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Theorem dvlog 23218
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 2402 . . . 4  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
21cnfldtop 21475 . . . . . 6  |-  ( TopOpen ` fld )  e.  Top
31cnfldtopon 21474 . . . . . . . 8  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
43toponunii 19617 . . . . . . 7  |-  CC  =  U. ( TopOpen ` fld )
54restid 14940 . . . . . 6  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
62, 5ax-mp 5 . . . . 5  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
76eqcomi 2415 . . . 4  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
8 cnelprrecn 9535 . . . . 5  |-  CC  e.  { RR ,  CC }
98a1i 11 . . . 4  |-  ( T. 
->  CC  e.  { RR ,  CC } )
10 logcn.d . . . . . 6  |-  D  =  ( CC  \  ( -oo (,] 0 ) )
1110logdmopn 23216 . . . . 5  |-  D  e.  ( TopOpen ` fld )
1211a1i 11 . . . 4  |-  ( T. 
->  D  e.  ( TopOpen
` fld
) )
13 logf1o 23136 . . . . . . . . 9  |-  log :
( CC  \  {
0 } ) -1-1-onto-> ran  log
14 f1of1 5754 . . . . . . . . 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 23209 . . . . . . . 8  |-  D  C_  ( CC  \  { 0 } )
17 f1ores 5769 . . . . . . . 8  |-  ( ( log : ( CC 
\  { 0 } ) -1-1-> ran  log  /\  D  C_  ( CC  \  { 0 } ) )  -> 
( log  |`  D ) : D -1-1-onto-> ( log " D
) )
1815, 16, 17mp2an 670 . . . . . . 7  |-  ( log  |`  D ) : D -1-1-onto-> ( log " D )
19 f1ocnv 5767 . . . . . . 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 23128 . . . . . . . . . . 11  |-  log  =  `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )
2221reseq1i 5211 . . . . . . . . . 10  |-  ( log  |`  D )  =  ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  |`  D )
2322cnveqi 5119 . . . . . . . . 9  |-  `' ( log  |`  D )  =  `' ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  |`  D )
24 eff 13918 . . . . . . . . . . 11  |-  exp : CC
--> CC
25 cnvimass 5298 . . . . . . . . . . . 12  |-  ( `' Im " ( -u pi (,] pi ) ) 
C_  dom  Im
26 imf 13002 . . . . . . . . . . . . 13  |-  Im : CC
--> RR
2726fdmi 5675 . . . . . . . . . . . 12  |-  dom  Im  =  CC
2825, 27sseqtri 3473 . . . . . . . . . . 11  |-  ( `' Im " ( -u pi (,] pi ) ) 
C_  CC
29 fssres 5690 . . . . . . . . . . 11  |-  ( ( exp : CC --> CC  /\  ( `' Im " ( -u pi (,] pi ) ) 
C_  CC )  -> 
( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) : ( `' Im " ( -u pi (,] pi ) ) --> CC )
3024, 28, 29mp2an 670 . . . . . . . . . 10  |-  ( exp  |`  ( `' Im "
( -u pi (,] pi ) ) ) : ( `' Im "
( -u pi (,] pi ) ) --> CC
31 ffun 5672 . . . . . . . . . 10  |-  ( ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) : ( `' Im " ( -u pi (,] pi ) ) --> CC  ->  Fun  ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) )
32 funcnvres2 5596 . . . . . . . . . 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 5298 . . . . . . . . . . 11  |-  ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) " D ) 
C_  dom  ( exp  |`  ( `' Im "
( -u pi (,] pi ) ) )
3530fdmi 5675 . . . . . . . . . . 11  |-  dom  ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  =  ( `' Im "
( -u pi (,] pi ) )
3634, 35sseqtri 3473 . . . . . . . . . 10  |-  ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) " D ) 
C_  ( `' Im " ( -u pi (,] pi ) )
37 resabs1 5243 . . . . . . . . . 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 2435 . . . . . . . 8  |-  `' ( log  |`  D )  =  ( exp  |`  ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) " D ) )
4021imaeq1i 5275 . . . . . . . . 9  |-  ( log " D )  =  ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) " D )
4140reseq2i 5212 . . . . . . . 8  |-  ( exp  |`  ( log " D
) )  =  ( exp  |`  ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )
" D ) )
4239, 41eqtr4i 2434 . . . . . . 7  |-  `' ( log  |`  D )  =  ( exp  |`  ( log " D ) )
43 f1oeq1 5746 . . . . . . 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 5119 . . . . . 6  |-  `' `' ( log  |`  D )  =  `' ( exp  |`  ( log " D ) )
48 relres 5242 . . . . . . 7  |-  Rel  ( log  |`  D )
49 dfrel2 5395 . . . . . . 7  |-  ( Rel  ( log  |`  D )  <->  `' `' ( log  |`  D )  =  ( log  |`  D ) )
5048, 49mpbi 208 . . . . . 6  |-  `' `' ( log  |`  D )  =  ( log  |`  D )
5147, 50eqtr3i 2433 . . . . 5  |-  `' ( exp  |`  ( log " D ) )  =  ( log  |`  D )
52 f1of 5755 . . . . . . 7  |-  ( ( log  |`  D ) : D -1-1-onto-> ( log " D
)  ->  ( log  |`  D ) : D --> ( log " D ) )
5318, 52mp1i 13 . . . . . 6  |-  ( T. 
