveq1d 7435	. 2 ⊢ (𝜑 → (((𝐹‘𝑋) + (𝐹‘𝑋)) / 2) = ((2 · (𝐹‘𝑋)) / 2))
46		2cnd 12421	. . 3 ⊢ (𝜑 → 2 ∈ ℂ)
47		2ne0 12449	. . . 4 ⊢ 2 ≠ 0
48	47	a1i 11	. . 3 ⊢ (𝜑 → 2 ≠ 0)
49	42, 46, 48	divcan3d 12098	. 2 ⊢ (𝜑 → ((2 · (𝐹‘𝑋)) / 2) = (𝐹‘𝑋))
50	40, 45, 49	3eqtrd 2800	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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 ∖ cdif 3896 ⊆ wss 3899 ∅c0 4279 ∪ cuni 4867 ↦ cmpt 5186 dom cdm 5651 ran crn 5652 ↾ cres 5653 ⟶wf 6534 ‘cfv 6538 (class class class)co 7420 Fincfn 8973 ℂcc 11198 ℝcr 11199 0cc0 11200 + caddc 11203 · cmul 11205 +∞cpnf 11340 -∞cmnf 11341 -cneg 11542 / cdiv 11973 ℕcn 12335 2c2 12397 ℕ₀cn0 12606 (,)cioo 13476 (,]cioc 13477 [,)cico 13478 Σcsu 15853 sincsin 16229 cosccos 16230 πcpi 16232 ↾_t crest 17591 TopOpenctopn 17592 topGenctg 17608 ℂ_fldccnfld 21678 Topctop 23211 CnP ccnp 23543 –cn→ccncf 25197 ∫citg 25939 lim_ℂ climc 26182 D cdv 26183

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751 ax-inf2 9642 ax-cc 10513 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-pre-sup 11278 ax-addf 11279

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-symdif 4199 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-disj 5071 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-ofr 7694 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8178 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-2o 8477 df-oadd 8480 df-omul 8481 df-er 8717 df-map 8849 df-pm 8850 df-ixp 8926 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fsupp 9354 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9504 df-dju 9982 df-card 10020 df-acn 10023 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-3 12406 df-4 12407 df-5 12408 df-6 12409 df-7 12410 df-8 12411 df-9 12412 df-n0 12607 df-xnn0 12680 df-z 12694 df-dec 12815 df-uz 12966 df-q 13076 df-rp 13121 df-xneg 13241 df-xadd 13242 df-xmul 13243 df-ioo 13480 df-ioc 13481 df-ico 13482 df-icc 13483 df-fz 13640 df-fzo 13789 df-fl 13932 df-mod 14010 df-seq 14145 df-exp 14205 df-fac 14418 df-bc 14447 df-hash 14475 df-shft 15220 df-cj 15266 df-re 15267 df-im 15268 df-sqrt 15402 df-abs 15403 df-limsup 15638 df-clim 15655 df-rlim 15656 df-sum 15854 df-ef 16233 df-sin 16235 df-cos 16236 df-pi 16238 df-struct 17325 df-sets 17342 df-slot 17360 df-ndx 17372 df-base 17388 df-ress 17409 df-plusg 17441 df-mulr 17442 df-starv 17443 df-sca 17444 df-vsca 17445 df-ip 17446 df-tset 17447 df-ple 17448 df-ds 17450 df-unif 17451 df-hom 17452 df-cco 17453 df-rest 17593 df-topn 17594 df-0g 17612 df-gsum 17613 df-topgen 17614 df-pt 17615 df-prds 17618 df-xrs 17674 df-qtop 17679 df-imas 17680 df-xps 17682 df-mre 17756 df-mrc 17757 df-acs 17759 df-mgm 18816 df-sgrp 18908 df-mnd 18924 df-submnd 18979 df-mulg 19278 df-cntz 19531 df-cmn 19996 df-psmet 21670 df-xmet 21671 df-met 21672 df-bl 21673 df-mopn 21674 df-fbas 21675 df-fg 21676 df-cnfld 21679 df-top 23212 df-topon 23229 df-topsp 23251 df-bases 23264 df-cld 23337 df-ntr 23338 df-cls 23339 df-nei 23416 df-lp 23454 df-perf 23455 df-cn 23545 df-cnp 23546 df-t1 23632 df-haus 23633 df-cmp 23705 df-tx 23881 df-hmeo 24074 df-fil 24165 df-fm 24257 df-flim 24258 df-flf 24259 df-xms 24639 df-ms 24640 df-tms 24641 df-cncf 25199 df-ovol 25785 df-vol 25786 df-mbf 25940 df-itg1 25941 df-itg2 25942 df-ibl 25943 df-itg 25944 df-0p 25991 df-ditg 26167 df-limc 26186 df-dv 26187

This theorem is used by: fouriercn 47241

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