fouriercn - Mathbox for Glauco Siliprandi

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Theorem fouriercn 47060

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 47050 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 5888	. . . . . 6 ⊢ dom 𝐺 = dom ((ℝ D 𝐹) ↾ (-π(,)π))
6		ioossre 13460	. . . . . . . 8 ⊢ (-π(,)π) ⊆ ℝ
7		fouriercn.dv	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (ℝ–cn→ℂ))
8		cncff 25121	. . . . . . . . 9 ⊢ ((ℝ D 𝐹) ∈ (ℝ–cn→ℂ) → (ℝ D 𝐹):ℝ⟶ℂ)
9		fdm 6712	. . . . . . . . 9 ⊢ ((ℝ D 𝐹):ℝ⟶ℂ → dom (ℝ D 𝐹) = ℝ)
10	7, 8, 9	3syl 19	. . . . . . . 8 ⊢ (𝜑 → dom (ℝ D 𝐹) = ℝ)
11	6, 10	sseqtrrid 3974	. . . . . . 7 ⊢ (𝜑 → (-π(,)π) ⊆ dom (ℝ D 𝐹))
12		ssdmres 6006	. . . . . . 7 ⊢ ((-π(,)π) ⊆ dom (ℝ D 𝐹) ↔ dom ((ℝ D 𝐹) ↾ (-π(,)π)) = (-π(,)π))
13	11, 12	sylib 221	. . . . . 6 ⊢ (𝜑 → dom ((ℝ D 𝐹) ↾ (-π(,)π)) = (-π(,)π))
14	5, 13	eqtrid 2807	. . . . 5 ⊢ (𝜑 → dom 𝐺 = (-π(,)π))
15	14	difeq2d 4074	. . . 4 ⊢ (𝜑 → ((-π(,)π) ∖ dom 𝐺) = ((-π(,)π) ∖ (-π(,)π)))
16		difid 4325	. . . 4 ⊢ ((-π(,)π) ∖ (-π(,)π)) = ∅
17	15, 16	eqtrdi 2811	. . 3 ⊢ (𝜑 → ((-π(,)π) ∖ dom 𝐺) = ∅)
18		0fi 9049	. . 3 ⊢ ∅ ∈ Fin
19	17, 18	eqeltrdi 2868	. 2 ⊢ (𝜑 → ((-π(,)π) ∖ dom 𝐺) ∈ Fin)
20		rescncf 25125	. . . 4 ⊢ ((-π(,)π) ⊆ ℝ → ((ℝ D 𝐹) ∈ (ℝ–cn→ℂ) → ((ℝ D 𝐹) ↾ (-π(,)π)) ∈ ((-π(,)π)–cn→ℂ)))
21	6, 7, 20	mpsyl 69	. . 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 7428	. . 3 ⊢ (𝜑 → (dom 𝐺–cn→ℂ) = ((-π(,)π)–cn→ℂ))
24	21, 22, 23	3eltr4d 2875	. 2 ⊢ (𝜑 → 𝐺 ∈ (dom 𝐺–cn→ℂ))
25		pire 26692	. . . . . 6 ⊢ π ∈ ℝ
26	25	renegcli 11543	. . . . 5 ⊢ -π ∈ ℝ
27	25	rexri 11291	. . . . 5 ⊢ π ∈ ℝ^*
28		icossre 13481	. . . . 5 ⊢ ((-π ∈ ℝ ∧ π ∈ ℝ^*) → (-π[,)π) ⊆ ℝ)
29	26, 27, 28	mp2an 705	. . . 4 ⊢ (-π[,)π) ⊆ ℝ
30		eldifi 4078	. . . 4 ⊢ (𝑥 ∈ ((-π[,)π) ∖ dom 𝐺) → 𝑥 ∈ (-π[,)π))
31	29, 30	sselid 3929	. . 3 ⊢ (𝑥 ∈ ((-π[,)π) ∖ dom 𝐺) → 𝑥 ∈ ℝ)
32		limcresi 26112	. . . . . 6 ⊢ ((ℝ D 𝐹) lim_ℂ 𝑥) ⊆ (((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (𝑥(,)+∞))) lim_ℂ 𝑥)
33	4	reseq1i 5968	. . . . . . . 8 ⊢ (𝐺 ↾ (𝑥(,)+∞)) = (((ℝ D 𝐹) ↾ (-π(,)π)) ↾ (𝑥(,)+∞))
34		resres 5985	. . . . . . . 8 ⊢ (((ℝ D 𝐹) ↾ (-π(,)π)) ↾ (𝑥(,)+∞)) = ((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (𝑥(,)+∞)))
35	33, 34	eqtr2i 2784	. . . . . . 7 ⊢ ((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (𝑥(,)+∞))) = (𝐺 ↾ (𝑥(,)+∞))
36	35	oveq1i 7423	. . . . . 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 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (ℝ D 𝐹) ∈ (ℝ–cn→ℂ))
39		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
40	38, 39	cnlimci 26116	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((ℝ D 𝐹)‘𝑥) ∈ ((ℝ D 𝐹) lim_ℂ 𝑥))
41	37, 40	sselid 3929	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((ℝ D 𝐹)‘𝑥) ∈ ((𝐺 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥))
42	41	ne0d 4288	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐺 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥) ≠ ∅)
43	31, 42	sylan2 605	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π[,)π) ∖ dom 𝐺)) → ((𝐺 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥) ≠ ∅)
44		negpitopissre 26777	. . . 4 ⊢ (-π(,]π) ⊆ ℝ
45		eldifi 4078	. . . 4 ⊢ (𝑥 ∈ ((-π(,]π) ∖ dom 𝐺) → 𝑥 ∈ (-π(,]π))
