fouriercn - Mathbox for Glauco Siliprandi

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Theorem fouriercn 46392

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 46382 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 5850	. . . . . 6 ⊢ dom 𝐺 = dom ((ℝ D 𝐹) ↾ (-π(,)π))
6		ioossre 13314	. . . . . . . 8 ⊢ (-π(,)π) ⊆ ℝ
7		fouriercn.dv	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (ℝ–cn→ℂ))
8		cncff 24833	. . . . . . . . 9 ⊢ ((ℝ D 𝐹) ∈ (ℝ–cn→ℂ) → (ℝ D 𝐹):ℝ⟶ℂ)
9		fdm 6668	. . . . . . . . 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 3974	. . . . . . 7 ⊢ (𝜑 → (-π(,)π) ⊆ dom (ℝ D 𝐹))
12		ssdmres 5969	. . . . . . 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 2780	. . . . 5 ⊢ (𝜑 → dom 𝐺 = (-π(,)π))
15	14	difeq2d 4075	. . . 4 ⊢ (𝜑 → ((-π(,)π) ∖ dom 𝐺) = ((-π(,)π) ∖ (-π(,)π)))
16		difid 4325	. . . 4 ⊢ ((-π(,)π) ∖ (-π(,)π)) = ∅
17	15, 16	eqtrdi 2784	. . 3 ⊢ (𝜑 → ((-π(,)π) ∖ dom 𝐺) = ∅)
18		0fi 8975	. . 3 ⊢ ∅ ∈ Fin
19	17, 18	eqeltrdi 2841	. 2 ⊢ (𝜑 → ((-π(,)π) ∖ dom 𝐺) ∈ Fin)
20		rescncf 24837	. . . 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 7370	. . 3 ⊢ (𝜑 → (dom 𝐺–cn→ℂ) = ((-π(,)π)–cn→ℂ))
24	21, 22, 23	3eltr4d 2848	. 2 ⊢ (𝜑 → 𝐺 ∈ (dom 𝐺–cn→ℂ))
25		pire 26413	. . . . . 6 ⊢ π ∈ ℝ
26	25	renegcli 11433	. . . . 5 ⊢ -π ∈ ℝ
27	25	rexri 11181	. . . . 5 ⊢ π ∈ ℝ^*
28		icossre 13335	. . . . 5 ⊢ ((-π ∈ ℝ ∧ π ∈ ℝ^*) → (-π[,)π) ⊆ ℝ)
29	26, 27, 28	mp2an 692	. . . 4 ⊢ (-π[,)π) ⊆ ℝ
30		eldifi 4080	. . . 4 ⊢ (𝑥 ∈ ((-π[,)π) ∖ dom 𝐺) → 𝑥 ∈ (-π[,)π))
31	29, 30	sselid 3928	. . 3 ⊢ (𝑥 ∈ ((-π[,)π) ∖ dom 𝐺) → 𝑥 ∈ ℝ)
32		limcresi 25833	. . . . . 6 ⊢ ((ℝ D 𝐹) lim_ℂ 𝑥) ⊆ (((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (𝑥(,)+∞))) lim_ℂ 𝑥)
33	4	reseq1i 5931	. . . . . . . 8 ⊢ (𝐺 ↾ (𝑥(,)+∞)) = (((ℝ D 𝐹) ↾ (-π(,)π)) ↾ (𝑥(,)+∞))
34		resres 5948	. . . . . . . 8 ⊢ (((ℝ D 𝐹) ↾ (-π(,)π)) ↾ (𝑥(,)+∞)) = ((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (𝑥(,)+∞)))
35	33, 34	eqtr2i 2757	. . . . . . 7 ⊢ ((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (𝑥(,)+∞))) = (𝐺 ↾ (𝑥(,)+∞))
36	35	oveq1i 7365	. . . . . 6 ⊢ (((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (𝑥(,)+∞))) lim_ℂ 𝑥) = ((𝐺 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥)
37	32, 36	sseqtri 3979	. . . . 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 25837	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((ℝ D 𝐹)‘𝑥) ∈ ((ℝ D 𝐹) lim_ℂ 𝑥))
41	37, 40	sselid 3928	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((ℝ D 𝐹)‘𝑥) ∈ ((𝐺 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥))
42	41	ne0d 4291	. . 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 26496	. . . 4 ⊢ (-π(,]π) ⊆ ℝ
