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Theorem limcicciooub 37717
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 11720 . . 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 37602 . . 3  |-  ( ph  ->  ( A [,] B
)  C_  RR )
7 ax-resscn 9596 . . 3  |-  RR  C_  CC
86, 7syl6ss 3444 . 2  |-  ( ph  ->  ( A [,] B
)  C_  CC )
9 eqid 2451 . 2  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
10 eqid 2451 . 2  |-  ( (
TopOpen ` fld )t  ( ( A [,] B )  u.  { B } ) )  =  ( ( TopOpen ` fld )t  ( ( A [,] B )  u. 
{ B } ) )
11 retop 21782 . . . . . . . . 9  |-  ( topGen ` 
ran  (,) )  e.  Top
1211a1i 11 . . . . . . . 8  |-  ( ph  ->  ( topGen `  ran  (,) )  e.  Top )
134rexrd 9690 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  RR* )
14 iocssre 11714 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( A (,] B )  C_  RR )
1513, 5, 14syl2anc 667 . . . . . . . . . 10  |-  ( ph  ->  ( A (,] B
)  C_  RR )
16 difssd 3561 . . . . . . . . . 10  |-  ( ph  ->  ( RR  \  ( A [,] B ) ) 
C_  RR )
1715, 16unssd 3610 . . . . . . . . 9  |-  ( ph  ->  ( ( A (,] B )  u.  ( RR  \  ( A [,] B ) ) ) 
C_  RR )
18 uniretop 21783 . . . . . . . . 9  |-  RR  =  U. ( topGen `  ran  (,) )
1917, 18syl6sseq 3478 . . . . . . . 8  |-  ( ph  ->  ( ( A (,] B )  u.  ( RR  \  ( A [,] B ) ) ) 
C_  U. ( topGen `  ran  (,) ) )
20 elioore 11666 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( A (,) +oo )  ->  x  e.  RR )
2120ad2antlr 733 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( A (,) +oo ) )  /\  x  <_  B )  ->  x  e.  RR )
22 simplr 762 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( A (,) +oo ) )  /\  x  <_  B )  ->  x  e.  ( A (,) +oo ) )
2313ad2antrr 732 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  x  e.  ( A (,) +oo ) )  /\  x  <_  B )  ->  A  e.  RR* )
24 pnfxr 11412 . . . . . . . . . . . . . . . . 17  |- +oo  e.  RR*
25 elioo2 11677 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  RR*  /\ +oo  e.  RR* )  ->  (
x  e.  ( A (,) +oo )  <->  ( x  e.  RR  /\  A  < 
x  /\  x  < +oo ) ) )
2623, 24, 25sylancl 668 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( A (,) +oo ) )  /\  x  <_  B )  ->  (
x  e.  ( A (,) +oo )  <->  ( x  e.  RR  /\  A  < 
x  /\  x  < +oo ) ) )
2722, 26mpbid 214 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  ( A (,) +oo ) )  /\  x  <_  B )  ->  (
x  e.  RR  /\  A  <  x  /\  x  < +oo ) )
2827simp2d 1021 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( A (,) +oo ) )  /\  x  <_  B )  ->  A  <  x )
29 simpr 463 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( A (,) +oo ) )  /\  x  <_  B )  ->  x  <_  B )
305ad2antrr 732 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  ( A (,) +oo ) )  /\  x  <_  B )  ->  B  e.  RR )
31 elioc2 11697 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  (
x  e.  ( A (,] B )  <->  ( x  e.  RR  /\  A  < 
x  /\  x  <_  B ) ) )
3223, 30, 31syl2anc 667 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( A (,) +oo ) )  /\  x  <_  B )  ->  (
