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Theorem limciccioolb 37798
Description: The limit of a function at the lower 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
limciccioolb.1  |-  ( ph  ->  A  e.  RR )
limciccioolb.2  |-  ( ph  ->  B  e.  RR )
limciccioolb.3  |-  ( ph  ->  A  <  B )
limciccioolb.4  |-  ( ph  ->  F : ( A [,] B ) --> CC )
Assertion
Ref Expression
limciccioolb  |-  ( ph  ->  ( ( F  |`  ( A (,) B ) ) lim CC  A )  =  ( F lim CC  A ) )

Proof of Theorem limciccioolb
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 limciccioolb.4 . 2  |-  ( ph  ->  F : ( A [,] B ) --> CC )
2 ioossicc 11745 . . 3  |-  ( A (,) B )  C_  ( A [,] B )
32a1i 11 . 2  |-  ( ph  ->  ( A (,) B
)  C_  ( A [,] B ) )
4 limciccioolb.1 . . . 4  |-  ( ph  ->  A  e.  RR )
5 limciccioolb.2 . . . 4  |-  ( ph  ->  B  e.  RR )
64, 5iccssred 37698 . . 3  |-  ( ph  ->  ( A [,] B
)  C_  RR )
7 ax-resscn 9614 . . 3  |-  RR  C_  CC
86, 7syl6ss 3430 . 2  |-  ( ph  ->  ( A [,] B
)  C_  CC )
9 eqid 2471 . 2  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
10 eqid 2471 . 2  |-  ( (
TopOpen ` fld )t  ( ( A [,] B )  u.  { A } ) )  =  ( ( TopOpen ` fld )t  ( ( A [,] B )  u. 
{ A } ) )
11 retop 21860 . . . . . . . . 9  |-  ( topGen ` 
ran  (,) )  e.  Top
1211a1i 11 . . . . . . . 8  |-  ( ph  ->  ( topGen `  ran  (,) )  e.  Top )
135rexrd 9708 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  RR* )
14 icossre 11740 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( A [,) B
)  C_  RR )
154, 13, 14syl2anc 673 . . . . . . . . . 10  |-  ( ph  ->  ( A [,) B
)  C_  RR )
16 difssd 3550 . . . . . . . . . 10  |-  ( ph  ->  ( RR  \  ( A [,] B ) ) 
C_  RR )
1715, 16unssd 3601 . . . . . . . . 9  |-  ( ph  ->  ( ( A [,) B )  u.  ( RR  \  ( A [,] B ) ) ) 
C_  RR )
18 uniretop 21861 . . . . . . . . 9  |-  RR  =  U. ( topGen `  ran  (,) )
1917, 18syl6sseq 3464 . . . . . . . 8  |-  ( ph  ->  ( ( A [,) B )  u.  ( RR  \  ( A [,] B ) ) ) 
C_  U. ( topGen `  ran  (,) ) )
20 elioore 11691 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( -oo (,) B )  ->  x  e.  RR )
2120ad2antlr 741 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  A  <_  x )  ->  x  e.  RR )
22 simpr 468 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  A  <_  x )  ->  A  <_  x )
23 simpr 468 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  ( -oo (,) B ) )  ->  x  e.  ( -oo (,) B ) )
24 mnfxr 11437 . . . . . . . . . . . . . . . . . . 19  |- -oo  e.  RR*
2524a1i 11 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  x  e.  ( -oo (,) B ) )  -> -oo  e.  RR* )
2613adantr 472 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  x  e.  ( -oo (,) B ) )  ->  B  e.  RR* )
27 elioo2 11702 . . . . . . . . . . . . . . . . . 18  |-  ( ( -oo  e.  RR*  /\  B  e.  RR* )  ->  (
x  e.  ( -oo (,) B )  <->  ( x  e.  RR  /\ -oo  <  x  /\  x  <  B
) ) )
2825, 26, 27syl2anc 673 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  ( -oo (,) B ) )  ->  ( x  e.  ( -oo (,) B
)  <->  ( x  e.  RR  /\ -oo  <  x  /\  x  <  B
) ) )
2923, 28mpbid 215 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( -oo (,) B ) )  ->  ( x  e.  RR  /\ -oo  <  x  /\  x  <  B
) )
3029simp3d 1044 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( -oo (,) B ) )  ->  x  <  B )
3130adantr 472 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  A  <_  x )  ->  x  <  B )
324ad2antrr 740 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  A  <_  x )  ->  A  e.  RR )
3313ad2antrr 740 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  A  <_  x )  ->  B  e.  RR* )
34 elico2 11723 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( x  e.  ( A [,) B )  <-> 
( x  e.  RR  /\  A  <_  x  /\  x  <  B ) ) )
3532, 33, 34syl2anc 673 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  A  <_  x )  ->  (
x  e.  ( A [,) B )  <->  ( x  e.  RR  /\  A  <_  x  /\  x  <  B
) ) )
3621, 22, 31, 35mpbir3and 1213 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  A  <_  x )  ->  x  e.  ( A [,) B
) )
3736orcd 399 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  A  <_  x )  ->  (
x  e.  ( A [,) B )  \/  x  e.  ( RR 
\  ( A [,] B ) ) ) )
3820ad2antlr 741 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  -.  A  <_  x )  ->  x  e.  RR )
39 simpr 468 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  -.  A  <_  x )  ->  -.  A  <_  x )
4039intnanrd 931 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  -.  A  <_  x )  ->  -.  ( A  <_  x  /\  x  <_  B ) )
414rexrd 9708 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  A  e.  RR* )
4241ad2antrr 740 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  -.  A  <_  x )  ->  A  e.  RR* )
4313ad2antrr 740 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  -.  A  <_  x )  ->  B  e.  RR* )
4438rexrd 9708 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  -.  A  <_  x )  ->  x  e.  RR* )
45 elicc4 11726 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  x  e. 
