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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 40859 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 5432 | . . . . . 6 ⊢ dom 𝐺 = dom ((ℝ D 𝐹) ↾ (-π(,)π)) |
| 6 | ioossre 12349 | . . . . . . . 8 ⊢ (-π(,)π) ⊆ ℝ | |
| 7 | fouriercn.dv | . . . . . . . . 9 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (ℝ–cn→ℂ)) | |
| 8 | cncff 22818 | . . . . . . . . 9 ⊢ ((ℝ D 𝐹) ∈ (ℝ–cn→ℂ) → (ℝ D 𝐹):ℝ⟶ℂ) | |
| 9 | fdm 6164 | . . . . . . . . 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 | syl5sseqr 3760 | . . . . . . 7 ⊢ (𝜑 → (-π(,)π) ⊆ dom (ℝ D 𝐹)) |
| 12 | ssdmres 5530 | . . . . . . 7 ⊢ ((-π(,)π) ⊆ dom (ℝ D 𝐹) ↔ dom ((ℝ D 𝐹) ↾ (-π(,)π)) = (-π(,)π)) | |
| 13 | 11, 12 | sylib 208 | . . . . . 6 ⊢ (𝜑 → dom ((ℝ D 𝐹) ↾ (-π(,)π)) = (-π(,)π)) |
| 14 | 5, 13 | syl5eq 2770 | . . . . 5 ⊢ (𝜑 → dom 𝐺 = (-π(,)π)) |
| 15 | 14 | difeq2d 3836 | . . . 4 ⊢ (𝜑 → ((-π(,)π) ∖ dom 𝐺) = ((-π(,)π) ∖ (-π(,)π))) |
| 16 | difid 4056 | . . . 4 ⊢ ((-π(,)π) ∖ (-π(,)π)) = ∅ | |
| 17 | 15, 16 | syl6eq 2774 | . . 3 ⊢ (𝜑 → ((-π(,)π) ∖ dom 𝐺) = ∅) |
| 18 | 0fin 8304 | . . 3 ⊢ ∅ ∈ Fin | |
| 19 | 17, 18 | syl6eqel 2811 | . 2 ⊢ (𝜑 → ((-π(,)π) ∖ dom 𝐺) ∈ Fin) |
| 20 | rescncf 22822 | . . . 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 6780 | . . 3 ⊢ (𝜑 → (dom 𝐺–cn→ℂ) = ((-π(,)π)–cn→ℂ)) |
| 24 | 21, 22, 23 | 3eltr4d 2818 | . 2 ⊢ (𝜑 → 𝐺 ∈ (dom 𝐺–cn→ℂ)) |
| 25 | pire 24330 | . . . . . 6 ⊢ π ∈ ℝ | |
| 26 | 25 | renegcli 10455 | . . . . 5 ⊢ -π ∈ ℝ |
| 27 | 25 | rexri 10210 | . . . . 5 ⊢ π ∈ ℝ* |
| 28 | icossre 12368 | . . . . 5 ⊢ ((-π ∈ ℝ ∧ π ∈ ℝ*) → (-π[,)π) ⊆ ℝ) | |
| 29 | 26, 27, 28 | mp2an 710 | . . . 4 ⊢ (-π[,)π) ⊆ ℝ |
| 30 | eldifi 3840 | . . . 4 ⊢ (𝑥 ∈ ((-π[,)π) ∖ dom 𝐺) → 𝑥 ∈ (-π[,)π)) | |
| 31 | 29, 30 | sseldi 3707 | . . 3 ⊢ (𝑥 ∈ ((-π[,)π) ∖ dom 𝐺) → 𝑥 ∈ ℝ) |
| 32 | limcresi 23769 | . . . . . 6 ⊢ ((ℝ D 𝐹) limℂ 𝑥) ⊆ (((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (𝑥(,)+∞))) limℂ 𝑥) | |
| 33 | 4 | reseq1i 5499 | . . . . . . . 8 ⊢ (𝐺 ↾ (𝑥(,)+∞)) = (((ℝ D 𝐹) ↾ (-π(,)π)) ↾ (𝑥(,)+∞)) |
| 34 | resres 5519 | . . . . . . . 8 ⊢ (((ℝ D 𝐹) ↾ (-π(,)π)) ↾ (𝑥(,)+∞)) = ((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (𝑥(,)+∞))) | |
