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Theorem dvlog 22222
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 2451 . . . 4  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
21cnfldtop 20488 . . . . . 6  |-  ( TopOpen ` fld )  e.  Top
31cnfldtopon 20487 . . . . . . . 8  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
43toponunii 18662 . . . . . . 7  |-  CC  =  U. ( TopOpen ` fld )
54restid 14483 . . . . . 6  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
62, 5ax-mp 5 . . . . 5  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
76eqcomi 2464 . . . 4  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
8 cnelprrecn 9479 . . . . 5  |-  CC  e.  { RR ,  CC }
98a1i 11 . . . 4  |-  ( T. 
->  CC  e.  { RR ,  CC } )
10 logcn.d . . . . . 6  |-  D  =  ( CC  \  ( -oo (,] 0 ) )
1110logdmopn 22220 . . . . 5  |-  D  e.  ( TopOpen ` fld )
1211a1i 11 . . . 4  |-  ( T. 
->  D  e.  ( TopOpen
` fld
) )
13 logf1o 22142 . . . . . . . . 9  |-  log :
( CC  \  {
0 } ) -1-1-onto-> ran  log
14 f1of1 5741 . . . . . . . . 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 22213 . . . . . . . 8  |-  D  C_  ( CC  \  { 0 } )
17 f1ores 5756 . . . . . . . 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 5754 . . . . . . 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 22134 . . . . . . . . . . 11  |-  log  =  `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )
2221reseq1i 5207 . . . . . . . . . 10  |-  ( log  |`  D )  =  ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  |`  D )
2322cnveqi 5115 . . . . . . . . 9  |-  `' ( log  |`  D )  =  `' ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  |`  D )
24 eff 13478 . . . . . . . . . . 11  |-  exp : CC
--> CC
25 cnvimass 5290 . . . . . . . . . . . 12  |-  ( `' Im " ( -u pi (,] pi ) ) 
C_  dom  Im
26 imf 12713 . . . . . . . . . . . . 13  |-  Im : CC
--> RR
2726fdmi 5665 . . . . . . . . . . . 12  |-  dom  Im  =  CC
2825, 27sseqtri 3489 . . . . . . . . . . 11  |-  ( `' Im " ( -u pi (,] pi ) ) 
C_  CC
29 fssres 5679 . . . . . . . . . . 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 5662 . . . . . . . . . 10  |-  ( ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) : ( `' Im " ( -u pi (,] pi ) ) --> CC  ->  Fun  ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) )
32 funcnvres2 5590 . . . . . . . . . 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 5290 . . . . . . . . . . 11  |-  ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) " D ) 
C_  dom  ( exp  |`  ( `' Im "
( -u pi (,] pi ) ) )
3530fdmi 5665 . . . . . . . . . . 11  |-  dom  ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  =  ( `' Im "
( -u pi (,] pi ) )
3634, 35sseqtri 3489 . . . . . . . . . 10  |-  ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) " D ) 
C_  ( `' Im " ( -u pi (,] pi ) )
37 resabs1 5240 . . . . . . . . . 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 2484 . . . . . . . 8  |-  `' ( log  |`  D )  =  ( exp  |`  ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) " D ) )
4021imaeq1i 5267 . . . . . . . . 9  |-  ( log " D )  =  ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) " D )
4140reseq2i 5208 . . . . . . . 8  |-  ( exp  |`  ( log " D
) )  =  ( exp  |`  ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )
" D ) )
4239, 41eqtr4i 2483 . . . . . . 7  |-  `' ( log  |`  D )  =  ( exp  |`  ( log " D ) )
43 f1oeq1 5733 . . . . . . 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 5115 . . . . . 6  |-  `' `' ( log  |`  D )  =  `' ( exp  |`  ( log " D ) )
48 relres 5239 . . . . . . 7  |-  Rel  ( log  |`  D )
49 dfrel2 5389 . . . . . . 7  |-  ( Rel  ( log  |`  D )  <->  `' `' ( log  |`  D )  =  ( log  |`  D ) )
5048, 49mpbi 208 . . . . . 6  |-  `' `' ( log  |`  D )  =  ( log  |`  D )
5147, 50eqtr3i 2482 . . . . 5  |-  `' ( exp  |`  ( log " D ) )  =  ( log  |`  D )
52 f1of 5742 . . . . . . 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 5281 . . . . . . . 8  |-  ( log " D )  C_  ran  log
55 logrncn 22140 . . . . . . . . 9  |-  ( x  e.  ran  log  ->  x  e.  CC )
5655ssriv 3461 . . . . . . . 8  |-  ran  log  C_  CC
5754, 56sstri 3466 . . . . . . 7  |-  ( log " D )  C_  CC
5810logcn 22218 . . . . . . 7  |-  ( log  |`  D )  e.  ( D -cn-> CC )
59 cncffvrn 20599 . . . . . . 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 2543 . . . 4  |-  ( T. 
