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Theorem dvlog 22071
Description: The derivative of the complex logarithm function. (Contributed by Mario Carneiro, 25-Feb-2015.)
Hypothesis
Ref Expression
logcn.d  |-  D  =  ( CC  \  ( -oo (,] 0 ) )
Assertion
Ref Expression
dvlog  |-  ( CC 
_D  ( log  |`  D ) )  =  ( x  e.  D  |->  ( 1  /  x ) )
Distinct variable group:    x, D

Proof of Theorem dvlog
StepHypRef Expression
1 eqid 2438 . . . 4  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
21cnfldtop 20338 . . . . . 6  |-  ( TopOpen ` fld )  e.  Top
31cnfldtopon 20337 . . . . . . . 8  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
43toponunii 18512 . . . . . . 7  |-  CC  =  U. ( TopOpen ` fld )
54restid 14364 . . . . . 6  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
62, 5ax-mp 5 . . . . 5  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
76eqcomi 2442 . . . 4  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
8 cnelprrecn 9367 . . . . 5  |-  CC  e.  { RR ,  CC }
98a1i 11 . . . 4  |-  ( T. 
->  CC  e.  { RR ,  CC } )
10 logcn.d . . . . . 6  |-  D  =  ( CC  \  ( -oo (,] 0 ) )
1110logdmopn 22069 . . . . 5  |-  D  e.  ( TopOpen ` fld )
1211a1i 11 . . . 4  |-  ( T. 
->  D  e.  ( TopOpen
` fld
) )
13 logf1o 21991 . . . . . . . . 9  |-  log :
( CC  \  {
0 } ) -1-1-onto-> ran  log
14 f1of1 5635 . . . . . . . . 9  |-  ( log
: ( CC  \  { 0 } ) -1-1-onto-> ran 
log  ->  log : ( CC 
\  { 0 } ) -1-1-> ran  log )
1513, 14ax-mp 5 . . . . . . . 8  |-  log :
( CC  \  {
0 } ) -1-1-> ran  log
1610logdmss 22062 . . . . . . . 8  |-  D  C_  ( CC  \  { 0 } )
17 f1ores 5650 . . . . . . . 8  |-  ( ( log : ( CC 
\  { 0 } ) -1-1-> ran  log  /\  D  C_  ( CC  \  { 0 } ) )  -> 
( log  |`  D ) : D -1-1-onto-> ( log " D
) )
1815, 16, 17mp2an 672 . . . . . . 7  |-  ( log  |`  D ) : D -1-1-onto-> ( log " D )
19 f1ocnv 5648 . . . . . . 7  |-  ( ( log  |`  D ) : D -1-1-onto-> ( log " D
)  ->  `' ( log  |`  D ) : ( log " D
)
-1-1-onto-> D )
2018, 19ax-mp 5 . . . . . 6  |-  `' ( log  |`  D ) : ( log " D
)
-1-1-onto-> D
21 df-log 21983 . . . . . . . . . . 11  |-  log  =  `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )
2221reseq1i 5101 . . . . . . . . . 10  |-  ( log  |`  D )  =  ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  |`  D )
2322cnveqi 5009 . . . . . . . . 9  |-  `' ( log  |`  D )  =  `' ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  |`  D )
24 eff 13359 . . . . . . . . . . 11  |-  exp : CC
--> CC
25 cnvimass 5184 . . . . . . . . . . . 12  |-  ( `' Im " ( -u pi (,] pi ) ) 
C_  dom  Im
26 imf 12594 . . . . . . . . . . . . 13  |-  Im : CC
--> RR
2726fdmi 5559 . . . . . . . . . . . 12  |-  dom  Im  =  CC
2825, 27sseqtri 3383 . . . . . . . . . . 11  |-  ( `' Im " ( -u pi (,] pi ) ) 
C_  CC
29 fssres 5573 . . . . . . . . . . 11  |-  ( ( exp : CC --> CC  /\  ( `' Im " ( -u pi (,] pi ) ) 
C_  CC )  -> 
( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) : ( `' Im " ( -u pi (,] pi ) ) --> CC )
3024, 28, 29mp2an 672 . . . . . . . . . 10  |-  ( exp  |`  ( `' Im "
( -u pi (,] pi ) ) ) : ( `' Im "
( -u pi (,] pi ) ) --> CC
