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Theorem fourierdlem33 38115
Description: Limit of a continuous function on an open subinterval. Upper bound version. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
fourierdlem33.1  |-  ( ph  ->  A  e.  RR )
fourierdlem33.2  |-  ( ph  ->  B  e.  RR )
fourierdlem33.3  |-  ( ph  ->  A  <  B )
fourierdlem33.4  |-  ( ph  ->  F  e.  ( ( A (,) B )
-cn-> CC ) )
fourierdlem33.5  |-  ( ph  ->  L  e.  ( F lim
CC  B ) )
fourierdlem33.6  |-  ( ph  ->  C  e.  RR )
fourierdlem33.7  |-  ( ph  ->  D  e.  RR )
fourierdlem33.8  |-  ( ph  ->  C  <  D )
fourierdlem33.ss  |-  ( ph  ->  ( C (,) D
)  C_  ( A (,) B ) )
fourierdlem33.y  |-  Y  =  if ( D  =  B ,  L , 
( F `  D
) )
fourierdlem33.10  |-  J  =  ( ( TopOpen ` fld )t  ( ( A (,) B )  u. 
{ B } ) )
Assertion
Ref Expression
fourierdlem33  |-  ( ph  ->  Y  e.  ( ( F  |`  ( C (,) D ) ) lim CC  D ) )

Proof of Theorem fourierdlem33
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fourierdlem33.5 . . . 4  |-  ( ph  ->  L  e.  ( F lim
CC  B ) )
21adantr 472 . . 3  |-  ( (
ph  /\  D  =  B )  ->  L  e.  ( F lim CC  B
) )
3 fourierdlem33.y . . . . 5  |-  Y  =  if ( D  =  B ,  L , 
( F `  D
) )
4 iftrue 3878 . . . . 5  |-  ( D  =  B  ->  if ( D  =  B ,  L ,  ( F `
 D ) )  =  L )
53, 4syl5req 2518 . . . 4  |-  ( D  =  B  ->  L  =  Y )
65adantl 473 . . 3  |-  ( (
ph  /\  D  =  B )  ->  L  =  Y )
7 oveq2 6316 . . . . 5  |-  ( D  =  B  ->  (
( F  |`  ( C (,) D ) ) lim
CC  D )  =  ( ( F  |`  ( C (,) D ) ) lim CC  B ) )
87adantl 473 . . . 4  |-  ( (
ph  /\  D  =  B )  ->  (
( F  |`  ( C (,) D ) ) lim
CC  D )  =  ( ( F  |`  ( C (,) D ) ) lim CC  B ) )
9 fourierdlem33.4 . . . . . . 7  |-  ( ph  ->  F  e.  ( ( A (,) B )
-cn-> CC ) )
10 cncff 22003 . . . . . . 7  |-  ( F  e.  ( ( A (,) B ) -cn-> CC )  ->  F :
( A (,) B
) --> CC )
119, 10syl 17 . . . . . 6  |-  ( ph  ->  F : ( A (,) B ) --> CC )
1211adantr 472 . . . . 5  |-  ( (
ph  /\  D  =  B )  ->  F : ( A (,) B ) --> CC )
13 fourierdlem33.ss . . . . . 6  |-  ( ph  ->  ( C (,) D
)  C_  ( A (,) B ) )
1413adantr 472 . . . . 5  |-  ( (
ph  /\  D  =  B )  ->  ( C (,) D )  C_  ( A (,) B ) )
15 ioosscn 37687 . . . . . 6  |-  ( A (,) B )  C_  CC
1615a1i 11 . . . . 5  |-  ( (
ph  /\  D  =  B )  ->  ( A (,) B )  C_  CC )
17 eqid 2471 . . . . 5  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
18 fourierdlem33.10 . . . . 5  |-  J  =  ( ( TopOpen ` fld )t  ( ( A (,) B )  u. 