45		eldifi 4080	. . . 4 ⊢ (𝑥 ∈ ((-π(,]π) ∖ dom 𝐺) → 𝑥 ∈ (-π(,]π))
46	44, 45	sselid 3928	. . 3 ⊢ (𝑥 ∈ ((-π(,]π) ∖ dom 𝐺) → 𝑥 ∈ ℝ)
47		limcresi 25833	. . . . . 6 ⊢ ((ℝ D 𝐹) lim_ℂ 𝑥) ⊆ (((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (-∞(,)𝑥))) lim_ℂ 𝑥)
48	4	reseq1i 5931	. . . . . . . 8 ⊢ (𝐺 ↾ (-∞(,)𝑥)) = (((ℝ D 𝐹) ↾ (-π(,)π)) ↾ (-∞(,)𝑥))
49		resres 5948	. . . . . . . 8 ⊢ (((ℝ D 𝐹) ↾ (-π(,)π)) ↾ (-∞(,)𝑥)) = ((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (-∞(,)𝑥)))
50	48, 49	eqtr2i 2757	. . . . . . 7 ⊢ ((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (-∞(,)𝑥))) = (𝐺 ↾ (-∞(,)𝑥))
51	50	oveq1i 7365	. . . . . 6 ⊢ (((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (-∞(,)𝑥))) lim_ℂ 𝑥) = ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥)
52	47, 51	sseqtri 3979	. . . . 5 ⊢ ((ℝ D 𝐹) lim_ℂ 𝑥) ⊆ ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥)
53	52, 40	sselid 3928	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((ℝ D 𝐹)‘𝑥) ∈ ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥))
54	53	ne0d 4291	. . 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 2733	. 2 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
57		ax-resscn 11074	. . . . . . 7 ⊢ ℝ ⊆ ℂ
58	57	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ⊆ ℂ)
59	1, 58	fssd 6676	. . . . . . 7 ⊢ (𝜑 → 𝐹:ℝ⟶ℂ)
60		ssid 3953	. . . . . . . 8 ⊢ ℝ ⊆ ℝ
61	60	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ⊆ ℝ)
62		dvcn 25870	. . . . . . 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 24838	. . . . . 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 2733	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
68		tgioo4 24740	. . . . . 6 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
69	67, 68, 68	cncfcn 24850	. . . . 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 2835	. . 3 ⊢ (𝜑 → 𝐹 ∈ ((topGen‘ran (,)) Cn (topGen‘ran (,))))
72		fouriercn.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ ℝ)
73		uniretop 24697	. . . 4 ⊢ ℝ = ∪ (topGen‘ran (,))
74	73	cncnpi 23213	. . 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 46386	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 2113 ≠ wne 2929 ∖ cdif 3895 ∩ cin 3897 ⊆ wss 3898 ∅c0 4282 ↦ cmpt 5176 dom cdm 5621 ran crn 5622 ↾ cres 5623 ⟶wf 6485 ‘cfv 6489 (class class class)co 7355 Fincfn 8879 ℂcc 11015 ℝcr 11016 0cc0 11017 + caddc 11020 · cmul 11022 +∞cpnf 11154 -∞cmnf 11155 ℝ^*cxr 11156 -cneg 11356 / cdiv 11785 ℕcn 12136 2c2 12191 ℕ₀cn0 12392 (,)cioo 13252 (,]cioc 13253 [,)cico 13254 Σcsu 15600 sincsin 15977 cosccos 15978 πcpi 15980 TopOpenctopn 17332 topGenctg 17348 ℂ_fldccnfld 21300 Cn ccn 23159 CnP ccnp 23160 –cn→ccncf 24816 ∫citg 25566 lim_ℂ climc 25810 D cdv 25811

