MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dvlog Structured version   Visualization version   GIF version

Theorem dvlog 24442
Description: The derivative of the complex logarithm function. (Contributed by Mario Carneiro, 25-Feb-2015.)
Hypothesis
Ref Expression
logcn.d 𝐷 = (ℂ ∖ (-∞(,]0))
Assertion
Ref Expression
dvlog (ℂ D (log ↾ 𝐷)) = (𝑥𝐷 ↦ (1 / 𝑥))
Distinct variable group:   𝑥,𝐷

Proof of Theorem dvlog
StepHypRef Expression
1 eqid 2651 . . . 4 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
21cnfldtop 22634 . . . . . 6 (TopOpen‘ℂfld) ∈ Top
31cnfldtopon 22633 . . . . . . . 8 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
43toponunii 20769 . . . . . . 7 ℂ = (TopOpen‘ℂfld)
54restid 16141 . . . . . 6 ((TopOpen‘ℂfld) ∈ Top → ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld))
62, 5ax-mp 5 . . . . 5 ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld)
76eqcomi 2660 . . . 4 (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
8 cnelprrecn 10067 . . . . 5 ℂ ∈ {ℝ, ℂ}
98a1i 11 . . . 4 (⊤ → ℂ ∈ {ℝ, ℂ})
10 logcn.d . . . . . 6 𝐷 = (ℂ ∖ (-∞(,]0))
1110logdmopn 24440 . . . . 5 𝐷 ∈ (TopOpen‘ℂfld)
1211a1i 11 . . . 4 (⊤ → 𝐷 ∈ (TopOpen‘ℂfld))
13 logf1o 24356 . . . . . . . . 9 log:(ℂ ∖ {0})–1-1-onto→ran log
14 f1of1 6174 . . . . . . . . 9 (log:(ℂ ∖ {0})–1-1-onto→ran log → log:(ℂ ∖ {0})–1-1→ran log)
1513, 14ax-mp 5 . . . . . . . 8 log:(ℂ ∖ {0})–1-1→ran log
1610logdmss 24433 . . . . . . . 8 𝐷 ⊆ (ℂ ∖ {0})
17 f1ores 6189 . . . . . . . 8 ((log:(ℂ ∖ {0})–1-1→ran log ∧ 𝐷 ⊆ (ℂ ∖ {0})) → (log ↾ 𝐷):𝐷1-1-onto→(log “ 𝐷))
1815, 16, 17mp2an 708 . . . . . . 7 (log ↾ 𝐷):𝐷1-1-onto→(log “ 𝐷)
19 f1ocnv 6187 . . . . . . 7 ((log ↾ 𝐷):𝐷1-1-onto→(log “ 𝐷) → (log ↾ 𝐷):(log “ 𝐷)–1-1-onto𝐷)
2018, 19ax-mp 5 . . . . . 6 (log ↾ 𝐷):(log “ 𝐷)–1-1-onto𝐷
21 df-log 24348 . . . . . . . . . . 11 log = (exp ↾ (ℑ “ (-π(,]π)))
2221reseq1i 5424 . . . . . . . . . 10 (log ↾ 𝐷) = ((exp ↾ (ℑ “ (-π(,]π))) ↾ 𝐷)
2322cnveqi 5329 . . . . . . . . 9 (log ↾ 𝐷) = ((exp ↾ (ℑ “ (-π(,]π))) ↾ 𝐷)
24 eff 14856 . . . . . . . . . . 11 exp:ℂ⟶ℂ
25 cnvimass 5520 . . . . . . . . . . . 12 (ℑ “ (-π(,]π)) ⊆ dom ℑ
26 imf 13897 . . . . . . . . . . . . 13 ℑ:ℂ⟶ℝ
