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Theorem itgioocnicc 37854
Description: The integral of a piecewise continuous function  F on an open interval is equal to the integral of the continuous function  G, in the corresponding closed interval.  G is equal to  F on the open interval, but it is continuous at the two boundaries, also. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
itgioocnicc.a  |-  ( ph  ->  A  e.  RR )
itgioocnicc.b  |-  ( ph  ->  B  e.  RR )
itgioocnicc.f  |-  ( ph  ->  F : dom  F --> CC )
itgioocnicc.fcn  |-  ( ph  ->  ( F  |`  ( A (,) B ) )  e.  ( ( A (,) B ) -cn-> CC ) )
itgioocnicc.fdom  |-  ( ph  ->  ( A [,] B
)  C_  dom  F )
itgioocnicc.r  |-  ( ph  ->  R  e.  ( ( F  |`  ( A (,) B ) ) lim CC  A ) )
itgioocnicc.l  |-  ( ph  ->  L  e.  ( ( F  |`  ( A (,) B ) ) lim CC  B ) )
itgioocnicc.g  |-  G  =  ( x  e.  ( A [,] B ) 
|->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) ) )
Assertion
Ref Expression
itgioocnicc  |-  ( ph  ->  ( G  e.  L^1  /\  S. ( A [,] B ) ( G `  x )  _d x  =  S. ( A [,] B
) ( F `  x )  _d x ) )
Distinct variable groups:    x, A    x, B    x, F    x, L    x, R    ph, x
Allowed substitution hint:    G( x)

Proof of Theorem itgioocnicc
StepHypRef Expression
1 itgioocnicc.a . . 3  |-  ( ph  ->  A  e.  RR )
2 itgioocnicc.b . . 3  |-  ( ph  ->  B  e.  RR )
3 itgioocnicc.g . . . . 5  |-  G  =  ( x  e.  ( A [,] B ) 
|->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) ) )
4 iftrue 3887 . . . . . . . . 9  |-  ( x  =  A  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  =  R )
5 iftrue 3887 . . . . . . . . 9  |-  ( x  =  A  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `  x ) ) )  =  R )
64, 5eqtr4d 2488 . . . . . . . 8  |-  ( x  =  A  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  =  if ( x  =  A ,  R ,  if ( x  =  B ,  L , 
( ( F  |`  ( A (,) B ) ) `  x ) ) ) )
76adantl 468 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  x  =  A )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  =  if ( x  =  A ,  R ,  if ( x  =  B ,  L , 
( ( F  |`  ( A (,) B ) ) `  x ) ) ) )
8 iftrue 3887 . . . . . . . . . . . 12  |-  ( x  =  B  ->  if ( x  =  B ,  L ,  ( F `
 x ) )  =  L )
9 iftrue 3887 . . . . . . . . . . . 12  |-  ( x  =  B  ->  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `  x ) )  =  L )
108, 9eqtr4d 2488 . . . . . . . . . . 11  |-  ( x  =  B  ->  if ( x  =  B ,  L ,  ( F `
 x ) )  =  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B
) ) `  x
) ) )
1110adantl 468 . . . . . . . . . 10  |-  ( ( -.  x  =  A  /\  x  =  B )  ->  if (
x  =  B ,  L ,  ( F `  x ) )  =  if ( x  =  B ,  L , 
( ( F  |`  ( A (,) B ) ) `  x ) ) )
1211ifeq2d 3900 . . . . . . . . 9  |-  ( ( -.  x  =  A  /\  x  =  B )  ->  if (
x  =  A ,  R ,  if (
x  =  B ,  L ,  ( F `  x ) ) )  =  if ( x  =  A ,  R ,  if ( x  =  B ,  L , 
( ( F  |`  ( A (,) B ) ) `  x ) ) ) )
1312adantll 720 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  x  =  B )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  =  if ( x  =  A ,  R ,  if ( x  =  B ,  L , 
( ( F  |`  ( A (,) B ) ) `  x ) ) ) )
14 iffalse 3890 . . . . . . . . . 10  |-  ( -.  x  =  A  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `
 x ) ) )  =  if ( x  =  B ,  L ,  ( F `  x ) ) )
1514ad2antlr 733 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  =  if ( x  =  B ,  L ,  ( F `  x ) ) )
16 iffalse 3890 . . . . . . . . . 10  |-  ( -.  x  =  B  ->  if ( x  =  B ,  L ,  ( F `  x ) )  =  ( F `
 x ) )
1716adantl 468 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  if ( x  =  B ,  L ,  ( F `
 x ) )  =  ( F `  x ) )
18 iffalse 3890 . . . . . . . . . . 11  |-  ( -.  x  =  A  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `  x ) ) )  =  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B
) ) `  x
) ) )
1918ad2antlr 733 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `  x ) ) )  =  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B
