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

Theorem limciccioolb 37695
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 11717 . . 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 37596 . . 3  |-  ( ph  ->  ( A [,] B
)  C_  RR )
7 ax-resscn 9593 . . 3  |-  RR  C_  CC
86, 7syl6ss 3443 . 2  |-  ( ph  ->  ( A [,] B
)  C_  CC )
9 eqid 2450 . 2  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
10 eqid 2450 . 2  |-  ( (
TopOpen ` fld )t  ( ( A [,] B )  u.  { A } ) )  =  ( ( TopOpen ` fld )t  ( ( A [,] B )  u. 
{ A } ) )
11 retop 21775 . . . . . . . . 9  |-  ( topGen ` 
ran  (,) )  e.  Top
1211a1i 11 . . . . . . . 8  |-  ( ph  ->  ( topGen `  ran  (,) )  e.  Top )
135rexrd 9687 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  RR* )
14 icossre 11712 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( A [,) B
)  C_  RR )
154, 13, 14syl2anc 666 . . . . . . . . . 10  |-  ( ph  ->  ( A [,) B
)  C_  RR )
16 difssd 3560 . . . . . . . . . 10  |-  ( ph  ->  ( RR  \  ( A [,] B ) ) 
C_  RR )
1715, 16unssd 3609 . . . . . . . . 9  |-  ( ph  ->  ( ( A [,) B )  u.  ( RR  \  ( A [,] B ) ) ) 
C_  RR )
18 uniretop 21776 . . . . . . . . 9  |-  RR  =  U. ( topGen `  ran  (,) )
1917, 18syl6sseq 3477 . . . . . . . 8  |-  ( ph  ->  ( ( A [,) B )  u.  ( RR  \  ( A [,] B ) ) ) 
C_  U. ( topGen `  ran  (,) ) )
20 elioore 11663 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( -oo (,) B )  ->  x  e.  RR )
2120ad2antlr 732 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  A  <_  x )  ->  x  e.  RR )
22 simpr 463 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  A  <_  x )  ->  A  <_  x )
23 simpr 463 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  ( -oo (,) B ) )  ->  x  e.  ( -oo (,) B ) )
24 mnfxr 11411 . . . . . . . . . . . . . . . . . . 19  |- -oo  e.  RR*
2524a1i 11 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  x  e.  ( -oo (,) B ) )  -> -oo  e.  RR* )
2613adantr 467 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  x  e.  ( -oo (,) B ) )  ->  B  e.  RR* )
27 elioo2 11674 . . . . . . . . . . . . . . . . . 18  |-  ( ( -oo  e.  RR*  /\  B  e.  RR* )  ->  (
x  e.  ( -oo (,) B )  <->  ( x  e.  RR  /\ -oo  <  x  /\  x  <  B
) ) )
2825, 26, 27syl2anc 666 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  ( -oo (,) B ) )  ->  ( x  e.  ( -oo (,) B
)  <->  ( x  e.  RR  /\ -oo  <  x  /\  x  <  B
) ) )
2923, 28mpbid 214 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( -oo (,) B ) )  ->  ( x  e.  RR  /\ -oo  <  x  /\  x  <  B
) )
3029simp3d 1021 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( -oo (,) B ) )  ->  x  <  B )
3130adantr 467 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  A  <_  x )  ->  x  <  B )
324ad2antrr 731 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  A  <_  x )  ->  A  e.  RR )
3313ad2antrr 731 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  A  <_  x )  ->  B  e.  RR* )
34 elico2 11695 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( x  e.  ( A [,) B )  <-> 
( x  e.  RR  /\  A  <_  x  /\  x  <  B ) ) )
3532, 33, 34syl2anc 666 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  A  <_  x )  ->  (
x  e.  ( A [,) B )  <->  ( x  e.  RR  /\  A  <_  x  /\  x  <  B
) ) )
3621, 22, 31, 35mpbir3and 1190 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  A  <_  x )  ->  x  e.  ( A [,) B
) )
3736orcd 394 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  A  <_  x )  ->  (
x  e.  ( A [,) B )  \/  x  e.  ( RR 
\  ( A [,] B ) ) ) )
3820ad2antlr 732 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  -.  A  <_  x )  ->  x  e.  RR )
39 simpr 463 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  -.  A  <_  x )  ->  -.  A  <_  x )
4039intnanrd 927 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  -.  A  <_  x )  ->  -.  ( A  <_  x  /\  x  <_  B ) )
414rexrd 9687 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  A  e.  RR* )
4241ad2antrr 731 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  -.  A  <_  x )  ->  A  e.  RR* )
4313ad2antrr 731 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  -.  A  <_  x )  ->  B  e.  RR* )
4438rexrd 9687 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  -.  A  <_  x )  ->  x  e.  RR* )
45 elicc4 11698 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  x  e. 