RR* )  ->  (
x  e.  ( A [,] B )  <->  ( A  <_  x  /\  x  <_  B ) ) )
4642, 43, 44, 45syl3anc 1292 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  -.  A  <_  x )  -> 
( x  e.  ( A [,] B )  <-> 
( A  <_  x  /\  x  <_  B ) ) )
4740, 46mtbird 308 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  -.  A  <_  x )  ->  -.  x  e.  ( A [,] B ) )
4838, 47eldifd 3401 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  -.  A  <_  x )  ->  x  e.  ( RR  \  ( A [,] B
) ) )
4948olcd 400 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  -.  A  <_  x )  -> 
( x  e.  ( A [,) B )  \/  x  e.  ( RR  \  ( A [,] B ) ) ) )
5037, 49pm2.61dan 808 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( -oo (,) B ) )  ->  ( x  e.  ( A [,) B
)  \/  x  e.  ( RR  \  ( A [,] B ) ) ) )
51 elun 3565 . . . . . . . . . . 11  |-  ( x  e.  ( ( A [,) B )  u.  ( RR  \  ( A [,] B ) ) )  <->  ( x  e.  ( A [,) B
)  \/  x  e.  ( RR  \  ( A [,] B ) ) ) )
5250, 51sylibr 217 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( -oo (,) B ) )  ->  x  e.  ( ( A [,) B )  u.  ( RR  \  ( A [,] B ) ) ) )
5352ralrimiva 2809 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  ( -oo (,) B ) x  e.  ( ( A [,) B )  u.  ( RR  \ 
( A [,] B
) ) ) )
54 dfss3 3408 . . . . . . . . 9  |-  ( ( -oo (,) B ) 
C_  ( ( A [,) B )  u.  ( RR  \  ( A [,] B ) ) )  <->  A. x  e.  ( -oo (,) B ) x  e.  ( ( A [,) B )  u.  ( RR  \ 
( A [,] B
) ) ) )
5553, 54sylibr 217 . . . . . . . 8  |-  ( ph  ->  ( -oo (,) B
)  C_  ( ( A [,) B )  u.  ( RR  \  ( A [,] B ) ) ) )
56 eqid 2471 . . . . . . . . 9  |-  U. ( topGen `
 ran  (,) )  =  U. ( topGen `  ran  (,) )
5756ntrss 20147 . . . . . . . 8  |-  ( ( ( topGen `  ran  (,) )  e.  Top  /\  ( ( A [,) B )  u.  ( RR  \ 
( A [,] B
) ) )  C_  U. ( topGen `  ran  (,) )  /\  ( -oo (,) B
)  C_  ( ( A [,) B )  u.  ( RR  \  ( A [,] B ) ) ) )  ->  (
( int `  ( topGen `
 ran  (,) )
) `  ( -oo (,) B ) )  C_  ( ( int `  ( topGen `
 ran  (,) )
) `  ( ( A [,) B )  u.  ( RR  \  ( A [,] B ) ) ) ) )
5812, 19, 55, 57syl3anc 1292 . . . . . . 7  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( -oo (,) B ) )  C_  ( ( int `  ( topGen `
 ran  (,) )
) `  ( ( A [,) B )  u.  ( RR  \  ( A [,] B ) ) ) ) )
5924a1i 11 . . . . . . . . 9  |-  ( ph  -> -oo  e.  RR* )
604mnfltd 11449 . . . . . . . . 9  |-  ( ph  -> -oo  <  A )
61 limciccioolb.3 . . . . . . . . 9  |-  ( ph  ->  A  <  B )
6259, 13, 4, 60, 61eliood 37691 . . . . . . . 8  |-  ( ph  ->  A  e.  ( -oo (,) B ) )
63 iooretop 21864 . . . . . . . . . 10  |-  ( -oo (,) B )  e.  (
topGen `  ran  (,) )
6463a1i 11 . . . . . . . . 9  |-  ( ph  ->  ( -oo (,) B
)  e.  ( topGen ` 
ran  (,) ) )
65 isopn3i 20175 . . . . . . . . 9  |-  ( ( ( topGen `  ran  (,) )  e.  Top  /\  ( -oo (,) B )  e.  (
