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Theorem dvlog 26693
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 2737 . . . 4 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
21cnfldtopon 24803 . . . . 5 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
32toponrestid 22927 . . . 4 (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
4 cnelprrecn 11248 . . . . 5 ℂ ∈ {ℝ, ℂ}
54a1i 11 . . . 4 (⊤ → ℂ ∈ {ℝ, ℂ})
6 logcn.d . . . . . 6 𝐷 = (ℂ ∖ (-∞(,]0))
76logdmopn 26691 . . . . 5 𝐷 ∈ (TopOpen‘ℂfld)
87a1i 11 . . . 4 (⊤ → 𝐷 ∈ (TopOpen‘ℂfld))
9 logf1o 26606 . . . . . . . . 9 log:(ℂ ∖ {0})–1-1-onto→ran log
10 f1of1 6847 . . . . . . . . 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 26684 . . . . . . . 8 𝐷 ⊆ (ℂ ∖ {0})
13 f1ores 6862 . . . . . . . 8 ((log:(ℂ ∖ {0})–1-1→ran log ∧ 𝐷 ⊆ (ℂ ∖ {0})) → (log ↾ 𝐷):𝐷1-1-onto→(log “ 𝐷))
1411, 12, 13mp2an 692 . . . . . . 7 (log ↾ 𝐷):𝐷1-1-onto→(log “ 𝐷)
15 f1ocnv 6860 . . . . . . 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 26598 . . . . . . . . . . 11 log = (exp ↾ (ℑ “ (-π(,]π)))
1817reseq1i 5993 . . . . . . . . . 10 (log ↾ 𝐷) = ((exp ↾ (ℑ “ (-π(,]π))) ↾ 𝐷)
1918cnveqi 5885 . . . . . . . . 9 (log ↾ 𝐷) = ((exp ↾ (ℑ “ (-π(,]π))) ↾ 𝐷)
20 eff 16117 . . . . . . . . . . 11 exp:ℂ⟶ℂ
21 cnvimass 6100 . . . . . . . . . . . 12 (ℑ “ (-π(,]π)) ⊆ dom ℑ
22 imf 15152 . . . . . . . . . . . . 13 ℑ:ℂ⟶ℝ
2322fdmi 6747 . . . . . . . . . . . 12 dom ℑ = ℂ
2421, 23sseqtri 4032 . . . . . . . . . . 11 (ℑ “ (-π(,]π)) ⊆ ℂ
25 fssres 6774 . . . . . . . . . . 11 ((exp:ℂ⟶ℂ ∧ (ℑ “ (-π(,]π)) ⊆ ℂ) → (exp ↾ (ℑ “ (-π(,]π))):(ℑ “ (-π(,]π))⟶ℂ)
2620, 24, 25mp2an 692 . . . . . . . . . 10 (exp ↾ (ℑ “ (-π(,]π))):(ℑ “ (-π(,]π))⟶ℂ
27 ffun 6739 . . . . . . . . . 10 ((exp ↾ (ℑ “ (-π(,]π))):(ℑ “ (-π(,]π))⟶ℂ → Fun (exp ↾ (ℑ “ (-π(,]π))))
28 funcnvres2 6646 . . . . . . . . . 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 6100 . . . . . . . . . . 11 ((exp ↾ (ℑ “ (-π(,]π))) “ 𝐷) ⊆ dom (exp ↾ (ℑ “ (-π(,]π)))
3126fdmi 6747 . . . . . . . . . . 11 dom (exp ↾ (ℑ “ (-π(,]π))) = (ℑ “ (-π(,]π))
3230, 31sseqtri 4032 . . . . . . . . . 10 ((exp ↾ (ℑ “ (-π(,]π))) “ 𝐷) ⊆ (ℑ “ (-π(,]π))
33 resabs1 6024 . . . . . . . . . 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 2769 . . . . . . . 8 (log ↾ 𝐷) = (exp ↾ ((exp ↾ (ℑ “ (-π(,]π))) “ 𝐷))
