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Theorem dvlog 24310
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 2621 . . . 4 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
21cnfldtop 22506 . . . . . 6 (TopOpen‘ℂfld) ∈ Top
31cnfldtopon 22505 . . . . . . . 8 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
43toponunii 20652 . . . . . . 7 ℂ = (TopOpen‘ℂfld)
54restid 16022 . . . . . 6 ((TopOpen‘ℂfld) ∈ Top → ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld))
62, 5ax-mp 5 . . . . 5 ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld)
76eqcomi 2630 . . . 4 (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
8 cnelprrecn 9980 . . . . 5 ℂ ∈ {ℝ, ℂ}
98a1i 11 . . . 4 (⊤ → ℂ ∈ {ℝ, ℂ})
10 logcn.d . . . . . 6 𝐷 = (ℂ ∖ (-∞(,]0))
1110logdmopn 24308 . . . . 5 𝐷 ∈ (TopOpen‘ℂfld)
1211a1i 11 . . . 4 (⊤ → 𝐷 ∈ (TopOpen‘ℂfld))
13 logf1o 24228 . . . . . . . . 9 log:(ℂ ∖ {0})–1-1-onto→ran log
14 f1of1 6098 . . . . . . . . 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 24301 . . . . . . . 8 𝐷 ⊆ (ℂ ∖ {0})
17 f1ores 6113 . . . . . . . 8 ((log:(ℂ ∖ {0})–1-1→ran log ∧ 𝐷 ⊆ (ℂ ∖ {0})) → (log ↾ 𝐷):𝐷1-1-onto→(log “ 𝐷))
1815, 16, 17mp2an 707 . . . . . . 7 (log ↾ 𝐷):𝐷1-1-onto→(log “ 𝐷)
19 f1ocnv 6111 . . . . . . 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 24220 . . . . . . . . . . 11 log = (exp ↾ (ℑ “ (-π(,]π)))
2221reseq1i 5357 . . . . . . . . . 10 (log ↾ 𝐷) = ((exp ↾ (ℑ “ (-π(,]π))) ↾ 𝐷)
2322cnveqi 5262 . . . . . . . . 9 (log ↾ 𝐷) = ((exp ↾ (ℑ “ (-π(,]π))) ↾ 𝐷)
24 eff 14744 . . . . . . . . . . 11 exp:ℂ⟶ℂ
25 cnvimass 5449 . . . . . . . . . . . 12 (ℑ “ (-π(,]π)) ⊆ dom ℑ
26 imf 13794 . . . . . . . . . . . . 13 ℑ:ℂ⟶ℝ
2726fdmi 6014 . . . . . . . . . . . 12 dom ℑ = ℂ
2825, 27sseqtri 3621 . . . . . . . . . . 11 (ℑ “ (-π(,]π)) ⊆ ℂ
29 fssres 6032 . . . . . . . . . . 11 ((exp:ℂ⟶ℂ ∧ (ℑ “ (-π(,]π)) ⊆ ℂ) → (exp ↾ (ℑ “ (-π(,]π))):(ℑ “ (-π(,]π))⟶ℂ)
3024, 28, 29mp2an 707 . . . . . . . . . 10 (exp ↾ (ℑ “ (-π(,]π))):(ℑ “ (-π(,]π))⟶ℂ
31 ffun 6010 . . . . . . . . . 10 ((exp ↾ (ℑ “ (-π(,]π))):(ℑ “ (-π(,]π))⟶ℂ → Fun (exp ↾ (ℑ “ (-π(,]π))))
32 funcnvres2 5932 . . . . . . . . . 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 5449 . . . . . . . . . . 11 ((exp ↾ (ℑ “ (-π(,]π))) “ 𝐷) ⊆ dom (exp ↾ (ℑ “ (-π(,]π)))
3530fdmi 6014 . . . . . . . . . . 11 dom (exp ↾ (ℑ “ (-π(,]π))) = (ℑ “ (-π(,]π))
3634, 35sseqtri 3621 . . . . . . . . . 10 ((exp ↾ (ℑ “ (-π(,]π))) “ 𝐷) ⊆ (ℑ “ (-π(,]π))
37 resabs1 5391 . . . . . . . . . 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 2647 . . . . . . . 8 (log ↾ 𝐷) = (exp ↾ ((exp ↾ (ℑ “ (-π(,]π))) “ 𝐷))
