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Theorem dvlog 25246
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 2801 . . . 4 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
21cnfldtopon 23392 . . . . 5 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
32toponrestid 21530 . . . 4 (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
4 cnelprrecn 10623 . . . . 5 ℂ ∈ {ℝ, ℂ}
54a1i 11 . . . 4 (⊤ → ℂ ∈ {ℝ, ℂ})
6 logcn.d . . . . . 6 𝐷 = (ℂ ∖ (-∞(,]0))
76logdmopn 25244 . . . . 5 𝐷 ∈ (TopOpen‘ℂfld)
87a1i 11 . . . 4 (⊤ → 𝐷 ∈ (TopOpen‘ℂfld))
9 logf1o 25160 . . . . . . . . 9 log:(ℂ ∖ {0})–1-1-onto→ran log
10 f1of1 6593 . . . . . . . . 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 25237 . . . . . . . 8 𝐷 ⊆ (ℂ ∖ {0})
13 f1ores 6608 . . . . . . . 8 ((log:(ℂ ∖ {0})–1-1→ran log ∧ 𝐷 ⊆ (ℂ ∖ {0})) → (log ↾ 𝐷):𝐷1-1-onto→(log “ 𝐷))
1411, 12, 13mp2an 691 . . . . . . 7 (log ↾ 𝐷):𝐷1-1-onto→(log “ 𝐷)
15 f1ocnv 6606 . . . . . . 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 25152 . . . . . . . . . . 11 log = (exp ↾ (ℑ “ (-π(,]π)))
1817reseq1i 5818 . . . . . . . . . 10 (log ↾ 𝐷) = ((exp ↾ (ℑ “ (-π(,]π))) ↾ 𝐷)
1918cnveqi 5713 . . . . . . . . 9 (log ↾ 𝐷) = ((exp ↾ (ℑ “ (-π(,]π))) ↾ 𝐷)
20 eff 15431 . . . . . . . . . . 11 exp:ℂ⟶ℂ
21 cnvimass 5920 . . . . . . . . . . . 12 (ℑ “ (-π(,]π)) ⊆ dom ℑ
22 imf 14468 . . . . . . . . . . . . 13 ℑ:ℂ⟶ℝ
2322fdmi 6502 . . . . . . . . . . . 12 dom ℑ = ℂ
2421, 23sseqtri 3954 . . . . . . . . . . 11 (ℑ “ (-π(,]π)) ⊆ ℂ
25 fssres 6522 . . . . . . . . . . 11 ((exp:ℂ⟶ℂ ∧ (ℑ “ (-π(,]π)) ⊆ ℂ) → (exp ↾ (ℑ “ (-π(,]π))):(ℑ “ (-π(,]π))⟶ℂ)
2620, 24, 25mp2an 691 . . . . . . . . . 10 (exp ↾ (ℑ “ (-π(,]π))):(ℑ “ (-π(,]π))⟶ℂ
27 ffun 6494 . . . . . . . . . 10 ((exp ↾ (ℑ “ (-π(,]π))):(ℑ “ (-π(,]π))⟶ℂ → Fun (exp ↾ (ℑ “ (-π(,]π))))
28 funcnvres2 6408 . . . . . . . . . 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 5920 . . . . . . . . . . 11 ((exp ↾ (ℑ “ (-π(,]π))) “ 𝐷) ⊆ dom (exp ↾ (ℑ “ (-π(,]π)))
3126fdmi 6502 . . . . . . . . . . 11 dom (exp ↾ (ℑ “ (-π(,]π))) = (ℑ “ (-π(,]π))
3230, 31sseqtri 3954 . . . . . . . . . 10 ((exp ↾ (ℑ “ (-π(,]π))) “ 𝐷) ⊆ (ℑ “ (-π(,]π))
33 resabs1 5852 . . . . . . . . . 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 2828 . . . . . . . 8 (log ↾ 𝐷) = (exp ↾ ((exp ↾ (ℑ “ (-π(,]π))) “ 𝐷))
3617imaeq1i 5897 . . . . . . . . 9 (log “ 𝐷) = ((exp ↾ (ℑ “ (-π(,]π))) “ 𝐷)
3736reseq2i 5819 . . . . . . . 8 (exp ↾ (log “ 𝐷)) = (exp ↾ ((exp ↾ (ℑ “ (-π(,]π))) “ 𝐷))