{ B } ) )
19 fourierdlem33.7 . . . . . . . . 9  |-  ( ph  ->  D  e.  RR )
20 fourierdlem33.8 . . . . . . . . 9  |-  ( ph  ->  C  <  D )
2119leidd 10201 . . . . . . . . 9  |-  ( ph  ->  D  <_  D )
22 fourierdlem33.6 . . . . . . . . . . 11  |-  ( ph  ->  C  e.  RR )
2322rexrd 9708 . . . . . . . . . 10  |-  ( ph  ->  C  e.  RR* )
24 elioc2 11722 . . . . . . . . . 10  |-  ( ( C  e.  RR*  /\  D  e.  RR )  ->  ( D  e.  ( C (,] D )  <->  ( D  e.  RR  /\  C  < 
D  /\  D  <_  D ) ) )
2523, 19, 24syl2anc 673 . . . . . . . . 9  |-  ( ph  ->  ( D  e.  ( C (,] D )  <-> 
( D  e.  RR  /\  C  <  D  /\  D  <_  D ) ) )
2619, 20, 21, 25mpbir3and 1213 . . . . . . . 8  |-  ( ph  ->  D  e.  ( C (,] D ) )
2726adantr 472 . . . . . . 7  |-  ( (
ph  /\  D  =  B )  ->  D  e.  ( C (,] D
) )
28 eqcom 2478 . . . . . . . . 9  |-  ( D  =  B  <->  B  =  D )
2928biimpi 199 . . . . . . . 8  |-  ( D  =  B  ->  B  =  D )
3029adantl 473 . . . . . . 7  |-  ( (
ph  /\  D  =  B )  ->  B  =  D )
3117cnfldtop 21882 . . . . . . . . . . 11  |-  ( TopOpen ` fld )  e.  Top
32 fourierdlem33.1 . . . . . . . . . . . . . 14  |-  ( ph  ->  A  e.  RR )
3332rexrd 9708 . . . . . . . . . . . . 13  |-  ( ph  ->  A  e.  RR* )
34 fourierdlem33.2 . . . . . . . . . . . . . 14  |-  ( ph  ->  B  e.  RR )
3534rexrd 9708 . . . . . . . . . . . . 13  |-  ( ph  ->  B  e.  RR* )
36 fourierdlem33.3 . . . . . . . . . . . . 13  |-  ( ph  ->  A  <  B )
37 snunioo2 37702 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( A (,) B
)  u.  { B } )  =  ( A (,] B ) )
3833, 35, 36, 37syl3anc 1292 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( A (,) B )  u.  { B } )  =  ( A (,] B ) )
39 ovex 6336 . . . . . . . . . . . . 13  |-  ( A (,] B )  e. 
_V
4039a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  ( A (,] B
)  e.  _V )
4138, 40eqeltrd 2549 . . . . . . . . . . 11  |-  ( ph  ->  ( ( A (,) B )  u.  { B } )  e.  _V )
42 resttop 20253 . . . . . . . . . . 11  |-  ( ( ( TopOpen ` fld )  e.  Top  /\  ( ( A (,) B )  u.  { B } )  e.  _V )  ->  ( ( TopOpen ` fld )t  (
( A (,) B
)  u.  { B } ) )  e. 
Top )
4331, 41, 42sylancr 676 . . . . . . . . . 10  |-  ( ph  ->  ( ( TopOpen ` fld )t  ( ( A (,) B )  u. 
