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Theorem dvlog 24197
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 2610 . . . 4 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
21cnfldtop 22397 . . . . . 6 (TopOpen‘ℂfld) ∈ Top
31cnfldtopon 22396 . . . . . . . 8 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
43toponunii 20547 . . . . . . 7 ℂ = (TopOpen‘ℂfld)
54restid 15917 . . . . . 6 ((TopOpen‘ℂfld) ∈ Top → ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld))
62, 5ax-mp 5 . . . . 5 ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld)
76eqcomi 2619 . . . 4 (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
8 cnelprrecn 9908 . . . . 5 ℂ ∈ {ℝ, ℂ}
98a1i 11 . . . 4 (⊤ → ℂ ∈ {ℝ, ℂ})
10 logcn.d . . . . . 6 𝐷 = (ℂ ∖ (-∞(,]0))
1110logdmopn 24195 . . . . 5 𝐷 ∈ (TopOpen‘ℂfld)
1211a1i 11 . . . 4 (⊤ → 𝐷 ∈ (TopOpen‘ℂfld))
13 logf1o 24115 . . . . . . . . 9 log:(ℂ ∖ {0})–1-1-onto→ran log
14 f1of1 6049 . . . . . . . . 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 24188 . . . . . . . 8 𝐷 ⊆ (ℂ ∖ {0})
17 f1ores 6064 . . . . . . . 8 ((log:(ℂ ∖ {0})–1-1→ran log ∧ 𝐷 ⊆ (ℂ ∖ {0})) → (log ↾ 𝐷):𝐷1-1-onto→(log “ 𝐷))
1815, 16, 17mp2an 704 . . . . . . 7 (log ↾ 𝐷):𝐷1-1-onto→(log “ 𝐷)
19 f1ocnv 6062 . . . . . . 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 24107 . . . . . . . . . . 11 log = (exp ↾ (ℑ “ (-π(,]π)))
2221reseq1i 5313 . . . . . . . . . 10 (log ↾ 𝐷) = ((exp ↾ (ℑ “ (-π(,]π))) ↾ 𝐷)
2322cnveqi 5219 . . . . . . . . 9 (log ↾ 𝐷) = ((exp ↾ (ℑ “ (-π(,]π))) ↾ 𝐷)
24 eff 14651 . . . . . . . . . . 11 exp:ℂ⟶ℂ
25 cnvimass 5404 . . . . . . . . . . . 12 (ℑ “ (-π(,]π)) ⊆ dom ℑ
26 imf 13701 . . . . . . . . . . . . 13 ℑ:ℂ⟶ℝ
2726fdmi 5965 . . . . . . . . . . . 12 dom ℑ = ℂ
2825, 27sseqtri 3600 . . . . . . . . . . 11 (ℑ “ (-π(,]π)) ⊆ ℂ
29 fssres 5983 . . . . . . . . . . 11 ((exp:ℂ⟶ℂ ∧ (ℑ “ (-π(,]π)) ⊆ ℂ) → (exp ↾ (ℑ “ (-π(,]π))):(ℑ “ (-π(,]π))⟶ℂ)
3024, 28, 29mp2an 704 . . . . . . . . . 10 (exp ↾ (ℑ “ (-π(,]π))):(ℑ “ (-π(,]π))⟶ℂ
31 ffun 5961 . . . . . . . . . 10 ((exp ↾ (ℑ “ (-π(,]π))):(ℑ “ (-π(,]π))⟶ℂ → Fun (exp ↾ (ℑ “ (-π(,]π))))
32 funcnvres2 5883 . . . . . . . . . 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 5404 . . . . . . . . . . 11 ((exp ↾ (ℑ “ (-π(,]π))) “ 𝐷) ⊆ dom (exp ↾ (ℑ “ (-π(,]π)))
3530fdmi 5965 . . . . . . . . . . 11 dom (exp ↾ (ℑ “ (-π(,]π))) = (ℑ “ (-π(,]π))