31 ffun 5556 . . . . . . . . . 10  |-  ( ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) : ( `' Im " ( -u pi (,] pi ) ) --> CC  ->  Fun  ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) )
32 funcnvres2 5484 . . . . . . . . . 10  |-  ( Fun  ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  ->  `' ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  |`  D )  =  ( ( exp  |`  ( `' Im "
( -u pi (,] pi ) ) )  |`  ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) " D ) ) )
3330, 31, 32mp2b 10 . . . . . . . . 9  |-  `' ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  |`  D )  =  ( ( exp  |`  ( `' Im "
( -u pi (,] pi ) ) )  |`  ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) " D ) )
34 cnvimass 5184 . . . . . . . . . . 11  |-  ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) " D ) 
C_  dom  ( exp  |`  ( `' Im "
( -u pi (,] pi ) ) )
3530fdmi 5559 . . . . . . . . . . 11  |-  dom  ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  =  ( `' Im "
( -u pi (,] pi ) )
3634, 35sseqtri 3383 . . . . . . . . . 10  |-  ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) " D ) 
C_  ( `' Im " ( -u pi (,] pi ) )
37 resabs1 5134 . . . . . . . . . 10  |-  ( ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) " D ) 
C_  ( `' Im " ( -u pi (,] pi ) )  ->  (
( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  |`  ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )
" D ) )  =  ( exp  |`  ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) " D ) ) )
3836, 37ax-mp 5 . . . . . . . . 9  |-  ( ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  |`  ( `' ( exp  |`  ( `' Im "
( -u pi (,] pi ) ) ) " D ) )  =  ( exp  |`  ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) " D ) )
3923, 33, 383eqtri 2462 . . . . . . . 8  |-  `' ( log  |`  D )  =  ( exp  |`  ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) " D ) )
4021imaeq1i 5161 . . . . . . . . 9  |-  ( log " D )  =  ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) " D )
4140reseq2i 5102 . . . . . . . 8  |-  ( exp  |`  ( log " D
) )  =  ( exp  |`  ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )
" D ) )
4239, 41eqtr4i 2461 . . . . . . 7  |-  `' ( log  |`  D )  =  ( exp  |`  ( log " D ) )
43 f1oeq1 5627 . . . . . . 7  |-  ( `' ( log  |`  D )  =  ( exp  |`  ( log " D ) )  ->  ( `' ( log  |`  D ) : ( log " D
)
-1-1-onto-> D 
<->  ( exp  |`  ( log " D ) ) : ( log " D
)
-1-1-onto-> D ) )
4442, 43ax-mp 5 . . . . . 6  |-  ( `' ( log  |`  D ) : ( log " D
)
-1-1-onto-> D 
<->  ( exp  |`  ( log " D ) ) : ( log " D
)
-1-1-onto-> D )
4520, 44mpbi 208 . . . . 5  |-  ( exp  |`  ( log " D
) ) : ( log " D ) -1-1-onto-> D
4645a1i 11 . . . 4  |-  ( T. 
->  ( exp  |`  ( log " D ) ) : ( log " D
)
-1-1-onto-> D )
4742cnveqi 5009 . . . . . 6  |-  `' `' ( log  |`  D )  =  `' ( exp  |`  ( log " D ) )
48 relres 5133 . . . . . . 7  |-  Rel  ( log  |`  D )
49 dfrel2 5283 . . . . . . 7  |-  ( Rel  ( log  |`  D )  <->  `' `' ( log  |`  D )  =  ( log  |`  D ) )
5048, 49mpbi 208 . . . . . 6  |-  `' `' ( log  |`  D )  =  ( log  |`  D )
5147, 50eqtr3i 2460 . . . . 5  |-  `' ( exp  |`  ( log " D ) )  =  ( log  |`  D )
52 f1of 5636 . . . . . . 7  |-  ( ( log  |`  D ) : D -1-1-onto-> ( log " D
)  ->  ( log  |`  D ) : D --> ( log " D ) )
5318, 52mp1i 12 . . . . . 6  |-  ( T. 