{ B } ) )  e.  Top )
4418, 43syl5eqel 2553 . . . . . . . . 9  |-  ( ph  ->  J  e.  Top )
4544adantr 472 . . . . . . . 8  |-  ( (
ph  /\  D  =  B )  ->  J  e.  Top )
46 oveq2 6316 . . . . . . . . . . 11  |-  ( D  =  B  ->  ( C (,] D )  =  ( C (,] B
) )
4746adantl 473 . . . . . . . . . 10  |-  ( (
ph  /\  D  =  B )  ->  ( C (,] D )  =  ( C (,] B
) )
4823adantr 472 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( C (,] B ) )  ->  C  e.  RR* )
49 pnfxr 11435 . . . . . . . . . . . . . . . . 17  |- +oo  e.  RR*
5049a1i 11 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( C (,] B ) )  -> +oo  e.  RR* )
51 simpr 468 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  x  e.  ( C (,] B ) )  ->  x  e.  ( C (,] B ) )
5234adantr 472 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  x  e.  ( C (,] B ) )  ->  B  e.  RR )
53 elioc2 11722 . . . . . . . . . . . . . . . . . . 19  |-  ( ( C  e.  RR*  /\  B  e.  RR )  ->  (
x  e.  ( C (,] B )  <->  ( x  e.  RR  /\  C  < 
x  /\  x  <_  B ) ) )
5448, 52, 53syl2anc 673 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  x  e.  ( C (,] B ) )  ->  ( x  e.  ( C (,] B
)  <->  ( x  e.  RR  /\  C  < 
x  /\  x  <_  B ) ) )
5551, 54mpbid 215 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  ( C (,] B ) )  ->  ( x  e.  RR  /\  C  < 
x  /\  x  <_  B ) )
5655simp1d 1042 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( C (,] B ) )  ->  x  e.  RR )
5755simp2d 1043 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( C (,] B ) )  ->  C  <  x )
5856ltpnfd 11446 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( C (,] B ) )  ->  x  < +oo )
5948, 50, 56, 57, 58eliood 37691 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( C (,] B ) )  ->  x  e.  ( C (,) +oo )
)
6032adantr 472 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  ( C (,] B ) )  ->  A  e.  RR )
6122adantr 472 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  ( C (,] B ) )  ->  C  e.  RR )
6232, 34, 22, 19, 20, 13fourierdlem10 38091 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( A  <_  C  /\  D  <_  B ) )
6362simpld 466 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  A  <_  C )
6463adantr 472 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  ( C (,] B ) )  ->  A  <_  C )
6560, 61, 56, 64, 57lelttrd 9810 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( C (,] B ) )  ->  A  <  x )
6655simp3d 1044 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( C (,] B ) )  ->  x  <_  B )
6733adantr 472 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  ( C (,] B ) )  ->  A  e.  RR* )
68 elioc2 11722 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  (
x  e.  ( A (,] B )  <->  ( x  e.  RR  /\  A  < 
x  /\  x  <_  B ) ) )
6967, 52, 68syl2anc 673 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( C (,] B ) )  ->  ( x  e.  ( A (,] B
)  <->  ( x  e.  RR  /\  A  < 
x  /\  x  <_  B ) ) )
7056, 65, 66, 69mpbir3and 1213 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( C (,] B ) )  ->  x  e.  ( A (,] B ) )
7159, 70elind 3609 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( C (,] B ) )  ->  x  e.  ( ( C (,) +oo )  i^i  ( A (,] B ) ) )
72 elinel1 3610 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( ( C (,) +oo )  i^i  ( A (,] B
) )  ->  x  e.  ( C (,) +oo ) )
73 elioore 11691 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( C (,) +oo )  ->  x  e.  RR )
7472, 73syl 17 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ( ( C (,) +oo )  i^i  ( A (,] B
) )  ->  x  e.  RR )
7574adantl 473 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( ( C (,) +oo )  i^i  ( A (,] B ) ) )  ->  x  e.  RR )
