Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  limcicciooub Structured version   Unicode version

Theorem limcicciooub 31889
Description: The limit of a function at the upper bound of a closed interval only depends on the values in the inner open interval (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
limcicciooub.1  |-  ( ph  ->  A  e.  RR )
limcicciooub.2  |-  ( ph  ->  B  e.  RR )
limcicciooub.3  |-  ( ph  ->  A  <  B )
limcicciooub.4  |-  ( ph  ->  F : ( A [,] B ) --> CC )
Assertion
Ref Expression
limcicciooub  |-  ( ph  ->  ( ( F  |`  ( A (,) B ) ) lim CC  B )  =  ( F lim CC  B ) )

Proof of Theorem limcicciooub
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 limcicciooub.4 . 2  |-  ( ph  ->  F : ( A [,] B ) --> CC )
2 ioossicc 11635 . . 3  |-  ( A (,) B )  C_  ( A [,] B )
32a1i 11 . 2  |-  ( ph  ->  ( A (,) B
)  C_  ( A [,] B ) )
4 limcicciooub.1 . . . 4  |-  ( ph  ->  A  e.  RR )
5 limcicciooub.2 . . . 4  |-  ( ph  ->  B  e.  RR )
64, 5iccssred 31785 . . 3  |-  ( ph  ->  ( A [,] B
)  C_  RR )
7 ax-resscn 9566 . . 3  |-  RR  C_  CC
86, 7syl6ss 3511 . 2  |-  ( ph  ->  ( A [,] B
)  C_  CC )
9 eqid 2457 . 2  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
10 eqid 2457 . 2  |-  ( (
TopOpen ` fld )t  ( ( A [,] B )  u.  { B } ) )  =  ( ( TopOpen ` fld )t  ( ( A [,] B )  u. 
{ B } ) )
11 retop 21485 . . . . . . . . 9  |-  ( topGen ` 
ran  (,) )  e.  Top
1211a1i 11 . . . . . . . 8  |-  ( ph  ->  ( topGen `  ran  (,) )  e.  Top )
134rexrd 9660 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  RR* )
14 iocssre 11629 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( A (,] B )  C_  RR )
1513, 5, 14syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  ( A (,] B
)  C_  RR )
16 difssd 3628 . . . . . . . . . 10  |-  ( ph  ->  ( RR  \  ( A [,] B ) ) 
C_  RR )
1715, 16unssd 3676 . . . . . . . . 9  |-  ( ph  ->  ( ( A (,] B )  u.  ( RR  \  ( A [,] B ) ) ) 
C_  RR )
18 uniretop 21486 . . . . . . . . 9  |-  RR  =  U. ( topGen `  ran  (,) )
1917, 18syl6sseq 3545 . . . . . . . 8  |-  ( ph  ->  ( ( A (,] B )  u.  ( RR  \  ( A [,] B ) ) ) 
C_  U. ( topGen `  ran  (,) ) )
20 elioore 11584 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( A (,) +oo )  ->  x  e.  RR )
2120ad2antlr 726 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( A (,) +oo ) )  /\  x  <_  B )  ->  x  e.  RR )
22 simplr 755 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( A (,) +oo ) )  /\  x  <_  B )  ->  x  e.  ( A (,) +oo ) )
2313ad2antrr 725 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  x  e.  ( A (,) +oo ) )  /\  x  <_  B )  ->  A  e.  RR* )
24 pnfxr 11346 . . . . . . . . . . . . . . . . 17  |- +oo  e.  RR*
