fouriercn - Mathbox for Glauco Siliprandi

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Theorem fouriercn 46249

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 46239 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 5842	. . . . . 6 ⊢ dom 𝐺 = dom ((ℝ D 𝐹) ↾ (-π(,)π))
6		ioossre 13299	. . . . . . . 8 ⊢ (-π(,)π) ⊆ ℝ
7		fouriercn.dv	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (ℝ–cn→ℂ))
8		cncff 24806	. . . . . . . . 9 ⊢ ((ℝ D 𝐹) ∈ (ℝ–cn→ℂ) → (ℝ D 𝐹):ℝ⟶ℂ)
9		fdm 6656	. . . . . . . . 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 3976	. . . . . . 7 ⊢ (𝜑 → (-π(,)π) ⊆ dom (ℝ D 𝐹))
12		ssdmres 5959	. . . . . . 7 ⊢ ((-π(,)π) ⊆ dom (ℝ D 𝐹) ↔ dom ((ℝ D 𝐹) ↾ (-π(,)π)) = (-π(,)π))
13	11, 12	sylib 218	. . . . . 6 ⊢ (𝜑 → dom ((ℝ D 𝐹) ↾ (-π(,)π)) = (-π(,)π))
14	5, 13	eqtrid 2777	. . . . 5 ⊢ (𝜑 → dom 𝐺 = (-π(,)π))
15	14	difeq2d 4074	. . . 4 ⊢ (𝜑 → ((-π(,)π) ∖ dom 𝐺) = ((-π(,)π) ∖ (-π(,)π)))
16		difid 4324	. . . 4 ⊢ ((-π(,)π) ∖ (-π(,)π)) = ∅
17	15, 16	eqtrdi 2781	. . 3 ⊢ (𝜑 → ((-π(,)π) ∖ dom 𝐺) = ∅)
18		0fi 8959	. . 3 ⊢ ∅ ∈ Fin
19	17, 18	eqeltrdi 2837	. 2 ⊢ (𝜑 → ((-π(,)π) ∖ dom 𝐺) ∈ Fin)
20		rescncf 24810	. . . 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 7356	. . 3 ⊢ (𝜑 → (dom 𝐺–cn→ℂ) = ((-π(,)π)–cn→ℂ))
24	21, 22, 23	3eltr4d 2844	. 2 ⊢ (𝜑 → 𝐺 ∈ (dom 𝐺–cn→ℂ))
25		pire 26386	. . . . . 6 ⊢ π ∈ ℝ
26	25	renegcli 11414	. . . . 5 ⊢ -π ∈ ℝ
27	25	rexri 11162	. . . . 5 ⊢ π ∈ ℝ^*
28		icossre 13320	. . . . 5 ⊢ ((-π ∈ ℝ ∧ π ∈ ℝ^*) → (-π[,)π) ⊆ ℝ)
29	26, 27, 28	mp2an 692	. . . 4 ⊢ (-π[,)π) ⊆ ℝ
30		eldifi 4079	. . . 4 ⊢ (𝑥 ∈ ((-π[,)π) ∖ dom 𝐺) → 𝑥 ∈ (-π[,)π))
31	29, 30	sselid 3930	. . 3 ⊢ (𝑥 ∈ ((-π[,)π) ∖ dom 𝐺) → 𝑥 ∈ ℝ)
32		limcresi 25806	. . . . . 6 ⊢ ((ℝ D 𝐹) lim_ℂ 𝑥) ⊆ (((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (𝑥(,)+∞))) lim_ℂ 𝑥)
33	4	reseq1i 5921	. . . . . . . 8 ⊢ (𝐺 ↾ (𝑥(,)+∞)) = (((ℝ D 𝐹) ↾ (-π(,)π)) ↾ (𝑥(,)+∞))
34		resres 5938	. . . . . . . 8 ⊢ (((ℝ D 𝐹) ↾ (-π(,)π)) ↾ (𝑥(,)+∞)) = ((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (𝑥(,)+∞)))
35	33, 34	eqtr2i 2754	. . . . . . 7 ⊢ ((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (𝑥(,)+∞))) = (𝐺 ↾ (𝑥(,)+∞))
36	35	oveq1i 7351	. . . . . 6 ⊢ (((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (𝑥(,)+∞))) lim_ℂ 𝑥) = ((𝐺 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥)
37	32, 36	sseqtri 3981	. . . . 5 ⊢ ((ℝ D 𝐹) lim_ℂ 𝑥) ⊆ ((𝐺 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥)
38	7	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (ℝ D 𝐹) ∈ (ℝ–cn→ℂ))
39		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
40	38, 39	cnlimci 25810	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((ℝ D 𝐹)‘𝑥) ∈ ((ℝ D 𝐹) lim_ℂ 𝑥))
41	37, 40	sselid 3930	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((ℝ D 𝐹)‘𝑥) ∈ ((𝐺 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥))
42	41	ne0d 4290	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐺 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥) ≠ ∅)
43	31, 42	sylan2 593	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π[,)π) ∖ dom 𝐺)) → ((𝐺 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥) ≠ ∅)
44		negpitopissre 26469	. . . 4 ⊢ (-π(,]π) ⊆ ℝ
45		eldifi 4079	. . . 4 ⊢ (𝑥 ∈ ((-π(,]π) ∖ dom 𝐺) → 𝑥 ∈ (-π(,]π))
46	44, 45	sselid 3930	. . 3 ⊢ (𝑥 ∈ ((-π(,]π) ∖ dom 𝐺) → 𝑥 ∈ ℝ)