7623adantr 472 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( ( C (,) +oo )  i^i  ( A (,] B ) ) )  ->  C  e.  RR* )
7749a1i 11 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( ( C (,) +oo )  i^i  ( A (,] B ) ) )  -> +oo  e.  RR* )
7872adantl 473 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( ( C (,) +oo )  i^i  ( A (,] B ) ) )  ->  x  e.  ( C (,) +oo )
)
79 ioogtlb 37688 . . . . . . . . . . . . . . . 16  |-  ( ( C  e.  RR*  /\ +oo  e.  RR*  /\  x  e.  ( C (,) +oo ) )  ->  C  <  x )
8076, 77, 78, 79syl3anc 1292 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( ( C (,) +oo )  i^i  ( A (,] B ) ) )  ->  C  <  x )
81 elinel2 3611 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  ( ( C (,) +oo )  i^i  ( A (,] B
) )  ->  x  e.  ( A (,] B
) )
8281adantl 473 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  ( ( C (,) +oo )  i^i  ( A (,] B ) ) )  ->  x  e.  ( A (,] B ) )
8333adantr 472 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  x  e.  ( ( C (,) +oo )  i^i  ( A (,] B ) ) )  ->  A  e.  RR* )
8434adantr 472 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  x  e.  ( ( C (,) +oo )  i^i  ( A (,] B ) ) )  ->  B  e.  RR )
8583, 84, 68syl2anc 673 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  ( ( C (,) +oo )  i^i  ( A (,] B ) ) )  ->  ( x  e.  ( A (,] B
)  <->  ( x  e.  RR  /\  A  < 
x  /\  x  <_  B ) ) )
8682, 85mpbid 215 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( ( C (,) +oo )  i^i  ( A (,] B ) ) )  ->  ( x  e.  RR  /\  A  < 
x  /\  x  <_  B ) )
8786simp3d 1044 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( ( C (,) +oo )  i^i  ( A (,] B ) ) )  ->  x  <_  B )
8876, 84, 53syl2anc 673 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( ( C (,) +oo )  i^i  ( A (,] B ) ) )  ->  ( x  e.  ( C (,] B
)  <->  ( x  e.  RR  /\  C  < 
x  /\  x  <_  B ) ) )
8975, 80, 87, 88mpbir3and 1213 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( ( C (,) +oo )  i^i  ( A (,] B ) ) )  ->  x  e.  ( C (,] B ) )
9071, 89impbida 850 . . . . . . . . . . . . 13  |-  ( ph  ->  ( x  e.  ( C (,] B )  <-> 
x  e.  ( ( C (,) +oo )  i^i  ( A (,] B
) ) ) )
9190eqrdv 2469 . . . . . . . . . . . 12  |-  ( ph  ->  ( C (,] B
)  =  ( ( C (,) +oo )  i^i  ( A (,] B
) ) )
92 retop 21860 . . . . . . . . . . . . . 14  |-  ( topGen ` 
ran  (,) )  e.  Top
9392a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  ( topGen `  ran  (,) )  e.  Top )
94 iooretop 21864 . . . . . . . . . . . . . 14  |-  ( C (,) +oo )  e.  ( topGen `  ran  (,) )
9594a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  ( C (,) +oo )  e.  ( topGen ` 
ran  (,) ) )
96 elrestr 15405 . . . . . . . . . . . . 13  |-  ( ( ( topGen `  ran  (,) )  e.  Top  /\  ( A (,] B )  e. 
_V  /\  ( C (,) +oo )  e.  (
topGen `  ran  (,) )
)  ->  ( ( C (,) +oo )  i^i  ( A (,] B
) )  e.  ( ( topGen `  ran  (,) )t  ( A (,] B ) ) )
9793, 40, 95, 96syl3anc 1292 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( C (,) +oo )  i^i  ( A (,] B ) )  e.  ( ( topGen ` 
ran  (,) )t  ( A (,] B ) ) )
9891, 97eqeltrd 2549 . . . . . . . . . . 11  |-  ( ph  ->  ( C (,] B
)  e.  ( (
topGen `  ran  (,) )t  ( A (,] B ) ) )
9998adantr 472 . . . . . . . . . 10  |-  ( (
ph  /\  D  =  B )  ->  ( C (,] B )  e.  ( ( topGen `  ran  (,) )t  ( A (,] B
) ) )
10047, 99eqeltrd 2549 . . . . . . . . 9  |-  ( (
ph  /\  D  =  B )  ->  ( C (,] D )  e.  ( ( topGen `  ran  (,) )t  ( A (,] B
) ) )
10118a1i 11 . . . . . . . . . . 11  |-  ( ph  ->  J  =  ( (
TopOpen ` fld )t  ( ( A (,) B )  u.  { B } ) ) )
10238oveq2d 6324 . . . . . . . . . . 11  |-  ( ph  ->  ( ( TopOpen ` fld )t  ( ( A (,) B )  u. 