) ) `  x
) ) )
20 iffalse 3890 . . . . . . . . . . 11  |-  ( -.  x  =  B  ->  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `
 x ) )  =  ( ( F  |`  ( A (,) B
) ) `  x
) )
2120adantl 468 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `  x ) )  =  ( ( F  |`  ( A (,) B ) ) `  x ) )
221adantr 467 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A  e.  RR )
2322rexrd 9690 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A  e.  RR* )
2423ad2antrr 732 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  A  e.  RR* )
252rexrd 9690 . . . . . . . . . . . . 13  |-  ( ph  ->  B  e.  RR* )
2625ad3antrrr 736 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  B  e.  RR* )
272adantr 467 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  B  e.  RR )
28 simpr 463 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  e.  ( A [,] B ) )
29 eliccre 37603 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  x  e.  ( A [,] B
) )  ->  x  e.  RR )
3022, 27, 28, 29syl3anc 1268 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  e.  RR )
3130ad2antrr 732 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  x  e.  RR )
321ad2antrr 732 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  A  e.  RR )
3330adantr 467 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  x  e.  RR )
3425adantr 467 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  B  e.  RR* )
35 iccgelb 11691 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  x  e.  ( A [,] B
) )  ->  A  <_  x )
3623, 34, 28, 35syl3anc 1268 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A  <_  x )
3736adantr 467 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  A  <_  x )
38 neqne 37374 . . . . . . . . . . . . . . 15  |-  ( -.  x  =  A  ->  x  =/=  A )
3938adantl 468 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  x  =/=  A )
4032, 33, 37, 39leneltd 9789 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  A  <  x )
4140adantr 467 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  A  <  x )
4230adantr 467 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  B )  ->  x  e.  RR )
432ad2antrr 732 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  B )  ->  B  e.  RR )
44 iccleub 11690 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  x  e.  ( A [,] B
) )  ->  x  <_  B )
4523, 34, 28, 44syl3anc 1268 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  <_  B )
4645adantr 467 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  B )  ->  x  <_  B )
47 eqcom 2458 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  B  <->  B  =  x )
4847notbii 298 . . . . . . . . . . . . . . . . 17  |-  ( -.  x  =  B  <->  -.  B  =  x )
4948biimpi 198 . . . . . . . . . . . . . . . 16  |-  ( -.  x  =  B  ->  -.  B  =  x
)
5049neqned 2631 . . . . . . . . . . . . . . 15  |-  ( -.  x  =  B  ->  B  =/=  x )
5150adantl 468 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  B )  ->  B  =/=  x )
5242, 43, 46, 51leneltd 9789 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  B )  ->  x  <  B )
5352adantlr 721 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  x  <  B )
5424, 26, 31, 41, 53eliood 37595 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  x  e.  ( A (,) B
) )
55 fvres 5879 . . . . . . . . . . 11  |-  ( x  e.  ( A (,) B )  ->  (
( F  |`  ( A (,) B ) ) `
 x )  =  ( F `  x
) )
5654, 55syl 17 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  (
( F  |`  ( A (,) B ) ) `
 x )  =  ( F `  x
) )
5719, 21, 563eqtrrd 2490 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  ( F `  x )  =  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `
 x ) ) ) )
5815, 17, 573eqtrd 2489 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  =  if ( x  =  A ,  R ,  if ( x  =  B ,  L , 
( ( F  |`  ( A (,) B ) ) `  x ) ) ) )
5913, 58pm2.61dan 800 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  =  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `  x ) ) ) )