47		limcresi 25806	. . . . . 6 ⊢ ((ℝ D 𝐹) lim_ℂ 𝑥) ⊆ (((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (-∞(,)𝑥))) lim_ℂ 𝑥)
48	4	reseq1i 5921	. . . . . . . 8 ⊢ (𝐺 ↾ (-∞(,)𝑥)) = (((ℝ D 𝐹) ↾ (-π(,)π)) ↾ (-∞(,)𝑥))
49		resres 5938	. . . . . . . 8 ⊢ (((ℝ D 𝐹) ↾ (-π(,)π)) ↾ (-∞(,)𝑥)) = ((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (-∞(,)𝑥)))
50	48, 49	eqtr2i 2754	. . . . . . 7 ⊢ ((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (-∞(,)𝑥))) = (𝐺 ↾ (-∞(,)𝑥))
51	50	oveq1i 7351	. . . . . 6 ⊢ (((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (-∞(,)𝑥))) lim_ℂ 𝑥) = ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥)
52	47, 51	sseqtri 3981	. . . . 5 ⊢ ((ℝ D 𝐹) lim_ℂ 𝑥) ⊆ ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥)
53	52, 40	sselid 3930	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((ℝ D 𝐹)‘𝑥) ∈ ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥))
54	53	ne0d 4290	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥) ≠ ∅)
55	46, 54	sylan2 593	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π(,]π) ∖ dom 𝐺)) → ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥) ≠ ∅)
56		eqid 2730	. 2 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
57		ax-resscn 11055	. . . . . . 7 ⊢ ℝ ⊆ ℂ
58	57	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ⊆ ℂ)
59	1, 58	fssd 6664	. . . . . . 7 ⊢ (𝜑 → 𝐹:ℝ⟶ℂ)
60		ssid 3955	. . . . . . . 8 ⊢ ℝ ⊆ ℝ
61	60	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ⊆ ℝ)
62		dvcn 25843	. . . . . . 7 ⊢ (((ℝ ⊆ ℂ ∧ 𝐹:ℝ⟶ℂ ∧ ℝ ⊆ ℝ) ∧ dom (ℝ D 𝐹) = ℝ) → 𝐹 ∈ (ℝ–cn→ℂ))
63	58, 59, 61, 10, 62	syl31anc 1375	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (ℝ–cn→ℂ))
64		cncfcdm 24811	. . . . . 6 ⊢ ((ℝ ⊆ ℂ ∧ 𝐹 ∈ (ℝ–cn→ℂ)) → (𝐹 ∈ (ℝ–cn→ℝ) ↔ 𝐹:ℝ⟶ℝ))
65	58, 63, 64	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝐹 ∈ (ℝ–cn→ℝ) ↔ 𝐹:ℝ⟶ℝ))
66	1, 65	mpbird 257	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (ℝ–cn→ℝ))
67		eqid 2730	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
68		tgioo4 24713	. . . . . 6 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
69	67, 68, 68	cncfcn 24823	. . . . 5 ⊢ ((ℝ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℝ–cn→ℝ) = ((topGen‘ran (,)) Cn (topGen‘ran (,))))
70	58, 58, 69	syl2anc 584	. . . 4 ⊢ (𝜑 → (ℝ–cn→ℝ) = ((topGen‘ran (,)) Cn (topGen‘ran (,))))
71	66, 70	eleqtrd 2831	. . 3 ⊢ (𝜑 → 𝐹 ∈ ((topGen‘ran (,)) Cn (topGen‘ran (,))))
72		fouriercn.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ ℝ)
73		uniretop 24670	. . . 4 ⊢ ℝ = ∪ (topGen‘ran (,))
74	73	cncnpi 23186	. . 3 ⊢ ((𝐹 ∈ ((topGen‘ran (,)) Cn (topGen‘ran (,))) ∧ 𝑋 ∈ ℝ) → 𝐹 ∈ (((topGen‘ran (,)) CnP (topGen‘ran (,)))‘𝑋))
75	71, 72, 74	syl2anc 584	. 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 46243	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 206 ∧ wa 395 = wceq 1541 ∈ wcel 2110 ≠ wne 2926 ∖ cdif 3897 ∩ cin 3899 ⊆ wss 3900 ∅c0 4281 ↦ cmpt 5170 dom cdm 5614 ran crn 5615 ↾ cres 5616 ⟶wf 6473 ‘cfv 6477 (class class class)co 7341 Fincfn 8864 ℂcc 10996 ℝcr 10997 0cc0 10998 + caddc 11001 · cmul 11003 +∞cpnf 11135 -∞cmnf 11136 ℝ^*cxr 11137 -cneg 11337 / cdiv 11766 ℕcn 12117 2c2 12172 ℕ₀cn0 12373 (,)cioo 13237 (,]cioc 13238 [,)cico 13239 Σcsu 15585 sincsin 15962 cosccos 15963 πcpi 15965 TopOpenctopn 17317 topGenctg 17333 ℂ_fldccnfld 21284 Cn ccn 23132 CnP ccnp 23133 –cn→ccncf 24789 ∫citg 25539 lim_ℂ climc 25783 D cdv 25784