{ B } ) )  =  ( (
TopOpen ` fld )t  ( A (,] B
) ) )
10317tgioo2 21899 . . . . . . . . . . . . . 14  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
104103eqcomi 2480 . . . . . . . . . . . . 13  |-  ( (
TopOpen ` fld )t  RR )  =  (
topGen `  ran  (,) )
105104oveq1i 6318 . . . . . . . . . . . 12  |-  ( ( ( TopOpen ` fld )t  RR )t  ( A (,] B ) )  =  ( ( topGen `  ran  (,) )t  ( A (,] B
) )
10631a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  ( TopOpen ` fld )  e.  Top )
107 iocssre 11739 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( A (,] B )  C_  RR )
10833, 34, 107syl2anc 673 . . . . . . . . . . . . 13  |-  ( ph  ->  ( A (,] B
)  C_  RR )
109 reex 9648 . . . . . . . . . . . . . 14  |-  RR  e.  _V
110109a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  RR  e.  _V )
111 restabs 20258 . . . . . . . . . . . . 13  |-  ( ( ( TopOpen ` fld )  e.  Top  /\  ( A (,] B
)  C_  RR  /\  RR  e.  _V )  ->  (
( ( TopOpen ` fld )t  RR )t  ( A (,] B ) )  =  ( ( TopOpen ` fld )t  ( A (,] B ) ) )
112106, 108, 110, 111syl3anc 1292 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( TopOpen ` fld )t  RR )t  ( A (,] B ) )  =  ( (
TopOpen ` fld )t  ( A (,] B
) ) )
113105, 112syl5reqr 2520 . . . . . . . . . . 11  |-  ( ph  ->  ( ( TopOpen ` fld )t  ( A (,] B ) )  =  ( ( topGen `  ran  (,) )t  ( A (,] B
) ) )
114101, 102, 1133eqtrrd 2510 . . . . . . . . . 10  |-  ( ph  ->  ( ( topGen `  ran  (,) )t  ( A (,] B
) )  =  J )
115114adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  D  =  B )  ->  (
( topGen `  ran  (,) )t  ( A (,] B ) )  =  J )
116100, 115eleqtrd 2551 . . . . . . . 8  |-  ( (
ph  /\  D  =  B )  ->  ( C (,] D )  e.  J )
117 isopn3i 20175 . . . . . . . 8  |-  ( ( J  e.  Top  /\  ( C (,] D )  e.  J )  -> 
( ( int `  J
) `  ( C (,] D ) )  =  ( C (,] D
) )
11845, 116, 117syl2anc 673 . . . . . . 7  |-  ( (
ph  /\  D  =  B )  ->  (
( int `  J
) `  ( C (,] D ) )  =  ( C (,] D
) )
11927, 30, 1183eltr4d 2564 . . . . . 6  |-  ( (
ph  /\  D  =  B )  ->  B  e.  ( ( int `  J
) `  ( C (,] D ) ) )
120 sneq 3969 . . . . . . . . . . 11  |-  ( D  =  B  ->  { D }  =  { B } )
121120eqcomd 2477 . . . . . . . . . 10  |-  ( D  =  B  ->  { B }  =  { D } )
122121uneq2d 3579 . . . . . . . . 9  |-  ( D  =  B  ->  (
( C (,) D
)  u.  { B } )  =  ( ( C (,) D
)  u.  { D } ) )
123122adantl 473 . . . . . . . 8  |-  ( (
ph  /\  D  =  B )  ->  (
( C (,) D
)  u.  { B } )  =  ( ( C (,) D
)  u.  { D } ) )
12419rexrd 9708 . . . . . . . . . 10  |-  ( ph  ->  D  e.  RR* )
125 snunioo2 37702 . . . . . . . . . 10  |-  ( ( C  e.  RR*  /\  D  e.  RR*  /\  C  < 
D )  ->  (
( C (,) D
)  u.  { D } )  =  ( C (,] D ) )
12623, 124, 20, 125syl3anc 1292 . . . . . . . . 9  |-  ( ph  ->  ( ( C (,) D )  u.  { D } )  =  ( C (,] D ) )
127126adantr 472 . . . . . . . 8  |-  ( (
ph  /\  D  =  B )  ->  (
( C (,) D
)  u.  { D } )  =  ( C (,] D ) )
128123, 127eqtr2d 2506 . . . . . . 7  |-  ( (
ph  /\  D  =  B )  ->  ( C (,] D )  =  ( ( C (,) D )  u.  { B } ) )
129128fveq2d 5883 . . . . . 6  |-  ( (
ph  /\  D  =  B )  ->  (
( int `  J
) `  ( C (,] D ) )  =  ( ( int `  J
) `  ( ( C (,) D )  u. 