46	44, 45	sselid 3929	. . 3 ⊢ (𝑥 ∈ ((-π(,]π) ∖ dom 𝐺) → 𝑥 ∈ ℝ)
47		limcresi 26112	. . . . . 6 ⊢ ((ℝ D 𝐹) lim_ℂ 𝑥) ⊆ (((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (-∞(,)𝑥))) lim_ℂ 𝑥)
48	4	reseq1i 5968	. . . . . . . 8 ⊢ (𝐺 ↾ (-∞(,)𝑥)) = (((ℝ D 𝐹) ↾ (-π(,)π)) ↾ (-∞(,)𝑥))
49		resres 5985	. . . . . . . 8 ⊢ (((ℝ D 𝐹) ↾ (-π(,)π)) ↾ (-∞(,)𝑥)) = ((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (-∞(,)𝑥)))
50	48, 49	eqtr2i 2784	. . . . . . 7 ⊢ ((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (-∞(,)𝑥))) = (𝐺 ↾ (-∞(,)𝑥))
51	50	oveq1i 7423	. . . . . 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 3929	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((ℝ D 𝐹)‘𝑥) ∈ ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥))
54	53	ne0d 4288	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥) ≠ ∅)
55	46, 54	sylan2 605	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π(,]π) ∖ dom 𝐺)) → ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥) ≠ ∅)
56		eqid 2760	. 2 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
57		ax-resscn 11181	. . . . . . 7 ⊢ ℝ ⊆ ℂ
58	57	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ⊆ ℂ)
59	1, 58	fssd 6720	. . . . . . 7 ⊢ (𝜑 → 𝐹:ℝ⟶ℂ)
60		ssid 3953	. . . . . . . 8 ⊢ ℝ ⊆ ℝ
61	60	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ⊆ ℝ)
62		dvcn 26148	. . . . . . 7 ⊢ (((ℝ ⊆ ℂ ∧ 𝐹:ℝ⟶ℂ ∧ ℝ ⊆ ℝ) ∧ dom (ℝ D 𝐹) = ℝ) → 𝐹 ∈ (ℝ–cn→ℂ))
63	58, 59, 61, 10, 62	syl31anc 1400	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (ℝ–cn→ℂ))
64		cncfcdm 25126	. . . . . 6 ⊢ ((ℝ ⊆ ℂ ∧ 𝐹 ∈ (ℝ–cn→ℂ)) → (𝐹 ∈ (ℝ–cn→ℝ) ↔ 𝐹:ℝ⟶ℝ))
65	58, 63, 64	syl2anc 596	. . . . 5 ⊢ (𝜑 → (𝐹 ∈ (ℝ–cn→ℝ) ↔ 𝐹:ℝ⟶ℝ))
66	1, 65	mpbird 260	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (ℝ–cn→ℝ))
67		eqid 2760	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
68		tgioo4 25031	. . . . . 6 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
69	67, 68, 68	cncfcn 25138	. . . . 5 ⊢ ((ℝ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℝ–cn→ℝ) = ((topGen‘ran (,)) Cn (topGen‘ran (,))))
70	58, 58, 69	syl2anc 596	. . . 4 ⊢ (𝜑 → (ℝ–cn→ℝ) = ((topGen‘ran (,)) Cn (topGen‘ran (,))))
71	66, 70	eleqtrd 2862	. . 3 ⊢ (𝜑 → 𝐹 ∈ ((topGen‘ran (,)) Cn (topGen‘ran (,))))
72		fouriercn.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ ℝ)
73		uniretop 24988	. . . 4 ⊢ ℝ = ∪ (topGen‘ran (,))
74	73	cncnpi 23503	. . 3 ⊢ ((𝐹 ∈ ((topGen‘ran (,)) Cn (topGen‘ran (,))) ∧ 𝑋 ∈ ℝ) → 𝐹 ∈ (((topGen‘ran (,)) CnP (topGen‘ran (,)))‘𝑋))
75	71, 72, 74	syl2anc 596	. 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 47054	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 2955 ∖ cdif 3896 ∩ cin 3898 ⊆ wss 3899 ∅c0 4279 ↦ cmpt 5186 dom cdm 5655 ran crn 5656 ↾ cres 5657 ⟶wf 6529 ‘cfv 6533 (class class class)co 7413 Fincfn 8952 ℂcc 11122 ℝcr 11123 0cc0 11124 + caddc 11127 · cmul 11129 +∞cpnf 11264 -∞cmnf 11265 ℝ^*cxr 11266 -cneg 11466 / cdiv 11895 ℕcn 12257 2c2 12319 ℕ₀cn0 12528 (,)cioo 13398 (,]cioc 13399 [,)cico 13400 Σcsu 15773 sincsin 16149 cosccos 16150 πcpi 16152 TopOpenctopn 17506 topGenctg 17522 ℂ_fldccnfld 21585 Cn ccn 23449 CnP ccnp 23450 –cn→ccncf 25104 ∫citg 25846 lim_ℂ climc 26089 D cdv 26090