3634, 35sseqtri 3600 . . . . . . . . . 10 ((exp ↾ (ℑ “ (-π(,]π))) “ 𝐷) ⊆ (ℑ “ (-π(,]π))
37 resabs1 5347 . . . . . . . . . 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 2636 . . . . . . . 8 (log ↾ 𝐷) = (exp ↾ ((exp ↾ (ℑ “ (-π(,]π))) “ 𝐷))
4021imaeq1i 5382 . . . . . . . . 9 (log “ 𝐷) = ((exp ↾ (ℑ “ (-π(,]π))) “ 𝐷)
4140reseq2i 5314 . . . . . . . 8 (exp ↾ (log “ 𝐷)) = (exp ↾ ((exp ↾ (ℑ “ (-π(,]π))) “ 𝐷))
4239, 41eqtr4i 2635 . . . . . . 7 (log ↾ 𝐷) = (exp ↾ (log “ 𝐷))
43 f1oeq1 6040 . . . . . . 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 219 . . . . 5 (exp ↾ (log “ 𝐷)):(log “ 𝐷)–1-1-onto𝐷
4645a1i 11 . . . 4 (⊤ → (exp ↾ (log “ 𝐷)):(log “ 𝐷)–1-1-onto𝐷)
4742cnveqi 5219 . . . . . 6 (log ↾ 𝐷) = (exp ↾ (log “ 𝐷))
48 relres 5346 . . . . . . 7 Rel (log ↾ 𝐷)
49 dfrel2 5502 . . . . . . 7 (Rel (log ↾ 𝐷) ↔ (log ↾ 𝐷) = (log ↾ 𝐷))
5048, 49mpbi 219 . . . . . 6 (log ↾ 𝐷) = (log ↾ 𝐷)
5147, 50eqtr3i 2634 . . . . 5 (exp ↾ (log “ 𝐷)) = (log ↾ 𝐷)
52 f1of 6050 . . . . . . 7 ((log ↾ 𝐷):𝐷1-1-onto→(log “ 𝐷) → (log ↾ 𝐷):𝐷⟶(log “ 𝐷))
5318, 52mp1i 13 . . . . . 6 (⊤ → (log ↾ 𝐷):𝐷⟶(log “ 𝐷))
54 imassrn 5396 . . . . . . . 8 (log “ 𝐷) ⊆ ran log
55 logrncn 24113 . . . . . . . . 9 (𝑥 ∈ ran log → 𝑥 ∈ ℂ)
5655ssriv 3572 . . . . . . . 8 ran log ⊆ ℂ
5754, 56sstri 3577 . . . . . . 7 (log “ 𝐷) ⊆ ℂ
5810logcn 24193 . . . . . . 7 (log ↾ 𝐷) ∈ (𝐷cn→ℂ)
59 cncffvrn 22509 . . . . . . 7 (((log “ 𝐷) ⊆ ℂ ∧ (log ↾ 𝐷) ∈ (𝐷cn→ℂ)) → ((log ↾ 𝐷) ∈ (𝐷cn→(log “ 𝐷)) ↔ (log ↾ 𝐷):𝐷⟶(log “ 𝐷)))
6057, 58, 59mp2an 704 . . . . . 6 ((log ↾ 𝐷) ∈ (𝐷cn→(log “ 𝐷)) ↔ (log ↾ 𝐷):𝐷⟶(log “ 𝐷))
6153, 60sylibr 223 . . . . 5 (⊤ → (log ↾ 𝐷) ∈ (𝐷cn→(log “ 𝐷)))
6251, 61syl5eqel 2692 . . . 4 (⊤ → (exp ↾ (log “ 𝐷)) ∈ (𝐷cn→(log “ 𝐷)))
63 ssid 3587 . . . . . . . . 9 ℂ ⊆ ℂ
641, 7dvres 23481 . . . . . . . . 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 705 . . . . . . . 8 (ℂ D (exp ↾ (log “ 𝐷))) = ((ℂ D exp) ↾ ((int‘(TopOpen‘ℂfld))‘(log “ 𝐷)))
66 dvef 23547 . . . . . . . . 9 (ℂ D exp) = exp
6710dvloglem 24194 . . . . . . . . . 10 (log “ 𝐷) ∈ (TopOpen‘ℂfld)
68 isopn3i 20696 . . . . . . . . . 10 (((TopOpen‘ℂfld) ∈ Top ∧ (log “ 𝐷) ∈ (TopOpen‘ℂfld)) → ((int‘(TopOpen‘ℂfld))‘(log “ 𝐷)) = (log “ 𝐷))
692, 67, 68mp2an 704 . . . . . . . . 9 ((int‘(TopOpen‘ℂfld))‘(log “ 𝐷)) = (log “ 𝐷)
