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Theorem dvlog 23157
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 2457 . . . 4  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
21cnfldtop 21416 . . . . . 6  |-  ( TopOpen ` fld )  e.  Top
31cnfldtopon 21415 . . . . . . . 8  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
43toponunii 19559 . . . . . . 7  |-  CC  =  U. ( TopOpen ` fld )
54restid 14850 . . . . . 6  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
62, 5ax-mp 5 . . . . 5  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
76eqcomi 2470 . . . 4  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
8 cnelprrecn 9602 . . . . 5  |-  CC  e.  { RR ,  CC }
98a1i 11 . . . 4  |-  ( T. 
->  CC  e.  { RR ,  CC } )
10 logcn.d . . . . . 6  |-  D  =  ( CC  \  ( -oo (,] 0 ) )
1110logdmopn 23155 . . . . 5  |-  D  e.  ( TopOpen ` fld )
1211a1i 11 . . . 4  |-  ( T. 
->  D  e.  ( TopOpen
` fld
) )
13 logf1o 23077 . . . . . . . . 9  |-  log :
( CC  \  {
0 } ) -1-1-onto-> ran  log
14 f1of1 5821 . . . . . . . . 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 23148 . . . . . . . 8  |-  D  C_  ( CC  \  { 0 } )
17 f1ores 5836 . . . . . . . 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 5834 . . . . . . 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 23069 . . . . . . . . . . 11  |-  log  =  `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )
2221reseq1i 5279 . . . . . . . . . 10  |-  ( log  |`  D )  =  ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  |`  D )
2322cnveqi 5187 . . . . . . . . 9  |-  `' ( log  |`  D )  =  `' ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  |`  D )
24 eff 13828 . . . . . . . . . . 11  |-  exp : CC
--> CC
25 cnvimass 5367 . . . . . . . . . . . 12  |-  ( `' Im " ( -u pi (,] pi ) ) 
C_  dom  Im
26 imf 12957 . . . . . . . . . . . . 13  |-  Im : CC
--> RR
2726fdmi 5742 . . . . . . . . . . . 12  |-  dom  Im  =  CC
2825, 27sseqtri 3531 . . . . . . . . . . 11  |-  ( `' Im " ( -u pi (,] pi ) ) 
C_  CC
29 fssres 5757 . . . . . . . . . . 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 5739 . . . . . . . . . 10  |-  ( ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) : ( `' Im " ( -u pi (,] pi ) ) --> CC  ->  Fun  ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) )
32 funcnvres2 5665 . . . . . . . . . 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 5367 . . . . . . . . . . 11  |-  ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) " D ) 
C_  dom  ( exp  |`  ( `' Im "
( -u pi (,] pi ) ) )
3530fdmi 5742 . . . . . . . . . . 11  |-  dom  ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  =  ( `' Im "
( -u pi (,] pi ) )
3634, 35sseqtri 3531 . . . . . . . . . 10  |-  ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) " D ) 
C_  ( `' Im " ( -u pi (,] pi ) )
37 resabs1 5312 . . . . . . . . . 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 2490 . . . . . . . 8  |-  `' ( log  |`  D )  =  ( exp  |`  ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) " D ) )
4021imaeq1i 5344 . . . . . . . . 9  |-  ( log " D )  =  ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) " D )
4140reseq2i 5280 . . . . . . . 8  |-  ( exp  |`  ( log " D
) )  =  ( exp  |`  ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )
" D ) )
4239, 41eqtr4i 2489 . . . . . . 7  |-  `' ( log  |`  D )  =  ( exp  |`  ( log " D ) )
43 f1oeq1 5813 . . . . . . 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 5187 . . . . . 6  |-  `' `' ( log  |`  D )  =  `' ( exp  |`  ( log " D ) )
48 relres 5311 . . . . . . 7  |-  Rel  ( log  |`  D )
49 dfrel2 5463 . . . . . . 7  |-  ( Rel  ( log  |`  D )  <->  `' `' ( log  |`  D )  =  ( log  |`  D ) )
5048, 49mpbi 208 . . . . . 6  |-  `' `' ( log  |`  D )  =  ( log  |`  D )
5147, 50eqtr3i 2488 . . . . 5  |-  `' ( exp  |`  ( log " D ) )  =  ( log  |`  D )
52 f1of 5822 . . . . . . 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 5358 . . . . . . . 8  |-  ( log " D )  C_  ran  log
55 logrncn 23075 . . . . . . . . 9  |-  ( x  e.  ran  log  ->  x  e.  CC )
5655ssriv 3503 . . . . . . . 8  |-  ran  log  C_  CC
5754, 56sstri 3508 . . . . . . 7  |-  ( log " D )  C_  CC
5810logcn 23153 . . . . . . 7  |-  ( log  |`  D )  e.  ( D -cn-> CC )
59 cncffvrn 21527 . . . . . . 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 2549 . . . 4  |-  ( T. 
