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

Theorem iblcncfioo 31980
Description: A continuous function  F on an open interval  ( A (,) B ) with a finite right limit  R in  A and a finite left limit  L in  B is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
iblcncfioo.a  |-  ( ph  ->  A  e.  RR )
iblcncfioo.b  |-  ( ph  ->  B  e.  RR )
iblcncfioo.f  |-  ( ph  ->  F  e.  ( ( A (,) B )
-cn-> CC ) )
iblcncfioo.l  |-  ( ph  ->  L  e.  ( F lim
CC  B ) )
iblcncfioo.r  |-  ( ph  ->  R  e.  ( F lim
CC  A ) )
Assertion
Ref Expression
iblcncfioo  |-  ( ph  ->  F  e.  L^1 )

Proof of Theorem iblcncfioo
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 iblcncfioo.f . . . . 5  |-  ( ph  ->  F  e.  ( ( A (,) B )
-cn-> CC ) )
2 cncff 21523 . . . . 5  |-  ( F  e.  ( ( A (,) B ) -cn-> CC )  ->  F :
( A (,) B
) --> CC )
31, 2syl 16 . . . 4  |-  ( ph  ->  F : ( A (,) B ) --> CC )
43feqmptd 5926 . . 3  |-  ( ph  ->  F  =  ( x  e.  ( A (,) B )  |->  ( F `
 x ) ) )
5 iblcncfioo.a . . . . . . . . . 10  |-  ( ph  ->  A  e.  RR )
65adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  A  e.  RR )
7 eliooord 11609 . . . . . . . . . . 11  |-  ( x  e.  ( A (,) B )  ->  ( A  <  x  /\  x  <  B ) )
87simpld 459 . . . . . . . . . 10  |-  ( x  e.  ( A (,) B )  ->  A  <  x )
98adantl 466 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  A  <  x )
106, 9gtned 9737 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  x  =/=  A )
1110neneqd 2659 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  -.  x  =  A )
1211iffalsed 3955 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  if (
x  =  A ,  R ,  if (
x  =  B ,  L ,  ( F `  x ) ) )  =  if ( x  =  B ,  L ,  ( F `  x ) ) )
13 elioore 11584 . . . . . . . . . 10  |-  ( x  e.  ( A (,) B )  ->  x  e.  RR )
1413adantl 466 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  x  e.  RR )
157simprd 463 . . . . . . . . . 10  |-  ( x  e.  ( A (,) B )  ->  x  <  B )
1615adantl 466 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  x  <  B )
1714, 16ltned 9738 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  x  =/=  B )
1817neneqd 2659 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  -.  x  =  B )
1918iffalsed 3955 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  if (
x  =  B ,  L ,  ( F `  x ) )  =  ( F `  x
) )
2012, 19eqtrd 2498 . . . . 5  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  if (
x  =  A ,  R ,  if (
x  =  B ,  L ,  ( F `  x ) ) )  =  ( F `  x ) )
2120eqcomd 2465 . . . 4  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( F `  x )  =  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `
 x ) ) ) )
2221mpteq2dva 4543 . . 3  |-  ( ph  ->  ( x  e.  ( A (,) B ) 
|->  ( F `  x
) )  =  ( x  e.  ( A (,) B )  |->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `
 x ) ) ) ) )
234, 22eqtrd 2498 . 2  |-  ( ph  ->  F  =  ( x  e.  ( A (,) B )  |->  if ( x  =  A ,  R ,  if (
x  =  B ,  L ,  ( F `  x ) ) ) ) )
24 ioossicc 11635 . . . 4  |-  ( A (,) B )  C_  ( A [,] B )
2524a1i 11 . . 3  |-  ( ph  ->  ( A (,) B
)  C_  ( A [,] B ) )
26 ioombl 22101 . . . 4  |-  ( A (,) B )  e. 
