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Theorem iblcncfioo 37855
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 21925 . . . . 5  |-  ( F  e.  ( ( A (,) B ) -cn-> CC )  ->  F :
( A (,) B
) --> CC )
31, 2syl 17 . . . 4  |-  ( ph  ->  F : ( A (,) B ) --> CC )
43feqmptd 5918 . . 3  |-  ( ph  ->  F  =  ( x  e.  ( A (,) B )  |->  ( F `
 x ) ) )
5 iblcncfioo.a . . . . . . . . . 10  |-  ( ph  ->  A  e.  RR )
65adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  A  e.  RR )
7 eliooord 11694 . . . . . . . . . . 11  |-  ( x  e.  ( A (,) B )  ->  ( A  <  x  /\  x  <  B ) )
87simpld 461 . . . . . . . . . 10  |-  ( x  e.  ( A (,) B )  ->  A  <  x )
98adantl 468 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  A  <  x )
106, 9gtned 9770 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  x  =/=  A )
1110neneqd 2629 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  -.  x  =  A )
1211iffalsed 3892 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  if (
x  =  A ,  R ,  if (
x  =  B ,  L ,  ( F `  x ) ) )  =  if ( x  =  B ,  L ,  ( F `  x ) ) )
13 elioore 11666 . . . . . . . . . 10  |-  ( x  e.  ( A (,) B )  ->  x  e.  RR )
1413adantl 468 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  x  e.  RR )
157simprd 465 . . . . . . . . . 10  |-  ( x  e.  ( A (,) B )  ->  x  <  B )
1615adantl 468 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  x  <  B )
1714, 16ltned 9771 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  x  =/=  B )
1817neneqd 2629 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  -.  x  =  B )
1918iffalsed 3892 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  if (
x  =  B ,  L ,  ( F `  x ) )  =  ( F `  x
) )
2012, 19eqtrd 2485 . . . . 5  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  if (
x  =  A ,  R ,  if (
x  =  B ,  L ,  ( F `  x ) ) )  =  ( F `  x ) )
2120eqcomd 2457 . . . 4  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( F `  x )  =  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `
 x ) ) ) )
2221mpteq2dva 4489 . . 3  |-  ( ph  ->  ( x  e.  ( A (,) B ) 
|->  ( F `  x
) )  =  ( x  e.  ( A (,) B )  |->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `
 x ) ) ) ) )
234, 22eqtrd 2485 . 2  |-  ( ph  ->  F  =  ( x  e.  ( A (,) B )  |->  if ( x  =  A ,  R ,  if (
x  =  B ,  L ,  ( F `  x ) ) ) ) )
24 ioossicc 11720 . . . 4  |-  ( A (,) B )  C_  ( A [,] B )
2524a1i 11 . . 3  |-  ( ph  ->  ( A (,) B
)  C_  ( A [,] B ) )
26 ioombl 22518 . . . 4  |-  ( A (,) B )  e. 
dom  vol
2726a1i 11 . . 3  |-  ( ph  ->  ( A (,) B
)  e.  dom  vol )
28 iftrue 3887 . . . . . . 7  |-  ( x  =  A  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  =  R )
2928adantl 468 . . . . . 6  |-  ( (
ph  /\  x  =  A )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  =  R )
30 limccl 22830 . . . . . . . 8  |-  ( F lim
CC  A )  C_  CC
31 iblcncfioo.r . . . . . . . 8  |-  ( ph  ->  R  e.  ( F lim
CC  A ) )
3230, 31sseldi 3430 . . . . . . 7  |-  ( ph  ->  R  e.  CC )
3332adantr 467 . . . . . 6  |-  ( (
ph  /\  x  =  A )  ->  R  e.  CC )
3429, 33eqeltrd 2529 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  e.  CC )
3534adantlr 721 . . . 4  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  x  =  A )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  e.  CC )
36 iffalse 3890 . . . . . . . . 9  |-  ( -.  x  =  A  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `
 x ) ) )  =  if ( x  =  B ,  L ,  ( F `  x ) ) )
3736ad2antlr 733 . . . . . . . 8  |-  ( ( ( ph  /\  -.  x  =  A )  /\  x  =  B
)  ->  if (
x  =  A ,  R ,  if (
x  =  B ,  L ,  ( F `  x ) ) )  =  if ( x  =  B ,  L ,  ( F `  x ) ) )
38 iftrue 3887 . . . . . . . . 9  |-  ( x  =  B  ->  if ( x  =  B ,  L ,  ( F `
 x ) )  =  L )
3938adantl 468 . . . . . . . 8  |-  ( ( ( ph  /\  -.  x  =  A )  /\  x  =  B
)  ->  if (
x  =  B ,  L ,  ( F `  x ) )  =  L )
4037, 39eqtrd 2485 . . . . . . 7  |-  ( ( ( ph  /\  -.  x  =  A )  /\  x  =  B
)  ->  if (
x  =  A ,  R ,  if (
x  =  B ,  L ,  ( F `  x ) ) )  =  L )
41 limccl 22830 . . . . . . . . 9  |-  ( F lim
CC  B )  C_  CC
42 iblcncfioo.l . . . . . . . . 9  |-  ( ph  ->  L  e.  ( F lim
CC  B ) )
