Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fouriercn | Structured version Visualization version GIF version |
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 43381 for a comparison). (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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)) |
Ref | Expression |
---|---|
fouriercn | ⊢ (𝜑 → (((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) = (𝐹‘𝑋)) |
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 5758 | . . . . . 6 ⊢ dom 𝐺 = dom ((ℝ D 𝐹) ↾ (-π(,)π)) |
6 | ioossre 12961 | . . . . . . . 8 ⊢ (-π(,)π) ⊆ ℝ | |
7 | fouriercn.dv | . . . . . . . . 9 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (ℝ–cn→ℂ)) | |
8 | cncff 23744 | . . . . . . . . 9 ⊢ ((ℝ D 𝐹) ∈ (ℝ–cn→ℂ) → (ℝ D 𝐹):ℝ⟶ℂ) | |
9 | fdm 6532 | . . . . . . . . 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 3940 | . . . . . . 7 ⊢ (𝜑 → (-π(,)π) ⊆ dom (ℝ D 𝐹)) |
12 | ssdmres 5859 | . . . . . . 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 | syl5eq 2783 | . . . . 5 ⊢ (𝜑 → dom 𝐺 = (-π(,)π)) |
15 | 14 | difeq2d 4023 | . . . 4 ⊢ (𝜑 → ((-π(,)π) ∖ dom 𝐺) = ((-π(,)π) ∖ (-π(,)π))) |
16 | difid 4271 | . . . 4 ⊢ ((-π(,)π) ∖ (-π(,)π)) = ∅ | |
17 | 15, 16 | eqtrdi 2787 | . . 3 ⊢ (𝜑 → ((-π(,)π) ∖ dom 𝐺) = ∅) |
18 | 0fin 8827 | . . 3 ⊢ ∅ ∈ Fin | |
19 | 17, 18 | eqeltrdi 2839 | . 2 ⊢ (𝜑 → ((-π(,)π) ∖ dom 𝐺) ∈ Fin) |
20 | rescncf 23748 | . . . 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 7206 | . . 3 ⊢ (𝜑 → (dom 𝐺–cn→ℂ) = ((-π(,)π)–cn→ℂ)) |
24 | 21, 22, 23 | 3eltr4d 2846 | . 2 ⊢ (𝜑 → 𝐺 ∈ (dom 𝐺–cn→ℂ)) |
25 | pire 25302 | . . . . . 6 ⊢ π ∈ ℝ | |
26 | 25 | renegcli 11104 | . . . . 5 ⊢ -π ∈ ℝ |
27 | 25 | rexri 10856 | . . . . 5 ⊢ π ∈ ℝ* |
28 | icossre 12981 | . . . . 5 ⊢ ((-π ∈ ℝ ∧ π ∈ ℝ*) → (-π[,)π) ⊆ ℝ) | |
29 | 26, 27, 28 | mp2an 692 | . . . 4 ⊢ (-π[,)π) ⊆ ℝ |
30 | eldifi 4027 | . . . 4 ⊢ (𝑥 ∈ ((-π[,)π) ∖ dom 𝐺) → 𝑥 ∈ (-π[,)π)) | |
31 | 29, 30 | sseldi 3885 | . . 3 ⊢ (𝑥 ∈ ((-π[,)π) ∖ dom 𝐺) → 𝑥 ∈ ℝ) |
32 | limcresi 24736 | . . . . . 6 ⊢ ((ℝ D 𝐹) limℂ 𝑥) ⊆ (((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (𝑥(,)+∞))) limℂ 𝑥) | |
33 | 4 | reseq1i 5832 | . . . . . . . 8 ⊢ (𝐺 ↾ (𝑥(,)+∞)) = (((ℝ D 𝐹) ↾ (-π(,)π)) ↾ (𝑥(,)+∞)) |
34 | resres 5849 | . . . . . . . 8 ⊢ (((ℝ D 𝐹) ↾ (-π(,)π)) ↾ (𝑥(,)+∞)) = ((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (𝑥(,)+∞))) | |
35 | 33, 34 | eqtr2i 2760 | . . . . . . 7 ⊢ ((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (𝑥(,)+∞))) = (𝐺 ↾ (𝑥(,)+∞)) |
36 | 35 | oveq1i 7201 | . . . . . 6 ⊢ (((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (𝑥(,)+∞))) limℂ 𝑥) = ((𝐺 ↾ (𝑥(,)+∞)) limℂ 𝑥) |
37 | 32, 36 | sseqtri 3923 | . . . . 5 ⊢ ((ℝ D 𝐹) limℂ 𝑥) ⊆ ((𝐺 ↾ (𝑥(,)+∞)) limℂ 𝑥) |
38 | 7 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (ℝ D 𝐹) ∈ (ℝ–cn→ℂ)) |
39 | simpr 488 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ) | |
40 | 38, 39 | cnlimci 24740 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((ℝ D 𝐹)‘𝑥) ∈ ((ℝ D 𝐹) limℂ 𝑥)) |
41 | 37, 40 | sseldi 3885 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((ℝ D 𝐹)‘𝑥) ∈ ((𝐺 ↾ (𝑥(,)+∞)) limℂ 𝑥)) |
