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Theorem itgioocnicc 37951
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 3878 . . . . . . . . 9  |-  ( x  =  A  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  =  R )
5 iftrue 3878 . . . . . . . . 9  |-  ( x  =  A  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `  x ) ) )  =  R )
64, 5eqtr4d 2508 . . . . . . . 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 473 . . . . . . 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 3878 . . . . . . . . . . . 12  |-  ( x  =  B  ->  if ( x  =  B ,  L ,  ( F `
 x ) )  =  L )
9 iftrue 3878 . . . . . . . . . . . 12  |-  ( x  =  B  ->  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `  x ) )  =  L )
108, 9eqtr4d 2508 . . . . . . . . . . 11  |-  ( x  =  B  ->  if ( x  =  B ,  L ,  ( F `
 x ) )  =  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B
) ) `  x
) ) )
1110adantl 473 . . . . . . . . . 10  |-  ( ( -.  x  =  A  /\  x  =  B )  ->  if (
x  =  B ,  L ,  ( F `  x ) )  =  if ( x  =  B ,  L , 
( ( F  |`  ( A (,) B ) ) `  x ) ) )
1211ifeq2d 3891 . . . . . . . . 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 728 . . . . . . . 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 3881 . . . . . . . . . 10  |-  ( -.  x  =  A  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `
 x ) ) )  =  if ( x  =  B ,  L ,  ( F `  x ) ) )
1514ad2antlr 741 . . . . . . . . 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 3881 . . . . . . . . . 10  |-  ( -.  x  =  B  ->  if ( x  =  B ,  L ,  ( F `  x ) )  =  ( F `
 x ) )
1716adantl 473 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  if ( x  =  B ,  L ,  ( F `
 x ) )  =  ( F `  x ) )
18 iffalse 3881 . . . . . . . . . . 11  |-  ( -.  x  =  A  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `  x ) ) )  =  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B
) ) `  x
) ) )
1918ad2antlr 741 . . . . . . . . . 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 3881 . . . . . . . . . . 11  |-  ( -.  x  =  B  ->  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `
 x ) )  =  ( ( F  |`  ( A (,) B
) ) `  x
) )
2120adantl 473 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `  x ) )  =  ( ( F  |`  ( A (,) B ) ) `  x ) )
221adantr 472 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A  e.  RR )
2322rexrd 9708 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A  e.  RR* )
2423ad2antrr 740 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  A  e.  RR* )
252rexrd 9708 . . . . . . . . . . . . 13  |-  ( ph  ->  B  e.  RR* )
2625ad3antrrr 744 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  B  e.  RR* )
272adantr 472 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  B  e.  RR )
28 simpr 468 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  e.  ( A [,] B ) )
29 eliccre 37699 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  x  e.  ( A [,] B
) )  ->  x  e.  RR )
3022, 27, 28, 29syl3anc 1292 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  e.  RR )
3130ad2antrr 740 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  x  e.  RR )
321ad2antrr 740 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  A  e.  RR )
3330adantr 472 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  x  e.  RR )
3425adantr 472 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  B  e.  RR* )
35 iccgelb 11716 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  x  e.  ( A [,] B
) )  ->  A  <_  x )
3623, 34, 28, 35syl3anc 1292 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A  <_  x )
3736adantr 472 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  A  <_  x )
38 neqne 2651 . . . . . . . . . . . . . . 15  |-  ( -.  x  =  A  ->  x  =/=  A )
3938adantl 473 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  x  =/=  A )
4032, 33, 37, 39leneltd 9806 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  A  <  x )
4140adantr 472 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  A  <  x )
4230adantr 472 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  B )  ->  x  e.  RR )
432ad2antrr 740 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  B )  ->  B  e.  RR )
44 iccleub 11715 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  x  e.  ( A [,] B
