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Theorem limciccioolb 37273
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 11720 . . 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 37187 . . 3  |-  ( ph  ->  ( A [,] B
)  C_  RR )
7 ax-resscn 9595 . . 3  |-  RR  C_  CC
86, 7syl6ss 3482 . 2  |-  ( ph  ->  ( A [,] B
)  C_  CC )
9 eqid 2429 . 2  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
10 eqid 2429 . 2  |-  ( (
TopOpen ` fld )t  ( ( A [,] B )  u.  { A } ) )  =  ( ( TopOpen ` fld )t  ( ( A [,] B )  u. 
{ A } ) )
11 retop 21693 . . . . . . . . 9  |-  ( topGen ` 
ran  (,) )  e.  Top
1211a1i 11 . . . . . . . 8  |-  ( ph  ->  ( topGen `  ran  (,) )  e.  Top )
135rexrd 9689 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  RR* )
14 icossre 11715 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( A [,) B
)  C_  RR )
154, 13, 14syl2anc 665 . . . . . . . . . 10  |-  ( ph  ->  ( A [,) B
)  C_  RR )
16 difssd 3599 . . . . . . . . . 10  |-  ( ph  ->  ( RR  \  ( A [,] B ) ) 
C_  RR )
1715, 16unssd 3648 . . . . . . . . 9  |-  ( ph  ->  ( ( A [,) B )  u.  ( RR  \  ( A [,] B ) ) ) 
C_  RR )
18 uniretop 21694 . . . . . . . . 9  |-  RR  =  U. ( topGen `  ran  (,) )
1917, 18syl6sseq 3516 . . . . . . . 8  |-  ( ph  ->  ( ( A [,) B )  u.  ( RR  \  ( A [,] B ) ) ) 
C_  U. ( topGen `  ran  (,) ) )
20 elioore 11666 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( -oo (,) B )  ->  x  e.  RR )
2120ad2antlr 731 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  A  <_  x )  ->  x  e.  RR )
22 simpr 462 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  A  <_  x )  ->  A  <_  x )
23 simpr 462 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  ( -oo (,) B ) )  ->  x  e.  ( -oo (,) B ) )
24 mnfxr 11414 . . . . . . . . . . . . . . . . . . 19  |- -oo  e.  RR*
2524a1i 11 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  x  e.  ( -oo (,) B ) )  -> -oo  e.  RR* )
2613adantr 466 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  x  e.  ( -oo (,) B ) )  ->  B  e.  RR* )
27 elioo2 11677 . . . . . . . . . . . . . . . . . 18  |-  ( ( -oo  e.  RR*  /\  B  e.  RR* )  ->  (
x  e.  ( -oo (,) B )  <->  ( x  e.  RR  /\ -oo  <  x  /\  x  <  B
) ) )
2825, 26, 27syl2anc 665 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  ( -oo (,) B ) )  ->  ( x  e.  ( -oo (,) B
)  <->  ( x  e.  RR  /\ -oo  <  x  /\  x  <  B
) ) )
2923, 28mpbid 213 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( -oo (,) B ) )  ->  ( x  e.  RR  /\ -oo  <  x  /\  x  <  B
) )
3029simp3d 1019 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( -oo (,) B ) )  ->  x  <  B )
3130adantr 466 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  A  <_  x )  ->  x  <  B )
324ad2antrr 730 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  A  <_  x )  ->  A  e.  RR )
3313ad2antrr 730 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  A  <_  x )  ->  B  e.  RR* )
34 elico2 11698 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( x  e.  ( A [,) B )  <-> 
( x  e.  RR  /\  A  <_  x  /\  x  <  B ) ) )
3532, 33, 34syl2anc 665 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  A  <_  x )  ->  (
x  e.  ( A [,) B )  <->  ( x  e.  RR  /\  A  <_  x  /\  x  <  B
) ) )
3621, 22, 31, 35mpbir3and 1188 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  A  <_  x )  ->  x  e.  ( A [,) B
) )
3736orcd 393 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  A  <_  x )  ->  (
x  e.  ( A [,) B )  \/  x  e.  ( RR 
\  ( A [,] B ) ) ) )
3820ad2antlr 731 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  -.  A  <_  x )  ->  x  e.  RR )
39 simpr 462 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  -.  A  <_  x )  ->  -.  A  <_  x )
4039intnanrd 925 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  -.  A  <_  x )  ->  -.  ( A  <_  x  /\  x  <_  B ) )
414rexrd 9689 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  A  e.  RR* )
4241ad2antrr 730 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  -.  A  <_  x )  ->  A  e.  RR* )
4313ad2antrr 730 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  -.  A  <_  x )  ->  B  e.  RR* )
4438rexrd 9689 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  -.  A  <_  x )  ->  x  e.  RR* )
45 elicc4 11701 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  x  e. 