2726fdmi 6090 . . . . . . . . . . . 12 dom ℑ = ℂ
2825, 27sseqtri 3670 . . . . . . . . . . 11 (ℑ “ (-π(,]π)) ⊆ ℂ
29 fssres 6108 . . . . . . . . . . 11 ((exp:ℂ⟶ℂ ∧ (ℑ “ (-π(,]π)) ⊆ ℂ) → (exp ↾ (ℑ “ (-π(,]π))):(ℑ “ (-π(,]π))⟶ℂ)
3024, 28, 29mp2an 708 . . . . . . . . . 10 (exp ↾ (ℑ “ (-π(,]π))):(ℑ “ (-π(,]π))⟶ℂ
31 ffun 6086 . . . . . . . . . 10 ((exp ↾ (ℑ “ (-π(,]π))):(ℑ “ (-π(,]π))⟶ℂ → Fun (exp ↾ (ℑ “ (-π(,]π))))
32 funcnvres2 6007 . . . . . . . . . 10 (Fun (exp ↾ (ℑ “ (-π(,]π))) → ((exp ↾ (ℑ “ (-π(,]π))) ↾ 𝐷) = ((exp ↾ (ℑ “ (-π(,]π))) ↾ ((exp ↾ (ℑ “ (-π(,]π))) “ 𝐷)))
3330, 31, 32mp2b 10 . . . . . . . . 9 ((exp ↾ (ℑ “ (-π(,]π))) ↾ 𝐷) = ((exp ↾ (ℑ “ (-π(,]π))) ↾ ((exp ↾ (ℑ “ (-π(,]π))) “ 𝐷))
34 cnvimass 5520 . . . . . . . . . . 11 ((exp ↾ (ℑ “ (-π(,]π))) “ 𝐷) ⊆ dom (exp ↾ (ℑ “ (-π(,]π)))
3530fdmi 6090 . . . . . . . . . . 11 dom (exp ↾ (ℑ “ (-π(,]π))) = (ℑ “ (-π(,]π))
3634, 35sseqtri 3670 . . . . . . . . . 10 ((exp ↾ (ℑ “ (-π(,]π))) “ 𝐷) ⊆ (ℑ “ (-π(,]π))
37 resabs1 5462 . . . . . . . . . 10 (((exp ↾ (ℑ “ (-π(,]π))) “ 𝐷) ⊆ (ℑ “ (-π(,]π)) → ((exp ↾ (ℑ “ (-π(,]π))) ↾ ((exp ↾ (ℑ “ (-π(,]π))) “ 𝐷)) = (exp ↾ ((exp ↾ (ℑ “ (-π(,]π))) “ 𝐷)))
3836, 37ax-mp 5 . . . . . . . . 9 ((exp ↾ (ℑ “ (-π(,]π))) ↾ ((exp ↾ (ℑ “ (-π(,]π))) “ 𝐷)) = (exp ↾ ((exp ↾ (ℑ “ (-π(,]π))) “ 𝐷))
3923, 33, 383eqtri 2677 . . . . . . . 8 (log ↾ 𝐷) = (exp ↾ ((exp ↾ (ℑ “ (-π(,]π))) “ 𝐷))
4021imaeq1i 5498 . . . . . . . . 9 (log “ 𝐷) = ((exp ↾ (ℑ “ (-π(,]π))) “ 𝐷)
4140reseq2i 5425 . . . . . . . 8 (exp ↾ (log “ 𝐷)) = (exp ↾ ((exp ↾ (ℑ “ (-π(,]π))) “ 𝐷))
4239, 41eqtr4i 2676 . . . . . . 7 (log ↾ 𝐷) = (exp ↾ (log “ 𝐷))
43 f1oeq1 6165 . . . . . . 7 ((log ↾ 𝐷) = (exp ↾ (log “ 𝐷)) → ((log ↾ 𝐷):(log “ 𝐷)–1-1-onto𝐷 ↔ (exp ↾ (log “ 𝐷)):(log “ 𝐷)–1-1-onto𝐷))
4442, 43ax-mp 5 . . . . . 6 ((log ↾ 𝐷):(log “ 𝐷)–1-1-onto𝐷 ↔ (exp ↾ (log “ 𝐷)):(log “ 𝐷)–1-1-onto𝐷)
4520, 44mpbi 220 . . . . 5 (exp ↾ (log “ 𝐷)):(log “ 𝐷)–1-1-onto𝐷
4645a1i 11 . . . 4 (⊤ → (exp ↾ (log “ 𝐷)):(log “ 𝐷)–1-1-onto𝐷)
4742cnveqi 5329 . . . . . 6 (log ↾ 𝐷) = (exp ↾ (log “ 𝐷))
48 relres 5461 . . . . . . 7 Rel (log ↾ 𝐷)
49 dfrel2 5618 . . . . . . 7 (Rel (log ↾ 𝐷) ↔ (log ↾ 𝐷) = (log ↾ 𝐷))
5048, 49mpbi 220 . . . . . 6 (log ↾ 𝐷) = (log ↾ 𝐷)