| 35 | 33, 34 | eqtr2i 2747 | . . . . . . 7 ⊢ ((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (𝑥(,)+∞))) = (𝐺 ↾ (𝑥(,)+∞)) |
| 36 | 35 | oveq1i 6775 | . . . . . 6 ⊢ (((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (𝑥(,)+∞))) limℂ 𝑥) = ((𝐺 ↾ (𝑥(,)+∞)) limℂ 𝑥) |
| 37 | 32, 36 | sseqtri 3743 | . . . . 5 ⊢ ((ℝ D 𝐹) limℂ 𝑥) ⊆ ((𝐺 ↾ (𝑥(,)+∞)) limℂ 𝑥) |
| 38 | 7 | adantr 472 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (ℝ D 𝐹) ∈ (ℝ–cn→ℂ)) |
| 39 | simpr 479 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ) | |
| 40 | 38, 39 | cnlimci 23773 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((ℝ D 𝐹)‘𝑥) ∈ ((ℝ D 𝐹) limℂ 𝑥)) |
| 41 | 37, 40 | sseldi 3707 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((ℝ D 𝐹)‘𝑥) ∈ ((𝐺 ↾ (𝑥(,)+∞)) limℂ 𝑥)) |
| 42 | ne0i 4029 | . . . 4 ⊢ (((ℝ D 𝐹)‘𝑥) ∈ ((𝐺 ↾ (𝑥(,)+∞)) limℂ 𝑥) → ((𝐺 ↾ (𝑥(,)+∞)) limℂ 𝑥) ≠ ∅) | |
| 43 | 41, 42 | syl 17 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐺 ↾ (𝑥(,)+∞)) limℂ 𝑥) ≠ ∅) |
| 44 | 31, 43 | sylan2 492 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π[,)π) ∖ dom 𝐺)) → ((𝐺 ↾ (𝑥(,)+∞)) limℂ 𝑥) ≠ ∅) |
| 45 | negpitopissre 24406 | . . . 4 ⊢ (-π(,]π) ⊆ ℝ | |
| 46 | eldifi 3840 | . . . 4 ⊢ (𝑥 ∈ ((-π(,]π) ∖ dom 𝐺) → 𝑥 ∈ (-π(,]π)) | |
| 47 | 45, 46 | sseldi 3707 | . . 3 ⊢ (𝑥 ∈ ((-π(,]π) ∖ dom 𝐺) → 𝑥 ∈ ℝ) |
| 48 | limcresi 23769 | . . . . . 6 ⊢ ((ℝ D 𝐹) limℂ 𝑥) ⊆ (((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (-∞(,)𝑥))) limℂ 𝑥) | |
| 49 | 4 | reseq1i 5499 | . . . . . . . 8 ⊢ (𝐺 ↾ (-∞(,)𝑥)) = (((ℝ D 𝐹) ↾ (-π(,)π)) ↾ (-∞(,)𝑥)) |
| 50 | resres 5519 | . . . . . . . 8 ⊢ (((ℝ D 𝐹) ↾ (-π(,)π)) ↾ (-∞(,)𝑥)) = ((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (-∞(,)𝑥))) | |
| 51 | 49, 50 | eqtr2i 2747 | . . . . . . 7 ⊢ ((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (-∞(,)𝑥))) = (𝐺 ↾ (-∞(,)𝑥)) |
| 52 | 51 | oveq1i 6775 | . . . . . 6 ⊢ (((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (-∞(,)𝑥))) limℂ 𝑥) = ((𝐺 ↾ (-∞(,)𝑥)) limℂ 𝑥) |
| 53 | 48, 52 | sseqtri 3743 | . . . . 5 ⊢ ((ℝ D 𝐹) limℂ 𝑥) ⊆ ((𝐺 ↾ (-∞(,)𝑥)) limℂ 𝑥) |
| 54 | 53, 40 | sseldi 3707 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((ℝ D 𝐹)‘𝑥) ∈ ((𝐺 ↾ (-∞(,)𝑥)) limℂ 𝑥)) |
| 55 | ne0i 4029 | . . . 4 ⊢ (((ℝ D 𝐹)‘𝑥) ∈ ((𝐺 ↾ (-∞(,)𝑥)) limℂ 𝑥) → ((𝐺 ↾ (-∞(,)𝑥)) limℂ 𝑥) ≠ ∅) | |