RR* )  ->  (
x  e.  ( A [,] B )  <->  ( A  <_  x  /\  x  <_  B ) ) )
4642, 43, 44, 45syl3anc 1267 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  -.  A  <_  x )  -> 
( x  e.  ( A [,] B )  <-> 
( A  <_  x  /\  x  <_  B ) ) )
4740, 46mtbird 303 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  -.  A  <_  x )  ->  -.  x  e.  ( A [,] B ) )
4838, 47eldifd 3414 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  -.  A  <_  x )  ->  x  e.  ( RR  \  ( A [,] B
) ) )
4948olcd 395 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  -.  A  <_  x )  -> 
( x  e.  ( A [,) B )  \/  x  e.  ( RR  \  ( A [,] B ) ) ) )
5037, 49pm2.61dan 799 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( -oo (,) B ) )  ->  ( x  e.  ( A [,) B
)  \/  x  e.  ( RR  \  ( A [,] B ) ) ) )
51 elun 3573 . . . . . . . . . . 11  |-  ( x  e.  ( ( A [,) B )  u.  ( RR  \  ( A [,] B ) ) )  <->  ( x  e.  ( A [,) B
)  \/  x  e.  ( RR  \  ( A [,] B ) ) ) )
5250, 51sylibr 216 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( -oo (,) B ) )  ->  x  e.  ( ( A [,) B )  u.  ( RR  \  ( A [,] B ) ) ) )
5352ralrimiva 2801 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  ( -oo (,) B ) x  e.  ( ( A [,) B )  u.  ( RR  \ 
( A [,] B
) ) ) )
54 dfss3 3421 . . . . . . . . 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 216 . . . . . . . 8  |-  ( ph  ->  ( -oo (,) B
)  C_  ( ( A [,) B )  u.  ( RR  \  ( A [,] B ) ) ) )
56 eqid 2450 . . . . . . . . 9  |-  U. ( topGen `
 ran  (,) )  =  U. ( topGen `  ran  (,) )
5756ntrss 20063 . . . . . . . 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 1267 . . . . . . 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 11423 . . . . . . . . 9  |-  ( ph  -> -oo  <  A )
61 limciccioolb.3 . . . . . . . . 9  |-  ( ph  ->  A  <  B )
6259, 13, 4, 60, 61eliood 37589 . . . . . . . 8  |-  ( ph  ->  A  e.  ( -oo (,) B ) )
63 iooretop 21779 . . . . . . . . . 10  |-  ( -oo (,) B )  e.  (
topGen `  ran  (,) )
6463a1i 11 . . . . . . . . 9  |-  ( ph  ->  ( -oo (,) B
)  e.  ( topGen ` 
ran  (,) ) )
65 isopn3i 20091 . . . . . . . . 9  |-  ( ( ( topGen `  ran  (,) )  e.  Top  /\  ( -oo (,) B )  e.  (
topGen `  ran  (,) )
)  ->  ( ( int `  ( topGen `  ran  (,) ) ) `  ( -oo (,) B ) )  =  ( -oo (,) B ) )
6612, 64, 65syl2anc 666 . . . . . . . 8  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( -oo (,) B ) )  =  ( -oo (,) B