) )  ->  x  <_  B )
4523, 34, 28, 44syl3anc 1292 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  <_  B )
4645adantr 472 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  B )  ->  x  <_  B )
47 eqcom 2478 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  B  <->  B  =  x )
4847notbii 303 . . . . . . . . . . . . . . . . 17  |-  ( -.  x  =  B  <->  -.  B  =  x )
4948biimpi 199 . . . . . . . . . . . . . . . 16  |-  ( -.  x  =  B  ->  -.  B  =  x
)
5049neqned 2650 . . . . . . . . . . . . . . 15  |-  ( -.  x  =  B  ->  B  =/=  x )
5150adantl 473 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  B )  ->  B  =/=  x )
5242, 43, 46, 51leneltd 9806 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  B )  ->  x  <  B )
5352adantlr 729 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  x  <  B )
5424, 26, 31, 41, 53eliood 37691 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  x  e.  ( A (,) B
) )
55 fvres 5893 . . . . . . . . . . 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 2510 . . . . . . . . 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 2509 . . . . . . . 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 808 . . . . . . 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 808 . . . . . 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 4482 . . . . 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 2517 . . . 4  |-  ( ph  ->  G  =  ( x  e.  ( A [,] B )  |->  if ( x  =  A ,  R ,  if (
x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `  x ) ) ) ) )
63 nfv 1769 . . . . 5  |-  F/ x ph
64 eqid 2471 . . . . 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 37869 . . . 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 2549 . . 3  |-  ( ph  ->  G  e.  ( ( A [,] B )
-cn-> CC ) )
70 cniccibl 22877 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  G  e.  ( ( A [,] B ) -cn-> CC ) )  ->  G  e.  L^1 )
711, 2, 69, 70syl3anc 1292 . 2  |-  ( ph  ->  G  e.  L^1 )
724adantl 473 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  x  =  A )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  =  R )
73 limccl 22909 . . . . . . . . . . 11  |-  ( ( F  |`  ( A (,) B ) ) lim CC  A )  C_  CC
7473, 67sseldi 3416 . . . . . . . . . 10  |-  ( ph  ->  R  e.  CC )
7574ad2antrr 740 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  x  =  A )  ->  R  e.  CC )
7672, 75eqeltrd 2549 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  x  =  A )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  e.  CC )
7714, 8sylan9eq 2525 . . . . . . . . . . 11  |-  ( ( -.  x  =  A  /\  x  =  B )  ->  if (
x  =  A ,  R ,  if (
x  =  B ,  L ,  ( F `  x ) ) )  =  L )
7877adantll 728 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  x  =  B )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  =  L )
79 limccl 22909 . . . . . . . . . . . 12  |-  ( ( F  |`  ( A (,) B ) ) lim CC  B )  C_  CC
8079, 66sseldi 3416 . . . . . . . . . . 11  |-  ( ph  ->  L  e.  CC )
8180ad3antrrr 744 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  x  =  B )  ->  L  e.  CC )
8278, 81eqeltrd 2549 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  x  =  B )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  e.  CC )
8314, 16sylan9eq 2525 . . . . . . . . . . 11  |-  ( ( -.  x  =  A  /\  -.  x  =  B )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  =  ( F `  x ) )
8483adantll 728 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  =  ( F `  x ) )
8556eqcomd 2477 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  ( F `  x )  =  ( ( F  |`  ( A (,) B
) ) `  x
) )
86 cncff 22003 . . . . . . . . . . . . . 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 744 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  ( F  |`  ( A (,) B ) ) : ( A (,) B
) --> CC )
8988, 54ffvelrnd 6038 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  (
( F  |`  ( A (,) B ) ) `
 x )  e.  CC )
9085, 89eqeltrd 2549 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  ( F `  x )  e.  CC )
9184, 90eqeltrd 2549 . . . . . . . . 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 808 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  e.  CC )
9376, 92pm2.61dan 808 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  if (
x  =  A ,  R ,  if (