topGen `  ran  (,) )
)  ->  ( ( int `  ( topGen `  ran  (,) ) ) `  ( -oo (,) B ) )  =  ( -oo (,) B ) )
6612, 64, 65syl2anc 673 . . . . . . . 8  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( -oo (,) B ) )  =  ( -oo (,) B
) )
6762, 66eleqtrrd 2552 . . . . . . 7  |-  ( ph  ->  A  e.  ( ( int `  ( topGen ` 
ran  (,) ) ) `  ( -oo (,) B ) ) )
6858, 67sseldd 3419 . . . . . 6  |-  ( ph  ->  A  e.  ( ( int `  ( topGen ` 
ran  (,) ) ) `  ( ( A [,) B )  u.  ( RR  \  ( A [,] B ) ) ) ) )
694leidd 10201 . . . . . . 7  |-  ( ph  ->  A  <_  A )
704, 5, 61ltled 9800 . . . . . . 7  |-  ( ph  ->  A  <_  B )
714, 5, 4, 69, 70eliccd 37697 . . . . . 6  |-  ( ph  ->  A  e.  ( A [,] B ) )
7268, 71elind 3609 . . . . 5  |-  ( ph  ->  A  e.  ( ( ( int `  ( topGen `
 ran  (,) )
) `  ( ( A [,) B )  u.  ( RR  \  ( A [,] B ) ) ) )  i^i  ( A [,] B ) ) )
73 icossicc 11746 . . . . . . 7  |-  ( A [,) B )  C_  ( A [,] B )
7473a1i 11 . . . . . 6  |-  ( ph  ->  ( A [,) B
)  C_  ( A [,] B ) )
75 eqid 2471 . . . . . . 7  |-  ( (
topGen `  ran  (,) )t  ( A [,] B ) )  =  ( ( topGen ` 
ran  (,) )t  ( A [,] B ) )
7618, 75restntr 20275 . . . . . 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 ) ) )
7712, 6, 74, 76syl3anc 1292 . . . . 5  |-  ( ph  ->  ( ( int `  (
( topGen `  ran  (,) )t  ( A [,] B ) ) ) `  ( A [,) B ) )  =  ( ( ( int `  ( topGen ` 
ran  (,) ) ) `  ( ( A [,) B )  u.  ( RR  \  ( A [,] B ) ) ) )  i^i  ( A [,] B ) ) )
7872, 77eleqtrrd 2552 . . . 4  |-  ( ph  ->  A  e.  ( ( int `  ( (
topGen `  ran  (,) )t  ( A [,] B ) ) ) `  ( A [,) B ) ) )
79 eqid 2471 . . . . . . . . 9  |-  ( topGen ` 
ran  (,) )  =  (
topGen `  ran  (,) )
809, 79rerest 21900 . . . . . . . 8  |-  ( ( A [,] B ) 
C_  RR  ->  ( (
TopOpen ` fld )t  ( A [,] B
) )  =  ( ( topGen `  ran  (,) )t  ( A [,] B ) ) )
816, 80syl 17 . . . . . . 7  |-  ( ph  ->  ( ( TopOpen ` fld )t  ( A [,] B ) )  =  ( ( topGen `  ran  (,) )t  ( A [,] B
) ) )
8281eqcomd 2477 . . . . . 6  |-  ( ph  ->  ( ( topGen `  ran  (,) )t  ( A [,] B
) )  =  ( ( TopOpen ` fld )t  ( A [,] B ) ) )
8382fveq2d 5883 . . . . 5  |-  ( ph  ->  ( int `  (
( topGen `  ran  (,) )t  ( A [,] B ) ) )  =  ( int `  ( ( TopOpen ` fld )t  ( A [,] B ) ) ) )
8483fveq1d 5881 . . . 4  |-  ( ph  ->  ( ( int `  (
( topGen `  ran  (,) )t  ( A [,] B ) ) ) `  ( A [,) B ) )  =  ( ( int `  ( ( TopOpen ` fld )t  ( A [,] B ) ) ) `
 ( A [,) B ) ) )
8578, 84eleqtrd 2551 . . 3  |-  ( ph  ->  A  e.  ( ( int `  ( (
TopOpen ` fld )t  ( A [,] B
) ) ) `  ( A [,) B ) ) )
8671snssd 4108 . . . . . . . 8  |-  ( ph  ->  { A }  C_  ( A [,] B ) )
87 ssequn2 3598 . . . . . . . 8  |-  ( { A }  C_  ( A [,] B )  <->  ( ( A [,] B )  u. 