->  `' ( exp  |`  ( log " D ) )  e.  ( D -cn-> ( log " D ) ) )
63 ssid 3476 . . . . . . . . 9  |-  CC  C_  CC
641, 7dvres 21512 . . . . . . . . 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 21578 . . . . . . . . 9  |-  ( CC 
_D  exp )  =  exp
6710dvloglem 22219 . . . . . . . . . 10  |-  ( log " D )  e.  (
TopOpen ` fld )
68 isopn3i 18811 . . . . . . . . . 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 5209 . . . . . . . 8  |-  ( ( CC  _D  exp )  |`  ( ( int `  ( TopOpen
` fld
) ) `  ( log " D ) ) )  =  ( exp  |`  ( log " D
) )
7165, 70eqtri 2480 . . . . . . 7  |-  ( CC 
_D  ( exp  |`  ( log " D ) ) )  =  ( exp  |`  ( log " D
) )
7271dmeqi 5142 . . . . . 6  |-  dom  ( CC  _D  ( exp  |`  ( log " D ) ) )  =  dom  ( exp  |`  ( log " D
) )
73 dmres 5232 . . . . . 6  |-  dom  ( exp  |`  ( log " D
) )  =  ( ( log " D
)  i^i  dom  exp )
7424fdmi 5665 . . . . . . . 8  |-  dom  exp  =  CC
7557, 74sseqtr4i 3490 . . . . . . 7  |-  ( log " D )  C_  dom  exp
76 df-ss 3443 . . . . . . 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 2484 . . . . 5  |-  dom  ( CC  _D  ( exp  |`  ( log " D ) ) )  =  ( log " D )
7978a1i 11 . . . 4  |-  ( T. 
->  dom  ( CC  _D  ( exp  |`  ( log " D ) ) )  =  ( log " D
) )
80 neirr 2653 . . . . . 6  |-  -.  0  =/=  0
81 resss 5235 . . . . . . . . . . . . 13  |-  ( ( CC  _D  exp )  |`  ( ( int `  ( TopOpen
` fld
) ) `  ( log " D ) ) )  C_  ( CC  _D  exp )
8265, 81eqsstri 3487 . . . . . . . . . . . 12  |-  ( CC 
_D  ( exp  |`  ( log " D ) ) )  C_  ( CC  _D  exp )
8382, 66sseqtri 3489 . . . . . . . . . . 11  |-  ( CC 
_D  ( exp  |`  ( log " D ) ) )  C_  exp
84 rnss 5169 . . . . . . . . . . 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 13494 . . . . . . . . . . 11  |-  exp : CC
--> ( CC  \  {
0 } )
87 frn 5666 . . . . . . . . . . 11  |-  ( exp
: CC --> ( CC 
\  { 0 } )  ->  ran  exp  C_  ( CC  \  { 0 } ) )
8886, 87ax-mp 5 . . . . . . . . . 10  |-  ran  exp  C_  ( CC  \  {
0 } )
8985, 88sstri 3466 . . . . . . . . 9  |-  ran  ( CC  _D  ( exp  |`  ( log " D ) ) )  C_  ( CC  \  { 0 } )
9089sseli 3453 . . . . . . . 8  |-  ( 0  e.  ran  ( CC 
_D  ( exp  |`  ( log " D ) ) )  ->  0  e.  ( CC  \  { 0 } ) )
91 eldifsn 4101 . . . . . . . 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 21575 . . 3  |-  ( T. 
