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 40439 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 5325 | . . . . . 6 ⊢ dom 𝐺 = dom ((ℝ D 𝐹) ↾ (-π(,)π)) |
6 | ioossre 12235 | . . . . . . . 8 ⊢ (-π(,)π) ⊆ ℝ | |
7 | fouriercn.dv | . . . . . . . . 9 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (ℝ–cn→ℂ)) | |
8 | cncff 22696 | . . . . . . . . 9 ⊢ ((ℝ D 𝐹) ∈ (ℝ–cn→ℂ) → (ℝ D 𝐹):ℝ⟶ℂ) | |
9 | fdm 6051 | . . . . . . . . 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 3654 | . . . . . . 7 ⊢ (𝜑 → (-π(,)π) ⊆ dom (ℝ D 𝐹)) |
12 | ssdmres 5420 | . . . . . . 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 2668 | . . . . 5 ⊢ (𝜑 → dom 𝐺 = (-π(,)π)) |
15 | 14 | difeq2d 3728 | . . . 4 ⊢ (𝜑 → ((-π(,)π) ∖ dom 𝐺) = ((-π(,)π) ∖ (-π(,)π))) |
16 | difid 3948 | . . . 4 ⊢ ((-π(,)π) ∖ (-π(,)π)) = ∅ | |
17 | 15, 16 | syl6eq 2672 | . . 3 ⊢ (𝜑 → ((-π(,)π) ∖ dom 𝐺) = ∅) |
18 | 0fin 8188 | . . 3 ⊢ ∅ ∈ Fin | |
19 | 17, 18 | syl6eqel 2709 | . 2 ⊢ (𝜑 → ((-π(,)π) ∖ dom 𝐺) ∈ Fin) |
20 | rescncf 22700 | . . . 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 6665 | . . 3 ⊢ (𝜑 → (dom 𝐺–cn→ℂ) = ((-π(,)π)–cn→ℂ)) |
24 | 21, 22, 23 | 3eltr4d 2716 | . 2 ⊢ (𝜑 → 𝐺 ∈ (dom 𝐺–cn→ℂ)) |
25 | pire 24210 | . . . . . 6 ⊢ π ∈ ℝ | |
26 | 25 | renegcli 10342 | . . . . 5 ⊢ -π ∈ ℝ |
27 | 25 | rexri 10097 | . . . . 5 ⊢ π ∈ ℝ* |
28 | icossre 12254 | . . . . 5 ⊢ ((-π ∈ ℝ ∧ π ∈ ℝ*) → (-π[,)π) ⊆ ℝ) | |
29 | 26, 27, 28 | mp2an 708 | . . . 4 ⊢ (-π[,)π) ⊆ ℝ |
30 | eldifi 3732 | . . . 4 ⊢ (𝑥 ∈ ((-π[,)π) ∖ dom 𝐺) → 𝑥 ∈ (-π[,)π)) | |
31 | 29, 30 | sseldi 3601 | . . 3 ⊢ (𝑥 ∈ ((-π[,)π) ∖ dom 𝐺) → 𝑥 ∈ ℝ) |
32 | limcresi 23649 | . . . . . 6 ⊢ ((ℝ D 𝐹) limℂ 𝑥) ⊆ (((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (𝑥(,)+∞))) limℂ 𝑥) | |
33 | 4 | reseq1i 5392 | . . . . . . . 8 ⊢ (𝐺 ↾ (𝑥(,)+∞)) = (((ℝ D 𝐹) ↾ (-π(,)π)) ↾ (𝑥(,)+∞)) |
34 | resres 5409 | . . . . . . . 8 ⊢ (((ℝ D 𝐹) ↾ (-π(,)π)) ↾ (𝑥(,)+∞)) = ((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (𝑥(,)+∞))) | |
35 | 33, 34 | eqtr2i 2645 | . . . . . . 7 ⊢ ((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (𝑥(,)+∞))) = (𝐺 ↾ (𝑥(,)+∞)) |
36 | 35 | oveq1i 6660 | . . . . . 6 ⊢ (((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (𝑥(,)+∞))) limℂ 𝑥) = ((𝐺 ↾ (𝑥(,)+∞)) limℂ 𝑥) |
37 | 32, 36 | sseqtri 3637 | . . . . 5 ⊢ ((ℝ D 𝐹) limℂ 𝑥) ⊆ ((𝐺 ↾ (𝑥(,)+∞)) limℂ 𝑥) |
38 | 7 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (ℝ D 𝐹) ∈ (ℝ–cn→ℂ)) |
39 | simpr 477 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ) | |