42 | 41 | ne0d 4236 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐺 ↾ (𝑥(,)+∞)) limℂ 𝑥) ≠ ∅) |
43 | 31, 42 | sylan2 596 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π[,)π) ∖ dom 𝐺)) → ((𝐺 ↾ (𝑥(,)+∞)) limℂ 𝑥) ≠ ∅) |
44 | negpitopissre 25383 | . . . 4 ⊢ (-π(,]π) ⊆ ℝ | |
45 | eldifi 4027 | . . . 4 ⊢ (𝑥 ∈ ((-π(,]π) ∖ dom 𝐺) → 𝑥 ∈ (-π(,]π)) | |
46 | 44, 45 | sseldi 3885 | . . 3 ⊢ (𝑥 ∈ ((-π(,]π) ∖ dom 𝐺) → 𝑥 ∈ ℝ) |
47 | limcresi 24736 | . . . . . 6 ⊢ ((ℝ D 𝐹) limℂ 𝑥) ⊆ (((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (-∞(,)𝑥))) limℂ 𝑥) | |
48 | 4 | reseq1i 5832 | . . . . . . . 8 ⊢ (𝐺 ↾ (-∞(,)𝑥)) = (((ℝ D 𝐹) ↾ (-π(,)π)) ↾ (-∞(,)𝑥)) |
49 | resres 5849 | . . . . . . . 8 ⊢ (((ℝ D 𝐹) ↾ (-π(,)π)) ↾ (-∞(,)𝑥)) = ((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (-∞(,)𝑥))) | |
50 | 48, 49 | eqtr2i 2760 | . . . . . . 7 ⊢ ((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (-∞(,)𝑥))) = (𝐺 ↾ (-∞(,)𝑥)) |
51 | 50 | oveq1i 7201 | . . . . . 6 ⊢ (((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (-∞(,)𝑥))) limℂ 𝑥) = ((𝐺 ↾ (-∞(,)𝑥)) limℂ 𝑥) |
52 | 47, 51 | sseqtri 3923 | . . . . 5 ⊢ ((ℝ D 𝐹) limℂ 𝑥) ⊆ ((𝐺 ↾ (-∞(,)𝑥)) limℂ 𝑥) |
53 | 52, 40 | sseldi 3885 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((ℝ D 𝐹)‘𝑥) ∈ ((𝐺 ↾ (-∞(,)𝑥)) limℂ 𝑥)) |
54 | 53 | ne0d 4236 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐺 ↾ (-∞(,)𝑥)) limℂ 𝑥) ≠ ∅) |
55 | 46, 54 | sylan2 596 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π(,]π) ∖ dom 𝐺)) → ((𝐺 ↾ (-∞(,)𝑥)) limℂ 𝑥) ≠ ∅) |
56 | eqid 2736 | . 2 ⊢ (topGen‘ran (,)) = (topGen‘ran (,)) | |
57 | ax-resscn 10751 | . . . . . . 7 ⊢ ℝ ⊆ ℂ | |
58 | 57 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ℝ ⊆ ℂ) |
59 | 1, 58 | fssd 6541 | . . . . . . 7 ⊢ (𝜑 → 𝐹:ℝ⟶ℂ) |
60 | ssid 3909 | . . . . . . . 8 ⊢ ℝ ⊆ ℝ | |
61 | 60 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → ℝ ⊆ ℝ) |
62 | dvcn 24772 | . . . . . . 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 | cncffvrn 23749 | . . . . . 6 ⊢ ((ℝ ⊆ ℂ ∧ 𝐹 ∈ (ℝ–cn→ℂ)) → (𝐹 ∈ (ℝ–cn→ℝ) ↔ 𝐹:ℝ⟶ℝ)) | |
65 | 58, 63, 64 | syl2anc 587 | . . . . 5 ⊢ (𝜑 → (𝐹 ∈ (ℝ–cn→ℝ) ↔ 𝐹:ℝ⟶ℝ)) |
66 | 1, 65 | mpbird 260 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (ℝ–cn→ℝ)) |
67 | eqid 2736 | . . . . . 6 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
68 | 67 | tgioo2 23654 | . . . . . 6 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ) |
69 | 67, 68, 68 | cncfcn 23761 | . . . . 5 ⊢ ((ℝ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℝ–cn→ℝ) = ((topGen‘ran (,)) Cn (topGen‘ran (,)))) |
70 | 58, 58, 69 | syl2anc 587 | . . . 4 ⊢ (𝜑 → (ℝ–cn→ℝ) = ((topGen‘ran (,)) Cn (topGen‘ran (,)))) |
71 | 66, 70 | eleqtrd 2833 | . . 3 ⊢ (𝜑 → 𝐹 ∈ ((topGen‘ran (,)) Cn (topGen‘ran (,)))) |
72 | fouriercn.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ ℝ) | |
73 | uniretop 23614 | . . . 4 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
74 | 73 | cncnpi 22129 | . . 3 ⊢ ((𝐹 ∈ ((topGen‘ran (,)) Cn (topGen‘ran (,))) ∧ 𝑋 ∈ ℝ) → 𝐹 ∈ (((topGen‘ran (,)) CnP (topGen‘ran (,)))‘𝑋)) |