25 elioo2 11595 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  RR*  /\ +oo  e.  RR* )  ->  (
x  e.  ( A (,) +oo )  <->  ( x  e.  RR  /\  A  < 
x  /\  x  < +oo ) ) )
2623, 24, 25sylancl 662 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( A (,) +oo ) )  /\  x  <_  B )  ->  (
x  e.  ( A (,) +oo )  <->  ( x  e.  RR  /\  A  < 
x  /\  x  < +oo ) ) )
2722, 26mpbid 210 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  ( A (,) +oo ) )  /\  x  <_  B )  ->  (
x  e.  RR  /\  A  <  x  /\  x  < +oo ) )
2827simp2d 1009 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( A (,) +oo ) )  /\  x  <_  B )  ->  A  <  x )
29 simpr 461 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( A (,) +oo ) )  /\  x  <_  B )  ->  x  <_  B )
305ad2antrr 725 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  ( A (,) +oo ) )  /\  x  <_  B )  ->  B  e.  RR )
31 elioc2 11612 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  (
x  e.  ( A (,] B )  <->  ( x  e.  RR  /\  A  < 
x  /\  x  <_  B ) ) )
3223, 30, 31syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( A (,) +oo ) )  /\  x  <_  B )  ->  (
x  e.  ( A (,] B )  <->  ( x  e.  RR  /\  A  < 
x  /\  x  <_  B ) ) )
3321, 28, 29, 32mpbir3and 1179 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( A (,) +oo ) )  /\  x  <_  B )  ->  x  e.  ( A (,] B
) )
3433orcd 392 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( A (,) +oo ) )  /\  x  <_  B )  ->  (
x  e.  ( A (,] B )  \/  x  e.  ( RR 
\  ( A [,] B ) ) ) )
3520ad2antlr 726 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( A (,) +oo ) )  /\  -.  x  <_  B )  ->  x  e.  RR )
36 3mix3 1167 . . . . . . . . . . . . . . . . 17  |-  ( -.  x  <_  B  ->  ( -.  x  e.  RR  \/  -.  A  <_  x  \/  -.  x  <_  B
) )
37 3ianor 990 . . . . . . . . . . . . . . . . 17  |-  ( -.  ( x  e.  RR  /\  A  <_  x  /\  x  <_  B )  <->  ( -.  x  e.  RR  \/  -.  A  <_  x  \/ 
-.  x  <_  B
) )
3836, 37sylibr 212 . . . . . . . . . . . . . . . 16  |-  ( -.  x  <_  B  ->  -.  ( x  e.  RR  /\  A  <_  x  /\  x  <_  B ) )
3938adantl 466 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  ( A (,) +oo ) )  /\  -.  x  <_  B )  ->  -.  ( x  e.  RR  /\  A  <_  x  /\  x  <_  B ) )
404ad2antrr 725 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( A (,) +oo ) )  /\  -.  x  <_  B )  ->  A  e.  RR )
415ad2antrr 725 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( A (,) +oo ) )  /\  -.  x  <_  B )  ->  B  e.  RR )
42 elicc2 11614 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( x  e.  ( A [,] B )  <-> 
( x  e.  RR  /\  A  <_  x  /\  x  <_  B ) ) )
4340, 41, 42syl2anc 661 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  ( A (,) +oo ) )  /\  -.  x  <_  B )  -> 
( x  e.  ( A [,] B )  <-> 
( x  e.  RR  /\  A  <_  x  /\  x  <_  B ) ) )
