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Theorem efopnlem2 23595
Description: Lemma for efopn 23596. (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 23507 . . . . . . . 8  |-  log :
( CC  \  {
0 } ) -1-1-onto-> ran  log
2 f1orn 5822 . . . . . . . . 9  |-  ( log
: ( CC  \  { 0 } ) -1-1-onto-> ran 
log 
<->  ( log  Fn  ( CC  \  { 0 } )  /\  Fun  `' log ) )
32simprbi 466 . . . . . . . 8  |-  ( log
: ( CC  \  { 0 } ) -1-1-onto-> ran 
log  ->  Fun  `' log )
4 funcnvres 5650 . . . . . . . 8  |-  ( Fun  `' log  ->  `' ( log  |`  ( CC  \ 
( -oo (,] 0 ) ) )  =  ( `' log  |`  ( log " ( CC  \  ( -oo (,] 0 ) ) ) ) )
51, 3, 4mp2b 10 . . . . . . 7  |-  `' ( log  |`  ( CC  \  ( -oo (,] 0
) ) )  =  ( `' log  |`  ( log " ( CC  \ 
( -oo (,] 0 ) ) ) )
6 df-log 23499 . . . . . . . . . 10  |-  log  =  `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )
76cnveqi 5008 . . . . . . . . 9  |-  `' log  =  `' `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )
8 relres 5131 . . . . . . . . . 10  |-  Rel  ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )
9 dfrel2 5285 . . . . . . . . . 10  |-  ( Rel  ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  <->  `' `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  =  ( exp  |`  ( `' Im "
( -u pi (,] pi ) ) ) )
108, 9mpbi 212 . . . . . . . . 9  |-  `' `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  =  ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )
117, 10eqtri 2472 . . . . . . . 8  |-  `' log  =  ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )
1211reseq1i 5100 . . . . . . 7  |-  ( `' log  |`  ( log " ( CC  \  ( -oo (,] 0 ) ) ) )  =  ( ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  |`  ( log " ( CC  \  ( -oo (,] 0 ) ) ) )
13 imassrn 5178 . . . . . . . . 9  |-  ( log " ( CC  \ 
( -oo (,] 0 ) ) )  C_  ran  log
14 logrn 23501 . . . . . . . . 9  |-  ran  log  =  ( `' Im " ( -u pi (,] pi ) )
1513, 14sseqtri 3463 . . . . . . . 8  |-  ( log " ( CC  \ 
( -oo (,] 0 ) ) )  C_  ( `' Im " ( -u pi (,] pi ) )
16 resabs1 5132 . . . . . . . 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 5 . . . . . . 7  |-  ( ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  |`  ( log " ( CC  \  ( -oo (,] 0 ) ) ) )  =  ( exp  |`  ( log " ( CC  \  ( -oo (,] 0 ) ) ) )
185, 12, 173eqtri 2476 . . . . . 6  |-  `' ( log  |`  ( CC  \  ( -oo (,] 0
) ) )  =  ( exp  |`  ( log " ( CC  \ 
( -oo (,] 0 ) ) ) )
1918imaeq1i 5164 . . . . 5  |-  ( `' ( log  |`  ( CC  \  ( -oo (,] 0 ) ) )
" ( 0 (
ball `  ( abs  o. 
-  ) ) R ) )  =  ( ( exp  |`  ( log " ( CC  \ 
( -oo (,] 0 ) ) ) ) "
( 0 ( ball `  ( abs  o.  -  ) ) R ) )
20 cnxmet 21786 . . . . . . . . . . . . 13  |-  ( abs 
o.  -  )  e.  ( *Met `  CC )
2120a1i 11 . . . . . . . . . . . 12  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  ( abs  o.  -  )  e.  ( *Met `  CC ) )
22 0cnd 9633 . . . . . . . . . . . 12  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  0  e.  CC )
23 rpxr 11306 . . . . . . . . . . . . 13  |-  ( R  e.  RR+  ->  R  e. 
