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Theorem efopnlem2 23506
Description: Lemma for efopn 23507. (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 23418 . . . . . . . 8  |-  log :
( CC  \  {
0 } ) -1-1-onto-> ran  log
2 f1orn 5832 . . . . . . . . 9  |-  ( log
: ( CC  \  { 0 } ) -1-1-onto-> ran 
log 
<->  ( log  Fn  ( CC  \  { 0 } )  /\  Fun  `' log ) )
32simprbi 465 . . . . . . . 8  |-  ( log
: ( CC  \  { 0 } ) -1-1-onto-> ran 
log  ->  Fun  `' log )
4 funcnvres 5661 . . . . . . . 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 23410 . . . . . . . . . 10  |-  log  =  `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )
76cnveqi 5020 . . . . . . . . 9  |-  `' log  =  `' `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )
8 relres 5143 . . . . . . . . . 10  |-  Rel  ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )
9 dfrel2 5297 . . . . . . . . . 10  |-  ( Rel  ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  <->  `' `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  =  ( exp  |`  ( `' Im "
( -u pi (,] pi ) ) ) )
108, 9mpbi 211 . . . . . . . . 9  |-  `' `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  =  ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )
117, 10eqtri 2449 . . . . . . . 8  |-  `' log  =  ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )
1211reseq1i 5112 . . . . . . 7  |-  ( `' log  |`  ( log " ( CC  \  ( -oo (,] 0 ) ) ) )  =  ( ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  |`  ( log " ( CC  \  ( -oo (,] 0 ) ) ) )
13 imassrn 5190 . . . . . . . . 9  |-  ( log " ( CC  \ 
( -oo (,] 0 ) ) )  C_  ran  log
14 logrn 23412 . . . . . . . . 9  |-  ran  log  =  ( `' Im " ( -u pi (,] pi ) )
1513, 14sseqtri 3493 . . . . . . . 8  |-  ( log " ( CC  \ 
( -oo (,] 0 ) ) )  C_  ( `' Im " ( -u pi (,] pi ) )
16 resabs1 5144 . . . . . . . 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 2453 . . . . . 6  |-  `' ( log  |`  ( CC  \  ( -oo (,] 0
) ) )  =  ( exp  |`  ( log " ( CC  \ 
( -oo (,] 0 ) ) ) )
1918imaeq1i 5176 . . . . 5  |-  ( `' ( log  |`  ( CC  \  ( -oo (,] 0 ) ) )
" ( 0 (
ball `  ( abs  o. 
-  ) ) R ) )  =  ( ( exp  |`  ( log " ( CC  \ 
( -oo (,] 0 ) ) ) ) "
( 0 ( ball `  ( abs  o.  -  ) ) R ) )
20 cnxmet 21730 . . . . . . . . . . . . 13  |-  ( abs 
o.  -  )  e.  ( *Met `  CC )
2120a1i 11 . . . . . . . . . . . 12  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  ( abs  o.  -  )  e.  ( *Met `  CC ) )
22 0cnd 9625 . . . . . . . . . . . 12  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  0  e.  CC )
23 rpxr 11298 . . . . . . . . . . . . 13  |-  ( R  e.  RR+  ->  R  e. 
