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Theorem dvlog 26670
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 2725 . . . 4 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
21cnfldtopon 24782 . . . . 5 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
32toponrestid 22906 . . . 4 (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
4 cnelprrecn 11247 . . . . 5 ℂ ∈ {ℝ, ℂ}
54a1i 11 . . . 4 (⊤ → ℂ ∈ {ℝ, ℂ})
6 logcn.d . . . . . 6 𝐷 = (ℂ ∖ (-∞(,]0))
76logdmopn 26668 . . . . 5 𝐷 ∈ (TopOpen‘ℂfld)
87a1i 11 . . . 4 (⊤ → 𝐷 ∈ (TopOpen‘ℂfld))
9 logf1o 26583 . . . . . . . . 9 log:(ℂ ∖ {0})–1-1-onto→ran log
10 f1of1 6841 . . . . . . . . 9 (log:(ℂ ∖ {0})–1-1-onto→ran log → log:(ℂ ∖ {0})–1-1→ran log)
119, 10ax-mp 5 . . . . . . . 8 log:(ℂ ∖ {0})–1-1→ran log
126logdmss 26661 . . . . . . . 8 𝐷 ⊆ (ℂ ∖ {0})
13 f1ores 6856 . . . . . . . 8 ((log:(ℂ ∖ {0})–1-1→ran log ∧ 𝐷 ⊆ (ℂ ∖ {0})) → (log ↾ 𝐷):𝐷1-1-onto→(log “ 𝐷))
1411, 12, 13mp2an 690 . . . . . . 7 (log ↾ 𝐷):𝐷1-1-onto→(log “ 𝐷)
15 f1ocnv 6854 . . . . . . 7 ((log ↾ 𝐷):𝐷1-1-onto→(log “ 𝐷) → (log ↾ 𝐷):(log “ 𝐷)–1-1-onto𝐷)
1614, 15ax-mp 5 . . . . . 6 (log ↾ 𝐷):(log “ 𝐷)–1-1-onto𝐷
17 df-log 26575 . . . . . . . . . . 11 log = (exp ↾ (ℑ “ (-π(,]π)))
1817reseq1i 5984 . . . . . . . . . 10 (log ↾ 𝐷) = ((exp ↾ (ℑ “ (-π(,]π))) ↾ 𝐷)
1918cnveqi 5880 . . . . . . . . 9 (log ↾ 𝐷) = ((exp ↾ (ℑ “ (-π(,]π))) ↾ 𝐷)
20 eff 16078 . . . . . . . . . . 11 exp:ℂ⟶ℂ
21 cnvimass 6090 . . . . . . . . . . . 12 (ℑ “ (-π(,]π)) ⊆ dom ℑ
22 imf 15113 . . . . . . . . . . . . 13 ℑ:ℂ⟶ℝ
2322fdmi 6738 . . . . . . . . . . . 12 dom ℑ = ℂ
2421, 23sseqtri 4015 . . . . . . . . . . 11 (ℑ “ (-π(,]π)) ⊆ ℂ
25 fssres 6767 . . . . . . . . . . 11 ((exp:ℂ⟶ℂ ∧ (ℑ “ (-π(,]π)) ⊆ ℂ) → (exp ↾ (ℑ “ (-π(,]π))):(ℑ “ (-π(,]π))⟶ℂ)
2620, 24, 25mp2an 690 . . . . . . . . . 10 (exp ↾ (ℑ “ (-π(,]π))):(ℑ “ (-π(,]π))⟶ℂ
27 ffun 6730 . . . . . . . . . 10 ((exp ↾ (ℑ “ (-π(,]π))):(ℑ “ (-π(,]π))⟶ℂ → Fun (exp ↾ (ℑ “ (-π(,]π))))
28 funcnvres2 6638 . . . . . . . . . 10 (Fun (exp ↾ (ℑ “ (-π(,]π))) → ((exp ↾ (ℑ “ (-π(,]π))) ↾ 𝐷) = ((exp ↾ (ℑ “ (-π(,]π))) ↾ ((exp ↾ (ℑ “ (-π(,]π))) “ 𝐷)))
2926, 27, 28mp2b 10 . . . . . . . . 9 ((exp ↾ (ℑ “ (-π(,]π))) ↾ 𝐷) = ((exp ↾ (ℑ “ (-π(,]π))) ↾ ((exp ↾ (ℑ “ (-π(,]π))) “ 𝐷))
30 cnvimass 6090 . . . . . . . . . . 11 ((exp ↾ (ℑ “ (-π(,]π))) “ 𝐷) ⊆ dom (exp ↾ (ℑ “ (-π(,]π)))
3126fdmi 6738 . . . . . . . . . . 11 dom (exp ↾ (ℑ “ (-π(,]π))) = (ℑ “ (-π(,]π))