{ B } ) ) )
130119, 129eleqtrd 2551 . . . . 5  |-  ( (
ph  /\  D  =  B )  ->  B  e.  ( ( int `  J
) `  ( ( C (,) D )  u. 
{ B } ) ) )
13112, 14, 16, 17, 18, 130limcres 22920 . . . 4  |-  ( (
ph  /\  D  =  B )  ->  (
( F  |`  ( C (,) D ) ) lim
CC  B )  =  ( F lim CC  B
) )
1328, 131eqtr2d 2506 . . 3  |-  ( (
ph  /\  D  =  B )  ->  ( F lim CC  B )  =  ( ( F  |`  ( C (,) D ) ) lim CC  D ) )
1332, 6, 1323eltr3d 2563 . 2  |-  ( (
ph  /\  D  =  B )  ->  Y  e.  ( ( F  |`  ( C (,) D ) ) lim CC  D ) )
134 limcresi 22919 . . 3  |-  ( F lim
CC  D )  C_  ( ( F  |`  ( C (,) D ) ) lim CC  D )
135 iffalse 3881 . . . . . 6  |-  ( -.  D  =  B  ->  if ( D  =  B ,  L ,  ( F `  D ) )  =  ( F `
 D ) )
1363, 135syl5eq 2517 . . . . 5  |-  ( -.  D  =  B  ->  Y  =  ( F `  D ) )
137136adantl 473 . . . 4  |-  ( (
ph  /\  -.  D  =  B )  ->  Y  =  ( F `  D ) )
138 ssid 3437 . . . . . . . . . . . . 13  |-  CC  C_  CC
139138a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  CC  C_  CC )
140 eqid 2471 . . . . . . . . . . . . 13  |-  ( (
TopOpen ` fld )t  ( A (,) B
) )  =  ( ( TopOpen ` fld )t  ( A (,) B ) )
141 unicntop 37433 . . . . . . . . . . . . . . . 16  |-  CC  =  U. ( TopOpen ` fld )
142141restid 15410 . . . . . . . . . . . . . . 15  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
14331, 142ax-mp 5 . . . . . . . . . . . . . 14  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
144143eqcomi 2480 . . . . . . . . . . . . 13  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
14517, 140, 144cncfcn 22019 . . . . . . . . . . . 12  |-  ( ( ( A (,) B
)  C_  CC  /\  CC  C_  CC )  ->  (
( A (,) B
) -cn-> CC )  =  ( ( ( TopOpen ` fld )t  ( A (,) B ) )  Cn  ( TopOpen ` fld ) ) )
14615, 139, 145sylancr 676 . . . . . . . . . . 11  |-  ( ph  ->  ( ( A (,) B ) -cn-> CC )  =  ( ( (
TopOpen ` fld )t  ( A (,) B
) )  Cn  ( TopOpen
` fld
) ) )
1479, 146eleqtrd 2551 . . . . . . . . . 10  |-  ( ph  ->  F  e.  ( ( ( TopOpen ` fld )t  ( A (,) B ) )  Cn  ( TopOpen ` fld ) ) )
14817cnfldtopon 21881 . . . . . . . . . . . 12  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
14915a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  ( A (,) B
)  C_  CC )
150 resttopon 20254 . . . . . . . . . . . 12  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  ( A (,) B )  C_  CC )  ->  ( (
TopOpen ` fld )t  ( A (,) B
) )  e.  (TopOn `  ( A (,) B
) ) )
151148, 149, 150sylancr 676 . . . . . . . . . . 11  |-  ( ph  ->  ( ( TopOpen ` fld )t  ( A (,) B ) )  e.  (TopOn `  ( A (,) B ) ) )
152148a1i 11 . . . . . . . . . . 11  |-  ( ph  ->  ( TopOpen ` fld )  e.  (TopOn `  CC ) )