dom  vol
2726a1i 11 . . 3  |-  ( ph  ->  ( A (,) B
)  e.  dom  vol )
28 iftrue 3950 . . . . . . 7  |-  ( x  =  A  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  =  R )
2928adantl 466 . . . . . 6  |-  ( (
ph  /\  x  =  A )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  =  R )
30 limccl 22405 . . . . . . . 8  |-  ( F lim
CC  A )  C_  CC
31 iblcncfioo.r . . . . . . . 8  |-  ( ph  ->  R  e.  ( F lim
CC  A ) )
3230, 31sseldi 3497 . . . . . . 7  |-  ( ph  ->  R  e.  CC )
3332adantr 465 . . . . . 6  |-  ( (
ph  /\  x  =  A )  ->  R  e.  CC )
3429, 33eqeltrd 2545 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  e.  CC )
3534adantlr 714 . . . 4  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  x  =  A )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  e.  CC )
36 iffalse 3953 . . . . . . . . 9  |-  ( -.  x  =  A  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `
 x ) ) )  =  if ( x  =  B ,  L ,  ( F `  x ) ) )
3736ad2antlr 726 . . . . . . . 8  |-  ( ( ( ph  /\  -.  x  =  A )  /\  x  =  B
)  ->  if (
x  =  A ,  R ,  if (
x  =  B ,  L ,  ( F `  x ) ) )  =  if ( x  =  B ,  L ,  ( F `  x ) ) )
38 iftrue 3950 . . . . . . . . 9  |-  ( x  =  B  ->  if ( x  =  B ,  L ,  ( F `
 x ) )  =  L )
3938adantl 466 . . . . . . . 8  |-  ( ( ( ph  /\  -.  x  =  A )  /\  x  =  B
)  ->  if (
x  =  B ,  L ,  ( F `  x ) )  =  L )
4037, 39eqtrd 2498 . . . . . . 7  |-  ( ( ( ph  /\  -.  x  =  A )  /\  x  =  B
)  ->  if (
x  =  A ,  R ,  if (
x  =  B ,  L ,  ( F `  x ) ) )  =  L )
41 limccl 22405 . . . . . . . . 9  |-  ( F lim
CC  B )  C_  CC
42 iblcncfioo.l . . . . . . . . 9  |-  ( ph  ->  L  e.  ( F lim
CC  B ) )
4341, 42sseldi 3497 . . . . . . . 8  |-  ( ph  ->  L  e.  CC )
4443ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  -.  x  =  A )  /\  x  =  B
)  ->  L  e.  CC )
4540, 44eqeltrd 2545 . . . . . 6  |-  ( ( ( ph  /\  -.  x  =  A )  /\  x  =  B
)  ->  if (
x  =  A ,  R ,  if (
x  =  B ,  L ,  ( F `  x ) ) )  e.  CC )
4645adantllr 718 . . . . 5  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  x  =  B )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  e.  CC )
47 simplll 759 . . . . . . 7  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  ph )
485rexrd 9660 . . . . . . . . . 10  |-  ( ph  ->  A  e.  RR* )
4947, 48syl 16 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  A  e.  RR* )
50 iblcncfioo.b . . . . . . . . . . 11  |-  ( ph  ->  B  e.  RR )
5150rexrd 9660 . . . . . . . . . 10  |-  ( ph  ->  B  e.  RR* )
5247, 51syl 16 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  B  e.  RR* )
53 eliccxr 31753 . . . . . . . . . 10  |-  ( x  e.  ( A [,] B )  ->  x  e.  RR* )
5453ad3antlr 730 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  x  e.  RR* )
5549, 52, 543jca 1176 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  ( A  e.  RR*  /\  B  e.  RR*  /\  x  e. 
RR* ) )
565ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  A  e.  RR )
575adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A  e.  RR )
5850adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  B  e.  RR )
59 simpr 461 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  e.  ( A [,] B ) )
60 eliccre 31743 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  x  e.  ( A [,] B
) )  ->  x  e.  RR )
6157, 58, 59, 60syl3anc 1228 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  e.  RR )
6261adantr 465 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  x  e.  RR )
635, 50jca 532 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( A  e.  RR  /\  B  e.  RR ) )
6463adantr 465 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( A  e.  RR  /\  B  e.  RR ) )
65 elicc2 11614 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( x  e.  ( A [,] B )  <-> 
( x  e.  RR  /\  A  <_  x  /\  x  <_  B ) ) )
6664, 65syl 16 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( x  e.  ( A [,] B
)  <->  ( x  e.  RR  /\  A  <_  x  /\  x  <_  B
) ) )
6759, 66mpbid 210 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( x  e.  RR  /\  A  <_  x  /\  x  <_  B
) )
6867simp2d 1009 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A  <_  x )
6968adantr 465 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  A  <_  x )
70 df-ne 2654 . . . . . . . . . . . . 13  |-  ( x  =/=  A  <->  -.  x  =  A )
7170biimpri 206 . . . . . . . . . . . 12  |-  ( -.  x  =  A  ->  x  =/=  A )
7271adantl 466 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  x  =/=  A )
7356, 62, 69, 72leneltd 31697 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  A  <  x )
7473adantr 465 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  A  <  x )
75 nesym 2729 . . . . . . . . . . . . 13  |-  ( B  =/=  x  <->  -.  x  =  B )
7675biimpri 206 . . . . . . . . . . . 12  |-  ( -.  x  =  B  ->  B  =/=  x )
7776adantl 466 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  B )  ->  B  =/=  x )
7867simp3d 1010 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  <_  B )
7961, 58, 783jca 1176 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( x  e.  RR  /\  B  e.  RR  /\  x  <_  B ) )
8079adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  B )  ->  ( x  e.  RR  /\  B  e.  RR  /\  x  <_  B ) )
81 leltne 9691 . . . . . . . . . . . 12  |-  ( ( x  e.  RR  /\  B  e.  RR  /\  x  <_  B )  ->  (
x  <  B  <->  B  =/=  x ) )
8280, 81syl 16 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  B )  ->  ( x  <  B  <->  B  =/=  x ) )
8377, 82mpbird 232 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  B )  ->  x  <  B )
8483adantlr 714 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  x  <  B )
8574, 84jca 532 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  ( A  <  x  /\  x  <  B ) )
86 elioo3g 11583 . . . . . . . 8  |-  ( x  e.  ( A (,) B )  <->  ( ( A  e.  RR*  /\  B  e.  RR*  /\  x  e. 
