fouriercn - Mathbox for Glauco Siliprandi

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Theorem fouriercn 42537

Description: If the derivative of 𝐹 is continuous, then the Fourier series for 𝐹 converges to 𝐹 everywhere and the hypothesis are simpler than those for the more general case of a piecewise smooth function (see fourierd 42527 for a comparison). (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fouriercn.f	⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
fouriercn.t	⊢ 𝑇 = (2 · π)
fouriercn.per	⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
fouriercn.dv	⊢ (𝜑 → (ℝ D 𝐹) ∈ (ℝ–cn→ℂ))
fouriercn.g	⊢ 𝐺 = ((ℝ D 𝐹) ↾ (-π(,)π))
fouriercn.x	⊢ (𝜑 → 𝑋 ∈ ℝ)
fouriercn.a	⊢ 𝐴 = (𝑛 ∈ ℕ₀ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π))
fouriercn.b	⊢ 𝐵 = (𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π))

**Assertion**
Ref	Expression
fouriercn	⊢ (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)

**Proof of Theorem fouriercn**
Step	Hyp	Ref	Expression
1		fouriercn.f	. 2 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
2		fouriercn.t	. 2 ⊢ 𝑇 = (2 · π)
3		fouriercn.per	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
4		fouriercn.g	. 2 ⊢ 𝐺 = ((ℝ D 𝐹) ↾ (-π(,)π))
5	4	dmeqi 5773	. . . . . 6 ⊢ dom 𝐺 = dom ((ℝ D 𝐹) ↾ (-π(,)π))
6		ioossre 12799	. . . . . . . 8 ⊢ (-π(,)π) ⊆ ℝ
7		fouriercn.dv	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (ℝ–cn→ℂ))
8		cncff 23501	. . . . . . . . 9 ⊢ ((ℝ D 𝐹) ∈ (ℝ–cn→ℂ) → (ℝ D 𝐹):ℝ⟶ℂ)
9		fdm 6522	. . . . . . . . 9 ⊢ ((ℝ D 𝐹):ℝ⟶ℂ → dom (ℝ D 𝐹) = ℝ)
10	7, 8, 9	3syl 18	. . . . . . . 8 ⊢ (𝜑 → dom (ℝ D 𝐹) = ℝ)
11	6, 10	sseqtrrid 4020	. . . . . . 7 ⊢ (𝜑 → (-π(,)π) ⊆ dom (ℝ D 𝐹))
12		ssdmres 5876	. . . . . . 7 ⊢ ((-π(,)π) ⊆ dom (ℝ D 𝐹) ↔ dom ((ℝ D 𝐹) ↾ (-π(,)π)) = (-π(,)π))
13	11, 12	sylib 220	. . . . . 6 ⊢ (𝜑 → dom ((ℝ D 𝐹) ↾ (-π(,)π)) = (-π(,)π))
14	5, 13	syl5eq 2868	. . . . 5 ⊢ (𝜑 → dom 𝐺 = (-π(,)π))
15	14	difeq2d 4099	. . . 4 ⊢ (𝜑 → ((-π(,)π) ∖ dom 𝐺) = ((-π(,)π) ∖ (-π(,)π)))
16		difid 4330	. . . 4 ⊢ ((-π(,)π) ∖ (-π(,)π)) = ∅
17	15, 16	syl6eq 2872	. . 3 ⊢ (𝜑 → ((-π(,)π) ∖ dom 𝐺) = ∅)
18		0fin 8746	. . 3 ⊢ ∅ ∈ Fin
19	17, 18	eqeltrdi 2921	. 2 ⊢ (𝜑 → ((-π(,)π) ∖ dom 𝐺) ∈ Fin)
20		rescncf 23505	. . . 4 ⊢ ((-π(,)π) ⊆ ℝ → ((ℝ D 𝐹) ∈ (ℝ–cn→ℂ) → ((ℝ D 𝐹) ↾ (-π(,)π)) ∈ ((-π(,)π)–cn→ℂ)))
21	6, 7, 20	mpsyl 68	. . 3 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (-π(,)π)) ∈ ((-π(,)π)–cn→ℂ))
22	4	a1i 11	. . 3 ⊢ (𝜑 → 𝐺 = ((ℝ D 𝐹) ↾ (-π(,)π)))
23	14	oveq1d 7171	. . 3 ⊢ (𝜑 → (dom 𝐺–cn→ℂ) = ((-π(,)π)–cn→ℂ))
24	21, 22, 23	3eltr4d 2928	. 2 ⊢ (𝜑 → 𝐺 ∈ (dom 𝐺–cn→ℂ))
25		pire 25044	. . . . . 6 ⊢ π ∈ ℝ
26	25	renegcli 10947	. . . . 5 ⊢ -π ∈ ℝ
27	25	rexri 10699	. . . . 5 ⊢ π ∈ ℝ^*
28		icossre 12818	. . . . 5 ⊢ ((-π ∈ ℝ ∧ π ∈ ℝ^*) → (-π[,)π) ⊆ ℝ)
29	26, 27, 28	mp2an 690	. . . 4 ⊢ (-π[,)π) ⊆ ℝ
30		eldifi 4103	. . . 4 ⊢ (𝑥 ∈ ((-π[,)π) ∖ dom 𝐺) → 𝑥 ∈ (-π[,)π))
31	29, 30	sseldi 3965	. . 3 ⊢ (𝑥 ∈ ((-π[,)π) ∖ dom 𝐺) → 𝑥 ∈ ℝ)
32		limcresi 24483	. . . . . 6 ⊢ ((ℝ D 𝐹) lim_ℂ 𝑥) ⊆ (((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (𝑥(,)+∞))) lim_ℂ 𝑥)
33	4	reseq1i 5849	. . . . . . . 8 ⊢ (𝐺 ↾ (𝑥(,)+∞)) = (((ℝ D 𝐹) ↾ (-π(,)π)) ↾ (𝑥(,)+∞))