->  ( log  |`  D ) : D --> ( log " D ) )
54 imassrn 5289 . . . . . . . 8  |-  ( log " D )  C_  ran  log
55 logrncn 23134 . . . . . . . . 9  |-  ( x  e.  ran  log  ->  x  e.  CC )
5655ssriv 3445 . . . . . . . 8  |-  ran  log  C_  CC
5754, 56sstri 3450 . . . . . . 7  |-  ( log " D )  C_  CC
5810logcn 23214 . . . . . . 7  |-  ( log  |`  D )  e.  ( D -cn-> CC )
59 cncffvrn 21586 . . . . . . 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 670 . . . . . 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 2494 . . . 4  |-  ( T. 
->  `' ( exp  |`  ( log " D ) )  e.  ( D -cn-> ( log " D ) ) )
63 ssid 3460 . . . . . . . . 9  |-  CC  C_  CC
641, 7dvres 22499 . . . . . . . . 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 671 . . . . . . . 8  |-  ( CC 
_D  ( exp  |`  ( log " D ) ) )  =  ( ( CC  _D  exp )  |`  ( ( int `  ( TopOpen
` fld
) ) `  ( log " D ) ) )
66 dvef 22565 . . . . . . . . 9  |-  ( CC 
_D  exp )  =  exp
6710dvloglem 23215 . . . . . . . . . 10  |-  ( log " D )  e.  (
TopOpen ` fld )
68 isopn3i 19768 . . . . . . . . . 10  |-  ( ( ( TopOpen ` fld )  e.  Top  /\  ( log " D
)  e.  ( TopOpen ` fld )
)  ->  ( ( int `  ( TopOpen ` fld ) ) `  ( log " D ) )  =  ( log " D
) )
692, 67, 68mp2an 670 . . . . . . . . 9  |-  ( ( int `  ( TopOpen ` fld )
) `  ( log " D ) )  =  ( log " D
)
7066, 69reseq12i 5213 . . . . . . . 8  |-  ( ( CC  _D  exp )  |`  ( ( int `  ( TopOpen
` fld
) ) `  ( log " D ) ) )  =  ( exp  |`  ( log " D
) )
7165, 70eqtri 2431 . . . . . . 7  |-  ( CC 
_D  ( exp  |`  ( log " D ) ) )  =  ( exp  |`  ( log " D
) )
7271dmeqi 5146 . . . . . 6  |-  dom  ( CC  _D  ( exp  |`  ( log " D ) ) )  =  dom  ( exp  |`  ( log " D
) )
73 dmres 5235 . . . . . 6  |-  dom  ( exp  |`  ( log " D
) )  =  ( ( log " D
)  i^i  dom  exp )
7424fdmi 5675 . . . . . . . 8  |-  dom  exp  =  CC
7557, 74sseqtr4i 3474 . . . . . . 7  |-  ( log " D )  C_  dom  exp
76 df-ss 3427 . . . . . . 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 2435 . . . . 5  |-  dom  ( CC  _D  ( exp  |`  ( log " D ) ) )  =  ( log " D )
7978a1i 11 . . . 4  |-  ( T. 