4021imaeq1i 5427 . . . . . . . . 9 (log “ 𝐷) = ((exp ↾ (ℑ “ (-π(,]π))) “ 𝐷)
4140reseq2i 5358 . . . . . . . 8 (exp ↾ (log “ 𝐷)) = (exp ↾ ((exp ↾ (ℑ “ (-π(,]π))) “ 𝐷))
4239, 41eqtr4i 2646 . . . . . . 7 (log ↾ 𝐷) = (exp ↾ (log “ 𝐷))
43 f1oeq1 6089 . . . . . . 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 5262 . . . . . 6 (log ↾ 𝐷) = (exp ↾ (log “ 𝐷))
48 relres 5390 . . . . . . 7 Rel (log ↾ 𝐷)
49 dfrel2 5547 . . . . . . 7 (Rel (log ↾ 𝐷) ↔ (log ↾ 𝐷) = (log ↾ 𝐷))
5048, 49mpbi 220 . . . . . 6 (log ↾ 𝐷) = (log ↾ 𝐷)
5147, 50eqtr3i 2645 . . . . 5 (exp ↾ (log “ 𝐷)) = (log ↾ 𝐷)
52 f1of 6099 . . . . . . 7 ((log ↾ 𝐷):𝐷1-1-onto→(log “ 𝐷) → (log ↾ 𝐷):𝐷⟶(log “ 𝐷))
5318, 52mp1i 13 . . . . . 6 (⊤ → (log ↾ 𝐷):𝐷⟶(log “ 𝐷))
54 imassrn 5441 . . . . . . . 8 (log “ 𝐷) ⊆ ran log
55 logrncn 24226 . . . . . . . . 9 (𝑥 ∈ ran log → 𝑥 ∈ ℂ)
5655ssriv 3591 . . . . . . . 8 ran log ⊆ ℂ
5754, 56sstri 3596 . . . . . . 7 (log “ 𝐷) ⊆ ℂ
5810logcn 24306 . . . . . . 7 (log ↾ 𝐷) ∈ (𝐷cn→ℂ)
59 cncffvrn 22620 . . . . . . 7 (((log “ 𝐷) ⊆ ℂ ∧ (log ↾ 𝐷) ∈ (𝐷cn→ℂ)) → ((log ↾ 𝐷) ∈ (𝐷cn→(log “ 𝐷)) ↔ (log ↾ 𝐷):𝐷⟶(log “ 𝐷)))
6057, 58, 59mp2an 707 . . . . . 6 ((log ↾ 𝐷) ∈ (𝐷cn→(log “ 𝐷)) ↔ (log ↾ 𝐷):𝐷⟶(log “ 𝐷))
6153, 60sylibr 224 . . . . 5 (⊤ → (log ↾ 𝐷) ∈ (𝐷cn→(log “ 𝐷)))
6251, 61syl5eqel 2702 . . . 4 (⊤ → (exp ↾ (log “ 𝐷)) ∈ (𝐷cn→(log “ 𝐷)))
63 ssid 3608 . . . . . . . . 9 ℂ ⊆ ℂ
641, 7dvres 23594 . . . . . . . . 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 708 . . . . . . . 8 (ℂ D (exp ↾ (log “ 𝐷))) = ((ℂ D exp) ↾ ((int‘(TopOpen‘ℂfld))‘(log “ 𝐷)))
66 dvef 23660 . . . . . . . . 9 (ℂ D exp) = exp
6710dvloglem 24307 . . . . . . . . . 10 (log “ 𝐷) ∈ (TopOpen‘ℂfld)
68 isopn3i 20805 . . . . . . . . . 10 (((TopOpen‘ℂfld) ∈ Top ∧ (log “ 𝐷) ∈ (TopOpen‘ℂfld)) → ((int‘(TopOpen‘ℂfld))‘(log “ 𝐷)) = (log “ 𝐷))
692, 67, 68mp2an 707 . . . . . . . . 9 ((int‘(TopOpen‘ℂfld))‘(log “ 𝐷)) = (log “ 𝐷)
7066, 69reseq12i 5359 . . . . . . . 8 ((ℂ D exp) ↾ ((int‘(TopOpen‘ℂfld))‘(log “ 𝐷))) = (exp ↾ (log “ 𝐷))
7165, 70eqtri 2643 . . . . . . 7 (ℂ D (exp ↾ (log “ 𝐷))) = (exp ↾ (log “ 𝐷))
7271dmeqi 5290 . . . . . 6 dom (ℂ D (exp ↾ (log “ 𝐷))) = dom (exp ↾ (log “ 𝐷))
73 dmres 5383 . . . . . 6 dom (exp ↾ (log “ 𝐷)) = ((log “ 𝐷) ∩ dom exp)
7424fdmi 6014 . . . . . . . 8 dom exp = ℂ
7557, 74sseqtr4i 3622 . . . . . . 7 (log “ 𝐷) ⊆ dom exp
76 df-ss 3573 . . . . . . 7 ((log “ 𝐷) ⊆ dom exp ↔ ((log “ 𝐷) ∩ dom exp) = (log “ 𝐷))
7775, 76mpbi 220 . . . . . 6 ((log “ 𝐷) ∩ dom exp) = (log “ 𝐷)
7872, 73, 773eqtri 2647 . . . . 5 dom (ℂ D (exp ↾ (log “ 𝐷))) = (log “ 𝐷)
7978a1i 11 . . . 4 (⊤ → dom (ℂ D (exp ↾ (log “ 𝐷))) = (log “ 𝐷))
80 neirr 2799 . . . . . 6 ¬ 0 ≠ 0