| 56 | 54, 55 | syl 17 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐺 ↾ (-∞(,)𝑥)) limℂ 𝑥) ≠ ∅) |
| 57 | 47, 56 | sylan2 492 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π(,]π) ∖ dom 𝐺)) → ((𝐺 ↾ (-∞(,)𝑥)) limℂ 𝑥) ≠ ∅) |
| 58 | eqid 2724 | . 2 ⊢ (topGen‘ran (,)) = (topGen‘ran (,)) | |
| 59 | ax-resscn 10106 | . . . . . . 7 ⊢ ℝ ⊆ ℂ | |
| 60 | 59 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ℝ ⊆ ℂ) |
| 61 | 1, 60 | fssd 6170 | . . . . . . 7 ⊢ (𝜑 → 𝐹:ℝ⟶ℂ) |
| 62 | ssid 3730 | . . . . . . . 8 ⊢ ℝ ⊆ ℝ | |
| 63 | 62 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → ℝ ⊆ ℝ) |
| 64 | dvcn 23804 | . . . . . . 7 ⊢ (((ℝ ⊆ ℂ ∧ 𝐹:ℝ⟶ℂ ∧ ℝ ⊆ ℝ) ∧ dom (ℝ D 𝐹) = ℝ) → 𝐹 ∈ (ℝ–cn→ℂ)) | |
| 65 | 60, 61, 63, 10, 64 | syl31anc 1442 | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (ℝ–cn→ℂ)) |
| 66 | cncffvrn 22823 | . . . . . 6 ⊢ ((ℝ ⊆ ℂ ∧ 𝐹 ∈ (ℝ–cn→ℂ)) → (𝐹 ∈ (ℝ–cn→ℝ) ↔ 𝐹:ℝ⟶ℝ)) | |
| 67 | 60, 65, 66 | syl2anc 696 | . . . . 5 ⊢ (𝜑 → (𝐹 ∈ (ℝ–cn→ℝ) ↔ 𝐹:ℝ⟶ℝ)) |
| 68 | 1, 67 | mpbird 247 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (ℝ–cn→ℝ)) |
| 69 | eqid 2724 | . . . . . 6 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
| 70 | 69 | tgioo2 22728 | . . . . . 6 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ) |
| 71 | 69, 70, 70 | cncfcn 22834 | . . . . 5 ⊢ ((ℝ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℝ–cn→ℝ) = ((topGen‘ran (,)) Cn (topGen‘ran (,)))) |
| 72 | 60, 60, 71 | syl2anc 696 | . . . 4 ⊢ (𝜑 → (ℝ–cn→ℝ) = ((topGen‘ran (,)) Cn (topGen‘ran (,)))) |
| 73 | 68, 72 | eleqtrd 2805 | . . 3 ⊢ (𝜑 → 𝐹 ∈ ((topGen‘ran (,)) Cn (topGen‘ran (,)))) |
| 74 | fouriercn.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ ℝ) | |
| 75 | uniretop 22688 | . . . 4 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
| 76 | 75 | cncnpi 21205 | . . 3 ⊢ ((𝐹 ∈ ((topGen‘ran (,)) Cn (topGen‘ran (,))) ∧ 𝑋 ∈ ℝ) → 𝐹 ∈ (((topGen‘ran (,)) CnP (topGen‘ran (,)))‘𝑋)) |
| 77 | 73, 74, 76 | syl2anc 696 | . 2 ⊢ (𝜑 → 𝐹 ∈ (((topGen‘ran (,)) CnP (topGen‘ran (,)))‘𝑋)) |
| 78 | fouriercn.a | . 2 ⊢ 𝐴 = (𝑛 ∈ ℕ0 ↦ (∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π)) | |
| 79 | fouriercn.b | . 2 ⊢ 𝐵 = (𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π)) | |