x  e.  ( A (,] B )  <->  ( x  e.  RR  /\  A  < 
x  /\  x  <_  B ) ) )
3321, 28, 29, 32mpbir3and 1191 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( A (,) +oo ) )  /\  x  <_  B )  ->  x  e.  ( A (,] B
) )
3433orcd 394 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( A (,) +oo ) )  /\  x  <_  B )  ->  (
x  e.  ( A (,] B )  \/  x  e.  ( RR 
\  ( A [,] B ) ) ) )
3520ad2antlr 733 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( A (,) +oo ) )  /\  -.  x  <_  B )  ->  x  e.  RR )
36 3mix3 1179 . . . . . . . . . . . . . . . . 17  |-  ( -.  x  <_  B  ->  ( -.  x  e.  RR  \/  -.  A  <_  x  \/  -.  x  <_  B
) )
37 3ianor 1002 . . . . . . . . . . . . . . . . 17  |-  ( -.  ( x  e.  RR  /\  A  <_  x  /\  x  <_  B )  <->  ( -.  x  e.  RR  \/  -.  A  <_  x  \/ 
-.  x  <_  B
) )
3836, 37sylibr 216 . . . . . . . . . . . . . . . 16  |-  ( -.  x  <_  B  ->  -.  ( x  e.  RR  /\  A  <_  x  /\  x  <_  B ) )
3938adantl 468 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  ( A (,) +oo ) )  /\  -.  x  <_  B )  ->  -.  ( x  e.  RR  /\  A  <_  x  /\  x  <_  B ) )
404ad2antrr 732 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( A (,) +oo ) )  /\  -.  x  <_  B )  ->  A  e.  RR )
415ad2antrr 732 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( A (,) +oo ) )  /\  -.  x  <_  B )  ->  B  e.  RR )
42 elicc2 11699 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( x  e.  ( A [,] B )  <-> 
( x  e.  RR  /\  A  <_  x  /\  x  <_  B ) ) )
4340, 41, 42syl2anc 667 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  ( A (,) +oo ) )  /\  -.  x  <_  B )  -> 
( x  e.  ( A [,] B )  <-> 
( x  e.  RR  /\  A  <_  x  /\  x  <_  B ) ) )
4439, 43mtbird 303 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( A (,) +oo ) )  /\  -.  x  <_  B )  ->  -.  x  e.  ( A [,] B ) )
4535, 44eldifd 3415 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( A (,) +oo ) )  /\  -.  x  <_  B )  ->  x  e.  ( RR  \  ( A [,] B
) ) )
4645olcd 395 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( A (,) +oo ) )  /\  -.  x  <_  B )  -> 
( x  e.  ( A (,] B )  \/  x  e.  ( RR  \  ( A [,] B ) ) ) )
4734, 46pm2.61dan 800 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A (,) +oo )
)  ->  ( x  e.  ( A (,] B
)  \/  x  e.  ( RR  \  ( A [,] B ) ) ) )
48 elun 3574 . . . . . . . . . . 11  |-  ( x  e.  ( ( A (,] B )  u.  ( RR  \  ( A [,] B ) ) )  <->  ( x  e.  ( A (,] B
)  \/  x  e.  ( RR  \  ( A [,] B ) ) ) )
4947, 48sylibr 216 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A (,) +oo )
)  ->  x  e.  ( ( A (,] B )  u.  ( RR  \  ( A [,] B ) ) ) )
5049ralrimiva 2802 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  ( A (,) +oo )
x  e.  ( ( A (,] B )  u.  ( RR  \ 
( A [,] B
) ) ) )
51 dfss3 3422 . . . . . . . . 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 216 . . . . . . . 8  |-  ( ph  ->  ( A (,) +oo )  C_  ( ( A (,] B )  u.  ( RR  \  ( A [,] B ) ) ) )
53 eqid 2451 . . . . . . . . 9  |-  U. ( topGen `
 ran  (,) )  =  U. ( topGen `  ran  (,) )