81 resss 5386 . . . . . . . . . . . . 13 ((ℂ D exp) ↾ ((int‘(TopOpen‘ℂfld))‘(log “ 𝐷))) ⊆ (ℂ D exp)
8265, 81eqsstri 3619 . . . . . . . . . . . 12 (ℂ D (exp ↾ (log “ 𝐷))) ⊆ (ℂ D exp)
8382, 66sseqtri 3621 . . . . . . . . . . 11 (ℂ D (exp ↾ (log “ 𝐷))) ⊆ exp
84 rnss 5319 . . . . . . . . . . 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 14761 . . . . . . . . . . 11 exp:ℂ⟶(ℂ ∖ {0})
87 frn 6015 . . . . . . . . . . 11 (exp:ℂ⟶(ℂ ∖ {0}) → ran exp ⊆ (ℂ ∖ {0}))
8886, 87ax-mp 5 . . . . . . . . . 10 ran exp ⊆ (ℂ ∖ {0})
8985, 88sstri 3596 . . . . . . . . 9 ran (ℂ D (exp ↾ (log “ 𝐷))) ⊆ (ℂ ∖ {0})
9089sseli 3583 . . . . . . . 8 (0 ∈ ran (ℂ D (exp ↾ (log “ 𝐷))) → 0 ∈ (ℂ ∖ {0}))
91 eldifsn 4292 . . . . . . . 8 (0 ∈ (ℂ ∖ {0}) ↔ (0 ∈ ℂ ∧ 0 ≠ 0))
9290, 91sylib 208 . . . . . . 7 (0 ∈ ran (ℂ D (exp ↾ (log “ 𝐷))) → (0 ∈ ℂ ∧ 0 ≠ 0))
9392simprd 479 . . . . . 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 23657 . . 3 (⊤ → (ℂ D (exp ↾ (log “ 𝐷))) = (𝑥𝐷 ↦ (1 / ((ℂ D (exp ↾ (log “ 𝐷)))‘((exp ↾ (log “ 𝐷))‘𝑥)))))
9796trud 1490 . 2 (ℂ D (exp ↾ (log “ 𝐷))) = (𝑥𝐷 ↦ (1 / ((ℂ D (exp ↾ (log “ 𝐷)))‘((exp ↾ (log “ 𝐷))‘𝑥))))
9851oveq2i 6621 . 2 (ℂ D (exp ↾ (log “ 𝐷))) = (ℂ D (log ↾ 𝐷))
9971fveq1i 6154 . . . . 5 ((ℂ D (exp ↾ (log “ 𝐷)))‘((exp ↾ (log “ 𝐷))‘𝑥)) = ((exp ↾ (log “ 𝐷))‘((exp ↾ (log “ 𝐷))‘𝑥))
100 f1ocnvfv2 6493 . . . . . 6 (((exp ↾ (log “ 𝐷)):(log “ 𝐷)–1-1-onto𝐷𝑥𝐷) → ((exp ↾ (log “ 𝐷))‘((exp ↾ (log “ 𝐷))‘𝑥)) = 𝑥)
10145, 100mpan 705 . . . . 5 (𝑥𝐷 → ((exp ↾ (log “ 𝐷))‘((exp ↾ (log “ 𝐷))‘𝑥)) = 𝑥)
10299, 101syl5eq 2667 . . . 4 (𝑥𝐷 → ((ℂ D (exp ↾ (log “ 𝐷)))‘((exp ↾ (log “ 𝐷))‘𝑥)) = 𝑥)
103102oveq2d 6626 . . 3 (𝑥𝐷 → (1 / ((ℂ D (exp ↾ (log “ 𝐷)))‘((exp ↾ (log “ 𝐷))‘𝑥))) = (1 / 𝑥))
104103mpteq2ia 4705 . 2 (𝑥𝐷 ↦ (1 / ((ℂ D (exp ↾ (log “ 𝐷)))‘((exp ↾ (log “ 𝐷))‘𝑥)))) = (𝑥𝐷 ↦ (1 / 𝑥))
10597, 98, 1043eqtr3i 2651 1 (ℂ D (log ↾ 𝐷)) = (𝑥𝐷 ↦ (1 / 𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 384   = wceq 1480  wtru 1481  wcel 1987  wne 2790  cdif 3556  cin 3558  wss 3559  {csn 4153  {cpr 4155  cmpt 4678  ccnv 5078  dom cdm 5079  ran crn 5080  cres 5081  cima 5082  Rel wrel 5084  Fun wfun 5846  wf 5848  1-1wf1 5849  1-1-ontowf1o 5851  cfv 5852  (class class class)co 6610  cc 9885  cr 9886  0cc0 9887  1c1 9888  -∞cmnf 10023  -cneg 10218   / cdiv 10635  (,]cioc 12125  cim 13779  expce 14724  πcpi 14729  t crest 16009  TopOpenctopn 16010  fldccnfld 19674  Topctop 20626  intcnt 20740  cnccncf 22598   D cdv 23546  logclog 24218
