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Theorem efopnlem2 22077
Description: Lemma for efopn 22078. (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 21991 . . . . . . . 8  |-  log :
( CC  \  {
0 } ) -1-1-onto-> ran  log
2 f1orn 5646 . . . . . . . . 9  |-  ( log
: ( CC  \  { 0 } ) -1-1-onto-> ran 
log 
<->  ( log  Fn  ( CC  \  { 0 } )  /\  Fun  `' log ) )
32simprbi 464 . . . . . . . 8  |-  ( log
: ( CC  \  { 0 } ) -1-1-onto-> ran 
log  ->  Fun  `' log )
4 funcnvres 5482 . . . . . . . 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 21983 . . . . . . . . . 10  |-  log  =  `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )
76cnveqi 5009 . . . . . . . . 9  |-  `' log  =  `' `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )
8 relres 5133 . . . . . . . . . 10  |-  Rel  ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )
9 dfrel2 5283 . . . . . . . . . 10  |-  ( Rel  ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  <->  `' `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  =  ( exp  |`  ( `' Im "
( -u pi (,] pi ) ) ) )
108, 9mpbi 208 . . . . . . . . 9  |-  `' `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  =  ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )
117, 10eqtri 2458 . . . . . . . 8  |-  `' log  =  ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )
1211reseq1i 5101 . . . . . . 7  |-  ( `' log  |`  ( log " ( CC  \  ( -oo (,] 0 ) ) ) )  =  ( ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  |`  ( log " ( CC  \  ( -oo (,] 0 ) ) ) )
13 imassrn 5175 . . . . . . . . 9  |-  ( log " ( CC  \ 
( -oo (,] 0 ) ) )  C_  ran  log
14 logrn 21985 . . . . . . . . 9  |-  ran  log  =  ( `' Im " ( -u pi (,] pi ) )
1513, 14sseqtri 3383 . . . . . . . 8  |-  ( log " ( CC  \ 
( -oo (,] 0 ) ) )  C_  ( `' Im " ( -u pi (,] pi ) )
16 resabs1 5134 . . . . . . . 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 2462 . . . . . 6  |-  `' ( log  |`  ( CC  \  ( -oo (,] 0
) ) )  =  ( exp  |`  ( log " ( CC  \ 
( -oo (,] 0 ) ) ) )
1918imaeq1i 5161 . . . . 5  |-  ( `' ( log  |`  ( CC  \  ( -oo (,] 0 ) ) )
" ( 0 (
ball `  ( abs  o. 
-  ) ) R ) )  =  ( ( exp  |`  ( log " ( CC  \ 
( -oo (,] 0 ) ) ) ) "
( 0 ( ball `  ( abs  o.  -  ) ) R ) )
20 cnxmet 20327 . . . . . . . . . . . . 13  |-  ( abs 
o.  -  )  e.  ( *Met `  CC )
2120a1i 11 . . . . . . . . . . . 12  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  ( abs  o.  -  )  e.  ( *Met `  CC ) )
22 0cnd 9371 . . . . . . . . . . . 12  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  0  e.  CC )
23 rpxr 10990 . . . . . . . . . . . . 13  |-  ( R  e.  RR+  ->  R  e. 
RR* )
2423adantr 465 . . . . . . . . . . . 12  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  R  e.  RR* )
25 blssm 19968 . . . . . . . . . . . 12  |-  ( ( ( abs  o.  -  )  e.  ( *Met `  CC )  /\  0  e.  CC  /\  R  e.  RR* )  ->  (
0 ( ball `  ( abs  o.  -  ) ) R )  C_  CC )
2621, 22, 24, 25syl3anc 1218 . . . . . . . . . . 11  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  (
0 ( ball `  ( abs  o.  -  ) ) R )  C_  CC )
2726sselda 3351 . . . . . . . . . 10  |-  ( ( ( R  e.  RR+  /\  R  <  pi )  /\  x  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R ) )  ->  x  e.  CC )
2827imcld 12676 . . . . . . . . . . 11  |-  ( ( ( R  e.  RR+  /\  R  <  pi )  /\  x  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R ) )  -> 
( Im `  x
)  e.  RR )
29 efopnlem1 22076 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  RR+  /\  R  <  pi )  /\  x  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R ) )  -> 
( abs `  (
Im `  x )
)  <  pi )
30 pire 21896 . . . . . . . . . . . . . 14  |-  pi  e.  RR
31 abslt 12794 . . . . . . . . . . . . . 14  |-  ( ( ( Im `  x
)  e.  RR  /\  pi  e.  RR )  -> 
( ( abs `  (
Im `  x )
)  <  pi  <->  ( -u pi  <  ( Im `  x