) )
6762, 66eleqtrrd 2531 . . . . . . 7  |-  ( ph  ->  A  e.  ( ( int `  ( topGen ` 
ran  (,) ) ) `  ( -oo (,) B ) ) )
6858, 67sseldd 3432 . . . . . 6  |-  ( ph  ->  A  e.  ( ( int `  ( topGen ` 
ran  (,) ) ) `  ( ( A [,) B )  u.  ( RR  \  ( A [,] B ) ) ) ) )
694leidd 10177 . . . . . . 7  |-  ( ph  ->  A  <_  A )
704, 5, 61ltled 9780 . . . . . . 7  |-  ( ph  ->  A  <_  B )
714, 5, 4, 69, 70eliccd 37595 . . . . . 6  |-  ( ph  ->  A  e.  ( A [,] B ) )
7268, 71elind 3617 . . . . 5  |-  ( ph  ->  A  e.  ( ( ( int `  ( topGen `
 ran  (,) )
) `  ( ( A [,) B )  u.  ( RR  \  ( A [,] B ) ) ) )  i^i  ( A [,] B ) ) )
73 icossicc 11718 . . . . . . 7  |-  ( A [,) B )  C_  ( A [,] B )
7473a1i 11 . . . . . 6  |-  ( ph  ->  ( A [,) B
)  C_  ( A [,] B ) )
75 eqid 2450 . . . . . . 7  |-  ( (
topGen `  ran  (,) )t  ( A [,] B ) )  =  ( ( topGen ` 
ran  (,) )t  ( A [,] B ) )
7618, 75restntr 20191 . . . . . 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 1267 . . . . 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 2531 . . . 4  |-  ( ph  ->  A  e.  ( ( int `  ( (
topGen `  ran  (,) )t  ( A [,] B ) ) ) `  ( A [,) B ) ) )
79 eqid 2450 . . . . . . . . 9  |-  ( topGen ` 
ran  (,) )  =  (
topGen `  ran  (,) )
809, 79rerest 21815 . . . . . . . 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 2456 . . . . . 6  |-  ( ph  ->  ( ( topGen `  ran  (,) )t  ( A [,] B
) )  =  ( ( TopOpen ` fld )t  ( A [,] B ) ) )
8382fveq2d 5867 . . . . 5  |-  ( ph  ->  ( int `  (
( topGen `  ran  (,) )t  ( A [,] B ) ) )  =  ( int `  ( ( TopOpen ` fld )t  ( A [,] B ) ) ) )
8483fveq1d 5865 . . . 4  |-  ( ph  ->  ( ( int `  (
( topGen `  ran  (,) )t  ( A [,] B ) ) ) `  ( A [,) B ) )  =  ( ( int `  ( ( TopOpen ` fld )t  ( A [,] B ) ) ) `
 ( A [,) B ) ) )
8578, 84eleqtrd 2530 . . 3  |-  ( ph  ->  A  e.  ( ( int `  ( (
TopOpen ` fld )t  ( A [,] B
) ) ) `  ( A [,) B ) ) )
8671snssd 4116 . . . . . . . 8  |-  ( ph  ->  { A }  C_  ( A [,] B ) )
87 ssequn2 3606 . . . . . . . 8  |-  ( { A }  C_  ( A [,] B )  <->  ( ( A [,] B )  u. 
{ A } )  =  ( A [,] B ) )
8886, 87sylib 200 . . . . . . 7  |-  ( ph  ->  ( ( A [,] B )  u.  { A } )  =  ( A [,] B ) )
8988eqcomd 2456 . . . . . 6  |-  ( ph  ->  ( A [,] B
)  =  ( ( A [,] B )  u.  { A }
) )
9089oveq2d 6304 . . . . 5  |-  ( ph  ->  ( ( TopOpen ` fld )t  ( A [,] B ) )  =  ( ( TopOpen ` fld )t  ( ( A [,] B )  u. 