| 80 | 1, 2, 3, 4, 19, 24, 44, 57, 58, 77, 78, 79 | fouriercnp 40863 | 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 196 ∧ wa 383 = wceq 1596 ∈ wcel 2103 ≠ wne 2896 ∖ cdif 3677 ∩ cin 3679 ⊆ wss 3680 ∅c0 4023 ↦ cmpt 4837 dom cdm 5218 ran crn 5219 ↾ cres 5220 ⟶wf 5997 ‘cfv 6001 (class class class)co 6765 Fincfn 8072 ℂcc 10047 ℝcr 10048 0cc0 10049 + caddc 10052 · cmul 10054 +∞cpnf 10184 -∞cmnf 10185 ℝ*cxr 10186 -cneg 10380 / cdiv 10797 ℕcn 11133 2c2 11183 ℕ0cn0 11405 (,)cioo 12289 (,]cioc 12290 [,)cico 12291 Σcsu 14536 sincsin 14914 cosccos 14915 πcpi 14917 TopOpenctopn 16205 topGenctg 16221 ℂfldccnfld 19869 Cn ccn 21151 CnP ccnp 21152 –cn→ccncf 22801 ∫citg 23507 limℂ climc 23746 D cdv 23747 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1835 ax-4 1850 ax-5 1952 ax-6 2018 ax-7 2054 ax-8 2105 ax-9 2112 ax-10 2132 ax-11 2147 ax-12 2160 ax-13 2355 ax-ext 2704 ax-rep 4879 ax-sep 4889 ax-nul 4897 ax-pow 4948 ax-pr 5011 ax-un 7066 ax-inf2 8651 ax-cc 9370 ax-cnex 10105 ax-resscn 10106 ax-1cn 10107 ax-icn 10108 ax-addcl 10109 ax-addrcl 10110 ax-mulcl 10111 ax-mulrcl 10112 ax-mulcom 10113 ax-addass 10114 ax-mulass 10115 ax-distr 10116 ax-i2m1 10117 ax-1ne0 10118 ax-1rid 10119 ax-rnegex 10120 ax-rrecex 10121 ax-cnre 10122 ax-pre-lttri 10123 ax-pre-lttrn 10124 ax-pre-ltadd 10125 ax-pre-mulgt0 10126 ax-pre-sup 10127 ax-addf 10128 ax-mulf 10129 |
| This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1599 df-fal 1602 df-ex 1818 df-nf 1823 df-sb 2011 df-eu 2575 df-mo 2576 df-clab 2711 df-cleq 2717 df-clel 2720 df-nfc 2855 df-ne 2897 df-nel 3000 df-ral 3019 df-rex 3020 df-reu 3021 df-rmo 3022 df-rab 3023 df-v 3306 df-sbc 3542 df-csb 3640 df-dif 3683 df-un 3685 df-in 3687 df-ss 3694 df-pss 3696 df-nul 4024 df-if 4195 df-pw 4268 df-sn 4286 df-pr 4288 df-tp 4290 df-op 4292 df-uni 4545 df-int 4584 df-iun 4630 df-iin 4631 df-disj 4729 df-br 4761 df-opab 4821 df-mpt 4838 df-tr 4861 df-id 5128 df-eprel 5133 df-po 5139 df-so 5140 df-fr 5177 df-se 5178 df-we 5179 df-xp 5224 df-rel 5225 df-cnv 5226 df-co 5227 df-dm 5228 df-rn 5229 df-res 5230 df-ima 5231 df-pred 5793 df-ord 5839 df-on 5840 df-lim 5841 df-suc 5842 df-iota 5964 df-fun 6003 df-fn 6004 df-f 6005 df-f1 6006 df-fo 6007 df-f1o 6008 df-fv 6009 df-isom 6010 df-riota 6726 df-ov 6768 df-oprab 6769 df-mpt2 6770 