3617imaeq1i 6075 . . . . . . . . 9 (log “ 𝐷) = ((exp ↾ (ℑ “ (-π(,]π))) “ 𝐷)
3736reseq2i 5994 . . . . . . . 8 (exp ↾ (log “ 𝐷)) = (exp ↾ ((exp ↾ (ℑ “ (-π(,]π))) “ 𝐷))
3835, 37eqtr4i 2768 . . . . . . 7 (log ↾ 𝐷) = (exp ↾ (log “ 𝐷))
39 f1oeq1 6836 . . . . . . 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 230 . . . . 5 (exp ↾ (log “ 𝐷)):(log “ 𝐷)–1-1-onto𝐷
4241a1i 11 . . . 4 (⊤ → (exp ↾ (log “ 𝐷)):(log “ 𝐷)–1-1-onto𝐷)
4338cnveqi 5885 . . . . . 6 (log ↾ 𝐷) = (exp ↾ (log “ 𝐷))
44 relres 6023 . . . . . . 7 Rel (log ↾ 𝐷)
45 dfrel2 6209 . . . . . . 7 (Rel (log ↾ 𝐷) ↔ (log ↾ 𝐷) = (log ↾ 𝐷))
4644, 45mpbi 230 . . . . . 6 (log ↾ 𝐷) = (log ↾ 𝐷)
4743, 46eqtr3i 2767 . . . . 5 (exp ↾ (log “ 𝐷)) = (log ↾ 𝐷)
48 f1of 6848 . . . . . . 7 ((log ↾ 𝐷):𝐷1-1-onto→(log “ 𝐷) → (log ↾ 𝐷):𝐷⟶(log “ 𝐷))
4914, 48mp1i 13 . . . . . 6 (⊤ → (log ↾ 𝐷):𝐷⟶(log “ 𝐷))
50 imassrn 6089 . . . . . . . 8 (log “ 𝐷) ⊆ ran log
51 logrncn 26604 . . . . . . . . 9 (𝑥 ∈ ran log → 𝑥 ∈ ℂ)
5251ssriv 3987 . . . . . . . 8 ran log ⊆ ℂ
5350, 52sstri 3993 . . . . . . 7 (log “ 𝐷) ⊆ ℂ
546logcn 26689 . . . . . . 7 (log ↾ 𝐷) ∈ (𝐷cn→ℂ)
55 cncfcdm 24924 . . . . . . 7 (((log “ 𝐷) ⊆ ℂ ∧ (log ↾ 𝐷) ∈ (𝐷cn→ℂ)) → ((log ↾ 𝐷) ∈ (𝐷cn→(log “ 𝐷)) ↔ (log ↾ 𝐷):𝐷⟶(log “ 𝐷)))
5653, 54, 55mp2an 692 . . . . . 6 ((log ↾ 𝐷) ∈ (𝐷cn→(log “ 𝐷)) ↔ (log ↾ 𝐷):𝐷⟶(log “ 𝐷))
5749, 56sylibr 234 . . . . 5 (⊤ → (log ↾ 𝐷) ∈ (𝐷cn→(log “ 𝐷)))
5847, 57eqeltrid 2845 . . . 4 (⊤ → (exp ↾ (log “ 𝐷)) ∈ (𝐷cn→(log “ 𝐷)))
59 ssid 4006 . . . . . . . . 9 ℂ ⊆ ℂ
601, 3dvres 25946 . . . . . . . . 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 693 . . . . . . . 8 (ℂ D (exp ↾ (log “ 𝐷))) = ((ℂ D exp) ↾ ((int‘(TopOpen‘ℂfld))‘(log “ 𝐷)))
62 dvef 26018 . . . . . . . . 9 (ℂ D exp) = exp
631cnfldtop 24804 . . . . . . . . . 10 (TopOpen‘ℂfld) ∈ Top
646dvloglem 26690 . . . . . . . . . 10 (log “ 𝐷) ∈ (TopOpen‘ℂfld)
65 isopn3i 23090 . . . . . . . . . 10 (((TopOpen‘ℂfld) ∈ Top ∧ (log “ 𝐷) ∈ (TopOpen‘ℂfld)) → ((int‘(TopOpen‘ℂfld))‘(log “ 𝐷)) = (log “ 𝐷))
6663, 64, 65mp2an 692 . . . . . . . . 9 ((int‘(TopOpen‘ℂfld))‘(log “ 𝐷)) = (log “ 𝐷)
6762, 66reseq12i 5995 . . . . . . . 8 ((ℂ D exp) ↾ ((int‘(TopOpen‘ℂfld))‘(log “ 𝐷))) = (exp ↾ (log “ 𝐷))
6861, 67eqtri 2765 . . . . . . 7 (ℂ D (exp ↾ (log “ 𝐷))) = (exp ↾ (log “ 𝐷))
6968dmeqi 5915 . . . . . 6 dom (ℂ D (exp ↾ (log “ 𝐷))) = dom (exp ↾ (log “ 𝐷))
