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Theorem limciccioolb 31486
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 11622 . . 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 )
6 iccssre 11618 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
74, 5, 6syl2anc 661 . . 3  |-  ( ph  ->  ( A [,] B
)  C_  RR )
8 ax-resscn 9561 . . . 4  |-  RR  C_  CC
98a1i 11 . . 3  |-  ( ph  ->  RR  C_  CC )
107, 9sstrd 3519 . 2  |-  ( ph  ->  ( A [,] B
)  C_  CC )
11 eqid 2467 . 2  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
12 eqid 2467 . 2  |-  ( (
TopOpen ` fld )t  ( ( A [,] B )  u.  { A } ) )  =  ( ( TopOpen ` fld )t  ( ( A [,] B )  u. 
{ A } ) )
13 mnflt 11345 . . . . . . . . . . . 12  |-  ( A  e.  RR  -> -oo  <  A )
144, 13syl 16 . . . . . . . . . . 11  |-  ( ph  -> -oo  <  A )
15 limciccioolb.3 . . . . . . . . . . 11  |-  ( ph  ->  A  <  B )
164, 14, 153jca 1176 . . . . . . . . . 10  |-  ( ph  ->  ( A  e.  RR  /\ -oo  <  A  /\  A  <  B ) )
17 mnfxr 11335 . . . . . . . . . . . 12  |- -oo  e.  RR*
1817a1i 11 . . . . . . . . . . 11  |-  ( ph  -> -oo  e.  RR* )
195rexrd 9655 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  RR* )
20 elioo2 11582 . . . . . . . . . . 11  |-  ( ( -oo  e.  RR*  /\  B  e.  RR* )  ->  ( A  e.  ( -oo (,) B )  <->  ( A  e.  RR  /\ -oo  <  A  /\  A  <  B
) ) )
2118, 19, 20syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  ( A  e.  ( -oo (,) B )  <-> 
( A  e.  RR  /\ -oo  <  A  /\  A  <  B ) ) )
2216, 21mpbird 232 . . . . . . . . 9  |-  ( ph  ->  A  e.  ( -oo (,) B ) )
23 retop 21136 . . . . . . . . . . . 12  |-  ( topGen ` 
ran  (,) )  e.  Top
2423a1i 11 . . . . . . . . . . 11  |-  ( ph  ->  ( topGen `  ran  (,) )  e.  Top )
25 iooretop 21141 . . . . . . . . . . . 12  |-  ( -oo (,) B )  e.  (
topGen `  ran  (,) )
2625a1i 11 . . . . . . . . . . 11  |-  ( ph  ->  ( -oo (,) B
)  e.  ( topGen ` 
ran  (,) ) )
27 isopn3i 19451 . . . . . . . . . . 11  |-  ( ( ( topGen `  ran  (,) )  e.  Top  /\  ( -oo (,) B )  e.  (
topGen `  ran  (,) )
)  ->  ( ( int `  ( topGen `  ran  (,) ) ) `  ( -oo (,) B ) )  =  ( -oo (,) B ) )
2824, 26, 27syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( -oo (,) B ) )  =  ( -oo (,) B
) )
2928eqcomd 2475 . . . . . . . . 9  |-  ( ph  ->  ( -oo (,) B
)  =  ( ( int `  ( topGen ` 
ran  (,) ) ) `  ( -oo (,) B ) ) )
3022, 29eleqtrd 2557 . . . . . . . 8  |-  ( ph  ->  A  e.  ( ( int `  ( topGen ` 
ran  (,) ) ) `  ( -oo (,) B ) ) )
31 icossre 11617 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( A [,) B
)  C_  RR )
324, 19, 31syl2anc 661 . . . . . . . . . . . . 13  |-  ( ph  ->  ( A [,) B
)  C_  RR )
33 difssd 3637 . . . . . . . . . . . . 13  |-  ( ph  ->  ( RR  \  ( A [,] B ) ) 
C_  RR )
34 unss12 3681 . . . . . . . . . . . . 13  |-  ( ( ( A [,) B
)  C_  RR  /\  ( RR  \  ( A [,] B ) )  C_  RR )  ->  ( ( A [,) B )  u.  ( RR  \ 
( A [,] B
) ) )  C_  ( RR  u.  RR ) )
3532, 33, 34syl2anc 661 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( A [,) B )  u.  ( RR  \  ( A [,] B ) ) ) 
C_  ( RR  u.  RR ) )
36 unidm 3652 . . . . . . . . . . . 12  |-  ( RR  u.  RR )  =  RR
3735, 36syl6sseq 3555 . . . . . . . . . . 11  |-  ( ph  ->  ( ( A [,) B )  u.  ( RR  \  ( A [,] B ) ) ) 
C_  RR )
