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Theorem efopnlem2 20020
Description: Lemma for efopn 20021. (Contributed by Mario Carneiro, 2-May-2015.)
Hypothesis
Ref Expression
efopn.j  |-  J  =  ( TopOpen ` fld )
Assertion
Ref Expression
efopnlem2  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  ( exp " ( 0 (
ball `  ( abs  o. 
-  ) ) R ) )  e.  J
)

Proof of Theorem efopnlem2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 logf1o 19938 . . . . . . . 8  |-  log :
( CC  \  {
0 } ) -1-1-onto-> ran  log
2 f1orn 5498 . . . . . . . . 9  |-  ( log
: ( CC  \  { 0 } ) -1-1-onto-> ran 
log 
<->  ( log  Fn  ( CC  \  { 0 } )  /\  Fun  `' log ) )
32simprbi 450 . . . . . . . 8  |-  ( log
: ( CC  \  { 0 } ) -1-1-onto-> ran 
log  ->  Fun  `' log )
4 funcnvres 5337 . . . . . . . 8  |-  ( Fun  `' log  ->  `' ( log  |`  ( CC  \ 
(  -oo (,] 0 ) ) )  =  ( `' log  |`  ( log " ( CC  \  (  -oo (,] 0 ) ) ) ) )
51, 3, 4mp2b 9 . . . . . . 7  |-  `' ( log  |`  ( CC  \  (  -oo (,] 0
) ) )  =  ( `' log  |`  ( log " ( CC  \ 
(  -oo (,] 0 ) ) ) )
6 df-log 19930 . . . . . . . . . 10  |-  log  =  `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )
76cnveqi 4872 . . . . . . . . 9  |-  `' log  =  `' `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )
8 relres 4999 . . . . . . . . . 10  |-  Rel  ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )
9 dfrel2 5140 . . . . . . . . . 10  |-  ( Rel  ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  <->  `' `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  =  ( exp  |`  ( `' Im "
( -u pi (,] pi ) ) ) )
108, 9mpbi 199 . . . . . . . . 9  |-  `' `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  =  ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )
117, 10eqtri 2316 . . . . . . . 8  |-  `' log  =  ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )
1211reseq1i 4967 . . . . . . 7  |-  ( `' log  |`  ( log " ( CC  \  (  -oo (,] 0 ) ) ) )  =  ( ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  |`  ( log " ( CC  \  (  -oo (,] 0 ) ) ) )
13 imassrn 5041 . . . . . . . . 9  |-  ( log " ( CC  \ 
(  -oo (,] 0 ) ) )  C_  ran  log
14 logrn 19932 . . . . . . . . 9  |-  ran  log  =  ( `' Im " ( -u pi (,] pi ) )
1513, 14sseqtri 3223 . . . . . . . 8  |-  ( log " ( CC  \ 
(  -oo (,] 0 ) ) )  C_  ( `' Im " ( -u pi (,] pi ) )
16 resabs1 5000 . . . . . . . 8  |-  ( ( log " ( CC 
\  (  -oo (,] 0 ) ) ) 
C_  ( `' Im " ( -u pi (,] pi ) )  ->  (
( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  |`  ( log " ( CC  \  (  -oo (,] 0 ) ) ) )  =  ( exp  |`  ( log " ( CC  \  (  -oo (,] 0 ) ) ) ) )
1715, 16ax-mp 8 . . . . . . 7  |-  ( ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  |`  ( log " ( CC  \  (  -oo (,] 0 ) ) ) )  =  ( exp  |`  ( log " ( CC  \  (  -oo (,] 0 ) ) ) )
185, 12, 173eqtri 2320 . . . . . 6  |-  `' ( log  |`  ( CC  \  (  -oo (,] 0
) ) )  =  ( exp  |`  ( log " ( CC  \ 
(  -oo (,] 0 ) ) ) )
1918imaeq1i 5025 . . . . 5  |-  ( `' ( log  |`  ( CC  \  (  -oo (,] 0 ) ) )
" ( 0 (
ball `  ( abs  o. 
