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Theorem ftc1anclem4 30298
Description: Lemma for ftc1anc 30303. (Contributed by Brendan Leahy, 17-Jun-2018.)
Assertion
Ref Expression
ftc1anclem4  |-  ( ( F  e.  dom  S.1  /\  G  e.  L^1 
/\  G : RR --> RR )  ->  ( S.2 `  ( t  e.  RR  |->  ( abs `  ( ( G `  t )  -  ( F `  t ) ) ) ) )  e.  RR )
Distinct variable groups:    t, F    t, G

Proof of Theorem ftc1anclem4
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ffvelrn 6030 . . . . . . . . . 10  |-  ( ( G : RR --> RR  /\  t  e.  RR )  ->  ( G `  t
)  e.  RR )
21recnd 9639 . . . . . . . . 9  |-  ( ( G : RR --> RR  /\  t  e.  RR )  ->  ( G `  t
)  e.  CC )
3 i1ff 22209 . . . . . . . . . . 11  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
43ffvelrnda 6032 . . . . . . . . . 10  |-  ( ( F  e.  dom  S.1  /\  t  e.  RR )  ->  ( F `  t )  e.  RR )
54recnd 9639 . . . . . . . . 9  |-  ( ( F  e.  dom  S.1  /\  t  e.  RR )  ->  ( F `  t )  e.  CC )
6 subcl 9838 . . . . . . . . 9  |-  ( ( ( G `  t
)  e.  CC  /\  ( F `  t )  e.  CC )  -> 
( ( G `  t )  -  ( F `  t )
)  e.  CC )
72, 5, 6syl2anr 478 . . . . . . . 8  |-  ( ( ( F  e.  dom  S.1 
/\  t  e.  RR )  /\  ( G : RR
--> RR  /\  t  e.  RR ) )  -> 
( ( G `  t )  -  ( F `  t )
)  e.  CC )
87anandirs 831 . . . . . . 7  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  (
( G `  t
)  -  ( F `
 t ) )  e.  CC )
98abscld 13279 . . . . . 6  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  ( abs `  ( ( G `
 t )  -  ( F `  t ) ) )  e.  RR )
109rexrd 9660 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  ( abs `  ( ( G `
 t )  -  ( F `  t ) ) )  e.  RR* )
118absge0d 13287 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  0  <_  ( abs `  (
( G `  t
)  -  ( F `
 t ) ) ) )
12 elxrge0 11654 . . . . 5  |-  ( ( abs `  ( ( G `  t )  -  ( F `  t ) ) )  e.  ( 0 [,] +oo )  <->  ( ( abs `  ( ( G `  t )  -  ( F `  t )
) )  e.  RR*  /\  0  <_  ( abs `  ( ( G `  t )  -  ( F `  t )
) ) ) )
1310, 11, 12sylanbrc 664 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  ( abs `  ( ( G `
 t )  -  ( F `  t ) ) )  e.  ( 0 [,] +oo )
)
14 eqid 2457 . . . 4  |-  ( t  e.  RR  |->  ( abs `  ( ( G `  t )  -  ( F `  t )
) ) )  =  ( t  e.  RR  |->  ( abs `  ( ( G `  t )  -  ( F `  t ) ) ) )
1513, 14fmptd 6056 . . 3  |-  ( ( F  e.  dom  S.1  /\  G : RR --> RR )  ->  ( t  e.  RR  |->  ( abs `  (
( G `  t
)  -  ( F `
 t ) ) ) ) : RR --> ( 0 [,] +oo ) )
16153adant2 1015 . 2  |-  ( ( F  e.  dom  S.1  /\  G  e.  L^1 
/\  G : RR --> RR )  ->  ( t  e.  RR  |->  ( abs `  ( ( G `  t )  -  ( F `  t )
) ) ) : RR --> ( 0 [,] +oo ) )
17 reex 9600 . . . . . . 7  |-  RR  e.  _V
1817a1i 11 . . . . . 6  |-  ( ( F  e.  dom  S.1  /\  G  e.  L^1 
/\  G : RR --> RR )  ->  RR  e.  _V )
19 fvex 5882 . . . . . . 7  |-  ( abs `  ( G `  t
) )  e.  _V
2019a1i 11 . . . . . 6  |-  ( ( ( F  e.  dom  S.1 
/\  G  e.  L^1  /\  G : RR --> RR )  /\  t  e.  RR )  ->  ( abs `  ( G `  t ) )  e. 