4439, 43mtbird 301 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( A (,) +oo ) )  /\  -.  x  <_  B )  ->  -.  x  e.  ( A [,] B ) )
4535, 44eldifd 3482 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( A (,) +oo ) )  /\  -.  x  <_  B )  ->  x  e.  ( RR  \  ( A [,] B
) ) )
4645olcd 393 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( A (,) +oo ) )  /\  -.  x  <_  B )  -> 
( x  e.  ( A (,] B )  \/  x  e.  ( RR  \  ( A [,] B ) ) ) )
4734, 46pm2.61dan 791 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A (,) +oo )
)  ->  ( x  e.  ( A (,] B
)  \/  x  e.  ( RR  \  ( A [,] B ) ) ) )
48 elun 3641 . . . . . . . . . . 11  |-  ( x  e.  ( ( A (,] B )  u.  ( RR  \  ( A [,] B ) ) )  <->  ( x  e.  ( A (,] B
)  \/  x  e.  ( RR  \  ( A [,] B ) ) ) )
4947, 48sylibr 212 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A (,) +oo )
)  ->  x  e.  ( ( A (,] B )  u.  ( RR  \  ( A [,] B ) ) ) )
5049ralrimiva 2871 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  ( A (,) +oo )
x  e.  ( ( A (,] B )  u.  ( RR  \ 
( A [,] B
) ) ) )
51 dfss3 3489 . . . . . . . . 9  |-  ( ( A (,) +oo )  C_  ( ( A (,] B )  u.  ( RR  \  ( A [,] B ) ) )  <->  A. x  e.  ( A (,) +oo ) x  e.  ( ( A (,] B )  u.  ( RR  \  ( A [,] B ) ) ) )
5250, 51sylibr 212 . . . . . . . 8  |-  ( ph  ->  ( A (,) +oo )  C_  ( ( A (,] B )  u.  ( RR  \  ( A [,] B ) ) ) )
53 eqid 2457 . . . . . . . . 9  |-  U. ( topGen `
 ran  (,) )  =  U. ( topGen `  ran  (,) )
5453ntrss 19774 . . . . . . . 8  |-  ( ( ( topGen `  ran  (,) )  e.  Top  /\  ( ( A (,] B )  u.  ( RR  \ 
( A [,] B
) ) )  C_  U. ( topGen `  ran  (,) )  /\  ( A (,) +oo )  C_  ( ( A (,] B )  u.  ( RR  \  ( A [,] B ) ) ) )  ->  (
( int `  ( topGen `
 ran  (,) )
) `  ( A (,) +oo ) )  C_  ( ( int `  ( topGen `
 ran  (,) )
) `  ( ( A (,] B )  u.  ( RR  \  ( A [,] B ) ) ) ) )
5512, 19, 52, 54syl3anc 1228 . . . . . . 7  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A (,) +oo ) )  C_  ( ( int `  ( topGen `
 ran  (,) )
) `  ( ( A (,] B )  u.  ( RR  \  ( A [,] B ) ) ) ) )
5624a1i 11 . . . . . . . . 9  |-  ( ph  -> +oo  e.  RR* )
57 limcicciooub.3 . . . . . . . . 9  |-  ( ph  ->  A  <  B )
585ltpnfd 31726 . . . . . . . . 9  |-  ( ph  ->  B  < +oo )
5913, 56, 5, 57, 58eliood 31777 . . . . . . . 8  |-  ( ph  ->  B  e.  ( A (,) +oo ) )
60 iooretop 21490 . . . . . . . . 9  |-  ( A (,) +oo )  e.  ( topGen `  ran  (,) )
61 isopn3i 19801 . . . . . . . . 9  |-  ( ( ( topGen `  ran  (,) )  e.  Top  /\  ( A (,) +oo )  e.  ( topGen `  ran  (,) )
)  ->  ( ( int `  ( topGen `  ran  (,) ) ) `  ( A (,) +oo ) )  =  ( A (,) +oo ) )
6212, 60, 61sylancl 662 . . . . . . . 8  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A (,) +oo ) )  =  ( A (,) +oo ) )
