fouriercnp - Mathbox for Glauco Siliprandi

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Theorem fouriercnp 47378

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 25096	. . . . . 6 ⊢ ℝ = ∪ (topGen‘ran (,))
11		fouriercnp.j	. . . . . . 7 ⊢ 𝐽 = (topGen‘ran (,))
12	11	unieqi 4882	. . . . . 6 ⊢ ∪ 𝐽 = ∪ (topGen‘ran (,))
13	10, 12	eqtr4i 2789	. . . . 5 ⊢ ℝ = ∪ 𝐽
14	13	cnprcl 23578	. . . 4 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐽)‘𝑋) → 𝑋 ∈ ℝ)
15	9, 14	syl 18	. . 3 ⊢ (𝜑 → 𝑋 ∈ ℝ)
16		limcresi 26220	. . . 4 ⊢ (𝐹 lim_ℂ 𝑋) ⊆ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)
17		tgioo4 25139	. . . . . . . . . . 11 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
18	11, 17	eqtri 2786	. . . . . . . . . 10 ⊢ 𝐽 = ((TopOpen‘ℂ_fld) ↾_t ℝ)
19	18	oveq2i 7441	. . . . . . . . 9 ⊢ (𝐽 CnP 𝐽) = (𝐽 CnP ((TopOpen‘ℂ_fld) ↾_t ℝ))
20	19	fveq1i 6897	. . . . . . . 8 ⊢ ((𝐽 CnP 𝐽)‘𝑋) = ((𝐽 CnP ((TopOpen‘ℂ_fld) ↾_t ℝ))‘𝑋)
21	9, 20	eleqtrdi 2874	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ ((𝐽 CnP ((TopOpen‘ℂ_fld) ↾_t ℝ))‘𝑋))
22		eqid 2763	. . . . . . . . . 10 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
23	22	cnfldtop 25117	. . . . . . . . 9 ⊢ (TopOpen‘ℂ_fld) ∈ Top
24	23	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (TopOpen‘ℂ_fld) ∈ Top)
25		ax-resscn 11270	. . . . . . . . 9 ⊢ ℝ ⊆ ℂ
26	25	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ ⊆ ℂ)
27		unicntop 25119	. . . . . . . . 9 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
28	13, 27	cnprest2 23623	. . . . . . . 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 26222	. . . . . . 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 3911	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋))
36		limcresi 26220	. . . 4 ⊢ (𝐹 lim_ℂ 𝑋) ⊆ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)
37	36, 34	sselid 3911	. . 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 47374	. 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 7096	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑋) ∈ ℝ)
42	41	recnd 11350	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑋) ∈ ℂ)
43	42	2timesd 12602	. . . 4 ⊢ (𝜑 → (2 · (𝐹‘𝑋)) = ((𝐹‘𝑋) + (𝐹‘𝑋)))
44	43	eqcomd 2769	. . 3 ⊢ (𝜑 → ((𝐹‘𝑋) + (𝐹‘𝑋)) = (2 · (𝐹‘𝑋)))
45	44	oveq1d 7445	. 2 ⊢ (𝜑 → (((𝐹‘𝑋) + (𝐹‘𝑋)) / 2) = ((2 · (𝐹‘𝑋)) / 2))
46		2cnd 12434	. . 3 ⊢ (𝜑 → 2 ∈ ℂ)
47		2ne0 12462	. . . 4 ⊢ 2 ≠ 0
48	47	a1i 11	. . 3 ⊢ (𝜑 → 2 ≠ 0)
49	42, 46, 48	divcan3d 12111	. 2 ⊢ (𝜑 → ((2 · (𝐹‘𝑋)) / 2) = (𝐹‘𝑋))
50	40, 45, 49	3eqtrd 2802	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 1569 ∈ wcel 2146 ≠ wne 2959 ⊆ wss 3900 ∖ cdif 4048 ∅c0 4281 ∪ cuni 4870 ↦ cmpt 5189 dom cdm 5655 ran crn 5656 ↾ cres 5657 ⟶wf 6543 ‘cfv 6547 (class class class)co 7430 Fincfn 8987 ℂcc 11211 ℝcr 11212 0cc0 11213 + caddc 11216 · cmul 11218 +∞cpnf 11353 -∞cmnf 11354 -cneg 11555 / cdiv 11986 ℕcn 12348 2c2 12410 ℕ₀cn0 12619 (,)cioo 13489 (,]cioc 13490 [,)cico 13491 Σcsu 15868 sincsin 16244 cosccos 16245 πcpi 16247 ↾_t crest 17606 TopOpenctopn 17607 topGenctg 17623 ℂ_fldccnfld 21693 Topctop 23226 CnP ccnp 23558 –cn→ccncf 25212 ∫citg 25954 lim_ℂ climc 26197 D cdv 26198