3230, 31sseqtri 4015 . . . . . . . . . 10 ((exp ↾ (ℑ “ (-π(,]π))) “ 𝐷) ⊆ (ℑ “ (-π(,]π))
33 resabs1 6015 . . . . . . . . . 10 (((exp ↾ (ℑ “ (-π(,]π))) “ 𝐷) ⊆ (ℑ “ (-π(,]π)) → ((exp ↾ (ℑ “ (-π(,]π))) ↾ ((exp ↾ (ℑ “ (-π(,]π))) “ 𝐷)) = (exp ↾ ((exp ↾ (ℑ “ (-π(,]π))) “ 𝐷)))
3432, 33ax-mp 5 . . . . . . . . 9 ((exp ↾ (ℑ “ (-π(,]π))) ↾ ((exp ↾ (ℑ “ (-π(,]π))) “ 𝐷)) = (exp ↾ ((exp ↾ (ℑ “ (-π(,]π))) “ 𝐷))
3519, 29, 343eqtri 2757 . . . . . . . 8 (log ↾ 𝐷) = (exp ↾ ((exp ↾ (ℑ “ (-π(,]π))) “ 𝐷))
3617imaeq1i 6065 . . . . . . . . 9 (log “ 𝐷) = ((exp ↾ (ℑ “ (-π(,]π))) “ 𝐷)
3736reseq2i 5985 . . . . . . . 8 (exp ↾ (log “ 𝐷)) = (exp ↾ ((exp ↾ (ℑ “ (-π(,]π))) “ 𝐷))
3835, 37eqtr4i 2756 . . . . . . 7 (log ↾ 𝐷) = (exp ↾ (log “ 𝐷))
39 f1oeq1 6830 . . . . . . 7 ((log ↾ 𝐷) = (exp ↾ (log “ 𝐷)) → ((log ↾ 𝐷):(log “ 𝐷)–1-1-onto𝐷 ↔ (exp ↾ (log “ 𝐷)):(log “ 𝐷)–1-1-onto𝐷))
4038, 39ax-mp 5 . . . . . 6 ((log ↾ 𝐷):(log “ 𝐷)–1-1-onto𝐷 ↔ (exp ↾ (log “ 𝐷)):(log “ 𝐷)–1-1-onto𝐷)
4116, 40mpbi 229 . . . . 5 (exp ↾ (log “ 𝐷)):(log “ 𝐷)–1-1-onto𝐷
4241a1i 11 . . . 4 (⊤ → (exp ↾ (log “ 𝐷)):(log “ 𝐷)–1-1-onto𝐷)
4338cnveqi 5880 . . . . . 6 (log ↾ 𝐷) = (exp ↾ (log “ 𝐷))
44 relres 6014 . . . . . . 7 Rel (log ↾ 𝐷)
45 dfrel2 6199 . . . . . . 7 (Rel (log ↾ 𝐷) ↔ (log ↾ 𝐷) = (log ↾ 𝐷))
4644, 45mpbi 229 . . . . . 6 (log ↾ 𝐷) = (log ↾ 𝐷)
4743, 46eqtr3i 2755 . . . . 5 (exp ↾ (log “ 𝐷)) = (log ↾ 𝐷)
48 f1of 6842 . . . . . . 7 ((log ↾ 𝐷):𝐷1-1-onto→(log “ 𝐷) → (log ↾ 𝐷):𝐷⟶(log “ 𝐷))
4914, 48mp1i 13 . . . . . 6 (⊤ → (log ↾ 𝐷):𝐷⟶(log “ 𝐷))
50 imassrn 6079 . . . . . . . 8 (log “ 𝐷) ⊆ ran log
51 logrncn 26581 . . . . . . . . 9 (𝑥 ∈ ran log → 𝑥 ∈ ℂ)
5251ssriv 3982 . . . . . . . 8 ran log ⊆ ℂ
5350, 52sstri 3988 . . . . . . 7 (log “ 𝐷) ⊆ ℂ
546logcn 26666 . . . . . . 7 (log ↾ 𝐷) ∈ (𝐷cn→ℂ)
55 cncfcdm 24901 . . . . . . 7 (((log “ 𝐷) ⊆ ℂ ∧ (log ↾ 𝐷) ∈ (𝐷cn→ℂ)) → ((log ↾ 𝐷) ∈ (𝐷cn→(log “ 𝐷)) ↔ (log ↾ 𝐷):𝐷⟶(log “ 𝐷)))
5653, 54, 55mp2an 690 . . . . . 6 ((log ↾ 𝐷) ∈ (𝐷cn→(log “ 𝐷)) ↔ (log ↾ 𝐷):𝐷⟶(log “ 𝐷))
5749, 56sylibr 233 . . . . 5 (⊤ → (log ↾ 𝐷) ∈ (𝐷cn→(log “ 𝐷)))
5847, 57eqeltrid 2829 . . . 4 (⊤ → (exp ↾ (log “ 𝐷)) ∈ (𝐷cn→(log “ 𝐷)))
59 ssid 4001 . . . . . . . . 9 ℂ ⊆ ℂ
601, 3dvres 25923 . . . . . . . . 9 (((ℂ ⊆ ℂ ∧ exp:ℂ⟶ℂ) ∧ (ℂ ⊆ ℂ ∧ (log “ 𝐷) ⊆ ℂ)) → (ℂ D (exp ↾ (log “ 𝐷))) = ((ℂ D exp) ↾ ((int‘(TopOpen‘ℂfld))‘(log “ 𝐷))))
6159, 20, 59, 53, 60mp4an 691 . . . . . . . 8 (ℂ D (exp ↾ (log “ 𝐷))) = ((ℂ D exp) ↾ ((int‘(TopOpen‘ℂfld))‘(log “ 𝐷)))