RR* )
2423adantr 466 . . . . . . . . . . . 12  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  R  e.  RR* )
25 blssm 21370 . . . . . . . . . . . 12  |-  ( ( ( abs  o.  -  )  e.  ( *Met `  CC )  /\  0  e.  CC  /\  R  e.  RR* )  ->  (
0 ( ball `  ( abs  o.  -  ) ) R )  C_  CC )
2621, 22, 24, 25syl3anc 1264 . . . . . . . . . . 11  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  (
0 ( ball `  ( abs  o.  -  ) ) R )  C_  CC )
2726sselda 3461 . . . . . . . . . 10  |-  ( ( ( R  e.  RR+  /\  R  <  pi )  /\  x  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R ) )  ->  x  e.  CC )
2827imcld 13226 . . . . . . . . . . 11  |-  ( ( ( R  e.  RR+  /\  R  <  pi )  /\  x  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R ) )  -> 
( Im `  x
)  e.  RR )
29 efopnlem1 23505 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  RR+  /\  R  <  pi )  /\  x  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R ) )  -> 
( abs `  (
Im `  x )
)  <  pi )
30 pire 23317 . . . . . . . . . . . . . 14  |-  pi  e.  RR
31 abslt 13345 . . . . . . . . . . . . . 14  |-  ( ( ( Im `  x
)  e.  RR  /\  pi  e.  RR )  -> 
( ( abs `  (
Im `  x )
)  <  pi  <->  ( -u pi  <  ( Im `  x
)  /\  ( Im `  x )  <  pi ) ) )
3228, 30, 31sylancl 666 . . . . . . . . . . . . 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 213 . . . . . . . . . . . 12  |-  ( ( ( R  e.  RR+  /\  R  <  pi )  /\  x  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R ) )  -> 
( -u pi  <  (
Im `  x )  /\  ( Im `  x
)  <  pi )
)
3433simpld 460 . . . . . . . . . . 11  |-  ( ( ( R  e.  RR+  /\  R  <  pi )  /\  x  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R ) )  ->  -u pi  <  ( Im
`  x ) )
3533simprd 464 . . . . . . . . . . 11  |-  ( ( ( R  e.  RR+  /\  R  <  pi )  /\  x  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R ) )  -> 
( Im `  x
)  <  pi )
3630renegcli 9924 . . . . . . . . . . . . 13  |-  -u pi  e.  RR
3736rexri 9682 . . . . . . . . . . . 12  |-  -u pi  e.  RR*
3830rexri 9682 . . . . . . . . . . . 12  |-  pi  e.  RR*
39 elioo2 11666 . . . . . . . . . . . 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 676 . . . . . . . . . . 11  |-  ( ( Im `  x )  e.  ( -u pi (,) pi )  <->  ( (
Im `  x )  e.  RR  /\  -u pi  <  ( Im `  x
)  /\  ( Im `  x )  <  pi ) )
4128, 34, 35, 40syl3anbrc 1189 . . . . . . . . . 10  |-  ( ( ( R  e.  RR+  /\  R  <  pi )  /\  x  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R ) )  -> 
( Im `  x
)  e.  ( -u pi (,) pi ) )
42 imf 13144 . . . . . . . . . . 11  |-  Im : CC
--> RR
43 ffn 5737 . . . . . . . . . . 11  |-  ( Im : CC --> RR  ->  Im  Fn  CC )
44 elpreima 6008 . . . . . . . . . . 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 668 . . . . . . . . 9  |-  ( ( ( R  e.  RR+  /\  R  <  pi )  /\  x  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R ) )  ->  x  e.  ( `' Im " ( -u pi (,) pi ) ) )
4746ex 435 . . . . . . . 8  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  (
x  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R )  ->  x  e.  ( `' Im "
( -u pi (,) pi ) ) ) )
4847ssrdv 3467 . . . . . . 7  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  (
0 ( ball `  ( abs  o.  -  ) ) R )  C_  ( `' Im " ( -u pi (,) pi ) ) )
49 df-ima 4858 . . . . . . . 8  |-  ( log " ( CC  \ 
( -oo (,] 0 ) ) )  =  ran  ( log  |`  ( CC  \  ( -oo (,] 0
) ) )
50 eqid 2420 . . . . . . . . . 10  |-  ( CC 
\  ( -oo (,] 0 ) )  =  ( CC  \  ( -oo (,] 0 ) )
5150logf1o2 23499 . . . . . . . . 9  |-  ( log  |`  ( CC  \  ( -oo (,] 0 ) ) ) : ( CC 
\  ( -oo (,] 0 ) ) -1-1-onto-> ( `' Im " ( -u pi (,) pi ) )
52 f1ofo 5829 . . . . . . . . 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 5804 . . . . . . . . 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 2449 . . . . . . 7  |-  ( log " ( CC  \ 
( -oo (,] 0 ) ) )  =  ( `' Im " ( -u pi (,) pi ) )
5648, 55syl6sseqr 3508 . . . . . 6  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  (
0 ( ball `  ( abs  o.  -  ) ) R )  C_  ( log " ( CC  \ 
( -oo (,] 0 ) ) ) )
57 resima2 5149 . . . . . 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 2473 . . . 4  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  ( `' ( log  |`  ( CC  \  ( -oo (,] 0 ) ) )
" ( 0 (
ball `  ( abs  o. 