)  /\  ( Im `  x )  <  pi ) ) )
3228, 30, 31sylancl 662 . . . . . . . . . . . . 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 210 . . . . . . . . . . . 12  |-  ( ( ( R  e.  RR+  /\  R  <  pi )  /\  x  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R ) )  -> 
( -u pi  <  (
Im `  x )  /\  ( Im `  x
)  <  pi )
)
3433simpld 459 . . . . . . . . . . 11  |-  ( ( ( R  e.  RR+  /\  R  <  pi )  /\  x  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R ) )  ->  -u pi  <  ( Im
`  x ) )
3533simprd 463 . . . . . . . . . . 11  |-  ( ( ( R  e.  RR+  /\  R  <  pi )  /\  x  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R ) )  -> 
( Im `  x
)  <  pi )
3630renegcli 9662 . . . . . . . . . . . . 13  |-  -u pi  e.  RR
3736rexri 9428 . . . . . . . . . . . 12  |-  -u pi  e.  RR*
3830rexri 9428 . . . . . . . . . . . 12  |-  pi  e.  RR*
39 elioo2 11333 . . . . . . . . . . . 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 672 . . . . . . . . . . 11  |-  ( ( Im `  x )  e.  ( -u pi (,) pi )  <->  ( (
Im `  x )  e.  RR  /\  -u pi  <  ( Im `  x
)  /\  ( Im `  x )  <  pi ) )
4128, 34, 35, 40syl3anbrc 1172 . . . . . . . . . 10  |-  ( ( ( R  e.  RR+  /\  R  <  pi )  /\  x  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R ) )  -> 
( Im `  x
)  e.  ( -u pi (,) pi ) )
42 imf 12594 . . . . . . . . . . 11  |-  Im : CC
--> RR
43 ffn 5554 . . . . . . . . . . 11  |-  ( Im : CC --> RR  ->  Im  Fn  CC )
44 elpreima 5818 . . . . . . . . . . 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 664 . . . . . . . . 9  |-  ( ( ( R  e.  RR+  /\  R  <  pi )  /\  x  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R ) )  ->  x  e.  ( `' Im " ( -u pi (,) pi ) ) )
4746ex 434 . . . . . . . 8  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  (
x  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R )  ->  x  e.  ( `' Im "
( -u pi (,) pi ) ) ) )
4847ssrdv 3357 . . . . . . 7  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  (
0 ( ball `  ( abs  o.  -  ) ) R )  C_  ( `' Im " ( -u pi (,) pi ) ) )
49 df-ima 4848 . . . . . . . 8  |-  ( log " ( CC  \ 
( -oo (,] 0 ) ) )  =  ran  ( log  |`  ( CC  \  ( -oo (,] 0
) ) )
50 eqid 2438 . . . . . . . . . 10  |-  ( CC 
\  ( -oo (,] 0 ) )  =  ( CC  \  ( -oo (,] 0 ) )
5150logf1o2 22070 . . . . . . . . 9  |-  ( log  |`  ( CC  \  ( -oo (,] 0 ) ) ) : ( CC 
\  ( -oo (,] 0 ) ) -1-1-onto-> ( `' Im " ( -u pi (,) pi ) )
52 f1ofo 5643 . . . . . . . . 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 5618 . . . . . . . . 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 2458 . . . . . . 7  |-  ( log " ( CC  \ 
( -oo (,] 0 ) ) )  =  ( `' Im " ( -u pi (,) pi ) )
5648, 55syl6sseqr 3398 . . . . . 6  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  (
0 ( ball `  ( abs  o.  -  ) ) R )  C_  ( log " ( CC  \ 
( -oo (,] 0 ) ) ) )
57 resima2 5138 . . . . . 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 16 . . . . 5  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  (
( exp  |`  ( log " ( CC  \ 
( -oo (,] 0 ) ) ) ) "
( 0 ( ball `  ( abs  o.  -  ) ) R ) )  =  ( exp " ( 0 (
ball `  ( abs  o. 
-  ) ) R ) ) )
5919, 58syl5eq 2482 . . . 4  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  ( `' ( log  |`  ( CC  \  ( -oo (,] 0 ) ) )
" ( 0 (
ball `  ( abs  o. 
-  ) ) R ) )  =  ( exp " ( 0 ( ball `  ( abs  o.  -  ) ) R ) ) )
6050logcn 22067 . . . . . 6  |-  ( log  |`  ( CC  \  ( -oo (,] 0 ) ) )  e.  ( ( CC  \  ( -oo (,] 0 ) ) -cn-> CC )
61 difss 3478 . . . . . . 7  |-  ( CC 
\  ( -oo (,] 0 ) )  C_  CC
62 ssid 3370 . . . . . . 7  |-  CC  C_  CC
63 efopn.j . . . . . . . 8  |-  J  =  ( TopOpen ` fld )
64 eqid 2438 . . . . . . . 8  |-  ( Jt  ( CC  \  ( -oo (,] 0 ) ) )  =  ( Jt  ( CC 
\  ( -oo (,] 0 ) ) )
6563cnfldtop 20338 . . . . . . . . . 10  |-  J  e. 
