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Theorem efopnlem2 20004
Description: Lemma for efopn 20005. (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 19922 . . . . . . . 8  |-  log :
( CC  \  {
0 } ) -1-1-onto-> ran  log
2 f1orn 5482 . . . . . . . . 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 5321 . . . . . . . 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 19914 . . . . . . . . . 10  |-  log  =  `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )
76cnveqi 4856 . . . . . . . . 9  |-  `' log  =  `' `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )
8 relres 4983 . . . . . . . . . 10  |-  Rel  ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )
9 dfrel2 5124 . . . . . . . . . 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 2303 . . . . . . . 8  |-  `' log  =  ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )
1211reseq1i 4951 . . . . . . 7  |-  ( `' log  |`  ( log " ( CC  \  (  -oo (,] 0 ) ) ) )  =  ( ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  |`  ( log " ( CC  \  (  -oo (,] 0 ) ) ) )
13 imassrn 5025 . . . . . . . . 9  |-  ( log " ( CC  \ 
(  -oo (,] 0 ) ) )  C_  ran  log
14 logrn 19916 . . . . . . . . 9  |-  ran  log  =  ( `' Im " ( -u pi (,] pi ) )
1513, 14sseqtri 3210 . . . . . . . 8  |-  ( log " ( CC  \ 
(  -oo (,] 0 ) ) )  C_  ( `' Im " ( -u pi (,] pi ) )
16 resabs1 4984 . . . . . . . 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 2307 . . . . . 6  |-  `' ( log  |`  ( CC  \  (  -oo (,] 0
) ) )  =  ( exp  |`  ( log " ( CC  \ 
(  -oo (,] 0 ) ) ) )
1918imaeq1i 5009 . . . . 5  |-  ( `' ( log  |`  ( CC  \  (  -oo (,] 0 ) ) )
" ( 0 (
ball `  ( abs  o. 
-  ) ) R ) )  =  ( ( exp  |`  ( log " ( CC  \ 
(  -oo (,] 0 ) ) ) ) "
( 0 ( ball `  ( abs  o.  -  ) ) R ) )
20 cnxmet 18282 . . . . . . . . . . . . 13  |-  ( abs 
o.  -  )  e.  ( * Met `  CC )
2120a1i 10 . . . . . . . . . . . 12  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  ( abs  o.  -  )  e.  ( * Met `  CC ) )
22 0cn 8831 . . . . . . . . . . . . 13  |-  0  e.  CC
2322a1i 10 . . . . . . . . . . . 12  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  0  e.  CC )
24 rpxr 10361 . . . . . . . . . . . . 13  |-  ( R  e.  RR+  ->  R  e. 
RR* )
2524adantr 451 . . . . . . . . . . . 12  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  R  e.  RR* )
26 blssm 17968 . . . . . . . . . . . 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 3180 . . . . . . . . . 10  |-  ( ( ( R  e.  RR+  /\  R  <  pi )  /\  x  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R ) )  ->  x  e.  CC )
2928imcld 11680 . . . . . . . . . . 11  |-  ( ( ( R  e.  RR+  /\  R  <  pi )  /\  x  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R ) )  -> 
( Im `  x
)  e.  RR )
30 efopnlem1 20003 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  RR+  /\  R  <  pi )  /\  x  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R ) )  -> 
( abs `  (
Im `  x )
)  <  pi )
31 pire 19832 . . . . . . . . . . . . . 14  |-  pi  e.  RR
32 abslt 11798 . . . . . . . . . . . . . 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 9108 . . . . . . . . . . . . 13  |-  -u pi  e.  RR
38 rexr 8877 . . . . . . . . . . . . 13  |-  ( -u pi  e.  RR  ->  -u pi  e.  RR* )
3937, 38ax-mp 8 . . . . . . . . . . . 12  |-  -u pi  e.  RR*
40 rexr 8877 . . . . . . . . . . . . 13  |-  ( pi  e.  RR  ->  pi  e.  RR* )
4131, 40ax-mp 8 . . . . . . . . . . . 12  |-  pi  e.  RR*
42 elioo2 10697 . . . . . . . . . . . 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 11598 . . . . . . . . . . 11  |-  Im : CC
--> RR
46 ffn 5389 . . . . . . . . . . 11  |-  ( Im : CC --> RR  ->  Im  Fn  CC )
47 elpreima 5645 . . . . . . . . . . 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 3185 . . . . . . 7  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  (
0 ( ball `  ( abs  o.  -  ) ) R )  C_  ( `' Im " ( -u pi (,) pi ) ) )
52 df-ima 4702 . . . . . . . 8  |-  ( log " ( CC  \ 
(  -oo (,] 0 ) ) )  =  ran  ( log  |`  ( CC  \  (  -oo (,] 0
) ) )
53 eqid 2283 . . . . . . . . . 10  |-  ( CC 
\  (  -oo (,] 0 ) )  =  ( CC  \  (  -oo (,] 0 ) )
5453logf1o2 19997 . . . . . . . . 9  |-  ( log  |`  ( CC  \  (  -oo (,] 0 ) ) ) : ( CC 
\  (  -oo (,] 0 ) ) -1-1-onto-> ( `' Im " ( -u pi (,) pi ) )
55 f1ofo 5479 . . . . . . . . 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 5454 . . . . . . . . 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 2303 . . . . . . 7  |-  ( log " ( CC  \ 
(  -oo (,] 0 ) ) )  =  ( `' Im " ( -u pi (,) pi ) )
5951, 58syl6sseqr 3225 . . . . . 6  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  (
0 ( ball `  ( abs  o.  -  ) ) R )  C_  ( log " ( CC  \ 
(  -oo (,] 0 ) ) ) )
60 resima2 4988 . . . . . 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 2327 . . . 4  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  ( `' ( log  |`  ( CC  \  (  -oo (,] 0 ) ) )
" ( 0 (
ball `  ( abs  o. 