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7761 ax-inf2 9655 ax-cc 10526 ax-cnex 11269 ax-resscn 11270 ax-1cn 11271 ax-icn 11272 ax-addcl 11273 ax-addrcl 11274 ax-mulcl 11275 ax-mulrcl 11276 ax-mulcom 11277 ax-addass 11278 ax-mulass 11279 ax-distr 11280 ax-i2m1 11281 ax-1ne0 11282 ax-1rid 11283 ax-rnegex 11284 ax-rrecex 11285 ax-cnre 11286 ax-pre-lttri 11287 ax-pre-lttrn 11288 ax-pre-ltadd 11289 ax-pre-mulgt0 11290 ax-pre-sup 11291 ax-addf 11292

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-symdif 4201 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-disj 5074 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6311 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6501 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-isom 6556 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7703 df-ofr 7704 df-om 7888 df-1st 8011 df-2nd 8012 df-supp 8191 df-frecs 8313 df-wrecs 8344 df-recs 8393 df-rdg 8432 df-1o 8490 df-2o 8491 df-oadd 8494 df-omul 8495 df-er 8731 df-map 8863 df-pm 8864 df-ixp 8940 df-en 8988 df-dom 8989 df-sdom 8990 df-fin 8991 df-fsupp 9368 df-fi 9416 df-sup 9447 df-inf 9448 df-oi 9517 df-dju 9995 df-card 10033 df-acn 10036 df-pnf 11358 df-mnf 11359 df-xr 11360 df-ltxr 11361 df-le 11362 df-sub 11556 df-neg 11557 df-div 11987 df-nn 12349 df-2 12418 df-3 12419 df-4 12420 df-5 12421 df-6 12422 df-7 12423 df-8 12424 df-9 12425 df-n0 12620 df-xnn0 12693 df-z 12707 df-dec 12828 df-uz 12979 df-q 13089 df-rp 13134 df-xneg 13254 df-xadd 13255 df-xmul 13256 df-ioo 13493 df-ioc 13494 df-ico 13495 df-icc 13496 df-fz 13653 df-fzo 13802 df-fl 13945 df-mod 14023 df-seq 14158 df-exp 14218 df-fac 14431 df-bc 14460 df-hash 14488 df-shft 15235 df-cj 15281 df-re 15282 df-im 15283 df-sqrt 15417 df-abs 15418 df-limsup 15653 df-clim 15670 df-rlim 15671 df-sum 15869 df-ef 16248 df-sin 16250 df-cos 16251 df-pi 16253 df-struct 17340 df-sets 17357 df-slot 17375 df-ndx 17387 df-base 17403 df-ress 17424 df-plusg 17456 df-mulr 17457 df-starv 17458 df-sca 17459 df-vsca 17460 df-ip 17461 df-tset 17462 df-ple 17463 df-ds 17465 df-unif 17466 df-hom 17467 df-cco 17468 df-rest 17608 df-topn 17609 df-0g 17627 df-gsum 17628 df-topgen 17629 df-pt 17630 df-prds 17633 df-xrs 17689 df-qtop 17694 df-imas 17695 df-xps 17697 df-mre 17771 df-mrc 17772 df-acs 17774 df-mgm 18831 df-sgrp 18923 df-mnd 18939 df-submnd 18994 df-mulg 19293 df-cntz 19546 df-cmn 20011 df-psmet 21685 df-xmet 21686 df-met 21687 df-bl 21688 df-mopn 21689 df-fbas 21690 df-fg 21691 df-cnfld 21694 df-top 23227 df-topon 23244 df-topsp 23266 df-bases 23279 df-cld 23352 df-ntr 23353 df-cls 23354 df-nei 23431 df-lp 23469 df-perf 23470 df-cn 23560 df-cnp 23561 df-t1 23647 df-haus 23648 df-cmp 23720 df-tx 23896 df-hmeo 24089 df-fil 24180 df-fm 24272 df-flim 24273 df-flf 24274 df-xms 24654 df-ms 24655 df-tms 24656 df-cncf 25214 df-ovol 25800 df-vol 25801 df-mbf 25955 df-itg1 25956 df-itg2 25957 df-ibl 25958 df-itg 25959 df-0p 26006 df-ditg 26182 df-limc 26201 df-dv 26202

This theorem is used by: fouriercn 47384

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