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 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-inf2 9542 ax-cc 10337 ax-cnex 11073 ax-resscn 11074 ax-1cn 11075 ax-icn 11076 ax-addcl 11077 ax-addrcl 11078 ax-mulcl 11079 ax-mulrcl 11080 ax-mulcom 11081 ax-addass 11082 ax-mulass 11083 ax-distr 11084 ax-i2m1 11085 ax-1ne0 11086 ax-1rid 11087 ax-rnegex 11088 ax-rrecex 11089 ax-cnre 11090 ax-pre-lttri 11091 ax-pre-lttrn 11092 ax-pre-ltadd 11093 ax-pre-mulgt0 11094 ax-pre-sup 11095 ax-addf 11096

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 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-symdif 4202 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4861 df-int 4900 df-iun 4945 df-iin 4946 df-disj 5063 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-isom 6498 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-of 7619 df-ofr 7620 df-om 7806 df-1st 7930 df-2nd 7931 df-supp 8100 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-2o 8395 df-oadd 8398 df-omul 8399 df-er 8631 df-map 8761 df-pm 8762 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9257 df-fi 9306 df-sup 9337 df-inf 9338 df-oi 9407 df-dju 9805 df-card 9843 df-acn 9846 df-pnf 11159 df-mnf 11160 df-xr 11161 df-ltxr 11162 df-le 11163 df-sub 11357 df-neg 11358 df-div 11786 df-nn 12137 df-2 12199 df-3 12200 df-4 12201 df-5 12202 df-6 12203 df-7 12204 df-8 12205 df-9 12206 df-n0 12393 df-xnn0 12466 df-z 12480 df-dec 12599 df-uz 12743 df-q 12853 df-rp 12897 df-xneg 13017 df-xadd 13018 df-xmul 13019 df-ioo 13256 df-ioc 13257 df-ico 13258 df-icc 13259 df-fz 13415 df-fzo 13562 df-fl 13703 df-mod 13781 df-seq 13916 df-exp 13976 df-fac 14188 df-bc 14217 df-hash 14245 df-shft 14981 df-cj 15013 df-re 15014 df-im 15015 df-sqrt 15149 df-abs 15150 df-limsup 15385 df-clim 15402 df-rlim 15403 df-sum 15601 df-ef 15981 df-sin 15983 df-cos 15984 df-pi 15986 df-struct 17065 df-sets 17082 df-slot 17100 df-ndx 17112 df-base 17128 df-ress 17149 df-plusg 17181 df-mulr 17182 df-starv 17183 df-sca 17184 df-vsca 17185 df-ip 17186 df-tset 17187 df-ple 17188 df-ds 17190 df-unif 17191 df-hom 17192 df-cco 17193 df-rest 17333 df-topn 17334 df-0g 17352 df-gsum 17353 df-topgen 17354 df-pt 17355 df-prds 17358 df-xrs 17414 df-qtop 17419 df-imas 17420 df-xps 17422 df-mre 17496 df-mrc 17497 df-acs 17499 df-mgm 18556 df-sgrp 18635 df-mnd 18651 df-submnd 18700 df-mulg 18989 df-cntz 19237 df-cmn 19702 df-psmet 21292 df-xmet 21293 df-met 21294 df-bl 21295 df-mopn 21296 df-fbas 21297 df-fg 21298 df-cnfld 21301 df-top 22829 df-topon 22846 df-topsp 22868 df-bases 22881 df-cld 22954 df-ntr 22955 df-cls 22956 df-nei 23033 df-lp 23071 df-perf 23072 df-cn 23162 df-cnp 23163 df-t1 23249 df-haus 23250 df-cmp 23322 df-tx 23497 df-hmeo 23690 df-fil 23781 df-fm 23873 df-flim 23874 df-flf 23875 df-xms 24255 df-ms 24256 df-tms 24257 df-cncf 24818 df-ovol 25412 df-vol 25413 df-mbf 25567 df-itg1 25568 df-itg2 25569 df-ibl 25570 df-itg 25571 df-0p 25618 df-ditg 25795 df-limc 25814 df-dv 25815

This theorem is referenced by: (None)

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