RR* )
2423adantr 467 . . . . . . . . . . . 12  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  R  e.  RR* )
25 blssm 21426 . . . . . . . . . . . 12  |-  ( ( ( abs  o.  -  )  e.  ( *Met `  CC )  /\  0  e.  CC  /\  R  e.  RR* )  ->  (
0 ( ball `  ( abs  o.  -  ) ) R )  C_  CC )
2621, 22, 24, 25syl3anc 1267 . . . . . . . . . . 11  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  (
0 ( ball `  ( abs  o.  -  ) ) R )  C_  CC )
2726sselda 3431 . . . . . . . . . 10  |-  ( ( ( R  e.  RR+  /\  R  <  pi )  /\  x  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R ) )  ->  x  e.  CC )
2827imcld 13251 . . . . . . . . . . 11  |-  ( ( ( R  e.  RR+  /\  R  <  pi )  /\  x  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R ) )  -> 
( Im `  x
)  e.  RR )
29 efopnlem1 23594 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  RR+  /\  R  <  pi )  /\  x  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R ) )  -> 
( abs `  (
Im `  x )
)  <  pi )
30 pire 23406 . . . . . . . . . . . . . 14  |-  pi  e.  RR
31 abslt 13370 . . . . . . . . . . . . . 14  |-  ( ( ( Im `  x
)  e.  RR  /\  pi  e.  RR )  -> 
( ( abs `  (
Im `  x )
)  <  pi  <->  ( -u pi  <  ( Im `  x
)  /\  ( Im `  x )  <  pi ) ) )
3228, 30, 31sylancl 667 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  RR+  /\  R  <  pi )  /\  x  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R ) )  -> 
( ( abs `  (
Im `  x )
)  <  pi  <->  ( -u pi  <  ( Im `  x
)  /\  ( Im `  x )  <  pi ) ) )
3329, 32mpbid 214 . . . . . . . . . . . 12  |-  ( ( ( R  e.  RR+  /\  R  <  pi )  /\  x  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R ) )  -> 
( -u pi  <  (
Im `  x )  /\  ( Im `  x
)  <  pi )
)
3433simpld 461 . . . . . . . . . . 11  |-  ( ( ( R  e.  RR+  /\  R  <  pi )  /\  x  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R ) )  ->  -u pi  <  ( Im
`  x ) )
3533simprd 465 . . . . . . . . . . 11  |-  ( ( ( R  e.  RR+  /\  R  <  pi )  /\  x  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R ) )  -> 
( Im `  x
)  <  pi )
3630renegcli 9932 . . . . . . . . . . . . 13  |-  -u pi  e.  RR
3736rexri 9690 . . . . . . . . . . . 12  |-  -u pi  e.  RR*
3830rexri 9690 . . . . . . . . . . . 12  |-  pi  e.  RR*
39 elioo2 11674 . . . . . . . . . . . 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 ) ) )
4037, 38, 39mp2an 677 . . . . . . . . . . 11  |-  ( ( Im `  x )  e.  ( -u pi (,) pi )  <->  ( (
Im `  x )  e.  RR  /\  -u pi  <  ( Im `  x
)  /\  ( Im `  x )  <  pi ) )
4128, 34, 35, 40syl3anbrc 1191 . . . . . . . . . 10  |-  ( ( ( R  e.  RR+  /\  R  <  pi )  /\  x  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R ) )  -> 
( Im `  x
)  e.  ( -u pi (,) pi ) )
42 imf 13169 . . . . . . . . . . 11  |-  Im : CC
--> RR
43 ffn 5726 . . . . . . . . . . 11  |-  ( Im : CC --> RR  ->  Im  Fn  CC )
44 elpreima 6000 . . . . . . . . . . 11  |-  ( Im  Fn  CC  ->  (
x  e.  ( `' Im " ( -u pi (,) pi ) )  <-> 
( x  e.  CC  /\  ( Im `  x
)  e.  ( -u pi (,) pi ) ) ) )
4542, 43, 44mp2b 10 . . . . . . . . . 10  |-  ( x  e.  ( `' Im " ( -u pi (,) pi ) )  <->  ( x  e.  CC  /\  ( Im
`  x )  e.  ( -u pi (,) pi ) ) )
4627, 41, 45sylanbrc 669 . . . . . . . . 9  |-  ( ( ( R  e.  RR+  /\  R  <  pi )  /\  x  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R ) )  ->  x  e.  ( `' Im " ( -u pi (,) pi ) ) )
4746ex 436 . . . . . . . 8  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  (
x  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R )  ->  x  e.  ( `' Im "
( -u pi (,) pi ) ) ) )
4847ssrdv 3437 . . . . . . 7  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  (
0 ( ball `  ( abs  o.  -  ) ) R )  C_  ( `' Im " ( -u pi (,) pi ) ) )
49 df-ima 4846 . . . . . . . 8  |-  ( log " ( CC  \ 
( -oo (,] 0 ) ) )  =  ran  ( log  |`  ( CC  \  ( -oo (,] 0
) ) )
50 eqid 2450 . . . . . . . . . 10  |-  ( CC 
\  ( -oo (,] 0 ) )  =  ( CC  \  ( -oo (,] 0 ) )
5150logf1o2 23588 . . . . . . . . 9  |-  ( log  |`  ( CC  \  ( -oo (,] 0 ) ) ) : ( CC 
\  ( -oo (,] 0 ) ) -1-1-onto-> ( `' Im " ( -u pi (,) pi ) )
52 f1ofo 5819 . . . . . . . . 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 ) ) )
53 forn 5794 . . . . . . . . 9  |-  ( ( log  |`  ( CC  \  ( -oo (,] 0
) ) ) : ( CC  \  ( -oo (,] 0 ) )
-onto-> ( `' Im "
( -u pi (,) pi ) )  ->  ran  ( log  |`  ( CC  \  ( -oo (,] 0
) ) )  =  ( `' Im "
( -u pi (,) pi ) ) )
5451, 52, 53mp2b 10 . . . . . . . 8  |-  ran  ( log  |`  ( CC  \ 
( -oo (,] 0 ) ) )  =  ( `' Im " ( -u pi (,) pi ) )
5549, 54eqtri 2472 . . . . . . 7  |-  ( log " ( CC  \ 
( -oo (,] 0 ) ) )  =  ( `' Im " ( -u pi (,) pi ) )
5648, 55syl6sseqr 3478 . . . . . 6  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  (
0 ( ball `  ( abs  o.  -  ) ) R )  C_  ( log " ( CC  \ 
( -oo (,] 0 ) ) ) )
57 resima2 5137 . . . . . 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 ) ) )
5856, 57syl 17 . . . . 5  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  (
( exp  |`  ( log " ( CC  \ 
( -oo (,] 0 ) ) ) ) "
( 0 ( ball `  ( abs  o.  -  ) ) R ) )  =  ( exp " ( 0 (
ball `  ( abs  o. 