->  `' ( exp  |`  ( log " D ) )  e.  ( D -cn-> ( log " D ) ) )
63 ssid 3518 . . . . . . . . 9  |-  CC  C_  CC
641, 7dvres 22440 . . . . . . . . 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 22506 . . . . . . . . 9  |-  ( CC 
_D  exp )  =  exp
6710dvloglem 23154 . . . . . . . . . 10  |-  ( log " D )  e.  (
TopOpen ` fld )
68 isopn3i 19709 . . . . . . . . . 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 5281 . . . . . . . 8  |-  ( ( CC  _D  exp )  |`  ( ( int `  ( TopOpen
` fld
) ) `  ( log " D ) ) )  =  ( exp  |`  ( log " D
) )
7165, 70eqtri 2486 . . . . . . 7  |-  ( CC 
_D  ( exp  |`  ( log " D ) ) )  =  ( exp  |`  ( log " D
) )
7271dmeqi 5214 . . . . . 6  |-  dom  ( CC  _D  ( exp  |`  ( log " D ) ) )  =  dom  ( exp  |`  ( log " D
) )
73 dmres 5304 . . . . . 6  |-  dom  ( exp  |`  ( log " D
) )  =  ( ( log " D
)  i^i  dom  exp )
7424fdmi 5742 . . . . . . . 8  |-  dom  exp  =  CC
7557, 74sseqtr4i 3532 . . . . . . 7  |-  ( log " D )  C_  dom  exp
76 df-ss 3485 . . . . . . 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 2490 . . . . 5  |-  dom  ( CC  _D  ( exp  |`  ( log " D ) ) )  =  ( log " D )
7978a1i 11 . . . 4  |-  ( T. 
->  dom  ( CC  _D  ( exp  |`  ( log " D ) ) )  =  ( log " D
) )
80 neirr 2661 . . . . . 6  |-  -.  0  =/=  0
81 resss 5307 . . . . . . . . . . . . 13  |-  ( ( CC  _D  exp )  |`  ( ( int `  ( TopOpen
` fld
) ) `  ( log " D ) ) )  C_  ( CC  _D  exp )
8265, 81eqsstri 3529 . . . . . . . . . . . 12  |-  ( CC 
_D  ( exp  |`  ( log " D ) ) )  C_  ( CC  _D  exp )
8382, 66sseqtri 3531 . . . . . . . . . . 11  |-  ( CC 
_D  ( exp  |`  ( log " D ) ) )  C_  exp
84 rnss 5241 . . . . . . . . . . 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 13845 . . . . . . . . . . 11  |-  exp : CC
--> ( CC  \  {
0 } )
87 frn 5743 . . . . . . . . . . 11  |-  ( exp
: CC --> ( CC 
\  { 0 } )  ->  ran  exp  C_  ( CC  \  { 0 } ) )
8886, 87ax-mp 5 . . . . . . . . . 10  |-  ran  exp  C_  ( CC  \  {
0 } )
8985, 88sstri 3508 . . . . . . . . 9  |-  ran  ( CC  _D  ( exp  |`  ( log " D ) ) )  C_  ( CC  \  { 0 } )
9089sseli 3495 . . . . . . . 8  |-  ( 0  e.  ran  ( CC 
_D  ( exp  |`  ( log " D ) ) )  ->  0  e.  ( CC  \  { 0 } ) )
91 eldifsn 4157 . . . . . . . 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 22503 . . 3  |-  ( T. 