{ A } ) ) )
9190fveq2d 5867 . . . 4  |-  ( ph  ->  ( int `  (
( TopOpen ` fld )t  ( A [,] B ) ) )  =  ( int `  (
( TopOpen ` fld )t  ( ( A [,] B )  u. 
{ A } ) ) ) )
92 uncom 3577 . . . . 5  |-  ( ( A (,) B )  u.  { A }
)  =  ( { A }  u.  ( A (,) B ) )
93 snunioo 11755 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  ( { A }  u.  ( A (,) B ) )  =  ( A [,) B ) )
9441, 13, 61, 93syl3anc 1267 . . . . 5  |-  ( ph  ->  ( { A }  u.  ( A (,) B
) )  =  ( A [,) B ) )
9592, 94syl5req 2497 . . . 4  |-  ( ph  ->  ( A [,) B
)  =  ( ( A (,) B )  u.  { A }
) )
9691, 95fveq12d 5869 . . 3  |-  ( ph  ->  ( ( int `  (
( TopOpen ` fld )t  ( A [,] B ) ) ) `
 ( A [,) B ) )  =  ( ( int `  (
( TopOpen ` fld )t  ( ( A [,] B )  u. 
{ A } ) ) ) `  (
( A (,) B
)  u.  { A } ) ) )
9785, 96eleqtrd 2530 . 2  |-  ( ph  ->  A  e.  ( ( int `  ( (
TopOpen ` fld )t  ( ( A [,] B )  u.  { A } ) ) ) `
 ( ( A (,) B )  u. 
{ A } ) ) )
981, 3, 8, 9, 10, 97limcres 22834 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 188    \/ wo 370    /\ wa 371    /\ w3a 984    = wceq 1443    e. wcel 1886   A.wral 2736    \ cdif 3400    u. cun 3401    i^i cin 3402    C_ wss 3403   {csn 3967   U.cuni 4197   class class class wbr 4401   ran crn 4834    |` cres 4835   -->wf 5577   ` cfv 5581  (class class class)co 6288   CCcc 9534   RRcr 9535   -oocmnf 9670   RR*cxr 9671    < clt 9672    <_ cle 9673   (,)cioo 11632   [,)cico 11634   [,]cicc 11635   ↾t crest 15312   TopOpenctopn 15313   topGenctg 15329  ℂfldccnfld 18963   Topctop 19910   intcnt 20025   lim CC climc 22810
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-iin 4280  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-oadd 7183  df-er 7360  df-map 7471  df-pm 7472  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-fi 7922  df-sup 7953  df-inf 7954  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-4 10667  df-5 10668  df-6 10669  df-7 10670  df-8 10671  df-9 10672  df-10 10673  df-n0 10867  df-z 10935  df-dec 11049  df-uz 11157  df-q 11262  df-rp 11300  df-xneg 11406  df-xadd 11407  df-xmul 11408  df-ioo 11636  df-ico 11638  df-icc 11639  df-fz 11782  df-seq 12211  df-exp 12270  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-plusg 15196  df-mulr 15197  df-starv 15198  df-tset 15202  df-ple 15203  df-ds 15205  df-unif 15206  df-rest 15314  df-topn 15315  df-topgen 15335  df-psmet 18955  df-xmet 18956  df-met 18957  df-bl 18958  df-mopn 18959  df-cnfld 18964  df-top 19914  df-bases 19915  df-topon 19916  df-topsp 19917  df-cld 20027  df-ntr 20028  df-cls 20029  df-cnp 20237  df-xms 21328  df-ms 21329  df-limc 22814
This theorem is referenced by:  cncfiooicclem1  37765  fourierdlem82  38046  fourierdlem93  38057  fourierdlem111  38075
  Copyright terms: Public domain W3C validator