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8489  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-mulcom 9951  ax-addass 9952  ax-mulass 9953  ax-distr 9954  ax-i2m1 9955  ax-1ne0 9956  ax-1rid 9957  ax-rnegex 9958  ax-rrecex 9959  ax-cnre 9960  ax-pre-lttri 9961  ax-pre-lttrn 9962  ax-pre-ltadd 9963  ax-pre-mulgt0 9964  ax-pre-sup 9965  ax-addf 9966  ax-mulf 9967
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-of 6857  df-om 7020  df-1st 7120  df-2nd 7121  df-supp 7248  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-2o 7513  df-oadd 7516  df-er 7694  df-map 7811  df-pm 7812  df-ixp 7860  df-en 7907  df-dom 7908  df-sdom 7909  df-fin 7910  df-fsupp 8227  df-fi 8268  df-sup 8299  df-inf 8300  df-oi 8366  df-card 8716  df-cda 8941  df-pnf 10027  df-mnf 10028  df-xr 10029  df-ltxr 10030  df-le 10031  df-sub 10219  df-neg 10220  df-div 10636  df-nn 10972  df-2 11030  df-3 11031  df-4 11032  df-5 11033  df-6 11034  df-7 11035  df-8 11036  df-9 11037  df-n0 11244  df-z 11329  df-dec 11445  df-uz 11639  df-q 11740  df-rp 11784  df-xneg 11897  df-xadd 11898  df-xmul 11899  df-ioo 12128  df-ioc 12129  df-ico 12130  df-icc 12131  df-fz 12276  df-fzo 12414  df-fl 12540  df-mod 12616  df-seq 12749  df-exp 12808  df-fac 13008  df-bc 13037  df-hash 13065  df-shft 13748  df-cj 13780  df-re 13781  df-im 13782  df-sqrt 13916  df-abs 13917  df-limsup 14143  df-clim 14160  df-rlim 14161  df-sum 14358  df-ef 14730  df-sin 14732  df-cos 14733  df-tan 14734  df-pi 14735  df-struct 15790  df-ndx 15791  df-slot 15792  df-base 15793  df-sets 15794  df-ress 15795  df-plusg 15882  df-mulr 15883  df-starv 15884  df-sca 15885  df-vsca 15886  df-ip 15887  df-tset 15888  df-ple 15889  df-ds 15892  df-unif 15893  df-hom 15894  df-cco 15895  df-rest 16011  df-topn 16012  df-0g 16030  df-gsum 16031  df-topgen 16032  df-pt 16033  df-prds 16036  df-xrs 16090  df-qtop 16095  df-imas 16096  df-xps 16098  df-mre 16174  df-mrc 16175  df-acs 16177  df-mgm 17170  df-sgrp 17212  df-mnd 17223  df-submnd 17264  df-mulg 17469  df-cntz 17678  df-cmn 18123  df-psmet 19666  df-xmet 19667  df-met 19668  df-bl 19669  df-mopn 19670  df-fbas 19671  df-fg 19672  df-cnfld 19675  df-top 20627  df-topon 20644  df-topsp 20657  df-bases 20670  df-cld 20742  df-ntr 20743  df-cls 20744  df-nei 20821  df-lp 20859  df-perf 20860  df-cn 20950  df-cnp 20951  df-haus 21038  df-cmp 21109  df-tx 21284  df-hmeo 21477  df-fil 21569  df-fm 21661  df-flim 21662  df-flf 21663  df-xms 22044  df-ms 22045  df-tms 22046  df-cncf 22600  df-limc 23549  df-dv 23550  df-log 24220
This theorem is referenced by:  dvlog2  24312  dvcncxp1  24397  dvatan  24575  lgamgulmlem2  24669  dvasin  33155
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