607, 59pm2.61dan 800 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  if (
x  =  A ,  R ,  if (
x  =  B ,  L ,  ( F `  x ) ) )  =  if ( x  =  A ,  R ,  if ( x  =  B ,  L , 
( ( F  |`  ( A (,) B ) ) `  x ) ) ) )
6160mpteq2dva 4489 . . . . 5  |-  ( ph  ->  ( 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  |`  ( A (,) B ) ) `
 x ) ) ) ) )
623, 61syl5eq 2497 . . . 4  |-  ( ph  ->  G  =  ( x  e.  ( A [,] B )  |->  if ( x  =  A ,  R ,  if (
x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `  x ) ) ) ) )
63 nfv 1761 . . . . 5  |-  F/ x ph
64 eqid 2451 . . . . 5  |-  ( x  e.  ( A [,] B )  |->  if ( x  =  A ,  R ,  if (
x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `  x ) ) ) )  =  ( x  e.  ( A [,] B )  |->  if ( x  =  A ,  R ,  if (
x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `  x ) ) ) )
65 itgioocnicc.fcn . . . . 5  |-  ( ph  ->  ( F  |`  ( A (,) B ) )  e.  ( ( A (,) B ) -cn-> CC ) )
66 itgioocnicc.l . . . . 5  |-  ( ph  ->  L  e.  ( ( F  |`  ( A (,) B ) ) lim CC  B ) )
67 itgioocnicc.r . . . . 5  |-  ( ph  ->  R  e.  ( ( F  |`  ( A (,) B ) ) lim CC  A ) )
6863, 64, 1, 2, 65, 66, 67cncfiooicc 37772 . . . 4  |-  ( ph  ->  ( x  e.  ( A [,] B ) 
|->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `
 x ) ) ) )  e.  ( ( A [,] B
) -cn-> CC ) )
6962, 68eqeltrd 2529 . . 3  |-  ( ph  ->  G  e.  ( ( A [,] B )
-cn-> CC ) )
70 cniccibl 22798 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  G  e.  ( ( A [,] B ) -cn-> CC ) )  ->  G  e.  L^1 )
711, 2, 69, 70syl3anc 1268 . 2  |-  ( ph  ->  G  e.  L^1 )
724adantl 468 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  x  =  A )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  =  R )
73 limccl 22830 . . . . . . . . . . 11  |-  ( ( F  |`  ( A (,) B ) ) lim CC  A )  C_  CC
7473, 67sseldi 3430 . . . . . . . . . 10  |-  ( ph  ->  R  e.  CC )
7574ad2antrr 732 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  x  =  A )  ->  R  e.  CC )
7672, 75eqeltrd 2529 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  x  =  A )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  e.  CC )
7714, 8sylan9eq 2505 . . . . . . . . . . 11  |-  ( ( -.  x  =  A  /\  x  =  B )  ->  if (
x  =  A ,  R ,  if (
x  =  B ,  L ,  ( F `  x ) ) )  =  L )
7877adantll 720 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  x  =  B )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  =  L )
79 limccl 22830 . . . . . . . . . . . 12  |-  ( ( F  |`  ( A (,) B ) ) lim CC  B )  C_  CC
8079, 66sseldi 3430 . . . . . . . . . . 11  |-  ( ph  ->  L  e.  CC )
8180ad3antrrr 736 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  x  =  B )  ->  L  e.  CC )
8278, 81eqeltrd 2529 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  x  =  B )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  e.  CC )
8314, 16sylan9eq 2505 . . . . . . . . . . 11  |-  ( ( -.  x  =  A  /\  -.  x  =  B )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  =  ( F `  x ) )
8483adantll 720 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  =  ( F `  x ) )
8556eqcomd 2457 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  ( F `  x )  =  ( ( F  |`  ( A (,) B
) ) `  x
) )
86 cncff 21925 . . . . . . . . . . . . . 14  |-  ( ( F  |`  ( A (,) B ) )  e.  ( ( A (,) B ) -cn-> CC )  ->  ( F  |`  ( A (,) B ) ) : ( A (,) B ) --> CC )
8765, 86syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  ( F  |`  ( A (,) B ) ) : ( A (,) B ) --> CC )
8887ad3antrrr 736 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  ( F  |`  ( A (,) B ) ) : ( A (,) B
) --> CC )
8988, 54ffvelrnd 6023 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  (
( F  |`  ( A (,) B ) ) `
 x )  e.  CC )
9085, 89eqeltrd 2529 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  ( F `  x )  e.  CC )
9184, 90eqeltrd 2529 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  e.  CC )
9282, 91pm2.61dan 800 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  e.  CC )