70 dmres 6030 . . . . . 6 dom (exp ↾ (log “ 𝐷)) = ((log “ 𝐷) ∩ dom exp)
7120fdmi 6747 . . . . . . . 8 dom exp = ℂ
7253, 71sseqtrri 4033 . . . . . . 7 (log “ 𝐷) ⊆ dom exp
73 dfss2 3969 . . . . . . 7 ((log “ 𝐷) ⊆ dom exp ↔ ((log “ 𝐷) ∩ dom exp) = (log “ 𝐷))
7472, 73mpbi 230 . . . . . 6 ((log “ 𝐷) ∩ dom exp) = (log “ 𝐷)
7569, 70, 743eqtri 2769 . . . . 5 dom (ℂ D (exp ↾ (log “ 𝐷))) = (log “ 𝐷)
7675a1i 11 . . . 4 (⊤ → dom (ℂ D (exp ↾ (log “ 𝐷))) = (log “ 𝐷))
77 neirr 2949 . . . . . 6 ¬ 0 ≠ 0
78 resss 6019 . . . . . . . . . . . . 13 ((ℂ D exp) ↾ ((int‘(TopOpen‘ℂfld))‘(log “ 𝐷))) ⊆ (ℂ D exp)
7961, 78eqsstri 4030 . . . . . . . . . . . 12 (ℂ D (exp ↾ (log “ 𝐷))) ⊆ (ℂ D exp)
8079, 62sseqtri 4032 . . . . . . . . . . 11 (ℂ D (exp ↾ (log “ 𝐷))) ⊆ exp
8180rnssi 5951 . . . . . . . . . 10 ran (ℂ D (exp ↾ (log “ 𝐷))) ⊆ ran exp
82 eff2 16135 . . . . . . . . . . 11 exp:ℂ⟶(ℂ ∖ {0})
83 frn 6743 . . . . . . . . . . 11 (exp:ℂ⟶(ℂ ∖ {0}) → ran exp ⊆ (ℂ ∖ {0}))
8482, 83ax-mp 5 . . . . . . . . . 10 ran exp ⊆ (ℂ ∖ {0})
8581, 84sstri 3993 . . . . . . . . 9 ran (ℂ D (exp ↾ (log “ 𝐷))) ⊆ (ℂ ∖ {0})
8685sseli 3979 . . . . . . . 8 (0 ∈ ran (ℂ D (exp ↾ (log “ 𝐷))) → 0 ∈ (ℂ ∖ {0}))
87 eldifsn 4786 . . . . . . . 8 (0 ∈ (ℂ ∖ {0}) ↔ (0 ∈ ℂ ∧ 0 ≠ 0))
8886, 87sylib 218 . . . . . . 7 (0 ∈ ran (ℂ D (exp ↾ (log “ 𝐷))) → (0 ∈ ℂ ∧ 0 ≠ 0))
8988simprd 495 . . . . . 6 (0 ∈ ran (ℂ D (exp ↾ (log “ 𝐷))) → 0 ≠ 0)
9077, 89mto 197 . . . . 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 26015 . . 3 (⊤ → (ℂ D (exp ↾ (log “ 𝐷))) = (𝑥𝐷 ↦ (1 / ((ℂ D (exp ↾ (log “ 𝐷)))‘((exp ↾ (log “ 𝐷))‘𝑥)))))
9392mptru 1547 . 2 (ℂ D (exp ↾ (log “ 𝐷))) = (𝑥𝐷 ↦ (1 / ((ℂ D (exp ↾ (log “ 𝐷)))‘((exp ↾ (log “ 𝐷))‘𝑥))))
9447oveq2i 7442 . 2 (ℂ D (exp ↾ (log “ 𝐷))) = (ℂ D (log ↾ 𝐷))
9568fveq1i 6907 . . . . 5 ((ℂ D (exp ↾ (log “ 𝐷)))‘((exp ↾ (log “ 𝐷))‘𝑥)) = ((exp ↾ (log “ 𝐷))‘((exp ↾ (log “ 𝐷))‘𝑥))
96 f1ocnvfv2 7297 . . . . . 6 (((exp ↾ (log “ 𝐷)):(log “ 𝐷)–1-1-onto𝐷𝑥𝐷) → ((exp ↾ (log “ 𝐷))‘((exp ↾ (log “ 𝐷))‘𝑥)) = 𝑥)
9741, 96mpan 690 . . . . 5 (𝑥𝐷 → ((exp ↾ (log “ 𝐷))‘((exp ↾ (log “ 𝐷))‘𝑥)) = 𝑥)
9895, 97eqtrid 2789 . . . 4 (𝑥𝐷 → ((ℂ D (exp ↾ (log “ 𝐷)))‘((exp ↾ (log “ 𝐷))‘𝑥)) = 𝑥)
9998oveq2d 7447 . . 3 (𝑥𝐷 → (1 / ((ℂ D (exp ↾ (log “ 𝐷)))‘((exp ↾ (log “ 𝐷))‘𝑥))) = (1 / 𝑥))
10099mpteq2ia 5245 . 2 (𝑥𝐷 ↦ (1 / ((ℂ D (exp ↾ (log “ 𝐷)))‘((exp ↾ (log “ 𝐷))‘𝑥)))) = (𝑥𝐷 ↦ (1 / 𝑥))
