fouriercn - Mathbox for Glauco Siliprandi

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Theorem fouriercn 46188

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 46178 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 5918	. . . . . 6 ⊢ dom 𝐺 = dom ((ℝ D 𝐹) ↾ (-π(,)π))
6		ioossre 13445	. . . . . . . 8 ⊢ (-π(,)π) ⊆ ℝ
7		fouriercn.dv	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (ℝ–cn→ℂ))
8		cncff 24933	. . . . . . . . 9 ⊢ ((ℝ D 𝐹) ∈ (ℝ–cn→ℂ) → (ℝ D 𝐹):ℝ⟶ℂ)
9		fdm 6746	. . . . . . . . 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 4049	. . . . . . 7 ⊢ (𝜑 → (-π(,)π) ⊆ dom (ℝ D 𝐹))
12		ssdmres 6033	. . . . . . 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 2787	. . . . 5 ⊢ (𝜑 → dom 𝐺 = (-π(,)π))
15	14	difeq2d 4136	. . . 4 ⊢ (𝜑 → ((-π(,)π) ∖ dom 𝐺) = ((-π(,)π) ∖ (-π(,)π)))
16		difid 4382	. . . 4 ⊢ ((-π(,)π) ∖ (-π(,)π)) = ∅
17	15, 16	eqtrdi 2791	. . 3 ⊢ (𝜑 → ((-π(,)π) ∖ dom 𝐺) = ∅)
18		0fi 9081	. . 3 ⊢ ∅ ∈ Fin
19	17, 18	eqeltrdi 2847	. 2 ⊢ (𝜑 → ((-π(,)π) ∖ dom 𝐺) ∈ Fin)
20		rescncf 24937	. . . 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 7446	. . 3 ⊢ (𝜑 → (dom 𝐺–cn→ℂ) = ((-π(,)π)–cn→ℂ))
24	21, 22, 23	3eltr4d 2854	. 2 ⊢ (𝜑 → 𝐺 ∈ (dom 𝐺–cn→ℂ))
25		pire 26515	. . . . . 6 ⊢ π ∈ ℝ
26	25	renegcli 11568	. . . . 5 ⊢ -π ∈ ℝ
27	25	rexri 11317	. . . . 5 ⊢ π ∈ ℝ^*
28		icossre 13465	. . . . 5 ⊢ ((-π ∈ ℝ ∧ π ∈ ℝ^*) → (-π[,)π) ⊆ ℝ)
29	26, 27, 28	mp2an 692	. . . 4 ⊢ (-π[,)π) ⊆ ℝ
30		eldifi 4141	. . . 4 ⊢ (𝑥 ∈ ((-π[,)π) ∖ dom 𝐺) → 𝑥 ∈ (-π[,)π))
31	29, 30	sselid 3993	. . 3 ⊢ (𝑥 ∈ ((-π[,)π) ∖ dom 𝐺) → 𝑥 ∈ ℝ)
32		limcresi 25935	. . . . . 6 ⊢ ((ℝ D 𝐹) lim_ℂ 𝑥) ⊆ (((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (𝑥(,)+∞))) lim_ℂ 𝑥)
33	4	reseq1i 5996	. . . . . . . 8 ⊢ (𝐺 ↾ (𝑥(,)+∞)) = (((ℝ D 𝐹) ↾ (-π(,)π)) ↾ (𝑥(,)+∞))
34		resres 6013	. . . . . . . 8 ⊢ (((ℝ D 𝐹) ↾ (-π(,)π)) ↾ (𝑥(,)+∞)) = ((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (𝑥(,)+∞)))
35	33, 34	eqtr2i 2764	. . . . . . 7 ⊢ ((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (𝑥(,)+∞))) = (𝐺 ↾ (𝑥(,)+∞))
36	35	oveq1i 7441	. . . . . 6 ⊢ (((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (𝑥(,)+∞))) lim_ℂ 𝑥) = ((𝐺 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥)
37	32, 36	sseqtri 4032	. . . . 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 25939	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((ℝ D 𝐹)‘𝑥) ∈ ((ℝ D 𝐹) lim_ℂ 𝑥))
41	37, 40	sselid 3993	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((ℝ D 𝐹)‘𝑥) ∈ ((𝐺 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥))
42	41	ne0d 4348	. . 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 26597	. . . 4 ⊢ (-π(,]π) ⊆ ℝ
45		eldifi 4141	. . . 4 ⊢ (𝑥 ∈ ((-π(,]π) ∖ dom 𝐺) → 𝑥 ∈ (-π(,]π))
46	44, 45	sselid 3993	. . 3 ⊢ (𝑥 ∈ ((-π(,]π) ∖ dom 𝐺) → 𝑥 ∈ ℝ)
47		limcresi 25935	. . . . . 6 ⊢ ((ℝ D 𝐹) lim_ℂ 𝑥) ⊆ (((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (-∞(,)𝑥))) lim_ℂ 𝑥)
48	4	reseq1i 5996	. . . . . . . 8 ⊢ (𝐺 ↾ (-∞(,)𝑥)) = (((ℝ D 𝐹) ↾ (-π(,)π)) ↾ (-∞(,)𝑥))
49		resres 6013	. . . . . . . 8 ⊢ (((ℝ D 𝐹) ↾ (-π(,)π)) ↾ (-∞(,)𝑥)) = ((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (-∞(,)𝑥)))
50	48, 49	eqtr2i 2764	. . . . . . 7 ⊢ ((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (-∞(,)𝑥))) = (𝐺 ↾ (-∞(,)𝑥))