-  ) ) R ) ) )
5919, 58syl5eq 2496 . . . 4  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  ( `' ( log  |`  ( CC  \  ( -oo (,] 0 ) ) )
" ( 0 (
ball `  ( abs  o. 
-  ) ) R ) )  =  ( exp " ( 0 ( ball `  ( abs  o.  -  ) ) R ) ) )
6050logcn 23585 . . . . . 6  |-  ( log  |`  ( CC  \  ( -oo (,] 0 ) ) )  e.  ( ( CC  \  ( -oo (,] 0 ) ) -cn-> CC )
61 difss 3559 . . . . . . 7  |-  ( CC 
\  ( -oo (,] 0 ) )  C_  CC
62 ssid 3450 . . . . . . 7  |-  CC  C_  CC
63 efopn.j . . . . . . . 8  |-  J  =  ( TopOpen ` fld )
64 eqid 2450 . . . . . . . 8  |-  ( Jt  ( CC  \  ( -oo (,] 0 ) ) )  =  ( Jt  ( CC 
\  ( -oo (,] 0 ) ) )
6563cnfldtop 21797 . . . . . . . . . 10  |-  J  e. 
Top
6663cnfldtopon 21796 . . . . . . . . . . . 12  |-  J  e.  (TopOn `  CC )
6766toponunii 19940 . . . . . . . . . . 11  |-  CC  =  U. J
6867restid 15325 . . . . . . . . . 10  |-  ( J  e.  Top  ->  ( Jt  CC )  =  J
)
6965, 68ax-mp 5 . . . . . . . . 9  |-  ( Jt  CC )  =  J
7069eqcomi 2459 . . . . . . . 8  |-  J  =  ( Jt  CC )
7163, 64, 70cncfcn 21934 . . . . . . 7  |-  ( ( ( CC  \  ( -oo (,] 0 ) ) 
C_  CC  /\  CC  C_  CC )  ->  ( ( CC  \  ( -oo (,] 0 ) ) -cn-> CC )  =  ( ( Jt  ( CC  \  ( -oo (,] 0 ) ) )  Cn  J ) )
7261, 62, 71mp2an 677 . . . . . 6  |-  ( ( CC  \  ( -oo (,] 0 ) ) -cn-> CC )  =  ( ( Jt  ( CC  \  ( -oo (,] 0 ) ) )  Cn  J )
7360, 72eleqtri 2526 . . . . 5  |-  ( log  |`  ( CC  \  ( -oo (,] 0 ) ) )  e.  ( ( Jt  ( CC  \  ( -oo (,] 0 ) ) )  Cn  J )
7463cnfldtopn 21795 . . . . . . 7  |-  J  =  ( MetOpen `  ( abs  o. 
-  ) )
7574blopn 21508 . . . . . 6  |-  ( ( ( abs  o.  -  )  e.  ( *Met `  CC )  /\  0  e.  CC  /\  R  e.  RR* )  ->  (
0 ( ball `  ( abs  o.  -  ) ) R )  e.  J
)
7621, 22, 24, 75syl3anc 1267 . . . . 5  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  (
0 ( ball `  ( abs  o.  -  ) ) R )  e.  J
)
77 cnima 20274 . . . . 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 ) ) ) )
7873, 76, 77sylancr 668 . . . 4  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  ( `' ( log  |`  ( CC  \  ( -oo (,] 0 ) ) )
" ( 0 (
ball `  ( abs  o. 