_V )
21 fvex 5882 . . . . . . 7  |-  ( abs `  ( F `  t
) )  e.  _V
2221a1i 11 . . . . . 6  |-  ( ( ( F  e.  dom  S.1 
/\  G  e.  L^1  /\  G : RR --> RR )  /\  t  e.  RR )  ->  ( abs `  ( F `  t ) )  e. 
_V )
23 eqidd 2458 . . . . . 6  |-  ( ( F  e.  dom  S.1  /\  G  e.  L^1 
/\  G : RR --> RR )  ->  ( t  e.  RR  |->  ( abs `  ( G `  t
) ) )  =  ( t  e.  RR  |->  ( abs `  ( G `
 t ) ) ) )
24 eqidd 2458 . . . . . 6  |-  ( ( F  e.  dom  S.1  /\  G  e.  L^1 
/\  G : RR --> RR )  ->  ( t  e.  RR  |->  ( abs `  ( F `  t
) ) )  =  ( t  e.  RR  |->  ( abs `  ( F `
 t ) ) ) )
2518, 20, 22, 23, 24offval2 6555 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  G  e.  L^1 
/\  G : RR --> RR )  ->  ( ( t  e.  RR  |->  ( abs `  ( G `
 t ) ) )  oF  +  ( t  e.  RR  |->  ( abs `  ( F `
 t ) ) ) )  =  ( t  e.  RR  |->  ( ( abs `  ( G `  t )
)  +  ( abs `  ( F `  t
) ) ) ) )
2625fveq2d 5876 . . . 4  |-  ( ( F  e.  dom  S.1  /\  G  e.  L^1 
/\  G : RR --> RR )  ->  ( S.2 `  ( ( t  e.  RR  |->  ( abs `  ( G `  t )
) )  oF  +  ( t  e.  RR  |->  ( abs `  ( F `  t )
) ) ) )  =  ( S.2 `  (
t  e.  RR  |->  ( ( abs `  ( G `  t )
)  +  ( abs `  ( F `  t
) ) ) ) ) )
27 id 22 . . . . . . . . . 10  |-  ( G : RR --> RR  ->  G : RR --> RR )
2827feqmptd 5926 . . . . . . . . 9  |-  ( G : RR --> RR  ->  G  =  ( t  e.  RR  |->  ( G `  t ) ) )
29 absf 13182 . . . . . . . . . . 11  |-  abs : CC
--> RR
3029a1i 11 . . . . . . . . . 10  |-  ( G : RR --> RR  ->  abs
: CC --> RR )
3130feqmptd 5926 . . . . . . . . 9  |-  ( G : RR --> RR  ->  abs  =  ( x  e.  CC  |->  ( abs `  x
) ) )
32 fveq2 5872 . . . . . . . . 9  |-  ( x  =  ( G `  t )  ->  ( abs `  x )  =  ( abs `  ( G `  t )
) )
332, 28, 31, 32fmptco 6065 . . . . . . . 8  |-  ( G : RR --> RR  ->  ( abs  o.  G )  =  ( t  e.  RR  |->  ( abs `  ( G `  t )
) ) )
3433adantl 466 . . . . . . 7  |-  ( ( G  e.  L^1 
/\  G : RR --> RR )  ->  ( abs 
o.  G )  =  ( t  e.  RR  |->  ( abs `  ( G `
 t ) ) ) )
35 iblmbf 22300 . . . . . . . . 9  |-  ( G  e.  L^1  ->  G  e. MblFn )
36 ftc1anclem1 30295 . . . . . . . . 9  |-  ( ( G : RR --> RR  /\  G  e. MblFn )  ->  ( abs  o.  G )  e. MblFn )
3735, 36sylan2 474 . . . . . . . 8  |-  ( ( G : RR --> RR  /\  G  e.  L^1
)  ->  ( abs  o.  G )  e. MblFn )
3837ancoms 453 . . . . . . 7  |-  ( ( G  e.  L^1 