->  ( CC  _D  `' ( exp  |`  ( log " D ) ) )  =  ( x  e.  D  |->  ( 1  / 
( ( CC  _D  ( exp  |`  ( log " D ) ) ) `
 ( `' ( exp  |`  ( log " D ) ) `  x ) ) ) ) )
9796trud 1404 . 2  |-  ( CC 
_D  `' ( exp  |`  ( log " D
) ) )  =  ( x  e.  D  |->  ( 1  /  (
( CC  _D  ( exp  |`  ( log " D
) ) ) `  ( `' ( exp  |`  ( log " D ) ) `
 x ) ) ) )
9851oveq2i 6307 . 2  |-  ( CC 
_D  `' ( exp  |`  ( log " D
) ) )  =  ( CC  _D  ( log  |`  D ) )
9971fveq1i 5873 . . . . 5  |-  ( ( CC  _D  ( exp  |`  ( log " D
) ) ) `  ( `' ( exp  |`  ( log " D ) ) `
 x ) )  =  ( ( exp  |`  ( log " D
) ) `  ( `' ( exp  |`  ( log " D ) ) `
 x ) )
100 f1ocnvfv2 6184 . . . . . 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 2510 . . . 4  |-  ( x  e.  D  ->  (
( CC  _D  ( exp  |`  ( log " D
) ) ) `  ( `' ( exp  |`  ( log " D ) ) `
 x ) )  =  x )
103102oveq2d 6312 . . 3  |-  ( x  e.  D  ->  (
1  /  ( ( CC  _D  ( exp  |`  ( log " D
) ) ) `  ( `' ( exp  |`  ( log " D ) ) `
 x ) ) )  =  ( 1  /  x ) )
104103mpteq2ia 4539 . 2  |-  ( x  e.  D  |->  ( 1  /  ( ( CC 
_D  ( exp  |`  ( log " D ) ) ) `  ( `' ( exp  |`  ( log " D ) ) `
 x ) ) ) )  =  ( x  e.  D  |->  ( 1  /  x ) )
10597, 98, 1043eqtr3i 2494 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 1395   T. wtru 1396    e. wcel 1819    =/= wne 2652    \ cdif 3468    i^i cin 3470    C_ wss 3471   {csn 4032   {cpr 4034    |-> cmpt 4515   `'ccnv 5007   dom cdm 5008   ran crn 5009    |` cres 5010   "cima 5011   Rel wrel 5013   Fun wfun 5588   -->wf 5590   -1-1->wf1 5591   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6296   CCcc 9507   RRcr 9508   0cc0 9509   1c1 9510   -oocmnf 9643   -ucneg 9825    / cdiv 10227   (,]cioc 11555   Imcim 12942   expce 13808   picpi 13813   ↾t crest 14837   TopOpenctopn 14838  ℂfldccnfld 18546   Topctop 19520   intcnt 19644   -cn->ccncf 21505    _D cdv 22392   logclog 23067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588  ax-mulf 9589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-fi 7889  df-sup 7919  df-oi 7953  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ioo 11558  df-ioc 11559  df-ico 11560  df-icc 11561  df-fz 11698  df-fzo 11821  df-fl 11931  df-mod 11999  df-seq 12110  df-exp 12169  df-fac 12356  df-bc 12383  df-hash 12408  df-shft 12911  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080  df-limsup 13305  df-clim 13322  df-rlim 13323  df-sum 13520  df-ef 13814  df-sin 13816  df-cos 13817  df-tan 13818  df-pi 13819  df-struct 14645  df-ndx 14646  df-slot 14647  df-base 14648  df-sets 14649  df-ress 14650  df-plusg 14724  df-mulr 14725  df-starv 14726  df-sca 14727  df-vsca 14728  df-ip 14729  df-tset 14730  df-ple 14731  df-ds 14733  df-unif 14734  df-hom 14735  df-cco 14736  df-rest 14839  df-topn 14840  df-0g 14858  df-gsum 14859  df-topgen 14860  df-pt 14861  df-prds 14864  df-xrs 14918  df-qtop 14923  df-imas 14924  df-xps 14926  df-mre 15002  df-mrc 15003  df-acs 15005  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-submnd 16093  df-mulg 16186  df-cntz 16481  df-cmn 16926  df-psmet 18537  df-xmet 18538  df-met 18539  df-bl 18540  df-mopn 18541  df-fbas 18542  df-fg 18543  df-cnfld 18547  df-top 19525  df-bases 19527  df-topon 19528  df-topsp 19529  df-cld 19646  df-ntr 19647  df-cls 19648  df-nei 19725  df-lp 19763  df-perf 19764  df-cn 19854  df-cnp 19855  df-haus 19942  df-cmp 20013  df-tx 20188  df-hmeo 20381  df-fil 20472  df-fm 20564  df-flim 20565  df-flf 20566  df-xms 20948  df-ms 20949  df-tms 20950  df-cncf 21507  df-limc 22395  df-dv 22396  df-log 23069
This theorem is referenced by:  dvlog2  23159  dvatan  23391  lgamgulmlem2  28747  dvcncxp1  30262  dvasin  30265
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