40 | 38, 39 | cnlimci 23653 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((ℝ D 𝐹)‘𝑥) ∈ ((ℝ D 𝐹) limℂ 𝑥)) |
41 | 37, 40 | sseldi 3601 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((ℝ D 𝐹)‘𝑥) ∈ ((𝐺 ↾ (𝑥(,)+∞)) limℂ 𝑥)) |
42 | ne0i 3921 | . . . 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 491 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π[,)π) ∖ dom 𝐺)) → ((𝐺 ↾ (𝑥(,)+∞)) limℂ 𝑥) ≠ ∅) |
45 | negpitopissre 24286 | . . . 4 ⊢ (-π(,]π) ⊆ ℝ | |
46 | eldifi 3732 | . . . 4 ⊢ (𝑥 ∈ ((-π(,]π) ∖ dom 𝐺) → 𝑥 ∈ (-π(,]π)) | |
47 | 45, 46 | sseldi 3601 | . . 3 ⊢ (𝑥 ∈ ((-π(,]π) ∖ dom 𝐺) → 𝑥 ∈ ℝ) |
48 | limcresi 23649 | . . . . . 6 ⊢ ((ℝ D 𝐹) limℂ 𝑥) ⊆ (((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (-∞(,)𝑥))) limℂ 𝑥) | |
49 | 4 | reseq1i 5392 | . . . . . . . 8 ⊢ (𝐺 ↾ (-∞(,)𝑥)) = (((ℝ D 𝐹) ↾ (-π(,)π)) ↾ (-∞(,)𝑥)) |
50 | resres 5409 | . . . . . . . 8 ⊢ (((ℝ D 𝐹) ↾ (-π(,)π)) ↾ (-∞(,)𝑥)) = ((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (-∞(,)𝑥))) | |
51 | 49, 50 | eqtr2i 2645 | . . . . . . 7 ⊢ ((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (-∞(,)𝑥))) = (𝐺 ↾ (-∞(,)𝑥)) |
52 | 51 | oveq1i 6660 | . . . . . 6 ⊢ (((ℝ D 𝐹) ↾ ((-π(,)π) ∩ (-∞(,)𝑥))) limℂ 𝑥) = ((𝐺 ↾ (-∞(,)𝑥)) limℂ 𝑥) |
53 | 48, 52 | sseqtri 3637 | . . . . 5 ⊢ ((ℝ D 𝐹) limℂ 𝑥) ⊆ ((𝐺 ↾ (-∞(,)𝑥)) limℂ 𝑥) |
54 | 53, 40 | sseldi 3601 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((ℝ D 𝐹)‘𝑥) ∈ ((𝐺 ↾ (-∞(,)𝑥)) limℂ 𝑥)) |
55 | ne0i 3921 | . . . 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 491 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π(,]π) ∖ dom 𝐺)) → ((𝐺 ↾ (-∞(,)𝑥)) limℂ 𝑥) ≠ ∅) |
58 | eqid 2622 | . 2 ⊢ (topGen‘ran (,)) = (topGen‘ran (,)) | |
59 | ax-resscn 9993 | . . . . . . 7 ⊢ ℝ ⊆ ℂ | |
60 | 59 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ℝ ⊆ ℂ) |
61 | 1, 60 | fssd 6057 | . . . . . . 7 ⊢ (𝜑 → 𝐹:ℝ⟶ℂ) |
62 | ssid 3624 | . . . . . . . 8 ⊢ ℝ ⊆ ℝ | |
63 | 62 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → ℝ ⊆ ℝ) |
64 | dvcn 23684 | . . . . . . 7 ⊢ (((ℝ ⊆ ℂ ∧ 𝐹:ℝ⟶ℂ ∧ ℝ ⊆ ℝ) ∧ dom (ℝ D 𝐹) = ℝ) → 𝐹 ∈ (ℝ–cn→ℂ)) | |
65 | 60, 61, 63, 10, 64 | syl31anc 1329 | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (ℝ–cn→ℂ)) |
66 | cncffvrn 22701 | . . . . . 6 ⊢ ((ℝ ⊆ ℂ ∧ 𝐹 ∈ (ℝ–cn→ℂ)) → (𝐹 ∈ (ℝ–cn→ℝ) ↔ 𝐹:ℝ⟶ℝ)) | |
67 | 60, 65, 66 | syl2anc 693 | . . . . 5 ⊢ (𝜑 → (𝐹 ∈ (ℝ–cn→ℝ) ↔ 𝐹:ℝ⟶ℝ)) |
68 | 1, 67 | mpbird 247 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (ℝ–cn→ℝ)) |