/\  G : RR --> RR )  ->  ( abs 
o.  G )  e. MblFn
)
3934, 38eqeltrrd 2546 . . . . . 6  |-  ( ( G  e.  L^1 
/\  G : RR --> RR )  ->  ( t  e.  RR  |->  ( abs `  ( G `  t
) ) )  e. MblFn
)
40393adant1 1014 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  G  e.  L^1 
/\  G : RR --> RR )  ->  ( t  e.  RR  |->  ( abs `  ( G `  t
) ) )  e. MblFn
)
412abscld 13279 . . . . . . . 8  |-  ( ( G : RR --> RR  /\  t  e.  RR )  ->  ( abs `  ( G `  t )
)  e.  RR )
422absge0d 13287 . . . . . . . 8  |-  ( ( G : RR --> RR  /\  t  e.  RR )  ->  0  <_  ( abs `  ( G `  t
) ) )
43 elrege0 11652 . . . . . . . 8  |-  ( ( abs `  ( G `
 t ) )  e.  ( 0 [,) +oo )  <->  ( ( abs `  ( G `  t
) )  e.  RR  /\  0  <_  ( abs `  ( G `  t
) ) ) )
4441, 42, 43sylanbrc 664 . . . . . . 7  |-  ( ( G : RR --> RR  /\  t  e.  RR )  ->  ( abs `  ( G `  t )
)  e.  ( 0 [,) +oo ) )
45 eqid 2457 . . . . . . 7  |-  ( t  e.  RR  |->  ( abs `  ( G `  t
) ) )  =  ( t  e.  RR  |->  ( abs `  ( G `
 t ) ) )
4644, 45fmptd 6056 . . . . . 6  |-  ( G : RR --> RR  ->  ( t  e.  RR  |->  ( abs `  ( G `
 t ) ) ) : RR --> ( 0 [,) +oo ) )
47463ad2ant3 1019 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  G  e.  L^1 
/\  G : RR --> RR )  ->  ( t  e.  RR  |->  ( abs `  ( G `  t
) ) ) : RR --> ( 0 [,) +oo ) )
48 iftrue 3950 . . . . . . . . 9  |-  ( t  e.  RR  ->  if ( t  e.  RR ,  ( abs `  ( G `  t )
) ,  0 )  =  ( abs `  ( G `  t )
) )
4948mpteq2ia 4539 . . . . . . . 8  |-  ( t  e.  RR  |->  if ( t  e.  RR , 
( abs `  ( G `  t )
) ,  0 ) )  =  ( t  e.  RR  |->  ( abs `  ( G `  t
) ) )
5049fveq2i 5875 . . . . . . 7  |-  ( S.2 `  ( t  e.  RR  |->  if ( t  e.  RR ,  ( abs `  ( G `  t )
) ,  0 ) ) )  =  ( S.2 `  ( t  e.  RR  |->  ( abs `  ( G `  t
) ) ) )
511adantll 713 . . . . . . . . . 10  |-  ( ( ( G  e.  L^1  /\  G : RR --> RR )  /\  t  e.  RR )  ->  ( G `  t )  e.  RR )
52 simpr 461 . . . . . . . . . . . 12  |-  ( ( G  e.  L^1 
/\  G : RR --> RR )  ->  G : RR
--> RR )
5352feqmptd 5926 . . . . . . . . . . 11  |-  ( ( G  e.  L^1 
/\  G : RR --> RR )  ->  G  =  ( t  e.  RR  |->  ( G `  t ) ) )
54 simpl 457 . . . . . . . . . . 11  |-  ( ( G  e.  L^1 
/\  G : RR --> RR )  ->  G  e.  L^1 )
5553, 54eqeltrrd 2546 . . . . . . . . . 10  |-  ( ( G  e.  L^1 