RR* )  ->  (
x  e.  ( A [,] B )  <->  ( A  <_  x  /\  x  <_  B ) ) )
4642, 43, 44, 45syl3anc 1264 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  -.  A  <_  x )  -> 
( x  e.  ( A [,] B )  <-> 
( A  <_  x  /\  x  <_  B ) ) )
4740, 46mtbird 302 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  -.  A  <_  x )  ->  -.  x  e.  ( A [,] B ) )
4838, 47eldifd 3453 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  -.  A  <_  x )  ->  x  e.  ( RR  \  ( A [,] B
) ) )
4948olcd 394 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  -.  A  <_  x )  -> 
( x  e.  ( A [,) B )  \/  x  e.  ( RR  \  ( A [,] B ) ) ) )
5037, 49pm2.61dan 798 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( -oo (,) B ) )  ->  ( x  e.  ( A [,) B
)  \/  x  e.  ( RR  \  ( A [,] B ) ) ) )
51 elun 3612 . . . . . . . . . . 11  |-  ( x  e.  ( ( A [,) B )  u.  ( RR  \  ( A [,] B ) ) )  <->  ( x  e.  ( A [,) B
)  \/  x  e.  ( RR  \  ( A [,] B ) ) ) )
5250, 51sylibr 215 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( -oo (,) B ) )  ->  x  e.  ( ( A [,) B )  u.  ( RR  \  ( A [,] B ) ) ) )
5352ralrimiva 2846 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  ( -oo (,) B ) x  e.  ( ( A [,) B )  u.  ( RR  \ 
( A [,] B
) ) ) )
54 dfss3 3460 . . . . . . . . 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 215 . . . . . . . 8  |-  ( ph  ->  ( -oo (,) B
)  C_  ( ( A [,) B )  u.  ( RR  \  ( A [,] B ) ) ) )
56 eqid 2429 . . . . . . . . 9  |-  U. ( topGen `
 ran  (,) )  =  U. ( topGen `  ran  (,) )
5756ntrss 20001 . . . . . . . 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 1264 . . . . . . 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 11426 . . . . . . . . 9  |-  ( ph  -> -oo  <  A )
61 limciccioolb.3 . . . . . . . . 9  |-  ( ph  ->  A  <  B )
6259, 13, 4, 60, 61eliood 37180 . . . . . . . 8  |-  ( ph  ->  A  e.  ( -oo (,) B ) )
63 iooretop 21697 . . . . . . . . . 10  |-  ( -oo (,) B )  e.  (
topGen `  ran  (,) )
6463a1i 11 . . . . . . . . 9  |-  ( ph  ->  ( -oo (,) B
)  e.  ( topGen ` 
ran  (,) ) )
65 isopn3i 20029 . . . . . . . . 9  |-  ( ( ( topGen `  ran  (,) )  e.  Top  /\  ( -oo (,) B )  e.  (
topGen `  ran  (,) )
)  ->  ( ( int `  ( topGen `  ran  (,) ) ) `  ( -oo (,) B ) )  =  ( -oo (,) B ) )
6612, 64, 65syl2anc 665 . . . . . . . 8  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( -oo (,) B ) )  =  ( -oo (,) B
) )
6762, 66eleqtrrd 2520 . . . . . . 7  |-  ( ph  ->  A  e.  ( ( int `  ( topGen ` 
ran  (,) ) ) `  ( -oo (,) B ) ) )
6858, 67sseldd 3471 . . . . . 6  |-  ( ph  ->  A  e.  ( ( int `  ( topGen ` 
ran  (,) ) ) `  ( ( A [,) B )  u.  ( RR  \  ( A [,] B ) ) ) ) )
694leidd 10179 . . . . . . 7  |-  ( ph  ->  A  <_  A )
704, 5, 61ltled 9782 . . . . . . 7  |-  ( ph  ->  A  <_  B )
714, 5, 4, 69, 70eliccd 37186 . . . . . 6  |-  ( ph  ->  A  e.  ( A [,] B ) )
7268, 71elind 3656 . . . . 5  |-  ( ph  ->  A  e.  ( ( ( int `  ( topGen `
 ran  (,) )
) `  ( ( A [,) B )  u.  ( RR  \  ( A [,] B ) ) ) )  i^i  ( A [,] B ) ) )
73 icossicc 11721 . . . . . . 7  |-  ( A [,) B )  C_  ( A [,] B )
7473a1i 11 . . . . . 6  |-  ( ph  ->  ( A [,) B
)  C_  ( A [,] B ) )
75 eqid 2429 . . . . . . 7  |-  ( (
topGen `  ran  (,) )t  ( A [,] B ) )  =  ( ( topGen ` 
ran  (,) )t  ( A [,] B ) )
7618, 75restntr 20129 . . . . . 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 1264 . . . . 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 2520 . . . 4  |-  ( ph  ->  A  e.  ( ( int `  ( (
topGen `  ran  (,) )t  ( A [,] B ) ) ) `  ( A [,) B ) ) )
79 eqid 2429 . . . . . . . . 9  |-  ( topGen ` 
ran  (,) )  =  (
topGen `  ran  (,) )
809, 79rerest 21733 . . . . . . . 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 2437 . . . . . 6  |-  ( ph  ->  ( ( topGen `  ran  (,) )t  ( A [,] B
) )  =  ( ( TopOpen ` fld )t  ( A [,] B ) ) )
8382fveq2d 5885 . . . . 5  |-  ( ph  ->  ( int `  (
( topGen `  ran  (,) )t  ( A [,] B ) ) )  =  ( int `  ( ( TopOpen ` fld )t  ( A [,] B ) ) ) )
8483fveq1d 5883 . . . 4  |-  ( ph  ->  ( ( int `  (
( topGen `  ran  (,) )t  ( A [,] B ) ) ) `  ( A [,) B ) )  =  ( ( int `  ( ( TopOpen ` fld )t  ( A [,] B ) ) ) `
 ( A [,) B ) ) )
8578, 84eleqtrd 2519 . . 3  |-  ( ph  ->  A  e.  ( ( int `  ( (
TopOpen ` fld )t  ( A [,] B
) ) ) `  ( A [,) B ) ) )
8671snssd 4148 . . . . . . . 8  |-  ( ph  ->  { A }  C_  ( A [,] B ) )
87 ssequn2 3645 . . . . . . . 8  |-  ( { A }  C_  ( A [,] B )  <->  ( ( A [,] B )  u. 