{ A } )  =  ( A [,] B ) )
8886, 87sylib 201 . . . . . . 7  |-  ( ph  ->  ( ( A [,] B )  u.  { A } )  =  ( A [,] B ) )
8988eqcomd 2477 . . . . . 6  |-  ( ph  ->  ( A [,] B
)  =  ( ( A [,] B )  u.  { A }
) )
9089oveq2d 6324 . . . . 5  |-  ( ph  ->  ( ( TopOpen ` fld )t  ( A [,] B ) )  =  ( ( TopOpen ` fld )t  ( ( A [,] B )  u. 
{ A } ) ) )
9190fveq2d 5883 . . . 4  |-  ( ph  ->  ( int `  (
( TopOpen ` fld )t  ( A [,] B ) ) )  =  ( int `  (
( TopOpen ` fld )t  ( ( A [,] B )  u. 
{ A } ) ) ) )
92 uncom 3569 . . . . 5  |-  ( ( A (,) B )  u.  { A }
)  =  ( { A }  u.  ( A (,) B ) )
93 snunioo 11784 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  ( { A }  u.  ( A (,) B ) )  =  ( A [,) B ) )
9441, 13, 61, 93syl3anc 1292 . . . . 5  |-  ( ph  ->  ( { A }  u.  ( A (,) B
) )  =  ( A [,) B ) )
9592, 94syl5req 2518 . . . 4  |-  ( ph  ->  ( A [,) B
)  =  ( ( A (,) B )  u.  { A }
) )
9691, 95fveq12d 5885 . . 3  |-  ( ph  ->  ( ( int `  (
( TopOpen ` fld )t  ( A [,] B ) ) ) `
 ( A [,) B ) )  =  ( ( int `  (
( TopOpen ` fld )t  ( ( A [,] B )  u. 
{ A } ) ) ) `  (
( A (,) B
)  u.  { A } ) ) )
9785, 96eleqtrd 2551 . 2  |-  ( ph  ->  A  e.  ( ( int `  ( (
TopOpen ` fld )t  ( ( A [,] B )  u.  { A } ) ) ) `
 ( ( A (,) B )  u. 
{ A } ) ) )
981, 3, 8, 9, 10, 97limcres 22920 1  |-  ( ph  ->  ( ( F  |`  ( A (,) B ) ) lim CC  A )  =  ( F lim CC  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    \/ wo 375    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   A.wral 2756    \ cdif 3387    u. cun 3388    i^i cin 3389    C_ wss 3390   {csn 3959   U.cuni 4190   class class class wbr 4395   ran crn 4840    |` cres 4841   -->wf 5585   ` cfv 5589  (class class class)co 6308   CCcc 9555   RRcr 9556   -oocmnf 9691   RR*cxr 9692    < clt 9693    <_ cle 9694   (,)cioo 11660   [,)cico 11662   [,]cicc 11663   ↾t crest 15397   TopOpenctopn 15398   topGenctg 15414  ℂfldccnfld 19047   Topctop 19994   intcnt 20109   lim CC climc 22896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fi 7943  df-sup 7974  df-inf 7975  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ico 11666  df-icc 11667  df-fz 11811  df-seq 12252  df-exp 12311  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-plusg 15281  df-mulr 15282  df-starv 15283  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-rest 15399  df-topn 15400  df-topgen 15420  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-cnp 20321  df-xms 21413  df-ms 21414  df-limc 22900
This theorem is referenced by:  cncfiooicclem1  37868  fourierdlem82  38164  fourierdlem93  38175  fourierdlem111  38193
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