3835, 37eqtr4i 2827 . . . . . . 7 (log ↾ 𝐷) = (exp ↾ (log “ 𝐷))
39 f1oeq1 6583 . . . . . . 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 233 . . . . 5 (exp ↾ (log “ 𝐷)):(log “ 𝐷)–1-1-onto𝐷
4241a1i 11 . . . 4 (⊤ → (exp ↾ (log “ 𝐷)):(log “ 𝐷)–1-1-onto𝐷)
4338cnveqi 5713 . . . . . 6 (log ↾ 𝐷) = (exp ↾ (log “ 𝐷))
44 relres 5851 . . . . . . 7 Rel (log ↾ 𝐷)
45 dfrel2 6017 . . . . . . 7 (Rel (log ↾ 𝐷) ↔ (log ↾ 𝐷) = (log ↾ 𝐷))
4644, 45mpbi 233 . . . . . 6 (log ↾ 𝐷) = (log ↾ 𝐷)
4743, 46eqtr3i 2826 . . . . 5 (exp ↾ (log “ 𝐷)) = (log ↾ 𝐷)
48 f1of 6594 . . . . . . 7 ((log ↾ 𝐷):𝐷1-1-onto→(log “ 𝐷) → (log ↾ 𝐷):𝐷⟶(log “ 𝐷))
4914, 48mp1i 13 . . . . . 6 (⊤ → (log ↾ 𝐷):𝐷⟶(log “ 𝐷))
50 imassrn 5911 . . . . . . . 8 (log “ 𝐷) ⊆ ran log
51 logrncn 25158 . . . . . . . . 9 (𝑥 ∈ ran log → 𝑥 ∈ ℂ)
5251ssriv 3922 . . . . . . . 8 ran log ⊆ ℂ
5350, 52sstri 3927 . . . . . . 7 (log “ 𝐷) ⊆ ℂ
546logcn 25242 . . . . . . 7 (log ↾ 𝐷) ∈ (𝐷cn→ℂ)
55 cncffvrn 23507 . . . . . . 7 (((log “ 𝐷) ⊆ ℂ ∧ (log ↾ 𝐷) ∈ (𝐷cn→ℂ)) → ((log ↾ 𝐷) ∈ (𝐷cn→(log “ 𝐷)) ↔ (log ↾ 𝐷):𝐷⟶(log “ 𝐷)))
5653, 54, 55mp2an 691 . . . . . 6 ((log ↾ 𝐷) ∈ (𝐷cn→(log “ 𝐷)) ↔ (log ↾ 𝐷):𝐷⟶(log “ 𝐷))
5749, 56sylibr 237 . . . . 5 (⊤ → (log ↾ 𝐷) ∈ (𝐷cn→(log “ 𝐷)))
5847, 57eqeltrid 2897 . . . 4 (⊤ → (exp ↾ (log “ 𝐷)) ∈ (𝐷cn→(log “ 𝐷)))
59 ssid 3940 . . . . . . . . 9 ℂ ⊆ ℂ
601, 3dvres 24518 . . . . . . . . 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 692 . . . . . . . 8 (ℂ D (exp ↾ (log “ 𝐷))) = ((ℂ D exp) ↾ ((int‘(TopOpen‘ℂfld))‘(log “ 𝐷)))
62 dvef 24587 . . . . . . . . 9 (ℂ D exp) = exp
631cnfldtop 23393 . . . . . . . . . 10 (TopOpen‘ℂfld) ∈ Top
646dvloglem 25243 . . . . . . . . . 10 (log “ 𝐷) ∈ (TopOpen‘ℂfld)
65 isopn3i 21691 . . . . . . . . . 10 (((TopOpen‘ℂfld) ∈ Top ∧ (log “ 𝐷) ∈ (TopOpen‘ℂfld)) → ((int‘(TopOpen‘ℂfld))‘(log “ 𝐷)) = (log “ 𝐷))
6663, 64, 65mp2an 691 . . . . . . . . 9 ((int‘(TopOpen‘ℂfld))‘(log “ 𝐷)) = (log “ 𝐷)
6762, 66reseq12i 5820 . . . . . . . 8 ((ℂ D exp) ↾ ((int‘(TopOpen‘ℂfld))‘(log “ 𝐷))) = (exp ↾ (log “ 𝐷))
6861, 67eqtri 2824 . . . . . . 7 (ℂ D (exp ↾ (log “ 𝐷))) = (exp ↾ (log “ 𝐷))
6968dmeqi 5741 . . . . . 6 dom (ℂ D (exp ↾ (log “ 𝐷))) = dom (exp ↾ (log “ 𝐷))
70 dmres 5844 . . . . . 6 dom (exp ↾ (log “ 𝐷)) = ((log “ 𝐷) ∩ dom exp)
7120fdmi 6502 . . . . . . . 8 dom exp = ℂ
7253, 71sseqtrri 3955 . . . . . . 7 (log “ 𝐷) ⊆ dom exp
73 df-ss 3901 . . . . . . 7 ((log “ 𝐷) ⊆ dom exp ↔ ((log “ 𝐷) ∩ dom exp) = (log “ 𝐷))
7472, 73mpbi 233 . . . . . 6 ((log “ 𝐷) ∩ dom exp) = (log “ 𝐷)