4341, 42sseldi 3430 . . . . . . . 8  |-  ( ph  ->  L  e.  CC )
4443ad2antrr 732 . . . . . . 7  |-  ( ( ( ph  /\  -.  x  =  A )  /\  x  =  B
)  ->  L  e.  CC )
4540, 44eqeltrd 2529 . . . . . 6  |-  ( ( ( ph  /\  -.  x  =  A )  /\  x  =  B
)  ->  if (
x  =  A ,  R ,  if (
x  =  B ,  L ,  ( F `  x ) ) )  e.  CC )
4645adantllr 725 . . . . 5  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  x  =  B )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  e.  CC )
47 simplll 768 . . . . . . 7  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  ph )
485rexrd 9690 . . . . . . . . . 10  |-  ( ph  ->  A  e.  RR* )
4947, 48syl 17 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  A  e.  RR* )
50 iblcncfioo.b . . . . . . . . . . 11  |-  ( ph  ->  B  e.  RR )
5150rexrd 9690 . . . . . . . . . 10  |-  ( ph  ->  B  e.  RR* )
5247, 51syl 17 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  B  e.  RR* )
53 eliccxr 37612 . . . . . . . . . 10  |-  ( x  e.  ( A [,] B )  ->  x  e.  RR* )
5453ad3antlr 737 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  x  e.  RR* )
5549, 52, 543jca 1188 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  ( A  e.  RR*  /\  B  e.  RR*  /\  x  e. 
RR* ) )
565ad2antrr 732 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  A  e.  RR )
575adantr 467 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A  e.  RR )
5850adantr 467 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  B  e.  RR )
59 simpr 463 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  e.  ( A [,] B ) )
60 eliccre 37603 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  x  e.  ( A [,] B
) )  ->  x  e.  RR )
6157, 58, 59, 60syl3anc 1268 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  e.  RR )
6261adantr 467 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  x  e.  RR )
635, 50jca 535 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( A  e.  RR  /\  B  e.  RR ) )
6463adantr 467 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( A  e.  RR  /\  B  e.  RR ) )
65 elicc2 11699 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( x  e.  ( A [,] B )  <-> 
( x  e.  RR  /\  A  <_  x  /\  x  <_  B ) ) )
6664, 65syl 17 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( x  e.  ( A [,] B
)  <->  ( x  e.  RR  /\  A  <_  x  /\  x  <_  B
) ) )
6759, 66mpbid 214 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( x  e.  RR  /\  A  <_  x  /\  x  <_  B
) )
6867simp2d 1021 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A  <_  x )
6968adantr 467 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  A  <_  x )
70 df-ne 2624 . . . . . . . . . . . . 13  |-  ( x  =/=  A  <->  -.  x  =  A )
7170biimpri 210 . . . . . . . . . . . 12  |-  ( -.  x  =  A  ->  x  =/=  A )
7271adantl 468 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  x  =/=  A )
7356, 62, 69, 72leneltd 9789 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  A  <  x )
7473adantr 467 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  A  <  x )
75 nesym 2680 . . . . . . . . . . . . 13  |-  ( B  =/=  x  <->  -.  x  =  B )
7675biimpri 210 . . . . . . . . . . . 12  |-  ( -.  x  =  B  ->  B  =/=  x )
7776adantl 468 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  B )  ->  B  =/=  x )
7867simp3d 1022 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  <_  B )
7961, 58, 783jca 1188 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( x  e.  RR  /\  B  e.  RR  /\  x  <_  B ) )
8079adantr 467 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  B )  ->  ( x  e.  RR  /\  B  e.  RR  /\  x  <_  B ) )
81 leltne 9723 . . . . . . . . . . . 12  |-  ( ( x  e.  RR  /\  B  e.  RR  /\  x  <_  B )  ->  (
x  <  B  <->  B  =/=  x ) )
8280, 81syl 17 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  B )  ->  ( x  <  B  <->  B  =/=  x ) )
8377, 82mpbird 236 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  B )  ->  x  <  B )
8483adantlr 721 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  x  <  B )
8574, 84jca 535 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  ( A  <  x  /\  x  <  B ) )
86 elioo3g 11665 . . . . . . . 8  |-  ( x  e.  ( A (,) B )  <->  ( ( A  e.  RR*  /\  B  e.  RR*  /\  x  e. 