153 cncnp 20373 . . . . . . . . . . 11  |-  ( ( ( ( TopOpen ` fld )t  ( A (,) B ) )  e.  (TopOn `  ( A (,) B ) )  /\  ( TopOpen ` fld )  e.  (TopOn `  CC ) )  -> 
( F  e.  ( ( ( TopOpen ` fld )t  ( A (,) B ) )  Cn  ( TopOpen ` fld ) )  <->  ( F : ( A (,) B ) --> CC  /\  A. x  e.  ( A (,) B ) F  e.  ( ( ( ( TopOpen ` fld )t  ( A (,) B ) )  CnP  ( TopOpen ` fld ) ) `  x
) ) ) )
154151, 152, 153syl2anc 673 . . . . . . . . . 10  |-  ( ph  ->  ( F  e.  ( ( ( TopOpen ` fld )t  ( A (,) B ) )  Cn  ( TopOpen ` fld ) )  <->  ( F : ( A (,) B ) --> CC  /\  A. x  e.  ( A (,) B ) F  e.  ( ( ( ( TopOpen ` fld )t  ( A (,) B ) )  CnP  ( TopOpen ` fld ) ) `  x
) ) ) )
155147, 154mpbid 215 . . . . . . . . 9  |-  ( ph  ->  ( F : ( A (,) B ) --> CC  /\  A. x  e.  ( A (,) B
) F  e.  ( ( ( ( TopOpen ` fld )t  ( A (,) B ) )  CnP  ( TopOpen ` fld ) ) `  x
) ) )
156155simprd 470 . . . . . . . 8  |-  ( ph  ->  A. x  e.  ( A (,) B ) F  e.  ( ( ( ( TopOpen ` fld )t  ( A (,) B ) )  CnP  ( TopOpen ` fld ) ) `  x
) )
157156adantr 472 . . . . . . 7  |-  ( (
ph  /\  -.  D  =  B )  ->  A. x  e.  ( A (,) B
) F  e.  ( ( ( ( TopOpen ` fld )t  ( A (,) B ) )  CnP  ( TopOpen ` fld ) ) `  x
) )
15833adantr 472 . . . . . . . 8  |-  ( (
ph  /\  -.  D  =  B )  ->  A  e.  RR* )
15935adantr 472 . . . . . . . 8  |-  ( (
ph  /\  -.  D  =  B )  ->  B  e.  RR* )
16019adantr 472 . . . . . . . 8  |-  ( (
ph  /\  -.  D  =  B )  ->  D  e.  RR )
16132, 22, 19, 63, 20lelttrd 9810 . . . . . . . . 9  |-  ( ph  ->  A  <  D )
162161adantr 472 . . . . . . . 8  |-  ( (
ph  /\  -.  D  =  B )  ->  A  <  D )
16334adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  -.  D  =  B )  ->  B  e.  RR )
16462simprd 470 . . . . . . . . . 10  |-  ( ph  ->  D  <_  B )
165164adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  -.  D  =  B )  ->  D  <_  B )
166 neqne 2651 . . . . . . . . . . 11  |-  ( -.  D  =  B  ->  D  =/=  B )
167166necomd 2698 . . . . . . . . . 10  |-  ( -.  D  =  B  ->  B  =/=  D )
168167adantl 473 . . . . . . . . 9  |-  ( (
ph  /\  -.  D  =  B )  ->  B  =/=  D )
169160, 163, 165, 168leneltd 9806 . . . . . . . 8  |-  ( (
ph  /\  -.  D  =  B )  ->  D  <  B )
170158, 159, 160, 162, 169eliood 37691 . . . . . . 7  |-  ( (
ph  /\  -.  D  =  B )  ->  D  e.  ( A (,) B
) )
171 fveq2 5879 . . . . . . . . 9  |-  ( x  =  D  ->  (
( ( ( TopOpen ` fld )t  ( A (,) B ) )  CnP  ( TopOpen ` fld ) ) `  x
)  =  ( ( ( ( TopOpen ` fld )t  ( A (,) B ) )  CnP  ( TopOpen ` fld ) ) `  D
) )
172171eleq2d 2534 . . . . . . . 8  |-  ( x  =  D  ->  ( F  e.  ( (