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 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-inf2 9620 ax-cc 10437 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 ax-addf 11203

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 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 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 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 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7678 df-ofr 7679 df-om 7863 df-1st 7986 df-2nd 7987 df-supp 8159 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-oadd 8459 df-omul 8460 df-er 8696 df-map 8828 df-pm 8829 df-ixp 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-fsupp 9332 df-fi 9381 df-sup 9412 df-inf 9413 df-oi 9482 df-dju 9906 df-card 9944 df-acn 9947 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-div 11896 df-nn 12258 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12529 df-xnn0 12602 df-z 12616 df-dec 12737 df-uz 12888 df-q 12998 df-rp 13043 df-xneg 13163 df-xadd 13164 df-xmul 13165 df-ioo 13402 df-ioc 13403 df-ico 13404 df-icc 13405 df-fz 13562 df-fzo 13710 df-fl 13853 df-mod 13931 df-seq 14066 df-exp 14126 df-fac 14338 df-bc 14367 df-hash 14395 df-shft 15140 df-cj 15186 df-re 15187 df-im 15188 df-sqrt 15322 df-abs 15323 df-limsup 15558 df-clim 15575 df-rlim 15576 df-sum 15774 df-ef 16153 df-sin 16155 df-cos 16156 df-pi 16158 df-struct 17239 df-sets 17256 df-slot 17274 df-ndx 17286 df-base 17302 df-ress 17323 df-plusg 17355 df-mulr 17356 df-starv 17357 df-sca 17358 df-vsca 17359 df-ip 17360 df-tset 17361 df-ple 17362 df-ds 17364 df-unif 17365 df-hom 17366 df-cco 17367 df-rest 17507 df-topn 17508 df-0g 17526 df-gsum 17527 df-topgen 17528 df-pt 17529 df-prds 17532 df-xrs 17588 df-qtop 17593 df-imas 17594 df-xps 17596 df-mre 17670 df-mrc 17671 df-acs 17673 df-mgm 18730 df-sgrp 18821 df-mnd 18837 df-submnd 18892 df-mulg 19191 df-cntz 19444 df-cmn 19909 df-psmet 21577 df-xmet 21578 df-met 21579 df-bl 21580 df-mopn 21581 df-fbas 21582 df-fg 21583 df-cnfld 21586 df-top 23119 df-topon 23136 df-topsp 23158 df-bases 23171 df-cld 23244 df-ntr 23245 df-cls 23246 df-nei 23323 df-lp 23361 df-perf 23362 df-cn 23452 df-cnp 23453 df-t1 23539 df-haus 23540 df-cmp 23612 df-tx 23788 df-hmeo 23981 df-fil 24072 df-fm 24164 df-flim 24165 df-flf 24166 df-xms 24546 df-ms 24547 df-tms 24548 df-cncf 25106 df-ovol 25692 df-vol 25693 df-mbf 25847 df-itg1 25848 df-itg2 25849 df-ibl 25850 df-itg 25851 df-0p 25898 df-ditg 26074 df-limc 26093 df-dv 26094

This theorem is used by: (None)

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