-  ) ) R ) )  =  ( exp " ( 0 ( ball `  ( abs  o.  -  ) ) R ) ) )
6353logcn 19994 . . . . . 6  |-  ( log  |`  ( CC  \  (  -oo (,] 0 ) ) )  e.  ( ( CC  \  (  -oo (,] 0 ) ) -cn-> CC )
64 difss 3303 . . . . . . 7  |-  ( CC 
\  (  -oo (,] 0 ) )  C_  CC
65 ssid 3197 . . . . . . 7  |-  CC  C_  CC
66 efopn.j . . . . . . . 8  |-  J  =  ( TopOpen ` fld )
67 eqid 2283 . . . . . . . 8  |-  ( Jt  ( CC  \  (  -oo (,] 0 ) ) )  =  ( Jt  ( CC 
\  (  -oo (,] 0 ) ) )
6866cnfldtop 18293 . . . . . . . . . 10  |-  J  e. 
Top
6966cnfldtopon 18292 . . . . . . . . . . . 12  |-  J  e.  (TopOn `  CC )
7069toponunii 16670 . . . . . . . . . . 11  |-  CC  =  U. J
7170restid 13338 . . . . . . . . . 10  |-  ( J  e.  Top  ->  ( Jt  CC )  =  J
)
7268, 71ax-mp 8 . . . . . . . . 9  |-  ( Jt  CC )  =  J
7372eqcomi 2287 . . . . . . . 8  |-  J  =  ( Jt  CC )
7466, 67, 73cncfcn 18413 . . . . . . 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 2355 . . . . 5  |-  ( log  |`  ( CC  \  (  -oo (,] 0 ) ) )  e.  ( ( Jt  ( CC  \  (  -oo (,] 0 ) ) )  Cn  J )
7766cnfldtopn 18291 . . . . . . 7  |-  J  =  ( MetOpen `  ( abs  o. 
-  ) )
7877blopn 18046 . . . . . 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 16994 . . . . 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 2358 . . 3  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  ( exp " ( 0 (
ball `  ( abs  o. 
-  ) ) R ) )  e.  ( Jt  ( CC  \  (  -oo (,] 0 ) ) ) )
8353logdmopn 19996 . . . . 5  |-  ( CC 
\  (  -oo (,] 0 ) )  e.  ( TopOpen ` fld )
8483, 66eleqtrri 2356 . . . 4  |-  ( CC 
\  (  -oo (,] 0 ) )  e.  J
85 restopn2 16908 . . . 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 1623    e. wcel 1684    \ cdif 3149    C_ wss 3152   {csn 3640   class class class wbr 4023   `'ccnv 4688   ran crn 4690    |` cres 4691   "cima 4692    o. ccom 4693   Rel wrel 4694   Fun wfun 5249    Fn wfn 5250   -->wf 5251   -onto->wfo 5253   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737    -oocmnf 8865   RR*cxr 8866    < clt 8867    - cmin 9037   -ucneg 9038   RR+crp 10354   (,)cioo 10656   (,]cioc 10657   Imcim 11583   abscabs 11719   expce 12343   picpi 12348   ↾t crest 13325   TopOpenctopn 13326   * Metcxmt 16369   ballcbl 16371  ℂfldccnfld 16377   Topctop 16631    Cn ccn 16954   -cn->ccncf 18380   logclog 19912
This theorem is referenced by:  efopn  20005
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351  df-cos 12352  df-tan 12353  df-pi 12354  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-cmp 17114  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217  df-log 19914
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