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Theorem efopnlem2 22910
Description: Lemma for efopn 22911. (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 22824 . . . . . . . 8  |-  log :
( CC  \  {
0 } ) -1-1-onto-> ran  log
2 f1orn 5816 . . . . . . . . 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 5647 . . . . . . . 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 22816 . . . . . . . . . 10  |-  log  =  `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )
76cnveqi 5167 . . . . . . . . 9  |-  `' log  =  `' `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )
8 relres 5291 . . . . . . . . . 10  |-  Rel  ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )
9 dfrel2 5447 . . . . . . . . . 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 2472 . . . . . . . 8  |-  `' log  =  ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )
1211reseq1i 5259 . . . . . . 7  |-  ( `' log  |`  ( log " ( CC  \  ( -oo (,] 0 ) ) ) )  =  ( ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  |`  ( log " ( CC  \  ( -oo (,] 0 ) ) ) )
13 imassrn 5338 . . . . . . . . 9  |-  ( log " ( CC  \ 
( -oo (,] 0 ) ) )  C_  ran  log
14 logrn 22818 . . . . . . . . 9  |-  ran  log  =  ( `' Im " ( -u pi (,] pi ) )
1513, 14sseqtri 3521 . . . . . . . 8  |-  ( log " ( CC  \ 
( -oo (,] 0 ) ) )  C_  ( `' Im " ( -u pi (,] pi ) )
16 resabs1 5292 . . . . . . . 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 5324 . . . . 5  |-  ( `' ( log  |`  ( CC  \  ( -oo (,] 0 ) ) )
" ( 0 (
ball `  ( abs  o. 
-  ) ) R ) )  =  ( ( exp  |`  ( log " ( CC  \ 
( -oo (,] 0 ) ) ) ) "
( 0 ( ball `  ( abs  o.  -  ) ) R ) )
20 cnxmet 21153 . . . . . . . . . . . . 13  |-  ( abs 
o.  -  )  e.  ( *Met `  CC )
2120a1i 11 . . . . . . . . . . . 12  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  ( abs  o.  -  )  e.  ( *Met `  CC ) )
22 0cnd 9592 . . . . . . . . . . . 12  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  0  e.  CC )
23 rpxr 11236 . . . . . . . . . . . . 13  |-  ( R  e.  RR+  ->  R  e. 
RR* )
2423adantr 465 . . . . . . . . . . . 12  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  R  e.  RR* )
25 blssm 20794 . . . . . . . . . . . 12  |-  ( ( ( abs  o.  -  )  e.  ( *Met `  CC )  /\  0  e.  CC  /\  R  e.  RR* )  ->  (
0 ( ball `  ( abs  o.  -  ) ) R )  C_  CC )
2621, 22, 24, 25syl3anc 1229 . . . . . . . . . . 11  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  (
0 ( ball `  ( abs  o.  -  ) ) R )  C_  CC )
2726sselda 3489 . . . . . . . . . 10  |-  ( ( ( R  e.  RR+  /\  R  <  pi )  /\  x  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R ) )  ->  x  e.  CC )
2827imcld 13007 . . . . . . . . . . 11  |-  ( ( ( R  e.  RR+  /\  R  <  pi )  /\  x  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R ) )  -> 
( Im `  x
)  e.  RR )
29 efopnlem1 22909 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  RR+  /\  R  <  pi )  /\  x  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R ) )  -> 
( abs `  (
Im `  x )
)  <  pi )
30 pire 22723 . . . . . . . . . . . . . 14  |-  pi  e.  RR
31 abslt 13126 . . . . . . . . . . . . . 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 9885 . . . . . . . . . . . . 13  |-  -u pi  e.  RR