/\  G : RR --> RR )  ->  ( t  e.  RR  |->  ( G `
 t ) )  e.  L^1 )
5651, 55, 39iblabsnc 30284 . . . . . . . . 9  |-  ( ( G  e.  L^1 
/\  G : RR --> RR )  ->  ( t  e.  RR  |->  ( abs `  ( G `  t
) ) )  e.  L^1 )
5741adantll 713 . . . . . . . . . 10  |-  ( ( ( G  e.  L^1  /\  G : RR --> RR )  /\  t  e.  RR )  ->  ( abs `  ( G `  t ) )  e.  RR )
5842adantll 713 . . . . . . . . . 10  |-  ( ( ( G  e.  L^1  /\  G : RR --> RR )  /\  t  e.  RR )  ->  0  <_  ( abs `  ( G `  t )
) )
5957, 58iblpos 22325 . . . . . . . . 9  |-  ( ( G  e.  L^1 
/\  G : RR --> RR )  ->  ( ( t  e.  RR  |->  ( abs `  ( G `
 t ) ) )  e.  L^1  <-> 
( ( t  e.  RR  |->  ( abs `  ( G `  t )
) )  e. MblFn  /\  ( S.2 `  ( t  e.  RR  |->  if ( t  e.  RR ,  ( abs `  ( G `
 t ) ) ,  0 ) ) )  e.  RR ) ) )
6056, 59mpbid 210 . . . . . . . 8  |-  ( ( G  e.  L^1 
/\  G : RR --> RR )  ->  ( ( t  e.  RR  |->  ( abs `  ( G `
 t ) ) )  e. MblFn  /\  ( S.2 `  ( t  e.  RR  |->  if ( t  e.  RR ,  ( abs `  ( G `
 t ) ) ,  0 ) ) )  e.  RR ) )
6160simprd 463 . . . . . . 7  |-  ( ( G  e.  L^1 
/\  G : RR --> RR )  ->  ( S.2 `  ( t  e.  RR  |->  if ( t  e.  RR ,  ( abs `  ( G `  t )
) ,  0 ) ) )  e.  RR )
6250, 61syl5eqelr 2550 . . . . . 6  |-  ( ( G  e.  L^1 
/\  G : RR --> RR )  ->  ( S.2 `  ( t  e.  RR  |->  ( abs `  ( G `
 t ) ) ) )  e.  RR )
63623adant1 1014 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  G  e.  L^1 
/\  G : RR --> RR )  ->  ( S.2 `  ( t  e.  RR  |->  ( abs `  ( G `
 t ) ) ) )  e.  RR )
645abscld 13279 . . . . . . . 8  |-  ( ( F  e.  dom  S.1  /\  t  e.  RR )  ->  ( abs `  ( F `  t )
)  e.  RR )
655absge0d 13287 . . . . . . . 8  |-  ( ( F  e.  dom  S.1  /\  t  e.  RR )  ->  0  <_  ( abs `  ( F `  t ) ) )
66 elrege0 11652 . . . . . . . 8  |-  ( ( abs `  ( F `
 t ) )  e.  ( 0 [,) +oo )  <->  ( ( abs `  ( F `  t
) )  e.  RR  /\  0  <_  ( abs `  ( F `  t
) ) ) )
6764, 65, 66sylanbrc 664 . . . . . . 7  |-  ( ( F  e.  dom  S.1  /\  t  e.  RR )  ->  ( abs `  ( F `  t )
)  e.  ( 0 [,) +oo ) )
68 eqid 2457 . . . . . . 7  |-  ( t  e.  RR  |->  ( abs `  ( F `  t
) ) )  =  ( t  e.  RR  |->  ( abs `  ( F `
 t ) ) )
6967, 68fmptd 6056 . . . . . 6  |-  ( F  e.  dom  S.1  ->  ( t  e.  RR  |->  ( abs `  ( F `
 t ) ) ) : RR --> ( 0 [,) +oo ) )