62 dvef 25995 . . . . . . . . 9 (ℂ D exp) = exp
631cnfldtop 24783 . . . . . . . . . 10 (TopOpen‘ℂfld) ∈ Top
646dvloglem 26667 . . . . . . . . . 10 (log “ 𝐷) ∈ (TopOpen‘ℂfld)
65 isopn3i 23069 . . . . . . . . . 10 (((TopOpen‘ℂfld) ∈ Top ∧ (log “ 𝐷) ∈ (TopOpen‘ℂfld)) → ((int‘(TopOpen‘ℂfld))‘(log “ 𝐷)) = (log “ 𝐷))
6663, 64, 65mp2an 690 . . . . . . . . 9 ((int‘(TopOpen‘ℂfld))‘(log “ 𝐷)) = (log “ 𝐷)
6762, 66reseq12i 5986 . . . . . . . 8 ((ℂ D exp) ↾ ((int‘(TopOpen‘ℂfld))‘(log “ 𝐷))) = (exp ↾ (log “ 𝐷))
6861, 67eqtri 2753 . . . . . . 7 (ℂ D (exp ↾ (log “ 𝐷))) = (exp ↾ (log “ 𝐷))
6968dmeqi 5910 . . . . . 6 dom (ℂ D (exp ↾ (log “ 𝐷))) = dom (exp ↾ (log “ 𝐷))
70 dmres 6020 . . . . . 6 dom (exp ↾ (log “ 𝐷)) = ((log “ 𝐷) ∩ dom exp)
7120fdmi 6738 . . . . . . . 8 dom exp = ℂ
7253, 71sseqtrri 4016 . . . . . . 7 (log “ 𝐷) ⊆ dom exp
73 dfss2 3964 . . . . . . 7 ((log “ 𝐷) ⊆ dom exp ↔ ((log “ 𝐷) ∩ dom exp) = (log “ 𝐷))
7472, 73mpbi 229 . . . . . 6 ((log “ 𝐷) ∩ dom exp) = (log “ 𝐷)
7569, 70, 743eqtri 2757 . . . . 5 dom (ℂ D (exp ↾ (log “ 𝐷))) = (log “ 𝐷)
7675a1i 11 . . . 4 (⊤ → dom (ℂ D (exp ↾ (log “ 𝐷))) = (log “ 𝐷))
77 neirr 2938 . . . . . 6 ¬ 0 ≠ 0
78 resss 6010 . . . . . . . . . . . . 13 ((ℂ D exp) ↾ ((int‘(TopOpen‘ℂfld))‘(log “ 𝐷))) ⊆ (ℂ D exp)
7961, 78eqsstri 4013 . . . . . . . . . . . 12 (ℂ D (exp ↾ (log “ 𝐷))) ⊆ (ℂ D exp)
8079, 62sseqtri 4015 . . . . . . . . . . 11 (ℂ D (exp ↾ (log “ 𝐷))) ⊆ exp
8180rnssi 5945 . . . . . . . . . 10 ran (ℂ D (exp ↾ (log “ 𝐷))) ⊆ ran exp
82 eff2 16096 . . . . . . . . . . 11 exp:ℂ⟶(ℂ ∖ {0})
83 frn 6734 . . . . . . . . . . 11 (exp:ℂ⟶(ℂ ∖ {0}) → ran exp ⊆ (ℂ ∖ {0}))
8482, 83ax-mp 5 . . . . . . . . . 10 ran exp ⊆ (ℂ ∖ {0})
8581, 84sstri 3988 . . . . . . . . 9 ran (ℂ D (exp ↾ (log “ 𝐷))) ⊆ (ℂ ∖ {0})
8685sseli 3974 . . . . . . . 8 (0 ∈ ran (ℂ D (exp ↾ (log “ 𝐷))) → 0 ∈ (ℂ ∖ {0}))
87 eldifsn 4794 . . . . . . . 8 (0 ∈ (ℂ ∖ {0}) ↔ (0 ∈ ℂ ∧ 0 ≠ 0))
8886, 87sylib 217 . . . . . . 7 (0 ∈ ran (ℂ D (exp ↾ (log “ 𝐷))) → (0 ∈ ℂ ∧ 0 ≠ 0))
8988simprd 494 . . . . . 6 (0 ∈ ran (ℂ D (exp ↾ (log “ 𝐷))) → 0 ≠ 0)
9077, 89mto 196 . . . . 5 ¬ 0 ∈ ran (ℂ D (exp ↾ (log “ 𝐷)))
9190a1i 11 . . . 4 (⊤ → ¬ 0 ∈ ran (ℂ D (exp ↾ (log “ 𝐷))))
921, 3, 5, 8, 42, 58, 76, 91dvcnv 25992 . . 3 (⊤ → (ℂ D (exp ↾ (log “ 𝐷))) = (𝑥𝐷 ↦ (1 / ((ℂ D (exp ↾ (log “ 𝐷)))‘((exp ↾ (log “ 𝐷))‘𝑥)))))