->  ( log  |`  D ) : D --> ( log " D ) )
54 imassrn 5175 . . . . . . . 8  |-  ( log " D )  C_  ran  log
55 logrncn 21989 . . . . . . . . 9  |-  ( x  e.  ran  log  ->  x  e.  CC )
5655ssriv 3355 . . . . . . . 8  |-  ran  log  C_  CC
5754, 56sstri 3360 . . . . . . 7  |-  ( log " D )  C_  CC
5810logcn 22067 . . . . . . 7  |-  ( log  |`  D )  e.  ( D -cn-> CC )
59 cncffvrn 20449 . . . . . . 7  |-  ( ( ( log " D
)  C_  CC  /\  ( log  |`  D )  e.  ( D -cn-> CC ) )  ->  ( ( log  |`  D )  e.  ( D -cn-> ( log " D ) )  <->  ( log  |`  D ) : D --> ( log " D ) ) )
6057, 58, 59mp2an 672 . . . . . 6  |-  ( ( log  |`  D )  e.  ( D -cn-> ( log " D ) )  <->  ( log  |`  D ) : D --> ( log " D ) )
6153, 60sylibr 212 . . . . 5  |-  ( T. 
->  ( log  |`  D )  e.  ( D -cn-> ( log " D ) ) )
6251, 61syl5eqel 2522 . . . 4  |-  ( T. 
->  `' ( exp  |`  ( log " D ) )  e.  ( D -cn-> ( log " D ) ) )
63 ssid 3370 . . . . . . . . 9  |-  CC  C_  CC
641, 7dvres 21361 . . . . . . . . 9  |-  ( ( ( CC  C_  CC  /\ 
exp : CC --> CC )  /\  ( CC  C_  CC  /\  ( log " D
)  C_  CC )
)  ->  ( CC  _D  ( exp  |`  ( log " D ) ) )  =  ( ( CC  _D  exp )  |`  ( ( int `  ( TopOpen
` fld
) ) `  ( log " D ) ) ) )
6563, 24, 63, 57, 64mp4an 673 . . . . . . . 8  |-  ( CC 
_D  ( exp  |`  ( log " D ) ) )  =  ( ( CC  _D  exp )  |`  ( ( int `  ( TopOpen
` fld
) ) `  ( log " D ) ) )
66 dvef 21427 . . . . . . . . 9  |-  ( CC 
_D  exp )  =  exp
6710dvloglem 22068 . . . . . . . . . 10  |-  ( log " D )  e.  (
TopOpen ` fld )
68 isopn3i 18661 . . . . . . . . . 10  |-  ( ( ( TopOpen ` fld )  e.  Top  /\  ( log " D
)  e.  ( TopOpen ` fld )
)  ->  ( ( int `  ( TopOpen ` fld ) ) `  ( log " D ) )  =  ( log " D
) )
692, 67, 68mp2an 672 . . . . . . . . 9  |-  ( ( int `  ( TopOpen ` fld )
) `  ( log " D ) )  =  ( log " D
)
7066, 69reseq12i 5103 . . . . . . . 8  |-  ( ( CC  _D  exp )  |`  ( ( int `  ( TopOpen
` fld
) ) `  ( log " D ) ) )  =  ( exp  |`  ( log " D
) )
7165, 70eqtri 2458 . . . . . . 7  |-  ( CC 
_D  ( exp  |`  ( log " D ) ) )  =  ( exp  |`  ( log " D
) )
7271dmeqi 5036 . . . . . 6  |-  dom  ( CC  _D  ( exp  |`  ( log " D ) ) )  =  dom  ( exp  |`  ( log " D
) )
73 dmres 5126 . . . . . 6  |-  dom  ( exp  |`  ( log " D
) )  =  ( ( log " D
)  i^i  dom  exp )
7424fdmi 5559 . . . . . . . 8  |-  dom  exp  =  CC
7557, 74sseqtr4i 3384 . . . . . . 7  |-  ( log " D )  C_  dom  exp
76 df-ss 3337 . . . . . . 7  |-  ( ( log " D ) 
C_  dom  exp  <->  ( ( log " D )  i^i 
dom  exp )  =  ( log " D ) )
7775, 76mpbi 208 . . . . . 6  |-  ( ( log " D )  i^i  dom  exp )  =  ( log " D
)
7872, 73, 773eqtri 2462 . . . . 5  |-  dom  ( CC  _D  ( exp  |`  ( log " D ) ) )  =  ( log " D )
7978a1i 11 . . . 4  |-  ( T. 