-  ) ) R ) )  =  ( exp " ( 0 ( ball `  ( abs  o.  -  ) ) R ) ) )
6050logcn 23496 . . . . . 6  |-  ( log  |`  ( CC  \  ( -oo (,] 0 ) ) )  e.  ( ( CC  \  ( -oo (,] 0 ) ) -cn-> CC )
61 difss 3589 . . . . . . 7  |-  ( CC 
\  ( -oo (,] 0 ) )  C_  CC
62 ssid 3480 . . . . . . 7  |-  CC  C_  CC
63 efopn.j . . . . . . . 8  |-  J  =  ( TopOpen ` fld )
64 eqid 2420 . . . . . . . 8  |-  ( Jt  ( CC  \  ( -oo (,] 0 ) ) )  =  ( Jt  ( CC 
\  ( -oo (,] 0 ) ) )
6563cnfldtop 21741 . . . . . . . . . 10  |-  J  e. 
Top
6663cnfldtopon 21740 . . . . . . . . . . . 12  |-  J  e.  (TopOn `  CC )
6766toponunii 19884 . . . . . . . . . . 11  |-  CC  =  U. J
6867restid 15292 . . . . . . . . . 10  |-  ( J  e.  Top  ->  ( Jt  CC )  =  J
)
6965, 68ax-mp 5 . . . . . . . . 9  |-  ( Jt  CC )  =  J
7069eqcomi 2433 . . . . . . . 8  |-  J  =  ( Jt  CC )
7163, 64, 70cncfcn 21863 . . . . . . 7  |-  ( ( ( CC  \  ( -oo (,] 0 ) ) 
C_  CC  /\  CC  C_  CC )  ->  ( ( CC  \  ( -oo (,] 0 ) ) -cn-> CC )  =  ( ( Jt  ( CC  \  ( -oo (,] 0 ) ) )  Cn  J ) )
7261, 62, 71mp2an 676 . . . . . 6  |-  ( ( CC  \  ( -oo (,] 0 ) ) -cn-> CC )  =  ( ( Jt  ( CC  \  ( -oo (,] 0 ) ) )  Cn  J )
7360, 72eleqtri 2506 . . . . 5  |-  ( log  |`  ( CC  \  ( -oo (,] 0 ) ) )  e.  ( ( Jt  ( CC  \  ( -oo (,] 0 ) ) )  Cn  J )
7463cnfldtopn 21739 . . . . . . 7  |-  J  =  ( MetOpen `  ( abs  o. 
-  ) )
7574blopn 21452 . . . . . 6  |-  ( ( ( abs  o.  -  )  e.  ( *Met `  CC )  /\  0  e.  CC  /\  R  e.  RR* )  ->  (
0 ( ball `  ( abs  o.  -  ) ) R )  e.  J
)
7621, 22, 24, 75syl3anc 1264 . . . . 5  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  (
0 ( ball `  ( abs  o.  -  ) ) R )  e.  J
)
77 cnima 20218 . . . . 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 667 . . . 4  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  ( `' ( log  |`  ( CC  \  ( -oo (,] 0 ) ) )
" ( 0 (
ball `  ( abs  o. 
-  ) ) R ) )  e.  ( Jt  ( CC  \  ( -oo (,] 0 ) ) ) )
7959, 78eqeltrrd 2509 . . 3  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  ( exp " ( 0 (
ball `  ( abs  o. 