6359, 62eleqtrrd 2548 . . . . . . 7  |-  ( ph  ->  B  e.  ( ( int `  ( topGen ` 
ran  (,) ) ) `  ( A (,) +oo )
) )
6455, 63sseldd 3500 . . . . . 6  |-  ( ph  ->  B  e.  ( ( int `  ( topGen ` 
ran  (,) ) ) `  ( ( A (,] B )  u.  ( RR  \  ( A [,] B ) ) ) ) )
655rexrd 9660 . . . . . . 7  |-  ( ph  ->  B  e.  RR* )
664, 5, 57ltled 9750 . . . . . . 7  |-  ( ph  ->  A  <_  B )
67 ubicc2 11662 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
6813, 65, 66, 67syl3anc 1228 . . . . . 6  |-  ( ph  ->  B  e.  ( A [,] B ) )
6964, 68elind 3684 . . . . 5  |-  ( ph  ->  B  e.  ( ( ( int `  ( topGen `
 ran  (,) )
) `  ( ( A (,] B )  u.  ( RR  \  ( A [,] B ) ) ) )  i^i  ( A [,] B ) ) )
70 iocssicc 11637 . . . . . . 7  |-  ( A (,] B )  C_  ( A [,] B )
7170a1i 11 . . . . . 6  |-  ( ph  ->  ( A (,] B
)  C_  ( A [,] B ) )
72 eqid 2457 . . . . . . 7  |-  ( (
topGen `  ran  (,) )t  ( A [,] B ) )  =  ( ( topGen ` 
ran  (,) )t  ( A [,] B ) )
7318, 72restntr 19901 . . . . . 6  |-  ( ( ( topGen `  ran  (,) )  e.  Top  /\  ( A [,] B )  C_  RR  /\  ( A (,] B )  C_  ( A [,] B ) )  ->  ( ( int `  ( ( topGen `  ran  (,) )t  ( A [,] B
) ) ) `  ( A (,] B ) )  =  ( ( ( int `  ( topGen `
 ran  (,) )
) `  ( ( A (,] B )  u.  ( RR  \  ( A [,] B ) ) ) )  i^i  ( A [,] B ) ) )
7412, 6, 71, 73syl3anc 1228 . . . . 5  |-  ( ph  ->  ( ( int `  (
( topGen `  ran  (,) )t  ( A [,] B ) ) ) `  ( A (,] B ) )  =  ( ( ( int `  ( topGen ` 
ran  (,) ) ) `  ( ( A (,] B )  u.  ( RR  \  ( A [,] B ) ) ) )  i^i  ( A [,] B ) ) )
7569, 74eleqtrrd 2548 . . . 4  |-  ( ph  ->  B  e.  ( ( int `  ( (
topGen `  ran  (,) )t  ( A [,] B ) ) ) `  ( A (,] B ) ) )
76 eqid 2457 . . . . . . . . 9  |-  ( topGen ` 
ran  (,) )  =  (
topGen `  ran  (,) )
779, 76rerest 21526 . . . . . . . 8  |-  ( ( A [,] B ) 
C_  RR  ->  ( (
TopOpen ` fld )t  ( A [,] B
) )  =  ( ( topGen `  ran  (,) )t  ( A [,] B ) ) )
786, 77syl 16 . . . . . . 7  |-  ( ph  ->  ( ( TopOpen ` fld )t  ( A [,] B ) )  =  ( ( topGen `  ran  (,) )t  ( A [,] B
) ) )
7978eqcomd 2465 . . . . . 6  |-  ( ph  ->  ( ( topGen `  ran  (,) )t  ( A [,] B
) )  =  ( ( TopOpen ` fld )t  ( A [,] B ) ) )
8079fveq2d 5876 . . . . 5  |-  ( ph  ->  ( int `  (
( topGen `  ran  (,) )t  ( A [,] B ) ) )  =  ( int `  ( ( TopOpen ` fld )t  ( A [,] B ) ) ) )
8180fveq1d 5874 . . . 4  |-  ( ph  ->  ( ( int `  (
( topGen `  ran  (,) )t  ( A [,] B ) ) ) `  ( A (,] B ) )  =  ( ( int `  ( ( TopOpen ` fld )t  ( A [,] B ) ) ) `
 ( A (,] B ) ) )
8275, 81eleqtrd 2547 . . 3  |-  ( ph  ->  B  e.  ( ( int `  ( (
TopOpen ` fld )t  ( A [,] B
) ) ) `  ( A (,] B ) ) )
8368snssd 4177 . . . . . . . 8  |-  ( ph  ->  { B }  C_  ( A [,] B ) )
84 ssequn2 3673 . . . . . . . 8  |-  ( { B }  C_  ( A [,] B )  <->  ( ( A [,] B )  u. 