9376, 92pm2.61dan 800 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  if (
x  =  A ,  R ,  if (
x  =  B ,  L ,  ( F `  x ) ) )  e.  CC )
943fvmpt2 5957 . . . . . . 7  |-  ( ( x  e.  ( A [,] B )  /\  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `
 x ) ) )  e.  CC )  ->  ( G `  x )  =  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `
 x ) ) ) )
9528, 93, 94syl2anc 667 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( G `  x )  =  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `
 x ) ) ) )
9695, 93eqeltrd 2529 . . . . 5  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( G `  x )  e.  CC )
971, 2, 96itgioo 22773 . . . 4  |-  ( ph  ->  S. ( A (,) B ) ( G `
 x )  _d x  =  S. ( A [,] B ) ( G `  x
)  _d x )
9897eqcomd 2457 . . 3  |-  ( ph  ->  S. ( A [,] B ) ( G `
 x )  _d x  =  S. ( A (,) B ) ( G `  x
)  _d x )
99 ioossicc 11720 . . . . . . 7  |-  ( A (,) B )  C_  ( A [,] B )
10099sseli 3428 . . . . . 6  |-  ( x  e.  ( A (,) B )  ->  x  e.  ( A [,] B
) )
101100, 95sylan2 477 . . . . 5  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( G `  x )  =  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `
 x ) ) ) )
1021adantr 467 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  A  e.  RR )
103 eliooord 11694 . . . . . . . . . 10  |-  ( x  e.  ( A (,) B )  ->  ( A  <  x  /\  x  <  B ) )
104103simpld 461 . . . . . . . . 9  |-  ( x  e.  ( A (,) B )  ->  A  <  x )
105104adantl 468 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  A  <  x )
106102, 105gtned 9770 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  x  =/=  A )
107106neneqd 2629 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  -.  x  =  A )
108107, 14syl 17 . . . . 5  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  if (
x  =  A ,  R ,  if (
x  =  B ,  L ,  ( F `  x ) ) )  =  if ( x  =  B ,  L ,  ( F `  x ) ) )
109100, 30sylan2 477 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  x  e.  RR )
110103simprd 465 . . . . . . . . 9  |-  ( x  e.  ( A (,) B )  ->  x  <  B )
111110adantl 468 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  x  <  B )
112109, 111ltned 9771 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  x  =/=  B )
113112neneqd 2629 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  -.  x  =  B )
114113, 16syl 17 . . . . 5  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  if (
x  =  B ,  L ,  ( F `  x ) )  =  ( F `  x
) )
115101, 108, 1143eqtrd 2489 . . . 4  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( G `  x )  =  ( F `  x ) )
116115itgeq2dv 22739 . . 3  |-  ( ph  ->  S. ( A (,) B ) ( G `
 x )  _d x  =  S. ( A (,) B ) ( F `  x
)  _d x )
117 itgioocnicc.f . . . . . 6  |-  ( ph  ->  F : dom  F --> CC )
118117adantr 467 . . . . 5  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  F : dom  F --> CC )
119 itgioocnicc.fdom . . . . . 6  |-  ( ph  ->  ( A [,] B
)  C_  dom  F )
120119sselda 3432 . . . . 5  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  e.  dom  F )
121118, 120ffvelrnd 6023 . . . 4  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  CC )
1221, 2, 121itgioo 22773 . . 3  |-  ( ph  ->  S. ( A (,) B ) ( F `
 x )  _d x  =  S. ( A [,] B ) ( F `  x
)  _d x )
12398, 116, 1223eqtrd 2489 . 2  |-  ( ph  ->  S. ( A [,] B ) ( G `
 x )  _d x  =  S. ( A [,] B ) ( F `  x
)  _d x )
12471, 123jca 535 1  |-  ( ph  ->  ( G  e.  L^1  /\  S. ( A [,] B ) ( G `  x )  _d x  =  S. ( A [,] B
) ( F `  x )  _d x ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371    = wceq 1444    e. wcel 1887    =/= wne 2622    C_ wss 3404   ifcif 3881   class class class wbr 4402    |-> cmpt 4461   dom cdm 4834    |` cres 4836   -->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   L^1cibl 22575   S.citg 22576   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-itg 22581  df-0p 22628  df-limc 22821
This theorem is referenced by:  fourierdlem81  38051
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