-  ) ) R ) )  e.  ( Jt  ( CC  \  ( -oo (,] 0 ) ) ) )
8050logdmopn 23498 . . . . 5  |-  ( CC 
\  ( -oo (,] 0 ) )  e.  ( TopOpen ` fld )
8180, 63eleqtrri 2507 . . . 4  |-  ( CC 
\  ( -oo (,] 0 ) )  e.  J
82 restopn2 20130 . . . 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 676 . . 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 199 . 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 460 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867    \ cdif 3430    C_ wss 3433   {csn 3993   class class class wbr 4417   `'ccnv 4844   ran crn 4846    |` cres 4847   "cima 4848    o. ccom 4849   Rel wrel 4850   Fun wfun 5586    Fn wfn 5587   -->wf 5588   -onto->wfo 5590   -1-1-onto->wf1o 5591   ` cfv 5592  (class class class)co 6296   CCcc 9526   RRcr 9527   0cc0 9528   -oocmnf 9662   RR*cxr 9663    < clt 9664    - cmin 9849   -ucneg 9850   RR+crp 11291   (,)cioo 11624   (,]cioc 11625   Imcim 13129   abscabs 13265   expce 14081   picpi 14086   ↾t crest 15279   TopOpenctopn 15280   *Metcxmt 18896   ballcbl 18898  ℂfldccnfld 18911   Topctop 19854    Cn ccn 20177   -cn->ccncf 21830   logclog 23408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-inf2 8137  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605  ax-pre-sup 9606  ax-addf 9607  ax-mulf 9608
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-iin 4296  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-se 4805  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-isom 5601  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6536  df-om 6698  df-1st 6798  df-2nd 6799  df-supp 6917  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-2o 7182  df-oadd 7185  df-er 7362  df-map 7473  df-pm 7474  df-ixp 7522  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-fsupp 7881  df-fi 7922  df-sup 7953  df-inf 7954  df-oi 8016  df-card 8363  df-cda 8587  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-div 10259  df-nn 10599  df-2 10657  df-3 10658  df-4 10659  df-5 10660  df-6 10661  df-7 10662  df-8 10663  df-9 10664  df-10 10665  df-n0 10859  df-z 10927  df-dec 11041  df-uz 11149  df-q 11254  df-rp 11292  df-xneg 11398  df-xadd 11399  df-xmul 11400  df-ioo 11628  df-ioc 11629  df-ico 11630  df-icc 11631  df-fz 11772  df-fzo 11903  df-fl 12014  df-mod 12083  df-seq 12200  df-exp 12259  df-fac 12446  df-bc 12474  df-hash 12502  df-shft 13098  df-cj 13130  df-re 13131  df-im 13132  df-sqrt 13266  df-abs 13267  df-limsup 13493  df-clim 13519  df-rlim 13520  df-sum 13720  df-ef 14088  df-sin 14090  df-cos 14091  df-tan 14092  df-pi 14093  df-struct 15083  df-ndx 15084  df-slot 15085  df-base 15086  df-sets 15087  df-ress 15088  df-plusg 15163  df-mulr 15164  df-starv 15165  df-sca 15166  df-vsca 15167  df-ip 15168  df-tset 15169  df-ple 15170  df-ds 15172  df-unif 15173  df-hom 15174  df-cco 15175  df-rest 15281  df-topn 15282  df-0g 15300  df-gsum 15301  df-topgen 15302  df-pt 15303  df-prds 15306  df-xrs 15360  df-qtop 15365  df-imas 15366  df-xps 15368  df-mre 15444  df-mrc 15445  df-acs 15447  df-mgm 16440  df-sgrp 16479  df-mnd 16489  df-submnd 16535  df-mulg 16628  df-cntz 16923  df-cmn 17373  df-psmet 18903  df-xmet 18904  df-met 18905  df-bl 18906  df-mopn 18907  df-fbas 18908  df-fg 18909  df-cnfld 18912  df-top 19858  df-bases 19859  df-topon 19860  df-topsp 19861  df-cld 19971  df-ntr 19972  df-cls 19973  df-nei 20051  df-lp 20089  df-perf 20090  df-cn 20180  df-cnp 20181  df-haus 20268  df-cmp 20339  df-tx 20514  df-hmeo 20707  df-fil 20798  df-fm 20890  df-flim 20891  df-flf 20892  df-xms 21272  df-ms 21273  df-tms 21274  df-cncf 21832  df-limc 22728  df-dv 22729  df-log 23410
This theorem is referenced by:  efopn  23507
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