9392mptru 1540 . 2 (ℂ D (exp ↾ (log “ 𝐷))) = (𝑥𝐷 ↦ (1 / ((ℂ D (exp ↾ (log “ 𝐷)))‘((exp ↾ (log “ 𝐷))‘𝑥))))
9447oveq2i 7434 . 2 (ℂ D (exp ↾ (log “ 𝐷))) = (ℂ D (log ↾ 𝐷))
9568fveq1i 6901 . . . . 5 ((ℂ D (exp ↾ (log “ 𝐷)))‘((exp ↾ (log “ 𝐷))‘𝑥)) = ((exp ↾ (log “ 𝐷))‘((exp ↾ (log “ 𝐷))‘𝑥))
96 f1ocnvfv2 7290 . . . . . 6 (((exp ↾ (log “ 𝐷)):(log “ 𝐷)–1-1-onto𝐷𝑥𝐷) → ((exp ↾ (log “ 𝐷))‘((exp ↾ (log “ 𝐷))‘𝑥)) = 𝑥)
9741, 96mpan 688 . . . . 5 (𝑥𝐷 → ((exp ↾ (log “ 𝐷))‘((exp ↾ (log “ 𝐷))‘𝑥)) = 𝑥)
9895, 97eqtrid 2777 . . . 4 (𝑥𝐷 → ((ℂ D (exp ↾ (log “ 𝐷)))‘((exp ↾ (log “ 𝐷))‘𝑥)) = 𝑥)
9998oveq2d 7439 . . 3 (𝑥𝐷 → (1 / ((ℂ D (exp ↾ (log “ 𝐷)))‘((exp ↾ (log “ 𝐷))‘𝑥))) = (1 / 𝑥))
10099mpteq2ia 5255 . 2 (𝑥𝐷 ↦ (1 / ((ℂ D (exp ↾ (log “ 𝐷)))‘((exp ↾ (log “ 𝐷))‘𝑥)))) = (𝑥𝐷 ↦ (1 / 𝑥))
10193, 94, 1003eqtr3i 2761 1 (ℂ D (log ↾ 𝐷)) = (𝑥𝐷 ↦ (1 / 𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 394   = wceq 1533  wtru 1534  wcel 2098  wne 2929  cdif 3943  cin 3945  wss 3946  {csn 4632  {cpr 4634  cmpt 5235  ccnv 5680  dom cdm 5681  ran crn 5682  cres 5683  cima 5684  Rel wrel 5686  Fun wfun 6547  wf 6549  1-1wf1 6550  1-1-ontowf1o 6552  cfv 6553  (class class class)co 7423  cc 11152  cr 11153  0cc0 11154  1c1 11155  -∞cmnf 11292  -cneg 11491   / cdiv 11917  (,]cioc 13374  cim 15098  expce 16058  πcpi 16063  TopOpenctopn 17431  fldccnfld 21335  Topctop 22878  intcnt 23004  cnccncf 24879   D cdv 25875  logclog 26573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5368  ax-pr 5432  ax-un 7745  ax-inf2 9680  ax-cnex 11210  ax-resscn 11211  ax-1cn 11212  ax-icn 11213  ax-addcl 11214  ax-addrcl 11215  ax-mulcl 11216  ax-mulrcl 11217  ax-mulcom 11218  ax-addass 11219  ax-mulass 11220  ax-distr 11221  ax-i2m1 11222  ax-1ne0 11223  ax-1rid 11224  ax-rnegex 11225  ax-rrecex 11226  ax-cnre 11227  ax-pre-lttri 11228  ax-pre-lttrn 11229  ax-pre-ltadd 11230  ax-pre-mulgt0 11231  ax-pre-sup 11232  ax-addf 11233
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4325  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-tp 4637  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-iin 5003  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5579  df-eprel 5585  df-po 5593  df-so 5594  df-fr 5636  df-se 5637  df-we 5638  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-rn 5692  df-res 5693  df-ima 5694  df-pred 6311  df-ord 6378  df-on 6379  df-lim 6380  df-suc 6381  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-isom 