70693ad2ant1 1017 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  G  e.  L^1 
/\  G : RR --> RR )  ->  ( t  e.  RR  |->  ( abs `  ( F `  t
) ) ) : RR --> ( 0 [,) +oo ) )
71 iftrue 3950 . . . . . . . . 9  |-  ( t  e.  RR  ->  if ( t  e.  RR ,  ( abs `  ( F `  t )
) ,  0 )  =  ( abs `  ( F `  t )
) )
7271mpteq2ia 4539 . . . . . . . 8  |-  ( t  e.  RR  |->  if ( t  e.  RR , 
( abs `  ( F `  t )
) ,  0 ) )  =  ( t  e.  RR  |->  ( abs `  ( F `  t
) ) )
7372fveq2i 5875 . . . . . . 7  |-  ( S.2 `  ( t  e.  RR  |->  if ( t  e.  RR ,  ( abs `  ( F `  t )
) ,  0 ) ) )  =  ( S.2 `  ( t  e.  RR  |->  ( abs `  ( F `  t
) ) ) )
743feqmptd 5926 . . . . . . . . . . 11  |-  ( F  e.  dom  S.1  ->  F  =  ( t  e.  RR  |->  ( F `  t ) ) )
75 i1fibl 22340 . . . . . . . . . . 11  |-  ( F  e.  dom  S.1  ->  F  e.  L^1 )
7674, 75eqeltrrd 2546 . . . . . . . . . 10  |-  ( F  e.  dom  S.1  ->  ( t  e.  RR  |->  ( F `  t ) )  e.  L^1 )
7729a1i 11 . . . . . . . . . . . . 13  |-  ( F  e.  dom  S.1  ->  abs
: CC --> RR )
7877feqmptd 5926 . . . . . . . . . . . 12  |-  ( F  e.  dom  S.1  ->  abs  =  ( x  e.  CC  |->  ( abs `  x
) ) )
79 fveq2 5872 . . . . . . . . . . . 12  |-  ( x  =  ( F `  t )  ->  ( abs `  x )  =  ( abs `  ( F `  t )
) )
805, 74, 78, 79fmptco 6065 . . . . . . . . . . 11  |-  ( F  e.  dom  S.1  ->  ( abs  o.  F )  =  ( t  e.  RR  |->  ( abs `  ( F `  t )
) ) )
81 i1fmbf 22208 . . . . . . . . . . . 12  |-  ( F  e.  dom  S.1  ->  F  e. MblFn )
82 ftc1anclem1 30295 . . . . . . . . . . . 12  |-  ( ( F : RR --> RR  /\  F  e. MblFn )  ->  ( abs  o.  F )  e. MblFn )
833, 81, 82syl2anc 661 . . . . . . . . . . 11  |-  ( F  e.  dom  S.1  ->  ( abs  o.  F )  e. MblFn )
8480, 83eqeltrrd 2546 . . . . . . . . . 10  |-  ( F  e.  dom  S.1  ->  ( t  e.  RR  |->  ( abs `  ( F `
 t ) ) )  e. MblFn )
854, 76, 84iblabsnc 30284 . . . . . . . . 9  |-  ( F  e.  dom  S.1  ->  ( t  e.  RR  |->  ( abs `  ( F `
 t ) ) )  e.  L^1 )
8664, 65iblpos 22325 . . . . . . . . 9  |-  ( F  e.  dom  S.1  ->  ( ( t  e.  RR  |->  ( abs `  ( F `
 t ) ) )  e.  L^1  <-> 
( ( t  e.  RR  |->  ( abs `  ( F `  t )
) )  e. MblFn  /\  ( S.2 `  ( t  e.  RR  |->  if ( t  e.  RR ,  ( abs `  ( F `
 t ) ) ,  0 ) ) )  e.  RR ) ) )
8785, 86mpbid 210 . . . . . . . 8  |-  ( F  e.  dom  S.1  ->  ( ( t  e.  RR  |->  ( abs `  ( F `