5453ntrss 20070 . . . . . . . 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 1268 . . . . . . 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 11423 . . . . . . . . 9  |-  ( ph  ->  B  < +oo )
5913, 56, 5, 57, 58eliood 37595 . . . . . . . 8  |-  ( ph  ->  B  e.  ( A (,) +oo ) )
60 iooretop 21786 . . . . . . . . 9  |-  ( A (,) +oo )  e.  ( topGen `  ran  (,) )
61 isopn3i 20098 . . . . . . . . 9  |-  ( ( ( topGen `  ran  (,) )  e.  Top  /\  ( A (,) +oo )  e.  ( topGen `  ran  (,) )
)  ->  ( ( int `  ( topGen `  ran  (,) ) ) `  ( A (,) +oo ) )  =  ( A (,) +oo ) )
6212, 60, 61sylancl 668 . . . . . . . 8  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A (,) +oo ) )  =  ( A (,) +oo ) )
6359, 62eleqtrrd 2532 . . . . . . 7  |-  ( ph  ->  B  e.  ( ( int `  ( topGen ` 
ran  (,) ) ) `  ( A (,) +oo )
) )
6455, 63sseldd 3433 . . . . . 6  |-  ( ph  ->  B  e.  ( ( int `  ( topGen ` 
ran  (,) ) ) `  ( ( A (,] B )  u.  ( RR  \  ( A [,] B ) ) ) ) )
655rexrd 9690 . . . . . . 7  |-  ( ph  ->  B  e.  RR* )
664, 5, 57ltled 9783 . . . . . . 7  |-  ( ph  ->  A  <_  B )
67 ubicc2 11749 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
6813, 65, 66, 67syl3anc 1268 . . . . . 6  |-  ( ph  ->  B  e.  ( A [,] B ) )
6964, 68elind 3618 . . . . 5  |-  ( ph  ->  B  e.  ( ( ( int `  ( topGen `
 ran  (,) )
) `  ( ( A (,] B )  u.  ( RR  \  ( A [,] B ) ) ) )  i^i  ( A [,] B ) ) )
70 iocssicc 11722 . . . . . . 7  |-  ( A (,] B )  C_  ( A [,] B )
7170a1i 11 . . . . . 6  |-  ( ph  ->  ( A (,] B
)  C_  ( A [,] B ) )
72 eqid 2451 . . . . . . 7  |-  ( (
topGen `  ran  (,) )t  ( A [,] B ) )  =  ( ( topGen ` 
ran  (,) )t  ( A [,] B ) )
7318, 72restntr 20198 . . . . . 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 1268 . . . . 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 2532 . . . 4  |-  ( ph  ->  B  e.  ( ( int `  ( (
topGen `  ran  (,) )t  ( A [,] B ) ) ) `  ( A (,] B ) ) )
76 eqid 2451 . . . . . . . . 9  |-  ( topGen ` 
ran  (,) )  =  (
topGen `  ran  (,) )
779, 76rerest 21822 . . . . . . . 8  |-  ( ( A [,] B ) 
C_  RR  ->  ( (
TopOpen ` fld )t  ( A [,] B
) )  =  ( ( topGen `  ran  (,) )t  ( A [,] B ) ) )
786, 77syl 17 . . . . . . 7  |-  ( ph  ->  ( ( TopOpen ` fld )t  ( A [,] B ) )  =  ( ( topGen `  ran  (,) )t  ( A [,] B
) ) )
7978eqcomd 2457 . . . . . 6  |-  ( ph  ->  ( ( topGen `  ran  (,) )t  ( A [,] B
) )  =  ( ( TopOpen ` fld )t  ( A [,] B ) ) )
8079fveq2d 5869 . . . . 5  |-  ( ph  ->  ( int `  (
( topGen `  ran  (,) )t  ( A [,] B ) ) )  =  ( int `  ( ( TopOpen ` fld )t  ( A [,] B ) ) ) )
8180fveq1d 5867 . . . 4  |-  ( ph  ->  ( ( int `  (
( topGen `  ran  (,) )t  ( A [,] B ) ) ) `  ( A (,] B ) )  =  ( ( int `  ( ( TopOpen ` fld )t  ( A [,] B ) ) ) `
 ( A (,] B ) ) )
8275, 81eleqtrd 2531 . . 3  |-  ( ph  ->  B  e.  ( ( int `  ( (
TopOpen ` fld )t  ( A [,] B
) ) ) `  ( A (,] B ) ) )
8368snssd 4117 . . . . . . . 8  |-  ( ph  ->  { B }  C_  ( A [,] B ) )
84 ssequn2 3607 . . . . . . . 8  |-  ( { B }  C_  ( A [,] B )  <->  ( ( A [,] B )  u. 