3736rexri 9649 . . . . . . . . . . . 12  |-  -u pi  e.  RR*
3830rexri 9649 . . . . . . . . . . . 12  |-  pi  e.  RR*
39 elioo2 11579 . . . . . . . . . . . 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 1181 . . . . . . . . . 10  |-  ( ( ( R  e.  RR+  /\  R  <  pi )  /\  x  e.  ( 0 ( ball `  ( abs  o.  -  ) ) R ) )  -> 
( Im `  x
)  e.  ( -u pi (,) pi ) )
42 imf 12925 . . . . . . . . . . 11  |-  Im : CC
--> RR
43 ffn 5721 . . . . . . . . . . 11  |-  ( Im : CC --> RR  ->  Im  Fn  CC )
44 elpreima 5992 . . . . . . . . . . 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 3495 . . . . . . 7  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  (
0 ( ball `  ( abs  o.  -  ) ) R )  C_  ( `' Im " ( -u pi (,) pi ) ) )
49 df-ima 5002 . . . . . . . 8  |-  ( log " ( CC  \ 
( -oo (,] 0 ) ) )  =  ran  ( log  |`  ( CC  \  ( -oo (,] 0
) ) )
50 eqid 2443 . . . . . . . . . 10  |-  ( CC 
\  ( -oo (,] 0 ) )  =  ( CC  \  ( -oo (,] 0 ) )
5150logf1o2 22903 . . . . . . . . 9  |-  ( log  |`  ( CC  \  ( -oo (,] 0 ) ) ) : ( CC 
\  ( -oo (,] 0 ) ) -1-1-onto-> ( `' Im " ( -u pi (,) pi ) )
52 f1ofo 5813 . . . . . . . . 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 5788 . . . . . . . . 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 3536 . . . . . 6  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  (
0 ( ball `  ( abs  o.  -  ) ) R )  C_  ( log " ( CC  \ 
( -oo (,] 0 ) ) ) )
57 resima2 5297 . . . . . 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 2496 . . . 4  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  ( `' ( log  |`  ( CC  \  ( -oo (,] 0 ) ) )
" ( 0 (
ball `  ( abs  o. 
-  ) ) R ) )  =  ( exp " ( 0 ( ball `  ( abs  o.  -  ) ) R ) ) )
6050logcn 22900 . . . . . 6  |-  ( log  |`  ( CC  \  ( -oo (,] 0 ) ) )  e.  ( ( CC  \  ( -oo (,] 0 ) ) -cn-> CC )
61 difss 3616 . . . . . . 7  |-  ( CC 
\  ( -oo (,] 0 ) )  C_  CC
62 ssid 3508 . . . . . . 7  |-  CC  C_  CC
63 efopn.j . . . . . . . 8  |-  J  =  ( TopOpen ` fld )
64 eqid 2443 . . . . . . . 8  |-  ( Jt  ( CC  \  ( -oo (,] 0 ) ) )  =  ( Jt  ( CC 
\  ( -oo (,] 0 ) ) )
6563cnfldtop 21164 . . . . . . . . . 10  |-  J  e. 
Top
6663cnfldtopon 21163 . . . . . . . . . . . 12  |-  J  e.  (TopOn `  CC )
6766toponunii 19306 . . . . . . . . . . 11  |-  CC  =  U. J
6867restid 14708 . . . . . . . . . 10  |-  ( J  e.  Top  ->  ( Jt  CC )  =  J
)
6965, 68ax-mp 5 . . . . . . . . 9  |-  ( Jt  CC )  =  J
7069eqcomi 2456 . . . . . . . 8  |-  J  =  ( Jt  CC )
7163, 64, 70cncfcn 21286 . . . . . . 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 2529 . . . . 5  |-  ( log  |`  ( CC  \  ( -oo (,] 0 ) ) )  e.  ( ( Jt  ( CC  \  ( -oo (,] 0 ) ) )  Cn  J )
7463cnfldtopn 21162 . . . . . . 7  |-  J  =  ( MetOpen `  ( abs  o. 
-  ) )
7574blopn 20876 . . . . . 6  |-  ( ( ( abs  o.  -  )  e.  ( *Met `  CC )  /\  0  e.  CC  /\  R  e.  RR* )  ->  (
0 ( ball `  ( abs  o.  -  ) ) R )  e.  J
)
7621, 22, 24, 75syl3anc 1229 . . . . 5  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  (
0 ( ball `  ( abs  o.  -  ) ) R )  e.  J
)
77 cnima 19639 . . . . 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 2532 . . 3  |-  ( ( R  e.  RR+  /\  R  <  pi )  ->  ( exp " ( 0 (
ball `  ( abs  o. 