->  dom  ( CC  _D  ( exp  |`  ( log " D ) ) )  =  ( log " D
) )
80 neirr 2608 . . . . . 6  |-  -.  0  =/=  0
81 resss 5129 . . . . . . . . . . . . 13  |-  ( ( CC  _D  exp )  |`  ( ( int `  ( TopOpen
` fld
) ) `  ( log " D ) ) )  C_  ( CC  _D  exp )
8265, 81eqsstri 3381 . . . . . . . . . . . 12  |-  ( CC 
_D  ( exp  |`  ( log " D ) ) )  C_  ( CC  _D  exp )
8382, 66sseqtri 3383 . . . . . . . . . . 11  |-  ( CC 
_D  ( exp  |`  ( log " D ) ) )  C_  exp
84 rnss 5063 . . . . . . . . . . 11  |-  ( ( CC  _D  ( exp  |`  ( log " D
) ) )  C_  exp  ->  ran  ( CC  _D  ( exp  |`  ( log " D ) ) )  C_  ran  exp )
8583, 84ax-mp 5 . . . . . . . . . 10  |-  ran  ( CC  _D  ( exp  |`  ( log " D ) ) )  C_  ran  exp
86 eff2 13375 . . . . . . . . . . 11  |-  exp : CC
--> ( CC  \  {
0 } )
87 frn 5560 . . . . . . . . . . 11  |-  ( exp
: CC --> ( CC 
\  { 0 } )  ->  ran  exp  C_  ( CC  \  { 0 } ) )
8886, 87ax-mp 5 . . . . . . . . . 10  |-  ran  exp  C_  ( CC  \  {
0 } )
8985, 88sstri 3360 . . . . . . . . 9  |-  ran  ( CC  _D  ( exp  |`  ( log " D ) ) )  C_  ( CC  \  { 0 } )
9089sseli 3347 . . . . . . . 8  |-  ( 0  e.  ran  ( CC 
_D  ( exp  |`  ( log " D ) ) )  ->  0  e.  ( CC  \  { 0 } ) )
91 eldifsn 3995 . . . . . . . 8  |-  ( 0  e.  ( CC  \  { 0 } )  <-> 
( 0  e.  CC  /\  0  =/=  0 ) )
9290, 91sylib 196 . . . . . . 7  |-  ( 0  e.  ran  ( CC 
_D  ( exp  |`  ( log " D ) ) )  ->  ( 0  e.  CC  /\  0  =/=  0 ) )
9392simprd 463 . . . . . 6  |-  ( 0  e.  ran  ( CC 
_D  ( exp  |`  ( log " D ) ) )  ->  0  =/=  0 )
9480, 93mto 176 . . . . 5  |-  -.  0  e.  ran  ( CC  _D  ( exp  |`  ( log " D ) ) )
9594a1i 11 . . . 4  |-  ( T. 
->  -.  0  e.  ran  ( CC  _D  ( exp  |`  ( log " D
) ) ) )
961, 7, 9, 12, 46, 62, 79, 95dvcnv 21424 . . 3  |-  ( T. 