38 uniretop 21137 . . . . . . . . . . 11  |-  RR  =  U. ( topGen `  ran  (,) )
3937, 38syl6sseq 3555 . . . . . . . . . 10  |-  ( ph  ->  ( ( A [,) B )  u.  ( RR  \  ( A [,] B ) ) ) 
C_  U. ( topGen `  ran  (,) ) )
40 elioore 11571 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  ( -oo (,) B )  ->  x  e.  RR )
4140ad2antlr 726 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  A  <_  x )  ->  x  e.  RR )
42 simpr 461 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  A  <_  x )  ->  A  <_  x )
43 simpr 461 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  x  e.  ( -oo (,) B ) )  ->  x  e.  ( -oo (,) B ) )
4417a1i 11 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  x  e.  ( -oo (,) B ) )  -> -oo  e.  RR* )
4519adantr 465 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  x  e.  ( -oo (,) B ) )  ->  B  e.  RR* )
46 elioo2 11582 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( -oo  e.  RR*  /\  B  e.  RR* )  ->  (
x  e.  ( -oo (,) B )  <->  ( x  e.  RR  /\ -oo  <  x  /\  x  <  B
) ) )
4744, 45, 46syl2anc 661 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  x  e.  ( -oo (,) B ) )  ->  ( x  e.  ( -oo (,) B
)  <->  ( x  e.  RR  /\ -oo  <  x  /\  x  <  B
) ) )
4843, 47mpbid 210 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  x  e.  ( -oo (,) B ) )  ->  ( x  e.  RR  /\ -oo  <  x  /\  x  <  B
) )
4948simp3d 1010 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  x  e.  ( -oo (,) B ) )  ->  x  <  B )
5049adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  A  <_  x )  ->  x  <  B )
5141, 42, 503jca 1176 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  A  <_  x )  ->  (
x  e.  RR  /\  A  <_  x  /\  x  <  B ) )
524ad2antrr 725 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  A  <_  x )  ->  A  e.  RR )
5319ad2antrr 725 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  A  <_  x )  ->  B  e.  RR* )
54 elico2 11600 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( x  e.  ( A [,) B )  <-> 
( x  e.  RR  /\  A  <_  x  /\  x  <  B ) ) )
5552, 53, 54syl2anc 661 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  A  <_  x )  ->  (
x  e.  ( A [,) B )  <->  ( x  e.  RR  /\  A  <_  x  /\  x  <  B
) ) )
5651, 55mpbird 232 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  A  <_  x )  ->  x  e.  ( A [,) B
) )
5756orcd 392 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  A  <_  x )  ->  (
x  e.  ( A [,) B )  \/  x  e.  ( RR 
\  ( A [,] B ) ) ) )
5840ad2antlr 726 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  -.  A  <_  x )  ->  x  e.  RR )
59 simpr 461 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  -.  A  <_  x )  ->  -.  A  <_  x )
6059intnanrd 915 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  -.  A  <_  x )  ->  -.  ( A  <_  x  /\  x  <_  B ) )
614rexrd 9655 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  A  e.  RR* )
6261ad2antrr 725 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  -.  A  <_  x )  ->  A  e.  RR* )
6345adantr 465 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  -.  A  <_  x )  ->  B  e.  RR* )
6458rexrd 9655 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  -.  A  <_  x )  ->  x  e.  RR* )
65 elicc4 11603 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  x  e. 