->  dom  ( CC  _D  ( exp  |`  ( log " D ) ) )  =  ( log " D
) )
80 neirr 2607 . . . . . 6  |-  -.  0  =/=  0
81 resss 5238 . . . . . . . . . . . . 13  |-  ( ( CC  _D  exp )  |`  ( ( int `  ( TopOpen
` fld
) ) `  ( log " D ) ) )  C_  ( CC  _D  exp )
8265, 81eqsstri 3471 . . . . . . . . . . . 12  |-  ( CC 
_D  ( exp  |`  ( log " D ) ) )  C_  ( CC  _D  exp )
8382, 66sseqtri 3473 . . . . . . . . . . 11  |-  ( CC 
_D  ( exp  |`  ( log " D ) ) )  C_  exp
84 rnss 5173 . . . . . . . . . . 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 13935 . . . . . . . . . . 11  |-  exp : CC
--> ( CC  \  {
0 } )
87 frn 5676 . . . . . . . . . . 11  |-  ( exp
: CC --> ( CC 
\  { 0 } )  ->  ran  exp  C_  ( CC  \  { 0 } ) )
8886, 87ax-mp 5 . . . . . . . . . 10  |-  ran  exp  C_  ( CC  \  {
0 } )
8985, 88sstri 3450 . . . . . . . . 9  |-  ran  ( CC  _D  ( exp  |`  ( log " D ) ) )  C_  ( CC  \  { 0 } )
9089sseli 3437 . . . . . . . 8  |-  ( 0  e.  ran  ( CC 
_D  ( exp  |`  ( log " D ) ) )  ->  0  e.  ( CC  \  { 0 } ) )
91 eldifsn 4096 . . . . . . . 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 461 . . . . . 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 22562 . . 3  |-  ( T. 
->  ( CC  _D  `' ( exp  |`  ( log " D ) ) )  =  ( x  e.  D  |->  ( 1  / 
( ( CC  _D  ( exp  |`  ( log " D ) ) ) `
 ( `' ( exp  |`  ( log " D ) ) `  x ) ) ) ) )
9796trud 1414 . 2  |-  ( CC 
_D  `' ( exp  |`  ( log " D
) ) )  =  ( x  e.  D  |->  ( 1  /  (
( CC  _D  ( exp  |`  ( log " D
) ) ) `  ( `' ( exp  |`  ( log " D ) ) `
 x ) ) ) )
9851oveq2i 6245 . 2  |-  ( CC 
_D  `' ( exp  |`  ( log " D
) ) )  =  ( CC  _D  ( log  |`  D ) )
9971fveq1i 5806 . . . . 5  |-  ( ( CC  _D  ( exp  |`  ( log " D
) ) ) `  ( `' ( exp  |`  ( log " D ) ) `
 x ) )  =  ( ( exp  |`  ( log " D
) ) `  ( `' ( exp  |`  ( log " D ) ) `
 x ) )
100 f1ocnvfv2 6120 . . . . . 6  |-  ( ( ( exp  |`  ( log " D ) ) : ( log " D
)
-1-1-onto-> D  /\  x  e.  D
)  ->  ( ( exp  |`  ( log " D
) ) `  ( `' ( exp  |`  ( log " D ) ) `
 x ) )  =  x )
10145, 100mpan 668 . . . . 5  |-  ( x  e.  D  ->  (
( exp  |`  ( log " D ) ) `  ( `' ( exp  |`  ( log " D ) ) `
 x ) )  =  x )
10299, 101syl5eq 2455 . . . 4  |-  ( x  e.  D  ->  (
( CC  _D  ( exp  |`  ( log " D
) ) ) `  ( `' ( exp  |`  ( log " D ) ) `
 x ) )  =  x )
103102oveq2d 6250 . . 3  |-  ( x  e.  D  ->  (
1  /  ( ( CC  _D  ( exp  |`  ( log " D
) ) ) `  ( `' ( exp  |`  ( log " D ) ) `
 x ) ) )  =  ( 1  /  x ) )
104103mpteq2ia 4476 . 2  |-  ( x  e.  D  |->  ( 1  /  ( ( CC 
_D  ( exp  |`  ( log " D ) ) ) `  ( `' ( exp  |`  ( log " D ) ) `
 x ) ) ) )  =  ( x  e.  D  |->  ( 1  /  x ) )
10597, 98, 1043eqtr3i 2439 1  |-  ( CC 