6562  df-riota 7379  df-ov 7426  df-oprab 7427  df-mpo 7428  df-of 7689  df-om 7876  df-1st 8002  df-2nd 8003  df-supp 8174  df-frecs 8295  df-wrecs 8326  df-recs 8400  df-rdg 8439  df-1o 8495  df-2o 8496  df-er 8733  df-map 8856  df-pm 8857  df-ixp 8926  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-fsupp 9402  df-fi 9450  df-sup 9481  df-inf 9482  df-oi 9549  df-card 9978  df-pnf 11296  df-mnf 11297  df-xr 11298  df-ltxr 11299  df-le 11300  df-sub 11492  df-neg 11493  df-div 11918  df-nn 12260  df-2 12322  df-3 12323  df-4 12324  df-5 12325  df-6 12326  df-7 12327  df-8 12328  df-9 12329  df-n0 12520  df-z 12606  df-dec 12725  df-uz 12870  df-q 12980  df-rp 13024  df-xneg 13141  df-xadd 13142  df-xmul 13143  df-ioo 13377  df-ioc 13378  df-ico 13379  df-icc 13380  df-fz 13534  df-fzo 13677  df-fl 13807  df-mod 13885  df-seq 14017  df-exp 14077  df-fac 14286  df-bc 14315  df-hash 14343  df-shft 15067  df-cj 15099  df-re 15100  df-im 15101  df-sqrt 15235  df-abs 15236  df-limsup 15468  df-clim 15485  df-rlim 15486  df-sum 15686  df-ef 16064  df-sin 16066  df-cos 16067  df-tan 16068  df-pi 16069  df-struct 17144  df-sets 17161  df-slot 17179  df-ndx 17191  df-base 17209  df-ress 17238  df-plusg 17274  df-mulr 17275  df-starv 17276  df-sca 17277  df-vsca 17278  df-ip 17279  df-tset 17280  df-ple 17281  df-ds 17283  df-unif 17284  df-hom 17285  df-cco 17286  df-rest 17432  df-topn 17433  df-0g 17451  df-gsum 17452  df-topgen 17453  df-pt 17454  df-prds 17457  df-xrs 17512  df-qtop 17517  df-imas 17518  df-xps 17520  df-mre 17594  df-mrc 17595  df-acs 17597  df-mgm 18628  df-sgrp 18707  df-mnd 18723  df-submnd 18769  df-mulg 19057  df-cntz 19306  df-cmn 19775  df-psmet 21327  df-xmet 21328  df-met 21329  df-bl 21330  df-mopn 21331  df-fbas 21332  df-fg 21333  df-cnfld 21336  df-top 22879  df-topon 22896  df-topsp 22918  df-bases 22932  df-cld 23006  df-ntr 23007  df-cls 23008  df-nei 23085  df-lp 23123  df-perf 23124  df-cn 23214  df-cnp 23215  df-haus 23302  df-cmp 23374  df-tx 23549  df-hmeo 23742  df-fil 23833  df-fm 23925  df-flim 23926  df-flf 23927  df-xms 24309  df-ms 24310  df-tms 24311  df-cncf 24881  df-limc 25878  df-dv 25879  df-log 26575
This theorem is referenced by:  dvlog2  26672  dvcncxp1  26762  dvatan  26955  lgamgulmlem2  27050  dvasin  37353
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