34		resres 5866	. . . . . . . 8 ⊢ (((ℝ D 𝐹) ↾ (-π(,)π)) ↾ (𝑥(,)+∞)) = ((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (𝑥(,)+∞)))
35	33, 34	eqtr2i 2845	. . . . . . 7 ⊢ ((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (𝑥(,)+∞))) = (𝐺 ↾ (𝑥(,)+∞))
36	35	oveq1i 7166	. . . . . 6 ⊢ (((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (𝑥(,)+∞))) lim_ℂ 𝑥) = ((𝐺 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥)
37	32, 36	sseqtri 4003	. . . . 5 ⊢ ((ℝ D 𝐹) lim_ℂ 𝑥) ⊆ ((𝐺 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥)
38	7	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (ℝ D 𝐹) ∈ (ℝ–cn→ℂ))
39		simpr 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
40	38, 39	cnlimci 24487	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((ℝ D 𝐹)‘𝑥) ∈ ((ℝ D 𝐹) lim_ℂ 𝑥))
41	37, 40	sseldi 3965	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((ℝ D 𝐹)‘𝑥) ∈ ((𝐺 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥))
42	41	ne0d 4301	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐺 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥) ≠ ∅)
43	31, 42	sylan2 594	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π[,)π) ∖ dom 𝐺)) → ((𝐺 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥) ≠ ∅)
44		negpitopissre 25124	. . . 4 ⊢ (-π(,]π) ⊆ ℝ
45		eldifi 4103	. . . 4 ⊢ (𝑥 ∈ ((-π(,]π) ∖ dom 𝐺) → 𝑥 ∈ (-π(,]π))
46	44, 45	sseldi 3965	. . 3 ⊢ (𝑥 ∈ ((-π(,]π) ∖ dom 𝐺) → 𝑥 ∈ ℝ)
47		limcresi 24483	. . . . . 6 ⊢ ((ℝ D 𝐹) lim_ℂ 𝑥) ⊆ (((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (-∞(,)𝑥))) lim_ℂ 𝑥)
48	4	reseq1i 5849	. . . . . . . 8 ⊢ (𝐺 ↾ (-∞(,)𝑥)) = (((ℝ D 𝐹) ↾ (-π(,)π)) ↾ (-∞(,)𝑥))
49		resres 5866	. . . . . . . 8 ⊢ (((ℝ D 𝐹) ↾ (-π(,)π)) ↾ (-∞(,)𝑥)) = ((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (-∞(,)𝑥)))
50	48, 49	eqtr2i 2845	. . . . . . 7 ⊢ ((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (-∞(,)𝑥))) = (𝐺 ↾ (-∞(,)𝑥))
51	50	oveq1i 7166	. . . . . 6 ⊢ (((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (-∞(,)𝑥))) lim_ℂ 𝑥) = ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥)
52	47, 51	sseqtri 4003	. . . . 5 ⊢ ((ℝ D 𝐹) lim_ℂ 𝑥) ⊆ ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥)
53	52, 40	sseldi 3965	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((ℝ D 𝐹)‘𝑥) ∈ ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥))
54	53	ne0d 4301	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥) ≠ ∅)
55	46, 54	sylan2 594	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π(,]π) ∖ dom 𝐺)) → ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥) ≠ ∅)
56		eqid 2821	. 2 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
57		ax-resscn 10594	. . . . . . 7 ⊢ ℝ ⊆ ℂ
58	57	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ⊆ ℂ)
59	1, 58	fssd 6528	. . . . . . 7 ⊢ (𝜑 → 𝐹:ℝ⟶ℂ)
60		ssid 3989	. . . . . . . 8 ⊢ ℝ ⊆ ℝ
61	60	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ⊆ ℝ)
62		dvcn 24518	. . . . . . 7 ⊢ (((ℝ ⊆ ℂ ∧ 𝐹:ℝ⟶ℂ ∧ ℝ ⊆ ℝ) ∧ dom (ℝ D 𝐹) = ℝ) → 𝐹 ∈ (ℝ–cn→ℂ))
63	58, 59, 61, 10, 62	syl31anc 1369	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (ℝ–cn→ℂ))
64		cncffvrn 23506	. . . . . 6 ⊢ ((ℝ ⊆ ℂ ∧ 𝐹 ∈ (ℝ–cn→ℂ)) → (𝐹 ∈ (ℝ–cn→ℝ) ↔ 𝐹:ℝ⟶ℝ))
65	58, 63, 64	syl2anc 586	. . . . 5 ⊢ (𝜑 → (𝐹 ∈ (ℝ–cn→ℝ) ↔ 𝐹:ℝ⟶ℝ))
66	1, 65	mpbird 259	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (ℝ–cn→ℝ))