 t ) ) )  e. MblFn  /\  ( S.2 `  ( t  e.  RR  |->  if ( t  e.  RR ,  ( abs `  ( F `
 t ) ) ,  0 ) ) )  e.  RR ) )
8887simprd 463 . . . . . . 7  |-  ( F  e.  dom  S.1  ->  ( S.2 `  ( t  e.  RR  |->  if ( t  e.  RR , 
( abs `  ( F `  t )
) ,  0 ) ) )  e.  RR )
8973, 88syl5eqelr 2550 . . . . . 6  |-  ( F  e.  dom  S.1  ->  ( S.2 `  ( t  e.  RR  |->  ( abs `  ( F `  t
) ) ) )  e.  RR )
90893ad2ant1 1017 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  G  e.  L^1 
/\  G : RR --> RR )  ->  ( S.2 `  ( t  e.  RR  |->  ( abs `  ( F `
 t ) ) ) )  e.  RR )
9140, 47, 63, 70, 90itg2addnc 30274 . . . 4  |-  ( ( F  e.  dom  S.1  /\  G  e.  L^1 
/\  G : RR --> RR )  ->  ( S.2 `  ( ( t  e.  RR  |->  ( abs `  ( G `  t )
) )  oF  +  ( t  e.  RR  |->  ( abs `  ( F `  t )
) ) ) )  =  ( ( S.2 `  ( t  e.  RR  |->  ( abs `  ( G `
 t ) ) ) )  +  ( S.2 `  ( t  e.  RR  |->  ( abs `  ( F `  t
) ) ) ) ) )
9226, 91eqtr3d 2500 . . 3  |-  ( ( F  e.  dom  S.1  /\  G  e.  L^1 
/\  G : RR --> RR )  ->  ( S.2 `  ( t  e.  RR  |->  ( ( abs `  ( G `  t )
)  +  ( abs `  ( F `  t
) ) ) ) )  =  ( ( S.2 `  ( t  e.  RR  |->  ( abs `  ( G `  t
) ) ) )  +  ( S.2 `  (
t  e.  RR  |->  ( abs `  ( F `
 t ) ) ) ) ) )
9363, 90readdcld 9640 . . 3  |-  ( ( F  e.  dom  S.1  /\  G  e.  L^1 
/\  G : RR --> RR )  ->  ( ( S.2 `  ( t  e.  RR  |->  ( abs `  ( G `  t
) ) ) )  +  ( S.2 `  (
t  e.  RR  |->  ( abs `  ( F `
 t ) ) ) ) )  e.  RR )
9492, 93eqeltrd 2545 . 2  |-  ( ( F  e.  dom  S.1  /\  G  e.  L^1 
/\  G : RR --> RR )  ->  ( S.2 `  ( t  e.  RR  |->  ( ( abs `  ( G `  t )
)  +  ( abs `  ( F `  t
) ) ) ) )  e.  RR )
95 readdcl 9592 . . . . . . . . 9  |-  ( ( ( abs `  ( G `  t )
)  e.  RR  /\  ( abs `  ( F `
 t ) )  e.  RR )  -> 
( ( abs `  ( G `  t )
)  +  ( abs `  ( F `  t
) ) )  e.  RR )
9641, 64, 95syl2anr 478 . . . . . . . 8  |-  ( ( ( F  e.  dom  S.1 
/\  t  e.  RR )  /\  ( G : RR
--> RR  /\  t  e.  RR ) )  -> 
( ( abs `  ( G `  t )
)  +  ( abs `  ( F `  t
) ) )  e.  RR )
9796anandirs 831 . . . . . . 7  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  (
( abs `  ( G `  t )
)  +  ( abs `  ( F `  t
) ) )  e.  RR )
9897rexrd 9660 . . . . . 6  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  (
( abs `  ( G `  t )
)  +  ( abs `  ( F `  t
) ) )  e. 