-  ) ) R ) )  e.  ( Jt  ( CC  \  ( -oo (,] 0 ) ) ) )
8050logdmopn 22902 . . . . 5  |-  ( CC 
\  ( -oo (,] 0 ) )  e.  ( TopOpen ` fld )
8180, 63eleqtrri 2530 . . . 4  |-  ( CC 
\  ( -oo (,] 0 ) )  e.  J
82 restopn2 19551 . . . 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 974    = wceq 1383    e. wcel 1804    \ cdif 3458    C_ wss 3461   {csn 4014   class class class wbr 4437   `'ccnv 4988   ran crn 4990    |` cres 4991   "cima 4992    o. ccom 4993   Rel wrel 4994   Fun wfun 5572    Fn wfn 5573   -->wf 5574   -onto->wfo 5576   -1-1-onto->wf1o 5577   ` cfv 5578  (class class class)co 6281   CCcc 9493   RRcr 9494   0cc0 9495   -oocmnf 9629   RR*cxr 9630    < clt 9631    - cmin 9810   -ucneg 9811   RR+crp 11229   (,)cioo 11538   (,]cioc 11539   Imcim 12910   abscabs 13046   expce 13675   picpi 13680   ↾t crest 14695   TopOpenctopn 14696   *Metcxmt 18277   ballcbl 18279  ℂfldccnfld 18294   Topctop 19267    Cn ccn 19598   -cn->ccncf 21253   logclog 22814
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574  ax-mulf 9575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-fi 7873  df-sup 7903  df-oi 7938  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-5 10603  df-6 10604  df-7 10605  df-8 10606  df-9 10607  df-10 10608  df-n0 10802  df-z 10871  df-dec 10985  df-uz 11091  df-q 11192  df-rp 11230  df-xneg 11327  df-xadd 11328  df-xmul 11329  df-ioo 11542  df-ioc 11543  df-ico 11544  df-icc 11545  df-fz 11682  df-fzo 11804  df-fl 11908  df-mod 11976  df-seq 12087  df-exp 12146  df-fac 12333  df-bc 12360  df-hash 12385  df-shft 12879  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-limsup 13273  df-clim 13290  df-rlim 13291  df-sum 13488  df-ef 13681  df-sin 13683  df-cos 13684  df-tan 13685  df-pi 13686  df-struct 14511  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-ress 14516  df-plusg 14587  df-mulr 14588  df-starv 14589  df-sca 14590  df-vsca 14591  df-ip 14592  df-tset 14593  df-ple 14594  df-ds 14596  df-unif 14597  df-hom 14598  df-cco 14599  df-rest 14697  df-topn 14698  df-0g 14716  df-gsum 14717  df-topgen 14718  df-pt 14719  df-prds 14722  df-xrs 14776  df-qtop 14781  df-imas 14782  df-xps 14784  df-mre 14860  df-mrc 14861  df-acs 14863  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15841  df-mulg 15934  df-cntz 16229  df-cmn 16674  df-psmet 18285  df-xmet 18286  df-met 18287  df-bl 18288  df-mopn 18289  df-fbas 18290  df-fg 18291  df-cnfld 18295  df-top 19272  df-bases 19274  df-topon 19275  df-topsp 19276  df-cld 19393  df-ntr 19394  df-cls 19395  df-nei 19472  df-lp 19510  df-perf 19511  df-cn 19601  df-cnp 19602  df-haus 19689  df-cmp 19760  df-tx 19936  df-hmeo 20129  df-fil 20220  df-fm 20312  df-flim 20313  df-flf 20314  df-xms 20696  df-ms 20697  df-tms 20698  df-cncf 21255  df-limc 22143  df-dv 22144  df-log 22816
This theorem is referenced by:  efopn  22911
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