67		eqid 2821	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
68	67	tgioo2 23411	. . . . . 6 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
69	67, 68, 68	cncfcn 23517	. . . . 5 ⊢ ((ℝ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℝ–cn→ℝ) = ((topGen‘ran (,)) Cn (topGen‘ran (,))))
70	58, 58, 69	syl2anc 586	. . . 4 ⊢ (𝜑 → (ℝ–cn→ℝ) = ((topGen‘ran (,)) Cn (topGen‘ran (,))))
71	66, 70	eleqtrd 2915	. . 3 ⊢ (𝜑 → 𝐹 ∈ ((topGen‘ran (,)) Cn (topGen‘ran (,))))
72		fouriercn.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ ℝ)
73		uniretop 23371	. . . 4 ⊢ ℝ = ∪ (topGen‘ran (,))
74	73	cncnpi 21886	. . 3 ⊢ ((𝐹 ∈ ((topGen‘ran (,)) Cn (topGen‘ran (,))) ∧ 𝑋 ∈ ℝ) → 𝐹 ∈ (((topGen‘ran (,)) CnP (topGen‘ran (,)))‘𝑋))
75	71, 72, 74	syl2anc 586	. 2 ⊢ (𝜑 → 𝐹 ∈ (((topGen‘ran (,)) CnP (topGen‘ran (,)))‘𝑋))
76		fouriercn.a	. 2 ⊢ 𝐴 = (𝑛 ∈ ℕ₀ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π))
77		fouriercn.b	. 2 ⊢ 𝐵 = (𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π))
78	1, 2, 3, 4, 19, 24, 43, 55, 56, 75, 76, 77	fouriercnp 42531	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 1537 ∈ wcel 2114 ≠ wne 3016 ∖ cdif 3933 ∩ cin 3935 ⊆ wss 3936 ∅c0 4291 ↦ cmpt 5146 dom cdm 5555 ran crn 5556 ↾ cres 5557 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 Fincfn 8509 ℂcc 10535 ℝcr 10536 0cc0 10537 + caddc 10540 · cmul 10542 +∞cpnf 10672 -∞cmnf 10673 ℝ^*cxr 10674 -cneg 10871 / cdiv 11297 ℕcn 11638 2c2 11693 ℕ₀cn0 11898 (,)cioo 12739 (,]cioc 12740 [,)cico 12741 Σcsu 15042 sincsin 15417 cosccos 15418 πcpi 15420 TopOpenctopn 16695 topGenctg 16711 ℂ_fldccnfld 20545 Cn ccn 21832 CnP ccnp 21833 –cn→ccncf 23484 ∫citg 24219 lim_ℂ climc 24460 D cdv 24461

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cc 9857 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3496 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 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-disj 5032 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-ofr 7410 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-omul 8107 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-dju 9330 df-card 9368 df-acn 9371 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-xnn0 11969 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ioo 12743 df-ioc 12744 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-fac 13635 df-bc 13664 df-hash 13692 df-shft 14426 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-limsup 14828 df-clim 14845 df-rlim 14846 df-sum 15043 df-ef 15421 df-sin 15423 df-cos 15424 df-pi 15426 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-fbas 20542 df-fg 20543 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-ntr 21628 df-cls 21629 df-nei 21706 df-lp 21744 df-perf 21745 df-cn 21835 df-cnp 21836 df-t1 21922 df-haus 21923 df-cmp 21995 df-tx 22170 df-hmeo 22363 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-xms 22930 df-ms 22931 df-tms 22932 df-cncf 23486 df-ovol 24065 df-vol 24066 df-mbf 24220 df-itg1 24221 df-itg2 24222 df-ibl 24223 df-itg 24224 df-0p 24271 df-ditg 24445 df-limc 24464 df-dv 24465

This theorem is referenced by: (None)

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