5147, 50eqtr3i 2675 . . . . 5 (exp ↾ (log “ 𝐷)) = (log ↾ 𝐷)
52 f1of 6175 . . . . . . 7 ((log ↾ 𝐷):𝐷1-1-onto→(log “ 𝐷) → (log ↾ 𝐷):𝐷⟶(log “ 𝐷))
5318, 52mp1i 13 . . . . . 6 (⊤ → (log ↾ 𝐷):𝐷⟶(log “ 𝐷))
54 imassrn 5512 . . . . . . . 8 (log “ 𝐷) ⊆ ran log
55 logrncn 24354 . . . . . . . . 9 (𝑥 ∈ ran log → 𝑥 ∈ ℂ)
5655ssriv 3640 . . . . . . . 8 ran log ⊆ ℂ
5754, 56sstri 3645 . . . . . . 7 (log “ 𝐷) ⊆ ℂ
5810logcn 24438 . . . . . . 7 (log ↾ 𝐷) ∈ (𝐷cn→ℂ)
59 cncffvrn 22748 . . . . . . 7 (((log “ 𝐷) ⊆ ℂ ∧ (log ↾ 𝐷) ∈ (𝐷cn→ℂ)) → ((log ↾ 𝐷) ∈ (𝐷cn→(log “ 𝐷)) ↔ (log ↾ 𝐷):𝐷⟶(log “ 𝐷)))
6057, 58, 59mp2an 708 . . . . . 6 ((log ↾ 𝐷) ∈ (𝐷cn→(log “ 𝐷)) ↔ (log ↾ 𝐷):𝐷⟶(log “ 𝐷))
6153, 60sylibr 224 . . . . 5 (⊤ → (log ↾ 𝐷) ∈ (𝐷cn→(log “ 𝐷)))
6251, 61syl5eqel 2734 . . . 4 (⊤ → (exp ↾ (log “ 𝐷)) ∈ (𝐷cn→(log “ 𝐷)))
63 ssid 3657 . . . . . . . . 9 ℂ ⊆ ℂ
641, 7dvres 23720 . . . . . . . . 9 (((ℂ ⊆ ℂ ∧ exp:ℂ⟶ℂ) ∧ (ℂ ⊆ ℂ ∧ (log “ 𝐷) ⊆ ℂ)) → (ℂ D (exp ↾ (log “ 𝐷))) = ((ℂ D exp) ↾ ((int‘(TopOpen‘ℂfld))‘(log “ 𝐷))))
6563, 24, 63, 57, 64mp4an 709 . . . . . . . 8 (ℂ D (exp ↾ (log “ 𝐷))) = ((ℂ D exp) ↾ ((int‘(TopOpen‘ℂfld))‘(log “ 𝐷)))
66 dvef 23788 . . . . . . . . 9 (ℂ D exp) = exp
6710dvloglem 24439 . . . . . . . . . 10 (log “ 𝐷) ∈ (TopOpen‘ℂfld)
68 isopn3i 20934 . . . . . . . . . 10 (((TopOpen‘ℂfld) ∈ Top ∧ (log “ 𝐷) ∈ (TopOpen‘ℂfld)) → ((int‘(TopOpen‘ℂfld))‘(log “ 𝐷)) = (log “ 𝐷))
692, 67, 68mp2an 708 . . . . . . . . 9 ((int‘(TopOpen‘ℂfld))‘(log “ 𝐷)) = (log “ 𝐷)
7066, 69reseq12i 5426 . . . . . . . 8 ((ℂ D exp) ↾ ((int‘(TopOpen‘ℂfld))‘(log “ 𝐷))) = (exp ↾ (log “ 𝐷))
7165, 70eqtri 2673 . . . . . . 7 (ℂ D (exp ↾ (log “ 𝐷))) = (exp ↾ (log “ 𝐷))
7271dmeqi 5357 . . . . . 6 dom (ℂ D (exp ↾ (log “ 𝐷))) = dom (exp ↾ (log “ 𝐷))
73 dmres 5454 . . . . . 6 dom (exp ↾ (log “ 𝐷)) = ((log “ 𝐷) ∩ dom exp)
7424fdmi 6090 . . . . . . . 8 dom exp = ℂ
7557, 74sseqtr4i 3671 . . . . . . 7 (log “ 𝐷) ⊆ dom exp
76 df-ss 3621 . . . . . . 7 ((log “ 𝐷) ⊆ dom exp ↔ ((log “ 𝐷) ∩ dom exp) = (log “ 𝐷))
7775, 76mpbi 220 . . . . . 6 ((log “ 𝐷) ∩ dom exp) = (log “ 𝐷)
7872, 73, 773eqtri 2677 . . . . 5 dom (ℂ D (exp ↾ (log “ 𝐷))) = (log “ 𝐷)
7978a1i 11 . . . 4 (⊤ → dom (ℂ D (exp ↾ (log “ 𝐷))) = (log “ 𝐷))