df-of 7014 df-ofr 7015 df-om 7183 df-1st 7285 df-2nd 7286 df-supp 7416 df-wrecs 7527 df-recs 7588 df-rdg 7626 df-1o 7680 df-2o 7681 df-oadd 7684 df-omul 7685 df-er 7862 df-map 7976 df-pm 7977 df-ixp 8026 df-en 8073 df-dom 8074 df-sdom 8075 df-fin 8076 df-fsupp 8392 df-fi 8433 df-sup 8464 df-inf 8465 df-oi 8531 df-card 8878 df-acn 8881 df-cda 9103 df-pnf 10189 df-mnf 10190 df-xr 10191 df-ltxr 10192 df-le 10193 df-sub 10381 df-neg 10382 df-div 10798 df-nn 11134 df-2 11192 df-3 11193 df-4 11194 df-5 11195 df-6 11196 df-7 11197 df-8 11198 df-9 11199 df-n0 11406 df-xnn0 11477 df-z 11491 df-dec 11607 df-uz 11801 df-q 11903 df-rp 11947 df-xneg 12060 df-xadd 12061 df-xmul 12062 df-ioo 12293 df-ioc 12294 df-ico 12295 df-icc 12296 df-fz 12441 df-fzo 12581 df-fl 12708 df-mod 12784 df-seq 12917 df-exp 12976 df-fac 13176 df-bc 13205 df-hash 13233 df-shft 13927 df-cj 13959 df-re 13960 df-im 13961 df-sqrt 14095 df-abs 14096 df-limsup 14322 df-clim 14339 df-rlim 14340 df-sum 14537 df-ef 14918 df-sin 14920 df-cos 14921 df-pi 14923 df-struct 15982 df-ndx 15983 df-slot 15984 df-base 15986 df-sets 15987 df-ress 15988 df-plusg 16077 df-mulr 16078 df-starv 16079 df-sca 16080 df-vsca 16081 df-ip 16082 df-tset 16083 df-ple 16084 df-ds 16087 df-unif 16088 df-hom 16089 df-cco 16090 df-rest 16206 df-topn 16207 df-0g 16225 df-gsum 16226 df-topgen 16227 df-pt 16228 df-prds 16231 df-xrs 16285 df-qtop 16290 df-imas 16291 df-xps 16293 df-mre 16369 df-mrc 16370 df-acs 16372 df-mgm 17364 df-sgrp 17406 df-mnd 17417 df-submnd 17458 df-mulg 17663 df-cntz 17871 df-cmn 18316 df-psmet 19861 df-xmet 19862 df-met 19863 df-bl 19864 df-mopn 19865 df-fbas 19866 df-fg 19867 df-cnfld 19870 df-top 20822 df-topon 20839 df-topsp 20860 df-bases 20873 df-cld 20946 df-ntr 20947 df-cls 20948 df-nei 21025 df-lp 21063 df-perf 21064 df-cn 21154 df-cnp 21155 df-t1 21241 df-haus 21242 df-cmp 21313 df-tx 21488 df-hmeo 21681 df-fil 21772 df-fm 21864 df-flim 21865 df-flf 21866 df-xms 22247 df-ms 22248 df-tms 22249 df-cncf 22803 df-ovol 23354 df-vol 23355 df-mbf 23508 df-itg1 23509 df-itg2 23510 df-ibl 23511 df-itg 23512 df-0p 23557 df-ditg 23731 df-limc 23750 df-dv 23751 |
| This theorem is referenced by: (None) |
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