7569, 70, 743eqtri 2828 . . . . 5 dom (ℂ D (exp ↾ (log “ 𝐷))) = (log “ 𝐷)
7675a1i 11 . . . 4 (⊤ → dom (ℂ D (exp ↾ (log “ 𝐷))) = (log “ 𝐷))
77 neirr 2999 . . . . . 6 ¬ 0 ≠ 0
78 resss 5847 . . . . . . . . . . . . 13 ((ℂ D exp) ↾ ((int‘(TopOpen‘ℂfld))‘(log “ 𝐷))) ⊆ (ℂ D exp)
7961, 78eqsstri 3952 . . . . . . . . . . . 12 (ℂ D (exp ↾ (log “ 𝐷))) ⊆ (ℂ D exp)
8079, 62sseqtri 3954 . . . . . . . . . . 11 (ℂ D (exp ↾ (log “ 𝐷))) ⊆ exp
8180rnssi 5778 . . . . . . . . . 10 ran (ℂ D (exp ↾ (log “ 𝐷))) ⊆ ran exp
82 eff2 15448 . . . . . . . . . . 11 exp:ℂ⟶(ℂ ∖ {0})
83 frn 6497 . . . . . . . . . . 11 (exp:ℂ⟶(ℂ ∖ {0}) → ran exp ⊆ (ℂ ∖ {0}))
8482, 83ax-mp 5 . . . . . . . . . 10 ran exp ⊆ (ℂ ∖ {0})
8581, 84sstri 3927 . . . . . . . . 9 ran (ℂ D (exp ↾ (log “ 𝐷))) ⊆ (ℂ ∖ {0})
8685sseli 3914 . . . . . . . 8 (0 ∈ ran (ℂ D (exp ↾ (log “ 𝐷))) → 0 ∈ (ℂ ∖ {0}))
87 eldifsn 4683 . . . . . . . 8 (0 ∈ (ℂ ∖ {0}) ↔ (0 ∈ ℂ ∧ 0 ≠ 0))
8886, 87sylib 221 . . . . . . 7 (0 ∈ ran (ℂ D (exp ↾ (log “ 𝐷))) → (0 ∈ ℂ ∧ 0 ≠ 0))
8988simprd 499 . . . . . 6 (0 ∈ ran (ℂ D (exp ↾ (log “ 𝐷))) → 0 ≠ 0)
9077, 89mto 200 . . . . 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 24584 . . 3 (⊤ → (ℂ D (exp ↾ (log “ 𝐷))) = (𝑥𝐷 ↦ (1 / ((ℂ D (exp ↾ (log “ 𝐷)))‘((exp ↾ (log “ 𝐷))‘𝑥)))))
9392mptru 1545 . 2 (ℂ D (exp ↾ (log “ 𝐷))) = (𝑥𝐷 ↦ (1 / ((ℂ D (exp ↾ (log “ 𝐷)))‘((exp ↾ (log “ 𝐷))‘𝑥))))
9447oveq2i 7150 . 2 (ℂ D (exp ↾ (log “ 𝐷))) = (ℂ D (log ↾ 𝐷))
9568fveq1i 6650 . . . . 5 ((ℂ D (exp ↾ (log “ 𝐷)))‘((exp ↾ (log “ 𝐷))‘𝑥)) = ((exp ↾ (log “ 𝐷))‘((exp ↾ (log “ 𝐷))‘𝑥))
96 f1ocnvfv2 7016 . . . . . 6 (((exp ↾ (log “ 𝐷)):(log “ 𝐷)–1-1-onto𝐷𝑥𝐷) → ((exp ↾ (log “ 𝐷))‘((exp ↾ (log “ 𝐷))‘𝑥)) = 𝑥)
9741, 96mpan 689 . . . . 5 (𝑥𝐷 → ((exp ↾ (log “ 𝐷))‘((exp ↾ (log “ 𝐷))‘𝑥)) = 𝑥)
9895, 97syl5eq 2848 . . . 4 (𝑥𝐷 → ((ℂ D (exp ↾ (log “ 𝐷)))‘((exp ↾ (log “ 𝐷))‘𝑥)) = 𝑥)
9998oveq2d 7155 . . 3 (𝑥𝐷 → (1 / ((ℂ D (exp ↾ (log “ 𝐷)))‘((exp ↾ (log “ 𝐷))‘𝑥))) = (1 / 𝑥))
10099mpteq2ia 5124 . 2 (𝑥𝐷 ↦ (1 / ((ℂ D (exp ↾ (log “ 𝐷)))‘((exp ↾ (log “ 𝐷))‘𝑥)))) = (𝑥𝐷 ↦ (1 / 𝑥))
10193, 94, 1003eqtr3i 2832 1 (ℂ D (log ↾ 𝐷)) = (𝑥𝐷 ↦ (1 / 𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 209  wa 399   = wceq 1538  wtru 1539  wcel 2112  wne 2990  cdif 3881  cin 3883  wss 3884  {csn 4528  {cpr 4530  cmpt 5113  ccnv 5522  dom cdm 5523  ran crn 5524  cres 5525  cima 5526  Rel wrel 5528  Fun wfun 6322  wf 6324  1-1wf1 6325  1-1-ontowf1o 6327  cfv 6328  (class class class)co 7139  cc 10528  cr 10529  0cc0 10530  1c1 10531  -∞cmnf 10666  -cneg 10864   / cdiv 11290  (,]cioc 12731  cim 14453  expce 15411  πcpi 15416  TopOpenctopn 16691  fldccnfld 20095  Topctop 21502  intcnt 21626  cnccncf 23485   D cdv 24470  logclog 25150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-inf2 9092  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-pre-sup 10608  ax-addf 10609  ax-mulf 10610