80 neirr 2832 . . . . . 6 ¬ 0 ≠ 0
81 resss 5457 . . . . . . . . . . . . 13 ((ℂ D exp) ↾ ((int‘(TopOpen‘ℂfld))‘(log “ 𝐷))) ⊆ (ℂ D exp)
8265, 81eqsstri 3668 . . . . . . . . . . . 12 (ℂ D (exp ↾ (log “ 𝐷))) ⊆ (ℂ D exp)
8382, 66sseqtri 3670 . . . . . . . . . . 11 (ℂ D (exp ↾ (log “ 𝐷))) ⊆ exp
84 rnss 5386 . . . . . . . . . . 11 ((ℂ D (exp ↾ (log “ 𝐷))) ⊆ exp → ran (ℂ D (exp ↾ (log “ 𝐷))) ⊆ ran exp)
8583, 84ax-mp 5 . . . . . . . . . 10 ran (ℂ D (exp ↾ (log “ 𝐷))) ⊆ ran exp
86 eff2 14873 . . . . . . . . . . 11 exp:ℂ⟶(ℂ ∖ {0})
87 frn 6091 . . . . . . . . . . 11 (exp:ℂ⟶(ℂ ∖ {0}) → ran exp ⊆ (ℂ ∖ {0}))
8886, 87ax-mp 5 . . . . . . . . . 10 ran exp ⊆ (ℂ ∖ {0})
8985, 88sstri 3645 . . . . . . . . 9 ran (ℂ D (exp ↾ (log “ 𝐷))) ⊆ (ℂ ∖ {0})
9089sseli 3632 . . . . . . . 8 (0 ∈ ran (ℂ D (exp ↾ (log “ 𝐷))) → 0 ∈ (ℂ ∖ {0}))
91 eldifsn 4350 . . . . . . . 8 (0 ∈ (ℂ ∖ {0}) ↔ (0 ∈ ℂ ∧ 0 ≠ 0))
9290, 91sylib 208 . . . . . . 7 (0 ∈ ran (ℂ D (exp ↾ (log “ 𝐷))) → (0 ∈ ℂ ∧ 0 ≠ 0))
9392simprd 478 . . . . . 6 (0 ∈ ran (ℂ D (exp ↾ (log “ 𝐷))) → 0 ≠ 0)
9480, 93mto 188 . . . . 5 ¬ 0 ∈ ran (ℂ D (exp ↾ (log “ 𝐷)))
9594a1i 11 . . . 4 (⊤ → ¬ 0 ∈ ran (ℂ D (exp ↾ (log “ 𝐷))))
961, 7, 9, 12, 46, 62, 79, 95dvcnv 23785 . . 3 (⊤ → (ℂ D (exp ↾ (log “ 𝐷))) = (𝑥𝐷 ↦ (1 / ((ℂ D (exp ↾ (log “ 𝐷)))‘((exp ↾ (log “ 𝐷))‘𝑥)))))
9796trud 1533 . 2 (ℂ D (exp ↾ (log “ 𝐷))) = (𝑥𝐷 ↦ (1 / ((ℂ D (exp ↾ (log “ 𝐷)))‘((exp ↾ (log “ 𝐷))‘𝑥))))
9851oveq2i 6701 . 2 (ℂ D (exp ↾ (log “ 𝐷))) = (ℂ D (log ↾ 𝐷))
9971fveq1i 6230 . . . . 5 ((ℂ D (exp ↾ (log “ 𝐷)))‘((exp ↾ (log “ 𝐷))‘𝑥)) = ((exp ↾ (log “ 𝐷))‘((exp ↾ (log “ 𝐷))‘𝑥))
100 f1ocnvfv2 6573 . . . . . 6 (((exp ↾ (log “ 𝐷)):(log “ 𝐷)–1-1-onto𝐷𝑥𝐷) → ((exp ↾ (log “ 𝐷))‘((exp ↾ (log “ 𝐷))‘𝑥)) = 𝑥)
10145, 100mpan 706 . . . . 5 (𝑥𝐷 → ((exp ↾ (log “ 𝐷))‘((exp ↾ (log “ 𝐷))‘𝑥)) = 𝑥)
10299, 101syl5eq 2697 . . . 4 (𝑥𝐷 → ((ℂ D (exp ↾ (log “ 𝐷)))‘((exp ↾ (log “ 𝐷))‘𝑥)) = 𝑥)
103102oveq2d 6706 . . 3 (𝑥𝐷 → (1 / ((ℂ D (exp ↾ (log “ 𝐷)))‘((exp ↾ (log “ 𝐷))‘𝑥))) = (1 / 𝑥))
104103mpteq2ia 4773 . 2 (𝑥𝐷 ↦ (1 / ((ℂ D (exp ↾ (log “ 𝐷)))‘((exp ↾ (log “ 𝐷))‘𝑥)))) = (𝑥𝐷 ↦ (1 / 𝑥))