-  ) ) R ) )  =  ( ( exp  |`  ( log " ( CC  \ 
(  -oo (,] 0 ) ) ) ) "
( 0 ( ball `  ( abs  o.  -  ) ) R ) )
20 cnxmet 18298 . . . . . . . . . . . . 13  |-  ( abs 
o.  -  )  e.  ( * Met `  CC )
2120a1i 10 . . . . . . . . . . . 12  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  ( abs  o.  -  )  e.  ( * Met `  CC ) )
22 0cn 8847 . . . . . . . . . . . . 13  |-  0  e.  CC
2322a1i 10 . . . . . . . . . . . 12  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  0  e.  CC )
24 rpxr 10377 . . . . . . . . . . . . 13  |-  ( R  e.  RR+  ->  R  e. 
RR* )
2524adantr 451 . . . . . . . . . . . 12  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  R  e.  RR* )
26 blssm 17984 . . . . . . . . . . . 12  |-  ( ( ( abs  o.  -  )  e.  ( * Met `  CC )  /\  0  e.  CC  /\  R  e.  RR* )  ->  (
0 ( ball `  ( abs  o.  -  ) ) R )  C_  CC )
2721, 23, 25, 26syl3anc 1182 . . . . . . . . . . 11  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  (
0 ( ball `  ( abs  o.  -  ) ) R )  C_  CC )
2827sselda 3193 . . . . . . . . . 10  |-  ( ( ( R  e.  RR+  /\  R  <  pi )  /\  x  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R ) )  ->  x  e.  CC )
2928imcld 11696 . . . . . . . . . . 11  |-  ( ( ( R  e.  RR+  /\  R  <  pi )  /\  x  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R ) )  -> 
( Im `  x
)  e.  RR )
30 efopnlem1 20019 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  RR+  /\  R  <  pi )  /\  x  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R ) )  -> 
( abs `  (
Im `  x )
)  <  pi )
31 pire 19848 . . . . . . . . . . . . . 14  |-  pi  e.  RR
32 abslt 11814 . . . . . . . . . . . . . 14  |-  ( ( ( Im `  x
)  e.  RR  /\  pi  e.  RR )  -> 
( ( abs `  (
Im `  x )
)  <  pi  <->  ( -u pi  <  ( Im `  x
)  /\  ( Im `  x )  <  pi ) ) )
3329, 31, 32sylancl 643 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  RR+  /\  R  <  pi )  /\  x  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R ) )  -> 
( ( abs `  (
Im `  x )
)  <  pi  <->  ( -u pi  <  ( Im `  x
)  /\  ( Im `  x )  <  pi ) ) )
3430, 33mpbid 201 . . . . . . . . . . . 12  |-  ( ( ( R  e.  RR+  /\  R  <  pi )  /\  x  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R ) )  -> 
( -u pi  <  (
Im `  x )  /\  ( Im `  x
)  <  pi )
)
3534simpld 445 . . . . . . . . . . 11  |-  ( ( ( R  e.  RR+  /\  R  <  pi )  /\  x  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R ) )  ->  -u pi  <  ( Im
`  x ) )
3634simprd 449 . . . . . . . . . . 11  |-  ( ( ( R  e.  RR+  /\  R  <  pi )  /\  x  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R ) )  -> 
( Im `  x
)  <  pi )
3731renegcli 9124 . . . . . . . . . . . . 13  |-  -u pi  e.  RR
38 rexr 8893 . . . . . . . . . . . . 13  |-  ( -u pi  e.  RR  ->  -u pi  e.  RR* )
3937, 38ax-mp 8 . . . . . . . . . . . 12  |-  -u pi  e.  RR*
40 rexr 8893 . . . . . . . . . . . . 13  |-  ( pi  e.  RR  ->  pi  e.  RR* )
4131, 40ax-mp 8 . . . . . . . . . . . 12  |-  pi  e.  RR*
42 elioo2 10713 . . . . . . . . . . . 12  |-  ( (