RR* )  ->  (
x  e.  ( A [,] B )  <->  ( A  <_  x  /\  x  <_  B ) ) )
6662, 63, 64, 65syl3anc 1228 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  -.  A  <_  x )  -> 
( x  e.  ( A [,] B )  <-> 
( A  <_  x  /\  x  <_  B ) ) )
6760, 66mtbird 301 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  -.  A  <_  x )  ->  -.  x  e.  ( A [,] B ) )
6858, 67jca 532 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  -.  A  <_  x )  -> 
( x  e.  RR  /\ 
-.  x  e.  ( A [,] B ) ) )
69 eldif 3491 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ( RR  \ 
( A [,] B
) )  <->  ( x  e.  RR  /\  -.  x  e.  ( A [,] B
) ) )
7068, 69sylibr 212 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  -.  A  <_  x )  ->  x  e.  ( RR  \  ( A [,] B
) ) )
7170olcd 393 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( -oo (,) B
) )  /\  -.  A  <_  x )  -> 
( x  e.  ( A [,) B )  \/  x  e.  ( RR  \  ( A [,] B ) ) ) )
7257, 71pm2.61dan 789 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( -oo (,) B ) )  ->  ( x  e.  ( A [,) B
)  \/  x  e.  ( RR  \  ( A [,] B ) ) ) )
73 elun 3650 . . . . . . . . . . . . 13  |-  ( x  e.  ( ( A [,) B )  u.  ( RR  \  ( A [,] B ) ) )  <->  ( x  e.  ( A [,) B
)  \/  x  e.  ( RR  \  ( A [,] B ) ) ) )
7472, 73sylibr 212 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( -oo (,) B ) )  ->  x  e.  ( ( A [,) B )  u.  ( RR  \  ( A [,] B ) ) ) )
7574ralrimiva 2881 . . . . . . . . . . 11  |-  ( ph  ->  A. x  e.  ( -oo (,) B ) x  e.  ( ( A [,) B )  u.  ( RR  \ 
( A [,] B
) ) ) )
76 dfss3 3499 . . . . . . . . . . 11  |-  ( ( -oo (,) B ) 
C_  ( ( A [,) B )  u.  ( RR  \  ( A [,] B ) ) )  <->  A. x  e.  ( -oo (,) B ) x  e.  ( ( A [,) B )  u.  ( RR  \ 
( A [,] B
) ) ) )
7775, 76sylibr 212 . . . . . . . . . 10  |-  ( ph  ->  ( -oo (,) B
)  C_  ( ( A [,) B )  u.  ( RR  \  ( A [,] B ) ) ) )
78 eqid 2467 . . . . . . . . . . 11  |-  U. ( topGen `
 ran  (,) )  =  U. ( topGen `  ran  (,) )
7978ntrss 19424 . . . . . . . . . 10  |-  ( ( ( 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 ) ) ) ) )
8024, 39, 77, 79syl3anc 1228 . . . . . . . . 9  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( -oo (,) B ) )  C_  ( ( int `  ( topGen `
 ran  (,) )
) `  ( ( A [,) B )  u.  ( RR  \  ( A [,] B ) ) ) ) )
8180sseld 3508 . . . . . . . 8  |-  ( ph  ->  ( A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( -oo (,) B ) )  ->  A  e.  ( ( int `  ( topGen `  ran  (,) ) ) `  (
( A [,) B
)  u.  ( RR 
\  ( A [,] B ) ) ) ) ) )
8230, 81mpd 15 . . . . . . 7  |-  ( ph  ->  A  e.  ( ( int `  ( topGen ` 
ran  (,) ) ) `  ( ( A [,) B )  u.  ( RR  \  ( A [,] B ) ) ) ) )
834leidd 10131 . . . . . . . . 9  |-  ( ph  ->  A  <_  A )
844, 5, 15ltled 9744 . . . . . . . . 9  |-  ( ph  ->  A  <_  B )
854, 83, 843jca 1176 . . . . . . . 8  |-  ( ph  ->  ( A  e.  RR  /\  A  <_  A  /\  A  <_  B ) )
86 elicc2 11601 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  e.  ( A [,] B )  <-> 
( A  e.  RR  /\  A  <_  A  /\  A  <_  B ) ) )