RR* )  /\  ( A  <  x  /\  x  <  B ) ) )
8755, 85, 86sylanbrc 664 . . . . . . 7  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  x  e.  ( A (,) B
) )
8847, 87jca 532 . . . . . 6  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  ( ph  /\  x  e.  ( A (,) B ) ) )
893ffvelrnda 6032 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( F `  x )  e.  CC )
9020, 89eqeltrd 2545 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  if (
x  =  A ,  R ,  if (
x  =  B ,  L ,  ( F `  x ) ) )  e.  CC )
9188, 90syl 16 . . . . 5  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  e.  CC )
9246, 91pm2.61dan 791 . . . 4  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  e.  CC )
9335, 92pm2.61dan 791 . . 3  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  if (
x  =  A ,  R ,  if (
x  =  B ,  L ,  ( F `  x ) ) )  e.  CC )
94 nfv 1708 . . . . 5  |-  F/ x ph
95 eqid 2457 . . . . 5  |-  ( x  e.  ( A [,] B )  |->  if ( x  =  A ,  R ,  if (
x  =  B ,  L ,  ( F `  x ) ) ) )  =  ( x  e.  ( A [,] B )  |->  if ( x  =  A ,  R ,  if (
x  =  B ,  L ,  ( F `  x ) ) ) )
9694, 95, 5, 50, 1, 42, 31cncfiooicc 31900 . . . 4  |-  ( ph  ->  ( x  e.  ( A [,] B ) 
|->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) ) )  e.  ( ( A [,] B ) -cn-> CC ) )
97 cniccibl 22373 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  (
x  e.  ( A [,] B )  |->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `
 x ) ) ) )  e.  ( ( A [,] B
) -cn-> CC ) )  -> 
( x  e.  ( A [,] B ) 
|->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) ) )  e.  L^1 )
985, 50, 96, 97syl3anc 1228 . . 3  |-  ( ph  ->  ( x  e.  ( A [,] B ) 
|->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) ) )  e.  L^1 )
9925, 27, 93, 98iblss 22337 . 2  |-  ( ph  ->  ( x  e.  ( A (,) B ) 
|->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) ) )  e.  L^1 )
10023, 99eqeltrd 2545 1  |-  ( ph  ->  F  e.  L^1 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652    C_ wss 3471   ifcif 3944   class class class wbr 4456    |-> cmpt 4515   dom cdm 5008   -->wf 5590   ` cfv 5594  (class class class)co 6296   CCcc 9507   RRcr 9508   RR*cxr 9644    < clt 9645    <_ cle 9646   (,)cioo 11554   [,]cicc 11557   -cn->ccncf 21506   volcvol 22001   L^1cibl 22152   lim CC climc 22392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cc 8832  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588  ax-mulf 9589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-disj 4428  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-ofr 6540  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-omul 7153  df-er 7329  df-map 7440  df-pm 7441  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-fi 7889  df-sup 7919  df-oi 7953  df-card 8337  df-acn 8340  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ioo 11558  df-ioc 11559  df-ico 11560  df-icc 11561  df-fz 11698  df-fzo 11822  df-fl 11932  df-mod 12000  df-seq 12111  df-exp 12170  df-hash 12409  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-limsup 13306  df-clim 13323  df-rlim 13324  df-sum 13521  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-starv 14727  df-sca 14728  df-vsca 14729  df-ip 14730  df-tset 14731  df-ple 14732  df-ds 14734  df-unif 14735  df-hom 14736  df-cco 14737  df-rest 14840  df-topn 14841  df-0g 14859  df-gsum 14860  df-topgen 14861  df-pt 14862  df-prds 14865  df-xrs 14919  df-qtop 14924  df-imas 14925  df-xps 14927  df-mre 15003  df-mrc 15004  df-acs 15006  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-submnd 16094  df-mulg 16187  df-cntz 16482  df-cmn 16927  df-psmet 18538  df-xmet 18539  df-met 18540  df-bl 18541  df-mopn 18542  df-cnfld 18548  df-top 19526  df-bases 19528  df-topon 19529  df-topsp 19530  df-cld 19647  df-ntr 19648  df-cls 19649  df-cn 19855  df-cnp 19856  df-cmp 20014  df-tx 20189  df-hmeo 20382  df-xms 20949  df-ms 20950  df-tms 20951  df-cncf 21508  df-ovol 22002  df-vol 22003  df-mbf 22154  df-itg1 22155  df-itg2 22156  df-ibl 22157  df-0p 22203  df-limc 22396
This theorem is referenced by:  fourierdlem69  32161  fourierdlem73  32165  fourierdlem81  32173  fourierdlem93  32185
  Copyright terms: Public domain W3C validator