( ( TopOpen ` fld )t  ( A (,) B ) )  CnP  ( TopOpen ` fld ) ) `  x
)  <->  F  e.  (
( ( ( TopOpen ` fld )t  ( A (,) B ) )  CnP  ( TopOpen ` fld ) ) `  D
) ) )
173172rspccva 3135 . . . . . . 7  |-  ( ( A. x  e.  ( A (,) B ) F  e.  ( ( ( ( TopOpen ` fld )t  ( A (,) B ) )  CnP  ( TopOpen ` fld ) ) `  x
)  /\  D  e.  ( A (,) B ) )  ->  F  e.  ( ( ( (
TopOpen ` fld )t  ( A (,) B
) )  CnP  ( TopOpen
` fld
) ) `  D
) )
174157, 170, 173syl2anc 673 . . . . . 6  |-  ( (
ph  /\  -.  D  =  B )  ->  F  e.  ( ( ( (
TopOpen ` fld )t  ( A (,) B
) )  CnP  ( TopOpen
` fld
) ) `  D
) )
17517, 140cnplimc 22921 . . . . . . 7  |-  ( ( ( A (,) B
)  C_  CC  /\  D  e.  ( A (,) B
) )  ->  ( F  e.  ( (
( ( TopOpen ` fld )t  ( A (,) B ) )  CnP  ( TopOpen ` fld ) ) `  D
)  <->  ( F :
( A (,) B
) --> CC  /\  ( F `  D )  e.  ( F lim CC  D
) ) ) )
17615, 170, 175sylancr 676 . . . . . 6  |-  ( (
ph  /\  -.  D  =  B )  ->  ( F  e.  ( (
( ( TopOpen ` fld )t  ( A (,) B ) )  CnP  ( TopOpen ` fld ) ) `  D
)  <->  ( F :
( A (,) B
) --> CC  /\  ( F `  D )  e.  ( F lim CC  D
) ) ) )
177174, 176mpbid 215 . . . . 5  |-  ( (
ph  /\  -.  D  =  B )  ->  ( F : ( A (,) B ) --> CC  /\  ( F `  D )  e.  ( F lim CC  D ) ) )
178177simprd 470 . . . 4  |-  ( (
ph  /\  -.  D  =  B )  ->  ( F `  D )  e.  ( F lim CC  D
) )
179137, 178eqeltrd 2549 . . 3  |-  ( (
ph  /\  -.  D  =  B )  ->  Y  e.  ( F lim CC  D
) )
180134, 179sseldi 3416 . 2  |-  ( (
ph  /\  -.  D  =  B )  ->  Y  e.  ( ( F  |`  ( C (,) D ) ) lim CC  D ) )
181133, 180pm2.61dan 808 1  |-  ( ph  ->  Y  e.  ( ( F  |`  ( C (,) D ) ) lim CC  D ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   _Vcvv 3031    u. cun 3388    i^i cin 3389    C_ wss 3390   ifcif 3872   {csn 3959   class class class wbr 4395   ran crn 4840    |` cres 4841   -->wf 5585   ` cfv 5589  (class class class)co 6308   CCcc 9555   RRcr 9556   +oocpnf 9690   RR*cxr 9692    < clt 9693    <_ cle 9694   (,)cioo 11660   (,]cioc 11661   ↾t crest 15397   TopOpenctopn 15398   topGenctg 15414  ℂfldccnfld 19047   Topctop 19994  TopOnctopon 19995   intcnt 20109    Cn ccn 20317    CnP ccnp 20318   -cn->ccncf 21986   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-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-ioc 11665  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-ntr 20112  df-cn 20320  df-cnp 20321  df-xms 21413  df-ms 21414  df-cncf 21988  df-limc 22900
This theorem is referenced by:  fourierdlem49  38131  fourierdlem76  38158  fourierdlem91  38173
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