7066, 69reseq12i 5315 . . . . . . . 8 ((ℂ D exp) ↾ ((int‘(TopOpen‘ℂfld))‘(log “ 𝐷))) = (exp ↾ (log “ 𝐷))
7165, 70eqtri 2632 . . . . . . 7 (ℂ D (exp ↾ (log “ 𝐷))) = (exp ↾ (log “ 𝐷))
7271dmeqi 5247 . . . . . 6 dom (ℂ D (exp ↾ (log “ 𝐷))) = dom (exp ↾ (log “ 𝐷))
73 dmres 5339 . . . . . 6 dom (exp ↾ (log “ 𝐷)) = ((log “ 𝐷) ∩ dom exp)
7424fdmi 5965 . . . . . . . 8 dom exp = ℂ
7557, 74sseqtr4i 3601 . . . . . . 7 (log “ 𝐷) ⊆ dom exp
76 df-ss 3554 . . . . . . 7 ((log “ 𝐷) ⊆ dom exp ↔ ((log “ 𝐷) ∩ dom exp) = (log “ 𝐷))
7775, 76mpbi 219 . . . . . 6 ((log “ 𝐷) ∩ dom exp) = (log “ 𝐷)
7872, 73, 773eqtri 2636 . . . . 5 dom (ℂ D (exp ↾ (log “ 𝐷))) = (log “ 𝐷)
7978a1i 11 . . . 4 (⊤ → dom (ℂ D (exp ↾ (log “ 𝐷))) = (log “ 𝐷))
80 neirr 2791 . . . . . 6 ¬ 0 ≠ 0
81 resss 5342 . . . . . . . . . . . . 13 ((ℂ D exp) ↾ ((int‘(TopOpen‘ℂfld))‘(log “ 𝐷))) ⊆ (ℂ D exp)
8265, 81eqsstri 3598 . . . . . . . . . . . 12 (ℂ D (exp ↾ (log “ 𝐷))) ⊆ (ℂ D exp)
8382, 66sseqtri 3600 . . . . . . . . . . 11 (ℂ D (exp ↾ (log “ 𝐷))) ⊆ exp
84 rnss 5275 . . . . . . . . . . 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 14668 . . . . . . . . . . 11 exp:ℂ⟶(ℂ ∖ {0})
87 frn 5966 . . . . . . . . . . 11 (exp:ℂ⟶(ℂ ∖ {0}) → ran exp ⊆ (ℂ ∖ {0}))
8886, 87ax-mp 5 . . . . . . . . . 10 ran exp ⊆ (ℂ ∖ {0})
8985, 88sstri 3577 . . . . . . . . 9 ran (ℂ D (exp ↾ (log “ 𝐷))) ⊆ (ℂ ∖ {0})
9089sseli 3564 . . . . . . . 8 (0 ∈ ran (ℂ D (exp ↾ (log “ 𝐷))) → 0 ∈ (ℂ ∖ {0}))
91 eldifsn 4260 . . . . . . . 8 (0 ∈ (ℂ ∖ {0}) ↔ (0 ∈ ℂ ∧ 0 ≠ 0))
9290, 91sylib 207 . . . . . . 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 187 . . . . 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 23544 . . 3 (⊤ → (ℂ D (exp ↾ (log “ 𝐷))) = (𝑥𝐷 ↦ (1 / ((ℂ D (exp ↾ (log “ 𝐷)))‘((exp ↾ (log “ 𝐷))‘𝑥)))))
9796trud 1484 . 2 (ℂ D (exp ↾ (log “ 𝐷))) = (𝑥𝐷 ↦ (1 / ((ℂ D (exp ↾ (log “ 𝐷)))‘((exp ↾ (log “ 𝐷))‘𝑥))))
9851oveq2i 6560 . 2 (ℂ D (exp ↾ (log “ 𝐷))) = (ℂ D (log ↾ 𝐷))
9971fveq1i 6104 . . . . 5 ((ℂ D (exp ↾ (log “ 𝐷)))‘((exp ↾ (log “ 𝐷))‘𝑥)) = ((exp ↾ (log “ 𝐷))‘((exp ↾ (log “ 𝐷))‘𝑥))
100 f1ocnvfv2 6433 . . . . . 6 (((exp ↾ (log “ 𝐷)):(log “ 𝐷)–1-1-onto𝐷𝑥𝐷) → ((exp ↾ (log “ 𝐷))‘((exp ↾ (log “ 𝐷))‘𝑥)) = 𝑥)
10145, 100mpan 702 . . . . 5 (𝑥𝐷 → ((exp ↾ (log “ 𝐷))‘((exp ↾ (log “ 𝐷))‘𝑥)) = 𝑥)
10299, 101syl5eq 2656 . . . 4 (𝑥𝐷 → ((ℂ D (exp ↾ (log “ 𝐷)))‘((exp ↾ (log “ 𝐷))‘𝑥)) = 𝑥)