874, 5, 86syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( A  e.  ( A [,] B )  <-> 
( A  e.  RR  /\  A  <_  A  /\  A  <_  B ) ) )
8885, 87mpbird 232 . . . . . . 7  |-  ( ph  ->  A  e.  ( A [,] B ) )
8982, 88jca 532 . . . . . 6  |-  ( ph  ->  ( A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( ( A [,) B )  u.  ( RR  \  ( A [,] B ) ) ) )  /\  A  e.  ( A [,] B
) ) )
90 elin 3692 . . . . . 6  |-  ( A  e.  ( ( ( int `  ( topGen ` 
ran  (,) ) ) `  ( ( A [,) B )  u.  ( RR  \  ( A [,] B ) ) ) )  i^i  ( A [,] B ) )  <-> 
( A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( ( A [,) B )  u.  ( RR  \  ( A [,] B ) ) ) )  /\  A  e.  ( A [,] B
) ) )
9189, 90sylibr 212 . . . . 5  |-  ( ph  ->  A  e.  ( ( ( int `  ( topGen `
 ran  (,) )
) `  ( ( A [,) B )  u.  ( RR  \  ( A [,] B ) ) ) )  i^i  ( A [,] B ) ) )
92 icossicc 11623 . . . . . . . . 9  |-  ( A [,) B )  C_  ( A [,] B )
9392a1i 11 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A [,) B )  C_  ( A [,] B ) )
9461, 19, 93syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( A [,) B
)  C_  ( A [,] B ) )
95 eqid 2467 . . . . . . . 8  |-  ( (
topGen `  ran  (,) )t  ( A [,] B ) )  =  ( ( topGen ` 
ran  (,) )t  ( A [,] B ) )
9638, 95restntr 19551 . . . . . . 7  |-  ( ( ( 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 ) ) )
9724, 7, 94, 96syl3anc 1228 . . . . . 6  |-  ( ph  ->  ( ( int `  (
( topGen `  ran  (,) )t  ( A [,] B ) ) ) `  ( A [,) B ) )  =  ( ( ( int `  ( topGen ` 
ran  (,) ) ) `  ( ( A [,) B )  u.  ( RR  \  ( A [,] B ) ) ) )  i^i  ( A [,] B ) ) )
9897eqcomd 2475 . . . . 5  |-  ( ph  ->  ( ( ( int `  ( topGen `  ran  (,) )
) `  ( ( A [,) B )  u.  ( RR  \  ( A [,] B ) ) ) )  i^i  ( A [,] B ) )  =  ( ( int `  ( ( topGen `  ran  (,) )t  ( A [,] B
) ) ) `  ( A [,) B ) ) )
9991, 98eleqtrd 2557 . . . 4  |-  ( ph  ->  A  e.  ( ( int `  ( (
topGen `  ran  (,) )t  ( A [,] B ) ) ) `  ( A [,) B ) ) )
100 eqid 2467 . . . . . . . . 9  |-  ( topGen ` 
ran  (,) )  =  (
topGen `  ran  (,) )
10111, 100rerest 21177 . . . . . . . 8  |-  ( ( A [,] B ) 
C_  RR  ->  ( (
TopOpen ` fld )t  ( A [,] B
) )  =  ( ( topGen `  ran  (,) )t  ( A [,] B ) ) )
1027, 101syl 16 . . . . . . 7  |-  ( ph  ->  ( ( TopOpen ` fld )t  ( A [,] B ) )  =  ( ( topGen `  ran  (,) )t  ( A [,] B
) ) )
103102eqcomd 2475 . . . . . 6  |-  ( ph  ->  ( ( topGen `  ran  (,) )t  ( A [,] B
) )  =  ( ( TopOpen ` fld )t  ( A [,] B ) ) )
104103fveq2d 5876 . . . . 5  |-  ( ph  ->  ( int `  (
( topGen `  ran  (,) )t  ( A [,] B ) ) )  =  ( int `  ( ( TopOpen ` fld )t  ( A [,] B ) ) ) )
105104fveq1d 5874 . . . 4  |-  ( ph  ->  ( ( int `  (
( topGen `  ran  (,) )t  ( A [,] B ) ) ) `  ( A [,) B ) )  =  ( ( int `  ( ( TopOpen ` fld )t  ( A [,] B ) ) ) `
 ( A [,) B ) ) )
10699, 105eleqtrd 2557 . . 3  |-  ( ph  ->  A  e.  ( ( int `  ( (
TopOpen ` fld )t  ( A [,] B
) ) ) `  ( A [,) B ) ) )
10788snssd 4178 . . . . . . . 8  |-  ( ph  ->  { A }  C_  ( A [,] B ) )
108 ssequn2 3682 . . . . . . . 8  |-  ( { A }  C_  ( A [,] B )  <->  ( ( A [,] B )  u. 