_D  ( log  |`  D ) )  =  ( x  e.  D  |->  ( 1  /  x ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ wa 367    = wceq 1405   T. wtru 1406    e. wcel 1842    =/= wne 2598    \ cdif 3410    i^i cin 3412    C_ wss 3413   {csn 3971   {cpr 3973    |-> cmpt 4452   `'ccnv 4941   dom cdm 4942   ran crn 4943    |` cres 4944   "cima 4945   Rel wrel 4947   Fun wfun 5519   -->wf 5521   -1-1->wf1 5522   -1-1-onto->wf1o 5524   ` cfv 5525  (class class class)co 6234   CCcc 9440   RRcr 9441   0cc0 9442   1c1 9443   -oocmnf 9576   -ucneg 9762    / cdiv 10167   (,]cioc 11501   Imcim 12987   expce 13898   picpi 13903   ↾t crest 14927   TopOpenctopn 14928  ℂfldccnfld 18632   Topctop 19578   intcnt 19702   -cn->ccncf 21564    _D cdv 22451   logclog 23126
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-inf2 8011  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518  ax-pre-mulgt0 9519  ax-pre-sup 9520  ax-addf 9521  ax-mulf 9522
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-iin 4273  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-of 6477  df-om 6639  df-1st 6738  df-2nd 6739  df-supp 6857  df-recs 6999  df-rdg 7033  df-1o 7087  df-2o 7088  df-oadd 7091  df-er 7268  df-map 7379  df-pm 7380  df-ixp 7428  df-en 7475  df-dom 7476  df-sdom 7477  df-fin 7478  df-fsupp 7784  df-fi 7825  df-sup 7855  df-oi 7889  df-card 8272  df-cda 8500  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-sub 9763  df-neg 9764  df-div 10168  df-nn 10497  df-2 10555  df-3 10556  df-4 10557  df-5 10558  df-6 10559  df-7 10560  df-8 10561  df-9 10562  df-10 10563  df-n0 10757  df-z 10826  df-dec 10940  df-uz 11046  df-q 11146  df-rp 11184  df-xneg 11289  df-xadd 11290  df-xmul 11291  df-ioo 11504  df-ioc 11505  df-ico 11506  df-icc 11507  df-fz 11644  df-fzo 11768  df-fl 11879  df-mod 11948  df-seq 12062  df-exp 12121  df-fac 12308  df-bc 12335  df-hash 12360  df-shft 12956  df-cj 12988  df-re 12989  df-im 12990  df-sqrt 13124  df-abs 13125  df-limsup 13350  df-clim 13367  df-rlim 13368  df-sum 13565  df-ef 13904  df-sin 13906  df-cos 13907  df-tan 13908  df-pi 13909  df-struct 14735  df-ndx 14736  df-slot 14737  df-base 14738  df-sets 14739  df-ress 14740  df-plusg 14814  df-mulr 14815  df-starv 14816  df-sca 14817  df-vsca 14818  df-ip 14819  df-tset 14820  df-ple 14821  df-ds 14823  df-unif 14824  df-hom 14825  df-cco 14826  df-rest 14929  df-topn 14930  df-0g 14948  df-gsum 14949  df-topgen 14950  df-pt 14951  df-prds 14954  df-xrs 15008  df-qtop 15013  df-imas 15014  df-xps 15016  df-mre 15092  df-mrc 15093  df-acs 15095  df-mgm 16088  df-sgrp 16127  df-mnd 16137  df-submnd 16183  df-mulg 16276  df-cntz 16571  df-cmn 17016  df-psmet 18623  df-xmet 18624  df-met 18625  df-bl 18626  df-mopn 18627  df-fbas 18628  df-fg 18629  df-cnfld 18633  df-top 19583  df-bases 19585  df-topon 19586  df-topsp 19587  df-cld 19704  df-ntr 19705  df-cls 19706  df-nei 19784  df-lp 19822  df-perf 19823  df-cn 19913  df-cnp 19914  df-haus 20001  df-cmp 20072  df-tx 20247  df-hmeo 20440  df-fil 20531  df-fm 20623  df-flim 20624  df-flf 20625  df-xms 21007  df-ms 21008  df-tms 21009  df-cncf 21566  df-limc 22454  df-dv 22455  df-log 23128
This theorem is referenced by:  dvlog2  23220  dvcncxp1  23305  dvatan  23483  lgamgulmlem2  23577  dvasin  31455
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