RR* )  /\  ( A  <  x  /\  x  <  B ) ) )
8755, 85, 86sylanbrc 670 . . . . . . 7  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  x  e.  ( A (,) B
) )
8847, 87jca 535 . . . . . 6  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  ( ph  /\  x  e.  ( A (,) B ) ) )
893ffvelrnda 6022 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( F `  x )  e.  CC )
9020, 89eqeltrd 2529 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  if (
x  =  A ,  R ,  if (
x  =  B ,  L ,  ( F `  x ) ) )  e.  CC )
9188, 90syl 17 . . . . 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 800 . . . 4  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  e.  CC )
9335, 92pm2.61dan 800 . . 3  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  if (
x  =  A ,  R ,  if (
x  =  B ,  L ,  ( F `  x ) ) )  e.  CC )
94 nfv 1761 . . . . 5  |-  F/ x ph
95 eqid 2451 . . . . 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 37772 . . . 4  |-  ( ph  ->  ( x  e.  ( A [,] B ) 
|->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) ) )  e.  ( ( A [,] B ) -cn-> CC ) )
97 cniccibl 22798 . . . 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 1268 . . 3  |-  ( ph  ->  ( x  e.  ( A [,] B ) 
|->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) ) )  e.  L^1 )
9925, 27, 93, 98iblss 22762 . 2  |-  ( ph  ->  ( x  e.  ( A (,) B ) 
|->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) ) )  e.  L^1 )
10023, 99eqeltrd 2529 1  |-  ( ph  ->  F  e.  L^1 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622    C_ wss 3404   ifcif 3881   class class class wbr 4402    |-> cmpt 4461   dom cdm 4834   -->wf 5578   ` cfv 5582  (class class class)co 6290   CCcc 9537   RRcr 9538   RR*cxr 9674    < clt 9675    <_ cle 9676   (,)cioo 11635   [,]cicc 11638   -cn->ccncf 21908   volcvol 22415   L^1cibl 22575   lim CC climc 22817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cc 8865  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-disj 4374  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-ofr 6532  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-omul 7187  df-er 7363  df-map 7474  df-pm 7475  df-ixp 7523  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-fi 7925  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-acn 8376  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  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-ioc 11640  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-fl 12028  df-mod 12097  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-limsup 13526  df-clim 13552  df-rlim 13553  df-sum 13753  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-starv 15205  df-sca 15206  df-vsca 15207  df-ip 15208  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-hom 15214  df-cco 15215  df-rest 15321  df-topn 15322  df-0g 15340  df-gsum 15341  df-topgen 15342  df-pt 15343  df-prds 15346  df-xrs 15400  df-qtop 15406  df-imas 15407  df-xps 15410  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-mulg 16676  df-cntz 16971  df-cmn 17432  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-cnfld 18971  df-top 19921  df-bases 19922  df-topon 19923  df-topsp 19924  df-cld 20034  df-ntr 20035  df-cls 20036  df-cn 20243  df-cnp 20244  df-cmp 20402  df-tx 20577  df-hmeo 20770  df-xms 21335  df-ms 21336  df-tms 21337  df-cncf 21910  df-ovol 22416  df-vol 22418  df-mbf 22577  df-itg1 22578  df-itg2 22579  df-ibl 22580  df-0p 22628  df-limc 22821
This theorem is referenced by:  fourierdlem69  38039  fourierdlem73  38043  fourierdlem81  38051  fourierdlem93  38063
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