10597, 98, 1043eqtr3i 2681 1 (ℂ D (log ↾ 𝐷)) = (𝑥𝐷 ↦ (1 / 𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 383   = wceq 1523  wtru 1524  wcel 2030  wne 2823  cdif 3604  cin 3606  wss 3607  {csn 4210  {cpr 4212  cmpt 4762  ccnv 5142  dom cdm 5143  ran crn 5144  cres 5145  cima 5146  Rel wrel 5148  Fun wfun 5920  wf 5922  1-1wf1 5923  1-1-ontowf1o 5925  cfv 5926  (class class class)co 6690  cc 9972  cr 9973  0cc0 9974  1c1 9975  -∞cmnf 10110  -cneg 10305   / cdiv 10722  (,]cioc 12214  cim 13882  expce 14836  πcpi 14841  t crest 16128  TopOpenctopn 16129  fldccnfld 19794  Topctop 20746  intcnt 20869  cnccncf 22726   D cdv 23672  logclog 24346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-addf 10053  ax-mulf 10054
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ioo 12217  df-ioc 12218  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-mod 12709  df-seq 12842  df-exp 12901  df-fac 13101  df-bc 13130  df-hash 13158  df-shft 13851  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-limsup 14246  df-clim 14263  df-rlim 14264  df-sum 14461  df-ef 14842  df-sin 14844  df-cos 14845  df-tan 14846  df-pi 14847  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-starv 16003  df-sca 16004  df-vsca 16005  df-ip 16006  df-tset 16007  df-ple 16008  df-ds 16011  df-unif 16012  df-hom 16013  df-cco 16014  df-rest 16130  df-topn 16131  df-0g 16149  df-gsum 16150  df-topgen 16151  df-pt 16152  df-prds 16155  df-xrs 16209  df-qtop 16214  df-imas 16215  df-xps 16217  df-mre 16293  df-mrc 16294  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-submnd 17383  df-mulg 17588  df-cntz 17796  df-cmn 18241  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-fbas 19791  df-fg 19792  df-cnfld 19795  df-top 20747  df-topon 20764  df-topsp 20785  df-bases 20798  df-cld 20871  df-ntr 20872  df-cls 20873  df-nei 20950  df-lp 20988  df-perf 20989  df-cn 21079  df-cnp 21080  df-haus 21167  df-cmp 21238  df-tx 21413  df-hmeo 21606  df-fil 21697  df-fm 21789  df-flim 21790  df-flf 21791  df-xms 22172  df-ms 22173  df-tms 22174  df-cncf 22728  df-limc 23675  df-dv 23676  df-log 24348
This theorem is referenced by:  dvlog2  24444  dvcncxp1  24529  dvatan  24707  lgamgulmlem2  24801  dvasin  33626
  Copyright terms: Public domain W3C validator