-  ) ) R ) )  e.  ( Jt  ( CC  \  ( -oo (,] 0 ) ) ) )
7959, 78eqeltrrd 2529 . . 3  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  ( exp " ( 0 (
ball `  ( abs  o. 
-  ) ) R ) )  e.  ( Jt  ( CC  \  ( -oo (,] 0 ) ) ) )
8050logdmopn 23587 . . . . 5  |-  ( CC 
\  ( -oo (,] 0 ) )  e.  ( TopOpen ` fld )
8180, 63eleqtrri 2527 . . . 4  |-  ( CC 
\  ( -oo (,] 0 ) )  e.  J
82 restopn2 20186 . . . 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 ) ) ) ) )
8365, 81, 82mp2an 677 . . 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 ) ) ) )
8479, 83sylib 200 . 2  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  (
( exp " (
0 ( ball `  ( abs  o.  -  ) ) R ) )  e.  J  /\  ( exp " ( 0 (
ball `  ( abs  o. 
-  ) ) R ) )  C_  ( CC  \  ( -oo (,] 0 ) ) ) )
8584simpld 461 1  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  ( exp " ( 0 (
ball `  ( abs  o. 
-  ) ) R ) )  e.  J
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 984    = wceq 1443    e. wcel 1886    \ cdif 3400    C_ wss 3403   {csn 3967   class class class wbr 4401   `'ccnv 4832   ran crn 4834    |` cres 4835   "cima 4836    o. ccom 4837   Rel wrel 4838   Fun wfun 5575    Fn wfn 5576   -->wf 5577   -onto->wfo 5579   -1-1-onto->wf1o 5580   ` cfv 5581  (class class class)co 6288   CCcc 9534   RRcr 9535   0cc0 9536   -oocmnf 9670   RR*cxr 9671    < clt 9672    - cmin 9857   -ucneg 9858   RR+crp 11299   (,)cioo 11632   (,]cioc 11633   Imcim 13154   abscabs 13290   expce 14107   picpi 14112   ↾t crest 15312   TopOpenctopn 15313   *Metcxmt 18948   ballcbl 18950  ℂfldccnfld 18963   Topctop 19910    Cn ccn 20233   -cn->ccncf 21901   logclog 23497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614  ax-addf 9615  ax-mulf 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-iin 4280  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-of 6528  df-om 6690  df-1st 6790  df-2nd 6791  df-supp 6912  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-2o 7180  df-oadd 7183  df-er 7360  df-map 7471  df-pm 7472  df-ixp 7520  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-fsupp 7881  df-fi 7922  df-sup 7953  df-inf 7954  df-oi 8022  df-card 8370  df-cda 8595  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-4 10667  df-5 10668  df-6 10669  df-7 10670  df-8 10671  df-9 10672  df-10 10673  df-n0 10867  df-z 10935  df-dec 11049  df-uz 11157  df-q 11262  df-rp 11300  df-xneg 11406  df-xadd 11407  df-xmul 11408  df-ioo 11636  df-ioc 11637  df-ico 11638  df-icc 11639  df-fz 11782  df-fzo 11913  df-fl 12025  df-mod 12094  df-seq 12211  df-exp 12270  df-fac 12457  df-bc 12485  df-hash 12513  df-shft 13123  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-limsup 13519  df-clim 13545  df-rlim 13546  df-sum 13746  df-ef 14114  df-sin 14116  df-cos 14117  df-tan 14118  df-pi 14119  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-mulr 15197  df-starv 15198  df-sca 15199  df-vsca 15200  df-ip 15201  df-tset 15202  df-ple 15203  df-ds 15205  df-unif 15206  df-hom 15207  df-cco 15208  df-rest 15314  df-topn 15315  df-0g 15333  df-gsum 15334  df-topgen 15335  df-pt 15336  df-prds 15339  df-xrs 15393  df-qtop 15399  df-imas 15400  df-xps 15403  df-mre 15485  df-mrc 15486  df-acs 15488  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-submnd 16576  df-mulg 16669  df-cntz 16964  df-cmn 17425  df-psmet 18955  df-xmet 18956  df-met 18957  df-bl 18958  df-mopn 18959  df-fbas 18960  df-fg 18961  df-cnfld 18964  df-top 19914  df-bases 19915  df-topon 19916  df-topsp 19917  df-cld 20027  df-ntr 20028  df-cls 20029  df-nei 20107  df-lp 20145  df-perf 20146  df-cn 20236  df-cnp 20237  df-haus 20324  df-cmp 20395  df-tx 20570  df-hmeo 20763  df-fil 20854  df-fm 20946  df-flim 20947  df-flf 20948  df-xms 21328  df-ms 21329  df-tms 21330  df-cncf 21903  df-limc 22814  df-dv 22815  df-log 23499
This theorem is referenced by:  efopn  23596
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