{ B } )  =  ( A [,] B ) )
8583, 84sylib 196 . . . . . . 7  |-  ( ph  ->  ( ( A [,] B )  u.  { B } )  =  ( A [,] B ) )
8685eqcomd 2465 . . . . . 6  |-  ( ph  ->  ( A [,] B
)  =  ( ( A [,] B )  u.  { B }
) )
8786oveq2d 6312 . . . . 5  |-  ( ph  ->  ( ( TopOpen ` fld )t  ( A [,] B ) )  =  ( ( TopOpen ` fld )t  ( ( A [,] B )  u. 
{ B } ) ) )
8887fveq2d 5876 . . . 4  |-  ( ph  ->  ( int `  (
( TopOpen ` fld )t  ( A [,] B ) ) )  =  ( int `  (
( TopOpen ` fld )t  ( ( A [,] B )  u. 
{ B } ) ) ) )
89 snunioo2 31790 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( A (,) B
)  u.  { B } )  =  ( A (,] B ) )
9013, 65, 57, 89syl3anc 1228 . . . . 5  |-  ( ph  ->  ( ( A (,) B )  u.  { B } )  =  ( A (,] B ) )
9190eqcomd 2465 . . . 4  |-  ( ph  ->  ( A (,] B
)  =  ( ( A (,) B )  u.  { B }
) )
9288, 91fveq12d 5878 . . 3  |-  ( ph  ->  ( ( int `  (
( TopOpen ` fld )t  ( A [,] B ) ) ) `
 ( A (,] B ) )  =  ( ( int `  (
( TopOpen ` fld )t  ( ( A [,] B )  u. 
{ B } ) ) ) `  (
( A (,) B
)  u.  { B } ) ) )
9382, 92eleqtrd 2547 . 2  |-  ( ph  ->  B  e.  ( ( int `  ( (
TopOpen ` fld )t  ( ( A [,] B )  u.  { B } ) ) ) `
 ( ( A (,) B )  u. 
{ B } ) ) )
941, 3, 8, 9, 10, 93limcres 22507 1  |-  ( ph  ->  ( ( F  |`  ( A (,) B ) ) lim CC  B )  =  ( F lim CC  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    \/ w3o 972    /\ w3a 973    = wceq 1395    e. wcel 1819   A.wral 2807    \ cdif 3468    u. cun 3469    i^i cin 3470    C_ wss 3471   {csn 4032   U.cuni 4251   class class class wbr 4456   ran crn 5009    |` cres 5010   -->wf 5590   ` cfv 5594  (class class class)co 6296   CCcc 9507   RRcr 9508   +oocpnf 9642   RR*cxr 9644    < clt 9645    <_ cle 9646   (,)cioo 11554   (,]cioc 11555   [,]cicc 11557   ↾t crest 14929   TopOpenctopn 14930   topGenctg 14946  ℂfldccnfld 18638   Topctop 19612   intcnt 19736   lim CC climc 22483
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fi 7889  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ioo 11558  df-ioc 11559  df-icc 11561  df-fz 11698  df-seq 12111  df-exp 12170  df-cj 13035  df-re 13036  df-im 13037  df-sqrt 13171  df-abs 13172  df-struct 14737  df-ndx 14738  df-slot 14739  df-base 14740  df-plusg 14816  df-mulr 14817  df-starv 14818  df-tset 14822  df-ple 14823  df-ds 14825  df-unif 14826  df-rest 14931  df-topn 14932  df-topgen 14952  df-psmet 18629  df-xmet 18630  df-met 18631  df-bl 18632  df-mopn 18633  df-cnfld 18639  df-top 19617  df-bases 19619  df-topon 19620  df-topsp 19621  df-cld 19738  df-ntr 19739  df-cls 19740  df-cnp 19947  df-xms 21040  df-ms 21041  df-limc 22487
This theorem is referenced by:  cncfiooicclem1  31942  fourierdlem82  32217  fourierdlem93  32228  fourierdlem111  32246
  Copyright terms: Public domain W3C validator