10193, 94, 1003eqtr3i 2773 1 (ℂ D (log ↾ 𝐷)) = (𝑥𝐷 ↦ (1 / 𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wa 395   = wceq 1540  wtru 1541  wcel 2108  wne 2940  cdif 3948  cin 3950  wss 3951  {csn 4626  {cpr 4628  cmpt 5225  ccnv 5684  dom cdm 5685  ran crn 5686  cres 5687  cima 5688  Rel wrel 5690  Fun wfun 6555  wf 6557  1-1wf1 6558  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  cc 11153  cr 11154  0cc0 11155  1c1 11156  -∞cmnf 11293  -cneg 11493   / cdiv 11920  (,]cioc 13388  cim 15137  expce 16097  πcpi 16102  TopOpenctopn 17466  fldccnfld 21364  Topctop 22899  intcnt 23025  cnccncf 24902   D cdv 25898  logclog 26596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233  ax-addf 11234
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-fi 9451  df-sup 9482  df-inf 9483  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-ioo 13391  df-ioc 13392  df-ico 13393  df-icc 13394  df-fz 13548  df-fzo 13695  df-fl 13832  df-mod 13910  df-seq 14043  df-exp 14103  df-fac 14313  df-bc 14342  df-hash 14370  df-shft 15106  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-limsup 15507  df-clim 15524  df-rlim 15525  df-sum 15723  df-ef 16103  df-sin 16105  df-cos 16106  df-tan 16107  df-pi 16108  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-starv 17312  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-hom 17321  df-cco 17322  df-rest 17467  df-topn 17468  df-0g 17486  df-gsum 17487  df-topgen 17488  df-pt 17489  df-prds 17492  df-xrs 17547  df-qtop 17552  df-imas 17553  df-xps 17555  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-mulg 19086  df-cntz 19335  df-cmn 19800  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-fbas 21361  df-fg 21362  df-cnfld 21365  df-top 22900  df-topon 22917  df-topsp 22939  df-bases 22953  df-cld 23027  df-ntr 23028  df-cls 23029  df-nei 23106  df-lp 23144  df-perf 23145  df-cn 23235  df-cnp 23236  df-haus 23323  df-cmp 23395  df-tx 23570  df-hmeo 23763  df-fil 23854  df-fm 23946  df-flim 23947  df-flf 23948  df-xms 24330  df-ms 24331  df-tms 24332  df-cncf 24904  df-limc 25901  df-dv 25902  df-log 26598
This theorem is referenced by:  dvlog2  26695  dvcncxp1  26785  dvatan  26978  lgamgulmlem2  27073  dvasin  37711  readvrec2  42391  readvrec  42392
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