x  =  B ,  L ,  ( F `  x ) ) )  e.  CC )
943fvmpt2 5972 . . . . . . 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 673 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( G `  x )  =  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `
 x ) ) ) )
9695, 93eqeltrd 2549 . . . . 5  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( G `  x )  e.  CC )
971, 2, 96itgioo 22852 . . . 4  |-  ( ph  ->  S. ( A (,) B ) ( G `
 x )  _d x  =  S. ( A [,] B ) ( G `  x
)  _d x )
9897eqcomd 2477 . . 3  |-  ( ph  ->  S. ( A [,] B ) ( G `
 x )  _d x  =  S. ( A (,) B ) ( G `  x
)  _d x )
99 ioossicc 11745 . . . . . . 7  |-  ( A (,) B )  C_  ( A [,] B )
10099sseli 3414 . . . . . 6  |-  ( x  e.  ( A (,) B )  ->  x  e.  ( A [,] B
) )
101100, 95sylan2 482 . . . . 5  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( G `  x )  =  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `
 x ) ) ) )
1021adantr 472 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  A  e.  RR )
103 eliooord 11719 . . . . . . . . . 10  |-  ( x  e.  ( A (,) B )  ->  ( A  <  x  /\  x  <  B ) )
104103simpld 466 . . . . . . . . 9  |-  ( x  e.  ( A (,) B )  ->  A  <  x )
105104adantl 473 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  A  <  x )
106102, 105gtned 9787 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  x  =/=  A )
107106neneqd 2648 . . . . . 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 482 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  x  e.  RR )
110103simprd 470 . . . . . . . . 9  |-  ( x  e.  ( A (,) B )  ->  x  <  B )
111110adantl 473 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  x  <  B )
112109, 111ltned 9788 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  x  =/=  B )
113112neneqd 2648 . . . . . 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 2509 . . . 4  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( G `  x )  =  ( F `  x ) )
116115itgeq2dv 22818 . . 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 472 . . . . 5  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  F : dom  F --> CC )
119 itgioocnicc.fdom . . . . . 6  |-  ( ph  ->  ( A [,] B
)  C_  dom  F )
120119sselda 3418 . . . . 5  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  e.  dom  F )
121118, 120ffvelrnd 6038 . . . 4  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  CC )
1221, 2, 121itgioo 22852 . . 3  |-  ( ph  ->  S. ( A (,) B ) ( F `
 x )  _d x  =  S. ( A [,] B ) ( F `  x
)  _d x )
12398, 116, 1223eqtrd 2509 . 2  |-  ( ph  ->  S. ( A [,] B ) ( G `
 x )  _d x  =  S. ( A [,] B ) ( F `  x
)  _d x )
12471, 123jca 541 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 376    = wceq 1452    e. wcel 1904    =/= wne 2641    C_ wss 3390   ifcif 3872   class class class wbr 4395    |-> cmpt 4454   dom cdm 4839    |` cres 4841   -->wf 5585   ` cfv 5589  (class class class)co 6308   CCcc 9555   RRcr 9556   RR*cxr 9692    < clt 9693    <_ cle 9694   (,)cioo 11660   [,]cicc 11663   -cn->ccncf 21986   L^1cibl 22654   S.citg 22655   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-inf2 8164  ax-cc 8883  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  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  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-iin 4272  df-disj 4367  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-se 4799  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-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-ofr 6551  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-omul 7205  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-acn 8394  df-cda 8616  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-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-limsup 13603  df-clim 13629  df-rlim 13630  df-sum 13830  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  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-cld 20111  df-ntr 20112  df-cls 20113  df-cn 20320  df-cnp 20321  df-cmp 20479  df-tx 20654  df-hmeo 20847  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-ovol 22494  df-vol 22496  df-mbf 22656  df-itg1 22657  df-itg2 22658  df-ibl 22659  df-itg 22660  df-0p 22707  df-limc 22900
This theorem is referenced by:  fourierdlem81  38163
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