-u pi  e.  RR*  /\  pi  e.  RR* )  ->  ( ( Im `  x )  e.  (
-u pi (,) pi ) 
<->  ( ( Im `  x )  e.  RR  /\  -u pi  <  ( Im
`  x )  /\  ( Im `  x )  <  pi ) ) )
4339, 41, 42mp2an 653 . . . . . . . . . . 11  |-  ( ( Im `  x )  e.  ( -u pi (,) pi )  <->  ( (
Im `  x )  e.  RR  /\  -u pi  <  ( Im `  x
)  /\  ( Im `  x )  <  pi ) )
4429, 35, 36, 43syl3anbrc 1136 . . . . . . . . . 10  |-  ( ( ( R  e.  RR+  /\  R  <  pi )  /\  x  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R ) )  -> 
( Im `  x
)  e.  ( -u pi (,) pi ) )
45 imf 11614 . . . . . . . . . . 11  |-  Im : CC
--> RR
46 ffn 5405 . . . . . . . . . . 11  |-  ( Im : CC --> RR  ->  Im  Fn  CC )
47 elpreima 5661 . . . . . . . . . . 11  |-  ( Im  Fn  CC  ->  (
x  e.  ( `' Im " ( -u pi (,) pi ) )  <-> 
( x  e.  CC  /\  ( Im `  x
)  e.  ( -u pi (,) pi ) ) ) )
4845, 46, 47mp2b 9 . . . . . . . . . 10  |-  ( x  e.  ( `' Im " ( -u pi (,) pi ) )  <->  ( x  e.  CC  /\  ( Im
`  x )  e.  ( -u pi (,) pi ) ) )
4928, 44, 48sylanbrc 645 . . . . . . . . 9  |-  ( ( ( R  e.  RR+  /\  R  <  pi )  /\  x  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R ) )  ->  x  e.  ( `' Im " ( -u pi (,) pi ) ) )
5049ex 423 . . . . . . . 8  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  (
x  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R )  ->  x  e.  ( `' Im "
( -u pi (,) pi ) ) ) )
5150ssrdv 3198 . . . . . . 7  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  (
0 ( ball `  ( abs  o.  -  ) ) R )  C_  ( `' Im " ( -u pi (,) pi ) ) )
52 df-ima 4718 . . . . . . . 8  |-  ( log " ( CC  \ 
(  -oo (,] 0 ) ) )  =  ran  ( log  |`  ( CC  \  (  -oo (,] 0
) ) )
53 eqid 2296 . . . . . . . . . 10  |-  ( CC 
\  (  -oo (,] 0 ) )  =  ( CC  \  (  -oo (,] 0 ) )
5453logf1o2 20013 . . . . . . . . 9  |-  ( log  |`  ( CC  \  (  -oo (,] 0 ) ) ) : ( CC 
\  (  -oo (,] 0 ) ) -1-1-onto-> ( `' Im " ( -u pi (,) pi ) )
55 f1ofo 5495 . . . . . . . . 9  |-  ( ( log  |`  ( CC  \  (  -oo (,] 0
) ) ) : ( CC  \  (  -oo (,] 0 ) ) -1-1-onto-> ( `' Im " ( -u pi (,) pi ) )  ->  ( log  |`  ( CC  \  (  -oo (,] 0 ) ) ) : ( CC  \ 
(  -oo (,] 0 ) ) -onto-> ( `' Im " ( -u pi (,) pi ) ) )
56 forn 5470 . . . . . . . . 9  |-  ( ( log  |`  ( CC  \  (  -oo (,] 0
) ) ) : ( CC  \  (  -oo (,] 0 ) )
-onto-> ( `' Im "
( -u pi (,) pi ) )  ->  ran  ( log  |`  ( CC  \  (  -oo (,] 0
) ) )  =  ( `' Im "
( -u pi (,) pi ) ) )
5754, 55, 56mp2b 9 . . . . . . . 8  |-  ran  ( log  |`  ( CC  \ 
(  -oo (,] 0 ) ) )  =  ( `' Im " ( -u pi (,) pi ) )
5852, 57eqtri 2316 . . . . . . 7  |-  ( log " ( CC  \ 
(  -oo (,] 0 ) ) )  =  ( `' Im " ( -u pi (,) pi ) )
5951, 58syl6sseqr 3238 . . . . . 6  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  (
0 ( ball `  ( abs  o.  -  ) ) R )  C_  ( log " ( CC  \ 
(  -oo (,] 0 ) ) ) )
60 resima2 5004 . . . . . 6  |-  ( ( 0 ( ball `  ( abs  o.  -  ) ) R )  C_  ( log " ( CC  \ 
(  -oo (,] 0 ) ) )  ->  (
( exp  |`  ( log " ( CC  \ 
(  -oo (,] 0 ) ) ) ) "
( 0 ( ball `  ( abs  o.  -  ) ) R ) )  =  ( exp " ( 0 (
ball `  ( abs  o. 