RR* )
9941adantll 713 . . . . . . 7  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  ( abs `  ( G `  t ) )  e.  RR )
10064adantlr 714 . . . . . . 7  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  ( abs `  ( F `  t ) )  e.  RR )
10142adantll 713 . . . . . . 7  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  0  <_  ( abs `  ( G `  t )
) )
10265adantlr 714 . . . . . . 7  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  0  <_  ( abs `  ( F `  t )
) )
10399, 100, 101, 102addge0d 10149 . . . . . 6  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  0  <_  ( ( abs `  ( G `  t )
)  +  ( abs `  ( F `  t
) ) ) )
104 elxrge0 11654 . . . . . 6  |-  ( ( ( abs `  ( G `  t )
)  +  ( abs `  ( F `  t
) ) )  e.  ( 0 [,] +oo ) 
<->  ( ( ( abs `  ( G `  t
) )  +  ( abs `  ( F `
 t ) ) )  e.  RR*  /\  0  <_  ( ( abs `  ( G `  t )
)  +  ( abs `  ( F `  t
) ) ) ) )
10598, 103, 104sylanbrc 664 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  (
( abs `  ( G `  t )
)  +  ( abs `  ( F `  t
) ) )  e.  ( 0 [,] +oo ) )
106 eqid 2457 . . . . 5  |-  ( t  e.  RR  |->  ( ( abs `  ( G `
 t ) )  +  ( abs `  ( F `  t )
) ) )  =  ( t  e.  RR  |->  ( ( abs `  ( G `  t )
)  +  ( abs `  ( F `  t
) ) ) )
107105, 106fmptd 6056 . . . 4  |-  ( ( F  e.  dom  S.1  /\  G : RR --> RR )  ->  ( t  e.  RR  |->  ( ( abs `  ( G `  t
) )  +  ( abs `  ( F `
 t ) ) ) ) : RR --> ( 0 [,] +oo ) )
1081073adant2 1015 . . 3  |-  ( ( F  e.  dom  S.1  /\  G  e.  L^1 
/\  G : RR --> RR )  ->  ( t  e.  RR  |->  ( ( abs `  ( G `
 t ) )  +  ( abs `  ( F `  t )
) ) ) : RR --> ( 0 [,] +oo ) )
109 abs2dif2 13178 . . . . . . . 8  |-  ( ( ( G `  t
)  e.  CC  /\  ( F `  t )  e.  CC )  -> 
( abs `  (
( G `  t
)  -  ( F `
 t ) ) )  <_  ( ( abs `  ( G `  t ) )  +  ( abs `  ( F `  t )
) ) )
1102, 5, 109syl2anr 478 . . . . . . 7  |-  ( ( ( F  e.  dom  S.1 
/\  t  e.  RR )  /\  ( G : RR
--> RR  /\  t  e.  RR ) )  -> 
( abs `  (
( G `  t
)  -  ( F `
 t ) ) )  <_  ( ( abs `  ( G `  t ) )  +  ( abs `  ( F `  t )
) ) )
111110anandirs 831 . . . . . 6  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  ( abs `  ( ( G `
 t )  -  ( F `  t ) ) )  <_  (
( abs `  ( G `  t )
)  +  ( abs `  ( F `  t
) ) ) )
112111ralrimiva 2871 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  G : RR --> RR )  ->  A. t  e.  RR  ( abs `  ( ( G `  t )  -  ( F `  t ) ) )  <_  ( ( abs `  ( G `  t
) )  +  ( abs `  ( F `
 t ) ) ) )
11317a1i 11 . . . . . 6  |-  ( ( F  e.  dom  S.1  /\  G : RR --> RR )  ->  RR  e.  _V )
114 eqidd 2458 . . . . . 6  |-  ( ( F  e.  dom  S.1  /\  G : RR --> RR )  ->  ( t  e.  RR  |->  ( abs `  (