75 | 71, 72, 74 | syl2anc 587 | . 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 43385 | 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 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 ∖ cdif 3850 ∩ cin 3852 ⊆ wss 3853 ∅c0 4223 ↦ cmpt 5120 dom cdm 5536 ran crn 5537 ↾ cres 5538 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 Fincfn 8604 ℂcc 10692 ℝcr 10693 0cc0 10694 + caddc 10697 · cmul 10699 +∞cpnf 10829 -∞cmnf 10830 ℝ*cxr 10831 -cneg 11028 / cdiv 11454 ℕcn 11795 2c2 11850 ℕ0cn0 12055 (,)cioo 12900 (,]cioc 12901 [,)cico 12902 Σcsu 15214 sincsin 15588 cosccos 15589 πcpi 15591 TopOpenctopn 16880 topGenctg 16896 ℂfldccnfld 20317 Cn ccn 22075 CnP ccnp 22076 –cn→ccncf 23727 ∫citg 24469 limℂ climc 24713 D cdv 24714 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-inf2 9234 ax-cc 10014 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 ax-addf 10773 ax-mulf 10774 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-symdif 4143 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-iin 4893 df-disj 5005 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-of 7447 df-ofr 7448 df-om 7623 df-1st 7739 df-2nd 7740 df-supp 7882 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-2o 8181 df-oadd 8184 df-omul 8185 df-er 8369 df-map 8488 df-pm 8489 df-ixp 8557 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-fsupp 8964 df-fi 9005 df-sup 9036 df-inf 9037 df-oi 9104 df-dju 9482 df-card 9520 df-acn 9523 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-xnn0 12128 df-z 12142 df-dec 12259 df-uz 12404 df-q 12510 df-rp 12552 df-xneg 12669 df-xadd 12670 df-xmul 12671 df-ioo 12904 df-ioc 12905 df-ico 12906 df-icc 12907 df-fz 13061 df-fzo 13204 df-fl 13332 df-mod 13408 df-seq 13540 df-exp 13601 df-fac 13805 df-bc 13834 df-hash 13862 df-shft 14595 df-cj 14627 df-re 14628 df-im 14629 df-sqrt 14763 df-abs 14764 df-limsup 14997 df-clim 15014 df-rlim 15015 df-sum 15215 df-ef 15592 df-sin 15594 df-cos 15595 df-pi 15597 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-mulr 16763 df-starv 16764 df-sca 16765 df-vsca 16766 df-ip 16767 df-tset 16768 df-ple 16769 df-ds 16771 df-unif 16772 df-hom 16773 df-cco 16774 df-rest 16881 df-topn 16882 df-0g 16900 df-gsum 16901 df-topgen 16902 df-pt 16903 df-prds 16906 df-xrs 16961 df-qtop 16966 df-imas 16967 df-xps 16969 df-mre 17043 df-mrc 17044 df-acs 17046 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-submnd 18173 df-mulg 18443 df-cntz 18665 df-cmn 19126 df-psmet 20309 df-xmet 20310 df-met 20311 df-bl 20312 df-mopn 20313 df-fbas 20314 df-fg 20315 df-cnfld 20318 df-top 21745 df-topon 21762 df-topsp 21784 df-bases 21797 df-cld 21870 df-ntr 21871 df-cls 21872 df-nei 21949 df-lp 21987 df-perf 21988 df-cn 22078 df-cnp 22079 df-t1 22165 df-haus 22166 df-cmp 22238 df-tx 22413 df-hmeo 22606 df-fil 22697 df-fm 22789 df-flim 22790 df-flf 22791 df-xms 23172 df-ms 23173 df-tms 23174 df-cncf 23729 df-ovol 24315 df-vol 24316 df-mbf 24470 df-itg1 24471 df-itg2 24472 df-ibl 24473 df-itg 24474 df-0p 24521 df-ditg 24698 df-limc 24717 df-dv 24718 |
This theorem is referenced by: (None) |
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