-  ) ) R ) ) )
6159, 60syl 15 . . . . 5  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  (
( exp  |`  ( log " ( CC  \ 
(  -oo (,] 0 ) ) ) ) "
( 0 ( ball `  ( abs  o.  -  ) ) R ) )  =  ( exp " ( 0 (
ball `  ( abs  o. 
-  ) ) R ) ) )
6219, 61syl5eq 2340 . . . 4  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  ( `' ( log  |`  ( CC  \  (  -oo (,] 0 ) ) )
" ( 0 (
ball `  ( abs  o. 
-  ) ) R ) )  =  ( exp " ( 0 ( ball `  ( abs  o.  -  ) ) R ) ) )
6353logcn 20010 . . . . . 6  |-  ( log  |`  ( CC  \  (  -oo (,] 0 ) ) )  e.  ( ( CC  \  (  -oo (,] 0 ) ) -cn-> CC )
64 difss 3316 . . . . . . 7  |-  ( CC 
\  (  -oo (,] 0 ) )  C_  CC
65 ssid 3210 . . . . . . 7  |-  CC  C_  CC
66 efopn.j . . . . . . . 8  |-  J  =  ( TopOpen ` fld )
67 eqid 2296 . . . . . . . 8  |-  ( Jt  ( CC  \  (  -oo (,] 0 ) ) )  =  ( Jt  ( CC 
\  (  -oo (,] 0 ) ) )
6866cnfldtop 18309 . . . . . . . . . 10  |-  J  e. 
Top
6966cnfldtopon 18308 . . . . . . . . . . . 12  |-  J  e.  (TopOn `  CC )
7069toponunii 16686 . . . . . . . . . . 11  |-  CC  =  U. J
7170restid 13354 . . . . . . . . . 10  |-  ( J  e.  Top  ->  ( Jt  CC )  =  J
)
7268, 71ax-mp 8 . . . . . . . . 9  |-  ( Jt  CC )  =  J
7372eqcomi 2300 . . . . . . . 8  |-  J  =  ( Jt  CC )
7466, 67, 73cncfcn 18429 . . . . . . 7  |-  ( ( ( CC  \  (  -oo (,] 0 ) ) 
C_  CC  /\  CC  C_  CC )  ->  ( ( CC  \  (  -oo (,] 0 ) ) -cn-> CC )  =  ( ( Jt  ( CC  \  (  -oo (,] 0 ) ) )  Cn  J ) )
7564, 65, 74mp2an 653 . . . . . 6  |-  ( ( CC  \  (  -oo (,] 0 ) ) -cn-> CC )  =  ( ( Jt  ( CC  \  (  -oo (,] 0 ) ) )  Cn  J )
7663, 75eleqtri 2368 . . . . 5  |-  ( log  |`  ( CC  \  (  -oo (,] 0 ) ) )  e.  ( ( Jt  ( CC  \  (  -oo (,] 0 ) ) )  Cn  J )
7766cnfldtopn 18307 . . . . . . 7  |-  J  =  ( MetOpen `  ( abs  o. 