{ A } )  =  ( A [,] B ) )
109107, 108sylib 196 . . . . . . 7  |-  ( ph  ->  ( ( A [,] B )  u.  { A } )  =  ( A [,] B ) )
110109eqcomd 2475 . . . . . 6  |-  ( ph  ->  ( A [,] B
)  =  ( ( A [,] B )  u.  { A }
) )
111110oveq2d 6311 . . . . 5  |-  ( ph  ->  ( ( TopOpen ` fld )t  ( A [,] B ) )  =  ( ( TopOpen ` fld )t  ( ( A [,] B )  u. 
{ A } ) ) )
112111fveq2d 5876 . . . 4  |-  ( ph  ->  ( int `  (
( TopOpen ` fld )t  ( A [,] B ) ) )  =  ( int `  (
( TopOpen ` fld )t  ( ( A [,] B )  u. 
{ A } ) ) ) )
113 uncom 3653 . . . . . . 7  |-  ( ( A (,) B )  u.  { A }
)  =  ( { A }  u.  ( A (,) B ) )
114113a1i 11 . . . . . 6  |-  ( ph  ->  ( ( A (,) B )  u.  { A } )  =  ( { A }  u.  ( A (,) B ) ) )
115 snunioo 11658 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  ( { A }  u.  ( A (,) B ) )  =  ( A [,) B ) )
11661, 19, 15, 115syl3anc 1228 . . . . . 6  |-  ( ph  ->  ( { A }  u.  ( A (,) B
) )  =  ( A [,) B ) )
117114, 116eqtrd 2508 . . . . 5  |-  ( ph  ->  ( ( A (,) B )  u.  { A } )  =  ( A [,) B ) )
118117eqcomd 2475 . . . 4  |-  ( ph  ->  ( A [,) B
)  =  ( ( A (,) B )  u.  { A }
) )
119112, 118fveq12d 5878 . . 3  |-  ( ph  ->  ( ( int `  (
( TopOpen ` fld )t  ( A [,] B ) ) ) `
 ( A [,) B ) )  =  ( ( int `  (
( TopOpen ` fld )t  ( ( A [,] B )  u. 
{ A } ) ) ) `  (
( A (,) B
)  u.  { A } ) ) )
120106, 119eleqtrd 2557 . 2  |-  ( ph  ->  A  e.  ( ( int `  ( (
TopOpen ` fld )t  ( ( A [,] B )  u.  { A } ) ) ) `
 ( ( A (,) B )  u. 
{ A } ) ) )
1211, 3, 10, 11, 12, 120limcres 22158 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 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2817    \ cdif 3478    u. cun 3479    i^i cin 3480    C_ wss 3481   {csn 4033   U.cuni 4251   class class class wbr 4453   ran crn 5006    |` cres 5007   -->wf 5590   ` cfv 5594  (class class class)co 6295   CCcc 9502   RRcr 9503   -oocmnf 9638   RR*cxr 9639    < clt 9640    <_ cle 9641   (,)cioo 11541   [,)cico 11543   [,]cicc 11544   ↾t crest 14693   TopOpenctopn 14694   topGenctg 14710  ℂfldccnfld 18290   Topctop 19263   intcnt 19386   lim CC climc 22134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fi 7883  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-ioo 11545  df-ico 11547  df-icc 11548  df-fz 11685  df-seq 12088  df-exp 12147  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-plusg 14585  df-mulr 14586  df-starv 14587  df-tset 14591  df-ple 14592  df-ds 14594  df-unif 14595  df-rest 14695  df-topn 14696  df-topgen 14716  df-psmet 18281  df-xmet 18282  df-met 18283  df-bl 18284  df-mopn 18285  df-cnfld 18291  df-top 19268  df-bases 19270  df-topon 19271  df-topsp 19272  df-cld 19388  df-ntr 19389  df-cls 19390  df-cnp 19597  df-xms 20691  df-ms 20692  df-limc 22138
This theorem is referenced by:  fourierdlem82  31812  fourierdlem93  31823  fourierdlem111  31841
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