{ B } )  =  ( A [,] B ) )
8583, 84sylib 200 . . . . . . 7  |-  ( ph  ->  ( ( A [,] B )  u.  { B } )  =  ( A [,] B ) )
8685eqcomd 2457 . . . . . 6  |-  ( ph  ->  ( A [,] B
)  =  ( ( A [,] B )  u.  { B }
) )
8786oveq2d 6306 . . . . 5  |-  ( ph  ->  ( ( TopOpen ` fld )t  ( A [,] B ) )  =  ( ( TopOpen ` fld )t  ( ( A [,] B )  u. 
{ B } ) ) )
8887fveq2d 5869 . . . 4  |-  ( ph  ->  ( int `  (
( TopOpen ` fld )t  ( A [,] B ) ) )  =  ( int `  (
( TopOpen ` fld )t  ( ( A [,] B )  u. 
{ B } ) ) ) )
89 snunioo2 37606 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( A (,) B
)  u.  { B } )  =  ( A (,] B ) )
9013, 65, 57, 89syl3anc 1268 . . . . 5  |-  ( ph  ->  ( ( A (,) B )  u.  { B } )  =  ( A (,] B ) )
9190eqcomd 2457 . . . 4  |-  ( ph  ->  ( A (,] B
)  =  ( ( A (,) B )  u.  { B }
) )
9288, 91fveq12d 5871 . . 3  |-  ( ph  ->  ( ( int `  (
( TopOpen ` fld )t  ( A [,] B ) ) ) `
 ( A (,] B ) )  =  ( ( int `  (
( TopOpen ` fld )t  ( ( A [,] B )  u. 
{ B } ) ) ) `  (
( A (,) B
)  u.  { B } ) ) )
9382, 92eleqtrd 2531 . 2  |-  ( ph  ->  B  e.  ( ( int `  ( (
TopOpen ` fld )t  ( ( A [,] B )  u.  { B } ) ) ) `
 ( ( A (,) B )  u. 
{ B } ) ) )
941, 3, 8, 9, 10, 93limcres 22841 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 188    \/ wo 370    /\ wa 371    \/ w3o 984    /\ w3a 985    = wceq 1444    e. wcel 1887   A.wral 2737    \ cdif 3401    u. cun 3402    i^i cin 3403    C_ wss 3404   {csn 3968   U.cuni 4198   class class class wbr 4402   ran crn 4835    |` cres 4836   -->wf 5578   ` cfv 5582  (class class class)co 6290   CCcc 9537   RRcr 9538   +oocpnf 9672   RR*cxr 9674    < clt 9675    <_ cle 9676   (,)cioo 11635   (,]cioc 11636   [,]cicc 11638   ↾t crest 15319   TopOpenctopn 15320   topGenctg 15336  ℂfldccnfld 18970   Topctop 19917   intcnt 20032   lim CC climc 22817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fi 7925  df-sup 7956  df-inf 7957  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ioc 11640  df-icc 11642  df-fz 11785  df-seq 12214  df-exp 12273  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-plusg 15203  df-mulr 15204  df-starv 15205  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-rest 15321  df-topn 15322  df-topgen 15342  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-cnfld 18971  df-top 19921  df-bases 19922  df-topon 19923  df-topsp 19924  df-cld 20034  df-ntr 20035  df-cls 20036  df-cnp 20244  df-xms 21335  df-ms 21336  df-limc 22821
This theorem is referenced by:  cncfiooicclem1  37771  fourierdlem82  38052  fourierdlem93  38063  fourierdlem111  38081
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