-  ) )
7877blopn 18062 . . . . . 6  |-  ( ( ( abs  o.  -  )  e.  ( * Met `  CC )  /\  0  e.  CC  /\  R  e.  RR* )  ->  (
0 ( ball `  ( abs  o.  -  ) ) R )  e.  J
)
7921, 23, 25, 78syl3anc 1182 . . . . 5  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  (
0 ( ball `  ( abs  o.  -  ) ) R )  e.  J
)
80 cnima 17010 . . . . 5  |-  ( ( ( log  |`  ( CC  \  (  -oo (,] 0 ) ) )  e.  ( ( Jt  ( CC  \  (  -oo (,] 0 ) ) )  Cn  J )  /\  ( 0 ( ball `  ( abs  o.  -  ) ) R )  e.  J )  -> 
( `' ( log  |`  ( CC  \  (  -oo (,] 0 ) ) ) " ( 0 ( ball `  ( abs  o.  -  ) ) R ) )  e.  ( Jt  ( CC  \ 
(  -oo (,] 0 ) ) ) )
8176, 79, 80sylancr 644 . . . 4  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  ( `' ( log  |`  ( CC  \  (  -oo (,] 0 ) ) )
" ( 0 (
ball `  ( abs  o. 
-  ) ) R ) )  e.  ( Jt  ( CC  \  (  -oo (,] 0 ) ) ) )
8262, 81eqeltrrd 2371 . . 3  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  ( exp " ( 0 (
ball `  ( abs  o. 
-  ) ) R ) )  e.  ( Jt  ( CC  \  (  -oo (,] 0 ) ) ) )
8353logdmopn 20012 . . . . 5  |-  ( CC 
\  (  -oo (,] 0 ) )  e.  ( TopOpen ` fld )
8483, 66eleqtrri 2369 . . . 4  |-  ( CC 
\  (  -oo (,] 0 ) )  e.  J
85 restopn2 16924 . . . 4  |-  ( ( J  e.  Top  /\  ( CC  \  (  -oo (,] 0 ) )  e.  J )  -> 
( ( exp " (
0 ( ball `  ( abs  o.  -  ) ) R ) )  e.  ( Jt  ( CC  \ 
(  -oo (,] 0 ) ) )  <->  ( ( exp " ( 0 (
ball `  ( abs  o. 
-  ) ) R ) )  e.  J  /\  ( exp " (
0 ( ball `  ( abs  o.  -  ) ) R ) )  C_  ( CC  \  (  -oo (,] 0 ) ) ) ) )
8668, 84, 85mp2an 653 . . 3  |-  ( ( exp " ( 0 ( ball `  ( abs  o.  -  ) ) R ) )  e.  ( Jt  ( CC  \ 
(  -oo (,] 0 ) ) )  <->  ( ( exp " ( 0 (
ball `  ( abs  o. 
-  ) ) R ) )  e.  J  /\  ( exp " (
0 ( ball `  ( abs  o.  -  ) ) R ) )  C_  ( CC  \  (  -oo (,] 0 ) ) ) )
8782, 86sylib 188 . 2  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  (
( exp " (
0 ( ball `  ( abs  o.  -  ) ) R ) )  e.  J  /\  ( exp " ( 0 (
ball `  ( abs  o. 
-  ) ) R ) )  C_  ( CC  \  (  -oo (,] 0 ) ) ) )
8887simpld 445 1  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  ( exp " ( 0 (
ball `  ( abs  o. 
-  ) ) R ) )  e.  J
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    \ cdif 3162    C_ wss 3165   {csn 3653   class class class wbr 4039   `'ccnv 4704   ran crn 4706    |` cres 4707   "cima 4708    o. ccom 4709   Rel wrel 4710   Fun wfun 5265    Fn wfn 5266   -->wf 5267   -onto->wfo 5269   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753    -oocmnf 8881   RR*cxr 8882    < clt 8883    - cmin 9053   -ucneg 9054   RR+crp 10370   (,)cioo 10672   (,]cioc 10673   Imcim 11599   abscabs 11735   expce 12359   picpi 12364   ↾t crest 13341   TopOpenctopn 13342   * Metcxmt 16385   ballcbl 16387  ℂfldccnfld 16393   Topctop 16647    Cn ccn 16970   -cn->ccncf 18396   logclog 19928
This theorem is referenced by:  efopn  20021
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ioc 10677  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175  df-ef 12365  df-sin 12367  df-cos 12368  df-tan 12369  df-pi 12370  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cn 16973  df-cnp 16974  df-haus 17059  df-cmp 17130  df-tx 17273  df-hmeo 17462  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-tms 17903  df-cncf 18398  df-limc 19232  df-dv 19233  df-log 19930
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