Top
6663cnfldtopon 20337 . . . . . . . . . . . 12  |-  J  e.  (TopOn `  CC )
6766toponunii 18512 . . . . . . . . . . 11  |-  CC  =  U. J
6867restid 14364 . . . . . . . . . 10  |-  ( J  e.  Top  ->  ( Jt  CC )  =  J
)
6965, 68ax-mp 5 . . . . . . . . 9  |-  ( Jt  CC )  =  J
7069eqcomi 2442 . . . . . . . 8  |-  J  =  ( Jt  CC )
7163, 64, 70cncfcn 20460 . . . . . . 7  |-  ( ( ( CC  \  ( -oo (,] 0 ) ) 
C_  CC  /\  CC  C_  CC )  ->  ( ( CC  \  ( -oo (,] 0 ) ) -cn-> CC )  =  ( ( Jt  ( CC  \  ( -oo (,] 0 ) ) )  Cn  J ) )
7261, 62, 71mp2an 672 . . . . . 6  |-  ( ( CC  \  ( -oo (,] 0 ) ) -cn-> CC )  =  ( ( Jt  ( CC  \  ( -oo (,] 0 ) ) )  Cn  J )
7360, 72eleqtri 2510 . . . . 5  |-  ( log  |`  ( CC  \  ( -oo (,] 0 ) ) )  e.  ( ( Jt  ( CC  \  ( -oo (,] 0 ) ) )  Cn  J )
7463cnfldtopn 20336 . . . . . . 7  |-  J  =  ( MetOpen `  ( abs  o. 
-  ) )
7574blopn 20050 . . . . . 6  |-  ( ( ( abs  o.  -  )  e.  ( *Met `  CC )  /\  0  e.  CC  /\  R  e.  RR* )  ->  (
0 ( ball `  ( abs  o.  -  ) ) R )  e.  J
)
7621, 22, 24, 75syl3anc 1218 . . . . 5  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  (
0 ( ball `  ( abs  o.  -  ) ) R )  e.  J
)
77 cnima 18844 . . . . 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 663 . . . 4  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  ( `' ( log  |`  ( CC  \  ( -oo (,] 0 ) ) )
" ( 0 (
ball `  ( abs  o. 
-  ) ) R ) )  e.  ( Jt  ( CC  \  ( -oo (,] 0 ) ) ) )
7959, 78eqeltrrd 2513 . . 3  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  ( exp " ( 0 (
ball `  ( abs  o. 
-  ) ) R ) )  e.  ( Jt  ( CC  \  ( -oo (,] 0 ) ) ) )
8050logdmopn 22069 . . . . 5  |-  ( CC 
\  ( -oo (,] 0 ) )  e.  ( TopOpen ` fld )
8180, 63eleqtrri 2511 . . . 4  |-  ( CC 
\  ( -oo (,] 0 ) )  e.  J
82 restopn2 18756 . . . 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 672 . . 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 196 . 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 459 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 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    \ cdif 3320    C_ wss 3323   {csn 3872   class class class wbr 4287   `'ccnv 4834   ran crn 4836    |` cres 4837   "cima 4838    o. ccom 4839   Rel wrel 4840   Fun wfun 5407    Fn wfn 5408   -->wf 5409   -onto->wfo 5411   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6086   CCcc 9272   RRcr 9273   0cc0 9274   -oocmnf 9408   RR*cxr 9409    < clt 9410    - cmin 9587   -ucneg 9588   RR+crp 10983   (,)cioo 11292   (,]cioc 11293   Imcim 12579   abscabs 12715   expce 13339   picpi 13344   ↾t crest 14351   TopOpenctopn 14352   *Metcxmt 17776   ballcbl 17778  ℂfldccnfld 17793   Topctop 18473    Cn ccn 18803   -cn->ccncf 20427   logclog 21981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-ioc 11297  df-ico 11298  df-icc 11299  df-fz 11430  df-fzo 11541  df-fl 11634  df-mod 11701  df-seq 11799  df-exp 11858  df-fac 12044  df-bc 12071  df-hash 12096  df-shft 12548  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-limsup 12941  df-clim 12958  df-rlim 12959  df-sum 13156  df-ef 13345  df-sin 13347  df-cos 13348  df-tan 13349  df-pi 13350  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-pt 14375  df-prds 14378  df-xrs 14432  df-qtop 14437  df-imas 14438  df-xps 14440  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-submnd 15457  df-mulg 15539  df-cntz 15826  df-cmn 16270  df-psmet 17784  df-xmet 17785  df-met 17786  df-bl 17787  df-mopn 17788  df-fbas 17789  df-fg 17790  df-cnfld 17794  df-top 18478  df-bases 18480  df-topon 18481  df-topsp 18482  df-cld 18598  df-ntr 18599  df-cls 18600  df-nei 18677  df-lp 18715  df-perf 18716  df-cn 18806  df-cnp 18807  df-haus 18894  df-cmp 18965  df-tx 19110  df-hmeo 19303  df-fil 19394  df-fm 19486  df-flim 19487  df-flf 19488  df-xms 19870  df-ms 19871  df-tms 19872  df-cncf 20429  df-limc 21316  df-dv 21317  df-log 21983
This theorem is referenced by:  efopn  22078
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