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 1968 ax-7 2009 ax-8 2112 ax-9 2120 ax-10 2143 ax-11 2159 ax-12 2179 ax-ext 2702 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7663 ax-inf2 9526 ax-cc 10318 ax-cnex 11054 ax-resscn 11055 ax-1cn 11056 ax-icn 11057 ax-addcl 11058 ax-addrcl 11059 ax-mulcl 11060 ax-mulrcl 11061 ax-mulcom 11062 ax-addass 11063 ax-mulass 11064 ax-distr 11065 ax-i2m1 11066 ax-1ne0 11067 ax-1rid 11068 ax-rnegex 11069 ax-rrecex 11070 ax-cnre 11071 ax-pre-lttri 11072 ax-pre-lttrn 11073 ax-pre-ltadd 11074 ax-pre-mulgt0 11075 ax-pre-sup 11076 ax-addf 11077

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3344 df-reu 3345 df-rab 3394 df-v 3436 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-symdif 4201 df-nul 4282 df-if 4474 df-pw 4550 df-sn 4575 df-pr 4577 df-tp 4579 df-op 4581 df-uni 4858 df-int 4896 df-iun 4941 df-iin 4942 df-disj 5057 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6244 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6433 df-fun 6479 df-fn 6480 df-f 6481 df-f1 6482 df-fo 6483 df-f1o 6484 df-fv 6485 df-isom 6486 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-of 7605 df-ofr 7606 df-om 7792 df-1st 7916 df-2nd 7917 df-supp 8086 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-2o 8381 df-oadd 8384 df-omul 8385 df-er 8617 df-map 8747 df-pm 8748 df-ixp 8817 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-fsupp 9241 df-fi 9290 df-sup 9321 df-inf 9322 df-oi 9391 df-dju 9786 df-card 9824 df-acn 9827 df-pnf 11140 df-mnf 11141 df-xr 11142 df-ltxr 11143 df-le 11144 df-sub 11338 df-neg 11339 df-div 11767 df-nn 12118 df-2 12180 df-3 12181 df-4 12182 df-5 12183 df-6 12184 df-7 12185 df-8 12186 df-9 12187 df-n0 12374 df-xnn0 12447 df-z 12461 df-dec 12581 df-uz 12725 df-q 12839 df-rp 12883 df-xneg 13003 df-xadd 13004 df-xmul 13005 df-ioo 13241 df-ioc 13242 df-ico 13243 df-icc 13244 df-fz 13400 df-fzo 13547 df-fl 13688 df-mod 13766 df-seq 13901 df-exp 13961 df-fac 14173 df-bc 14202 df-hash 14230 df-shft 14966 df-cj 14998 df-re 14999 df-im 15000 df-sqrt 15134 df-abs 15135 df-limsup 15370 df-clim 15387 df-rlim 15388 df-sum 15586 df-ef 15966 df-sin 15968 df-cos 15969 df-pi 15971 df-struct 17050 df-sets 17067 df-slot 17085 df-ndx 17097 df-base 17113 df-ress 17134 df-plusg 17166 df-mulr 17167 df-starv 17168 df-sca 17169 df-vsca 17170 df-ip 17171 df-tset 17172 df-ple 17173 df-ds 17175 df-unif 17176 df-hom 17177 df-cco 17178 df-rest 17318 df-topn 17319 df-0g 17337 df-gsum 17338 df-topgen 17339 df-pt 17340 df-prds 17343 df-xrs 17398 df-qtop 17403 df-imas 17404 df-xps 17406 df-mre 17480 df-mrc 17481 df-acs 17483 df-mgm 18540 df-sgrp 18619 df-mnd 18635 df-submnd 18684 df-mulg 18973 df-cntz 19222 df-cmn 19687 df-psmet 21276 df-xmet 21277 df-met 21278 df-bl 21279 df-mopn 21280 df-fbas 21281 df-fg 21282 df-cnfld 21285 df-top 22802 df-topon 22819 df-topsp 22841 df-bases 22854 df-cld 22927 df-ntr 22928 df-cls 22929 df-nei 23006 df-lp 23044 df-perf 23045 df-cn 23135 df-cnp 23136 df-t1 23222 df-haus 23223 df-cmp 23295 df-tx 23470 df-hmeo 23663 df-fil 23754 df-fm 23846 df-flim 23847 df-flf 23848 df-xms 24228 df-ms 24229 df-tms 24230 df-cncf 24791 df-ovol 25385 df-vol 25386 df-mbf 25540 df-itg1 25541 df-itg2 25542 df-ibl 25543 df-itg 25544 df-0p 25591 df-ditg 25768 df-limc 25787 df-dv 25788

This theorem is referenced by: (None)

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