103102oveq2d 6565 . . 3 (𝑥𝐷 → (1 / ((ℂ D (exp ↾ (log “ 𝐷)))‘((exp ↾ (log “ 𝐷))‘𝑥))) = (1 / 𝑥))
104103mpteq2ia 4668 . 2 (𝑥𝐷 ↦ (1 / ((ℂ D (exp ↾ (log “ 𝐷)))‘((exp ↾ (log “ 𝐷))‘𝑥)))) = (𝑥𝐷 ↦ (1 / 𝑥))
10597, 98, 1043eqtr3i 2640 1 (ℂ D (log ↾ 𝐷)) = (𝑥𝐷 ↦ (1 / 𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 195  wa 383   = wceq 1475  wtru 1476  wcel 1977  wne 2780  cdif 3537  cin 3539  wss 3540  {csn 4125  {cpr 4127  cmpt 4643  ccnv 5037  dom cdm 5038  ran crn 5039  cres 5040  cima 5041  Rel wrel 5043  Fun wfun 5798  wf 5800  1-1wf1 5801  1-1-ontowf1o 5803  cfv 5804  (class class class)co 6549  cc 9813  cr 9814  0cc0 9815  1c1 9816  -∞cmnf 9951  -cneg 10146   / cdiv 10563  (,]cioc 12047  cim 13686  expce 14631  πcpi 14636  t crest 15904  TopOpenctopn 15905  fldccnfld 19567  Topctop 20517  intcnt 20631  cnccncf 22487   D cdv 23433  logclog 24105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894  ax-mulf 9895
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ioc 12051  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-seq 12664  df-exp 12723  df-fac 12923  df-bc 12952  df-hash 12980  df-shft 13655  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-limsup 14050  df-clim 14067  df-rlim 14068  df-sum 14265  df-ef 14637  df-sin 14639  df-cos 14640  df-tan 14641  df-pi 14642  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-hom 15793  df-cco 15794  df-rest 15906  df-topn 15907  df-0g 15925  df-gsum 15926  df-topgen 15927  df-pt 15928  df-prds 15931  df-xrs 15985  df-qtop 15990  df-imas 15991  df-xps 15993  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-mulg 17364  df-cntz 17573  df-cmn 18018  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-fbas 19564  df-fg 19565  df-cnfld 19568  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cld 20633  df-ntr 20634  df-cls 20635  df-nei 20712  df-lp 20750  df-perf 20751  df-cn 20841  df-cnp 20842  df-haus 20929  df-cmp 21000  df-tx 21175  df-hmeo 21368  df-fil 21460  df-fm 21552  df-flim 21553  df-flf 21554  df-xms 21935  df-ms 21936  df-tms 21937  df-cncf 22489  df-limc 23436  df-dv 23437  df-log 24107
This theorem is referenced by:  dvlog2  24199  dvcncxp1  24284  dvatan  24462  lgamgulmlem2  24556  dvasin  32666
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