->  ( CC  _D  `' ( exp  |`  ( log " D ) ) )  =  ( x  e.  D  |->  ( 1  / 
( ( CC  _D  ( exp  |`  ( log " D ) ) ) `
 ( `' ( exp  |`  ( log " D ) ) `  x ) ) ) ) )
9796trud 1378 . 2  |-  ( CC 
_D  `' ( exp  |`  ( log " D
) ) )  =  ( x  e.  D  |->  ( 1  /  (
( CC  _D  ( exp  |`  ( log " D
) ) ) `  ( `' ( exp  |`  ( log " D ) ) `
 x ) ) ) )
9851oveq2i 6097 . 2  |-  ( CC 
_D  `' ( exp  |`  ( log " D
) ) )  =  ( CC  _D  ( log  |`  D ) )
9971fveq1i 5687 . . . . 5  |-  ( ( CC  _D  ( exp  |`  ( log " D
) ) ) `  ( `' ( exp  |`  ( log " D ) ) `
 x ) )  =  ( ( exp  |`  ( log " D
) ) `  ( `' ( exp  |`  ( log " D ) ) `
 x ) )
100 f1ocnvfv2 5979 . . . . . 6  |-  ( ( ( exp  |`  ( log " D ) ) : ( log " D
)
-1-1-onto-> D  /\  x  e.  D
)  ->  ( ( exp  |`  ( log " D
) ) `  ( `' ( exp  |`  ( log " D ) ) `
 x ) )  =  x )
10145, 100mpan 670 . . . . 5  |-  ( x  e.  D  ->  (
( exp  |`  ( log " D ) ) `  ( `' ( exp  |`  ( log " D ) ) `
 x ) )  =  x )
10299, 101syl5eq 2482 . . . 4  |-  ( x  e.  D  ->  (
( CC  _D  ( exp  |`  ( log " D
) ) ) `  ( `' ( exp  |`  ( log " D ) ) `
 x ) )  =  x )
103102oveq2d 6102 . . 3  |-  ( x  e.  D  ->  (
1  /  ( ( CC  _D  ( exp  |`  ( log " D
) ) ) `  ( `' ( exp  |`  ( log " D ) ) `
 x ) ) )  =  ( 1  /  x ) )
104103mpteq2ia 4369 . 2  |-  ( x  e.  D  |->  ( 1  /  ( ( CC 
_D  ( exp  |`  ( log " D ) ) ) `  ( `' ( exp  |`  ( log " D ) ) `
 x ) ) ) )  =  ( x  e.  D  |->  ( 1  /  x ) )
10597, 98, 1043eqtr3i 2466 1  |-  ( CC 
_D  ( log  |`  D ) )  =  ( x  e.  D  |->  ( 1  /  x ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ wa 369    = wceq 1369   T. wtru 1370    e. wcel 1756    =/= wne 2601    \ cdif 3320    i^i cin 3322    C_ wss 3323   {csn 3872   {cpr 3874    e. cmpt 4345   `'ccnv 4834   dom cdm 4835   ran crn 4836    |` cres 4837   "cima 4838   Rel wrel 4840   Fun wfun 5407   -->wf 5409   -1-1->wf1 5410   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6086   CCcc 9272   RRcr 9273   0cc0 9274   1c1 9275   -oocmnf 9408   -ucneg 9588    / cdiv 9985   (,]cioc 11293   Imcim 12579   expce 13339   picpi 13344   ↾t crest 14351   TopOpenctopn 14352  ℂfldccnfld 17793   Topctop 18473   intcnt 18596   -cn->ccncf 20427    _D cdv 21313   logclog 21981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-ioc 11297  df-ico 11298  df-icc 11299  df-fz 11430  df-fzo 11541  df-fl 11634  df-mod 11701  df-seq 11799  df-exp 11858  df-fac 12044  df-bc 12071  df-hash 12096  df-shft 12548  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-limsup 12941  df-clim 12958  df-rlim 12959  df-sum 13156  df-ef 13345  df-sin 13347  df-cos 13348  df-tan 13349  df-pi 13350  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-pt 14375  df-prds 14378  df-xrs 14432  df-qtop 14437  df-imas 14438  df-xps 14440  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-submnd 15457  df-mulg 15539  df-cntz 15826  df-cmn 16270  df-psmet 17784  df-xmet 17785  df-met 17786  df-bl 17787  df-mopn 17788  df-fbas 17789  df-fg 17790  df-cnfld 17794  df-top 18478  df-bases 18480  df-topon 18481  df-topsp 18482  df-cld 18598  df-ntr 18599  df-cls 18600  df-nei 18677  df-lp 18715  df-perf 18716  df-cn 18806  df-cnp 18807  df-haus 18894  df-cmp 18965  df-tx 19110  df-hmeo 19303  df-fil 19394  df-fm 19486  df-flim 19487  df-flf 19488  df-xms 19870  df-ms 19871  df-tms 19872  df-cncf 20429  df-limc 21316  df-dv 21317  df-log 21983
This theorem is referenced by:  dvlog2  22073  dvatan  22305  lgamgulmlem2  26968  dvcncxp1  28430  dvasin  28433
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