69 | eqid 2622 | . . . . . 6 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
70 | 69 | tgioo2 22606 | . . . . . 6 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ) |
71 | 69, 70, 70 | cncfcn 22712 | . . . . 5 ⊢ ((ℝ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℝ–cn→ℝ) = ((topGen‘ran (,)) Cn (topGen‘ran (,)))) |
72 | 60, 60, 71 | syl2anc 693 | . . . 4 ⊢ (𝜑 → (ℝ–cn→ℝ) = ((topGen‘ran (,)) Cn (topGen‘ran (,)))) |
73 | 68, 72 | eleqtrd 2703 | . . 3 ⊢ (𝜑 → 𝐹 ∈ ((topGen‘ran (,)) Cn (topGen‘ran (,)))) |
74 | fouriercn.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ ℝ) | |
75 | uniretop 22566 | . . . 4 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
76 | 75 | cncnpi 21082 | . . 3 ⊢ ((𝐹 ∈ ((topGen‘ran (,)) Cn (topGen‘ran (,))) ∧ 𝑋 ∈ ℝ) → 𝐹 ∈ (((topGen‘ran (,)) CnP (topGen‘ran (,)))‘𝑋)) |
77 | 73, 74, 76 | syl2anc 693 | . 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 40443 | 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 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∖ cdif 3571 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 ↦ cmpt 4729 dom cdm 5114 ran crn 5115 ↾ cres 5116 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 ℂcc 9934 ℝcr 9935 0cc0 9936 + caddc 9939 · cmul 9941 +∞cpnf 10071 -∞cmnf 10072 ℝ*cxr 10073 -cneg 10267 / cdiv 10684 ℕcn 11020 2c2 11070 ℕ0cn0 11292 (,)cioo 12175 (,]cioc 12176 [,)cico 12177 Σcsu 14416 sincsin 14794 cosccos 14795 πcpi 14797 TopOpenctopn 16082 topGenctg 16098 ℂfldccnfld 19746 Cn ccn 21028 CnP ccnp 21029 –cn→ccncf 22679 ∫citg 23387 limℂ climc 23626 D cdv 23627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cc 9257 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-disj 4621 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-ofr 6898 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-omul 7565 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-acn 8768 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-xnn0 11364 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-ef 14798 df-sin 14800 df-cos 14801 df-pi 14803 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-t1 21118 df-haus 21119 df-cmp 21190 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-ovol 23233 df-vol 23234 df-mbf 23388 df-itg1 23389 df-itg2 23390 df-ibl 23391 df-itg 23392 df-0p 23437 df-ditg 23611 df-limc 23630 df-dv 23631 |
This theorem is referenced by: (None) |
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