51	50	oveq1i 7441	. . . . . 6 ⊢ (((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (-∞(,)𝑥))) lim_ℂ 𝑥) = ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥)
52	47, 51	sseqtri 4032	. . . . 5 ⊢ ((ℝ D 𝐹) lim_ℂ 𝑥) ⊆ ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥)
53	52, 40	sselid 3993	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((ℝ D 𝐹)‘𝑥) ∈ ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥))
54	53	ne0d 4348	. . 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 2735	. 2 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
57		ax-resscn 11210	. . . . . . 7 ⊢ ℝ ⊆ ℂ
58	57	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ⊆ ℂ)
59	1, 58	fssd 6754	. . . . . . 7 ⊢ (𝜑 → 𝐹:ℝ⟶ℂ)
60		ssid 4018	. . . . . . . 8 ⊢ ℝ ⊆ ℝ
61	60	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ⊆ ℝ)
62		dvcn 25972	. . . . . . 7 ⊢ (((ℝ ⊆ ℂ ∧ 𝐹:ℝ⟶ℂ ∧ ℝ ⊆ ℝ) ∧ dom (ℝ D 𝐹) = ℝ) → 𝐹 ∈ (ℝ–cn→ℂ))
63	58, 59, 61, 10, 62	syl31anc 1372	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (ℝ–cn→ℂ))
64		cncfcdm 24938	. . . . . 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 2735	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
68	67	tgioo2 24839	. . . . . 6 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
69	67, 68, 68	cncfcn 24950	. . . . 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 2841	. . 3 ⊢ (𝜑 → 𝐹 ∈ ((topGen‘ran (,)) Cn (topGen‘ran (,))))
72		fouriercn.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ ℝ)
73		uniretop 24799	. . . 4 ⊢ ℝ = ∪ (topGen‘ran (,))
74	73	cncnpi 23302	. . 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 46182	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 1537 ∈ wcel 2106 ≠ wne 2938 ∖ cdif 3960 ∩ cin 3962 ⊆ wss 3963 ∅c0 4339 ↦ cmpt 5231 dom cdm 5689 ran crn 5690 ↾ cres 5691 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 Fincfn 8984 ℂcc 11151 ℝcr 11152 0cc0 11153 + caddc 11156 · cmul 11158 +∞cpnf 11290 -∞cmnf 11291 ℝ^*cxr 11292 -cneg 11491 / cdiv 11918 ℕcn 12264 2c2 12319 ℕ₀cn0 12524 (,)cioo 13384 (,]cioc 13385 [,)cico 13386 Σcsu 15719 sincsin 16096 cosccos 16097 πcpi 16099 TopOpenctopn 17468 topGenctg 17484 ℂ_fldccnfld 21382 Cn ccn 23248 CnP ccnp 23249 –cn→ccncf 24916 ∫citg 25667 lim_ℂ climc 25912 D cdv 25913

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cc 10473 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-symdif 4259 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-disj 5116 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-omul 8510 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-dju 9939 df-card 9977 df-acn 9980 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 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 12525 df-xnn0 12598 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-fac 14310 df-bc 14339 df-hash 14367 df-shft 15103 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-sum 15720 df-ef 16100 df-sin 16102 df-cos 16103 df-pi 16105 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-t1 23338 df-haus 23339 df-cmp 23411 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-xms 24346 df-ms 24347 df-tms 24348 df-cncf 24918 df-ovol 25513 df-vol 25514 df-mbf 25668 df-itg1 25669 df-itg2 25670 df-ibl 25671 df-itg 25672 df-0p 25719 df-ditg 25897 df-limc 25916 df-dv 25917

This theorem is referenced by: (None)

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