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-of 7393  df-om 7565  df-1st 7675  df-2nd 7676  df-supp 7818  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-er 8276  df-map 8395  df-pm 8396  df-ixp 8449  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fsupp 8822  df-fi 8863  df-sup 8894  df-inf 8895  df-oi 8962  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-nn 11630  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-q 12341  df-rp 12382  df-xneg 12499  df-xadd 12500  df-xmul 12501  df-ioo 12734  df-ioc 12735  df-ico 12736  df-icc 12737  df-fz 12890  df-fzo 13033  df-fl 13161  df-mod 13237  df-seq 13369  df-exp 13430  df-fac 13634  df-bc 13663  df-hash 13691  df-shft 14422  df-cj 14454  df-re 14455  df-im 14456  df-sqrt 14590  df-abs 14591  df-limsup 14824  df-clim 14841  df-rlim 14842  df-sum 15039  df-ef 15417  df-sin 15419  df-cos 15420  df-tan 15421  df-pi 15422  df-struct 16481  df-ndx 16482  df-slot 16483  df-base 16485  df-sets 16486  df-ress 16487  df-plusg 16574  df-mulr 16575  df-starv 16576  df-sca 16577  df-vsca 16578  df-ip 16579  df-tset 16580  df-ple 16581  df-ds 16583  df-unif 16584  df-hom 16585  df-cco 16586  df-rest 16692  df-topn 16693  df-0g 16711  df-gsum 16712  df-topgen 16713  df-pt 16714  df-prds 16717  df-xrs 16771  df-qtop 16776  df-imas 16777  df-xps 16779  df-mre 16853  df-mrc 16854  df-acs 16856  df-mgm 17848  df-sgrp 17897  df-mnd 17908  df-submnd 17953  df-mulg 18221  df-cntz 18443  df-cmn 18904  df-psmet 20087  df-xmet 20088  df-met 20089  df-bl 20090  df-mopn 20091  df-fbas 20092  df-fg 20093  df-cnfld 20096  df-top 21503  df-topon 21520  df-topsp 21542  df-bases 21555  df-cld 21628  df-ntr 21629  df-cls 21630  df-nei 21707  df-lp 21745  df-perf 21746  df-cn 21836  df-cnp 21837  df-haus 21924  df-cmp 21996  df-tx 22171  df-hmeo 22364  df-fil 22455  df-fm 22547  df-flim 22548  df-flf 22549  df-xms 22931  df-ms 22932  df-tms 22933  df-cncf 23487  df-limc 24473  df-dv 24474  df-log 25152
This theorem is referenced by:  dvlog2  25248  dvcncxp1  25336  dvatan  25525  lgamgulmlem2  25619  dvasin  35140
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