->  ( CC  _D  `' ( exp  |`  ( log " D ) ) )  =  ( x  e.  D  |->  ( 1  / 
( ( CC  _D  ( exp  |`  ( log " D ) ) ) `
 ( `' ( exp  |`  ( log " D ) ) `  x ) ) ) ) )
9796trud 1379 . 2  |-  ( CC 
_D  `' ( exp  |`  ( log " D
) ) )  =  ( x  e.  D  |->  ( 1  /  (
( CC  _D  ( exp  |`  ( log " D
) ) ) `  ( `' ( exp  |`  ( log " D ) ) `
 x ) ) ) )
9851oveq2i 6204 . 2  |-  ( CC 
_D  `' ( exp  |`  ( log " D
) ) )  =  ( CC  _D  ( log  |`  D ) )
9971fveq1i 5793 . . . . 5  |-  ( ( CC  _D  ( exp  |`  ( log " D
) ) ) `  ( `' ( exp  |`  ( log " D ) ) `
 x ) )  =  ( ( exp  |`  ( log " D
) ) `  ( `' ( exp  |`  ( log " D ) ) `
 x ) )
100 f1ocnvfv2 6086 . . . . . 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 2504 . . . 4  |-  ( x  e.  D  ->  (
( CC  _D  ( exp  |`  ( log " D
) ) ) `  ( `' ( exp  |`  ( log " D ) ) `
 x ) )  =  x )
103102oveq2d 6209 . . 3  |-  ( x  e.  D  ->  (
1  /  ( ( CC  _D  ( exp  |`  ( log " D
) ) ) `  ( `' ( exp  |`  ( log " D ) ) `
 x ) ) )  =  ( 1  /  x ) )
104103mpteq2ia 4475 . 2  |-  ( x  e.  D  |->  ( 1  /  ( ( CC 
_D  ( exp  |`  ( log " D ) ) ) `  ( `' ( exp  |`  ( log " D ) ) `
 x ) ) ) )  =  ( x  e.  D  |->  ( 1  /  x ) )
10597, 98, 1043eqtr3i 2488 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 1370   T. wtru 1371    e. wcel 1758    =/= wne 2644    \ cdif 3426    i^i cin 3428    C_ wss 3429   {csn 3978   {cpr 3980    |-> cmpt 4451   `'ccnv 4940   dom cdm 4941   ran crn 4942    |` cres 4943   "cima 4944   Rel wrel 4946   Fun wfun 5513   -->wf 5515   -1-1->wf1 5516   -1-1-onto->wf1o 5518   ` cfv 5519  (class class class)co 6193   CCcc 9384   RRcr 9385   0cc0 9386   1c1 9387   -oocmnf 9520   -ucneg 9700    / cdiv 10097   (,]cioc 11405   Imcim 12698   expce 13458   picpi 13463   ↾t crest 14470   TopOpenctopn 14471  ℂfldccnfld 17936   Topctop 18623   intcnt 18746   -cn->ccncf 20577    _D cdv 21464   logclog 22132
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464  ax-addf 9465  ax-mulf 9466
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-iin 4275  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-of 6423  df-om 6580  df-1st 6680  df-2nd 6681  df-supp 6794  df-recs 6935  df-rdg 6969  df-1o 7023  df-2o 7024  df-oadd 7027  df-er 7204  df-map 7319  df-pm 7320  df-ixp 7367  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-fsupp 7725  df-fi 7765  df-sup 7795  df-oi 7828  df-card 8213  df-cda 8441  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-4 10486  df-5 10487  df-6 10488  df-7 10489  df-8 10490  df-9 10491  df-10 10492  df-n0 10684  df-z 10751  df-dec 10860  df-uz 10966  df-q 11058  df-rp 11096  df-xneg 11193  df-xadd 11194  df-xmul 11195  df-ioo 11408  df-ioc 11409  df-ico 11410  df-icc 11411  df-fz 11548  df-fzo 11659  df-fl 11752  df-mod 11819  df-seq 11917  df-exp 11976  df-fac 12162  df-bc 12189  df-hash 12214  df-shft 12667  df-cj 12699  df-re 12700  df-im 12701  df-sqr 12835  df-abs 12836  df-limsup 13060  df-clim 13077  df-rlim 13078  df-sum 13275  df-ef 13464  df-sin 13466  df-cos 13467  df-tan 13468  df-pi 13469  df-struct 14287  df-ndx 14288  df-slot 14289  df-base 14290  df-sets 14291  df-ress 14292  df-plusg 14362  df-mulr 14363  df-starv 14364  df-sca 14365  df-vsca 14366  df-ip 14367  df-tset 14368  df-ple 14369  df-ds 14371  df-unif 14372  df-hom 14373  df-cco 14374  df-rest 14472  df-topn 14473  df-0g 14491  df-gsum 14492  df-topgen 14493  df-pt 14494  df-prds 14497  df-xrs 14551  df-qtop 14556  df-imas 14557  df-xps 14559  df-mre 14635  df-mrc 14636  df-acs 14638  df-mnd 15526  df-submnd 15576  df-mulg 15659  df-cntz 15946  df-cmn 16392  df-psmet 17927  df-xmet 17928  df-met 17929  df-bl 17930  df-mopn 17931  df-fbas 17932  df-fg 17933  df-cnfld 17937  df-top 18628  df-bases 18630  df-topon 18631  df-topsp 18632  df-cld 18748  df-ntr 18749  df-cls 18750  df-nei 18827  df-lp 18865  df-perf 18866  df-cn 18956  df-cnp 18957  df-haus 19044  df-cmp 19115  df-tx 19260  df-hmeo 19453  df-fil 19544  df-fm 19636  df-flim 19637  df-flf 19638  df-xms 20020  df-ms 20021  df-tms 20022  df-cncf 20579  df-limc 21467  df-dv 21468  df-log 22134
This theorem is referenced by:  dvlog2  22224  dvatan  22456  lgamgulmlem2  27153  dvcncxp1  28618  dvasin  28621
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