{ A } )  =  ( A [,] B ) )
8886, 87sylib 199 . . . . . . 7  |-  ( ph  ->  ( ( A [,] B )  u.  { A } )  =  ( A [,] B ) )
8988eqcomd 2437 . . . . . 6  |-  ( ph  ->  ( A [,] B
)  =  ( ( A [,] B )  u.  { A }
) )
9089oveq2d 6321 . . . . 5  |-  ( ph  ->  ( ( TopOpen ` fld )t  ( A [,] B ) )  =  ( ( TopOpen ` fld )t  ( ( A [,] B )  u. 
{ A } ) ) )
9190fveq2d 5885 . . . 4  |-  ( ph  ->  ( int `  (
( TopOpen ` fld )t  ( A [,] B ) ) )  =  ( int `  (
( TopOpen ` fld )t  ( ( A [,] B )  u. 
{ A } ) ) ) )
92 uncom 3616 . . . . 5  |-  ( ( A (,) B )  u.  { A }
)  =  ( { A }  u.  ( A (,) B ) )
93 snunioo 11756 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  ( { A }  u.  ( A (,) B ) )  =  ( A [,) B ) )
9441, 13, 61, 93syl3anc 1264 . . . . 5  |-  ( ph  ->  ( { A }  u.  ( A (,) B
) )  =  ( A [,) B ) )
9592, 94syl5req 2483 . . . 4  |-  ( ph  ->  ( A [,) B
)  =  ( ( A (,) B )  u.  { A }
) )
9691, 95fveq12d 5887 . . 3  |-  ( ph  ->  ( ( int `  (
( TopOpen ` fld )t  ( A [,] B ) ) ) `
 ( A [,) B ) )  =  ( ( int `  (
( TopOpen ` fld )t  ( ( A [,] B )  u. 
{ A } ) ) ) `  (
( A (,) B
)  u.  { A } ) ) )
9785, 96eleqtrd 2519 . 2  |-  ( ph  ->  A  e.  ( ( int `  ( (
TopOpen ` fld )t  ( ( A [,] B )  u.  { A } ) ) ) `
 ( ( A (,) B )  u. 
{ A } ) ) )
981, 3, 8, 9, 10, 97limcres 22718 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 187    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   A.wral 2782    \ cdif 3439    u. cun 3440    i^i cin 3441    C_ wss 3442   {csn 4002   U.cuni 4222   class class class wbr 4426   ran crn 4855    |` cres 4856   -->wf 5597   ` cfv 5601  (class class class)co 6305   CCcc 9536   RRcr 9537   -oocmnf 9672   RR*cxr 9673    < clt 9674    <_ cle 9675   (,)cioo 11635   [,)cico 11637   [,]cicc 11638   ↾t crest 15278   TopOpenctopn 15279   topGenctg 15295  ℂfldccnfld 18905   Topctop 19848   intcnt 19963   lim CC climc 22694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fi 7931  df-sup 7962  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  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-ico 11641  df-icc 11642  df-fz 11783  df-seq 12211  df-exp 12270  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-plusg 15165  df-mulr 15166  df-starv 15167  df-tset 15171  df-ple 15172  df-ds 15174  df-unif 15175  df-rest 15280  df-topn 15281  df-topgen 15301  df-psmet 18897  df-xmet 18898  df-met 18899  df-bl 18900  df-mopn 18901  df-cnfld 18906  df-top 19852  df-bases 19853  df-topon 19854  df-topsp 19855  df-cld 19965  df-ntr 19966  df-cls 19967  df-cnp 20175  df-xms 21266  df-ms 21267  df-limc 22698
This theorem is referenced by:  cncfiooicclem1  37343  fourierdlem82  37620  fourierdlem93  37631  fourierdlem111  37649
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