( G `  t
)  -  ( F `
 t ) ) ) )  =  ( t  e.  RR  |->  ( abs `  ( ( G `  t )  -  ( F `  t ) ) ) ) )
115 eqidd 2458 . . . . . 6  |-  ( ( F  e.  dom  S.1  /\  G : RR --> RR )  ->  ( t  e.  RR  |->  ( ( abs `  ( G `  t
) )  +  ( abs `  ( F `
 t ) ) ) )  =  ( t  e.  RR  |->  ( ( abs `  ( G `  t )
)  +  ( abs `  ( F `  t
) ) ) ) )
116113, 9, 97, 114, 115ofrfval2 6556 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  G : RR --> RR )  ->  ( ( t  e.  RR  |->  ( abs `  ( ( G `  t )  -  ( F `  t )
) ) )  oR  <_  ( t  e.  RR  |->  ( ( abs `  ( G `  t
) )  +  ( abs `  ( F `
 t ) ) ) )  <->  A. t  e.  RR  ( abs `  (
( G `  t
)  -  ( F `
 t ) ) )  <_  ( ( abs `  ( G `  t ) )  +  ( abs `  ( F `  t )
) ) ) )
117112, 116mpbird 232 . . . 4  |-  ( ( F  e.  dom  S.1  /\  G : RR --> RR )  ->  ( t  e.  RR  |->  ( abs `  (
( G `  t
)  -  ( F `
 t ) ) ) )  oR  <_  ( t  e.  RR  |->  ( ( abs `  ( G `  t
) )  +  ( abs `  ( F `
 t ) ) ) ) )
1181173adant2 1015 . . 3  |-  ( ( F  e.  dom  S.1  /\  G  e.  L^1 
/\  G : RR --> RR )  ->  ( t  e.  RR  |->  ( abs `  ( ( G `  t )  -  ( F `  t )
) ) )  oR  <_  ( t  e.  RR  |->  ( ( abs `  ( G `  t
) )  +  ( abs `  ( F `
 t ) ) ) ) )
119 itg2le 22272 . . 3  |-  ( ( ( t  e.  RR  |->  ( abs `  ( ( G `  t )  -  ( F `  t ) ) ) ) : RR --> ( 0 [,] +oo )  /\  ( t  e.  RR  |->  ( ( abs `  ( G `  t )
)  +  ( abs `  ( F `  t
) ) ) ) : RR --> ( 0 [,] +oo )  /\  ( t  e.  RR  |->  ( abs `  ( ( G `  t )  -  ( F `  t ) ) ) )  oR  <_ 
( t  e.  RR  |->  ( ( abs `  ( G `  t )
)  +  ( abs `  ( F `  t
) ) ) ) )  ->  ( S.2 `  ( t  e.  RR  |->  ( abs `  ( ( G `  t )  -  ( F `  t ) ) ) ) )  <_  ( S.2 `  ( t  e.  RR  |->  ( ( abs `  ( G `  t
) )  +  ( abs `  ( F `
 t ) ) ) ) ) )
12016, 108, 118, 119syl3anc 1228 . 2  |-  ( ( F  e.  dom  S.1  /\  G  e.  L^1 
/\  G : RR --> RR )  ->  ( S.2 `  ( t  e.  RR  |->  ( abs `  ( ( G `  t )  -  ( F `  t ) ) ) ) )  <_  ( S.2 `  ( t  e.  RR  |->  ( ( abs `  ( G `  t
) )  +  ( abs `  ( F `
 t ) ) ) ) ) )
121 itg2lecl 22271 . 2  |-  ( ( ( t  e.  RR  |->  ( abs `  ( ( G `  t )  -  ( F `  t ) ) ) ) : RR --> ( 0 [,] +oo )  /\  ( S.2 `  ( t  e.  RR  |->  ( ( abs `  ( G `
 t ) )  +  ( abs `  ( F `  t )
) ) ) )  e.  RR  /\  ( S.2 `  ( t  e.  RR  |->  ( abs `  (
( G `  t
)  -  ( F `
 t ) ) ) ) )  <_ 
( S.2 `  ( t  e.  RR  |->  ( ( abs `  ( G `
 t ) )  +  ( abs `  ( F `  t )
) ) ) ) )  ->  ( S.2 `  ( t  e.  RR  |->  ( abs `  ( ( G `  t )  -  ( F `  t ) ) ) ) )  e.  RR )
12216, 94, 120, 121syl3anc 1228 1  |-  ( ( F  e.  dom  S.1  /\  G  e.  L^1 
/\  G : RR --> RR )  ->  ( S.2 `  ( t  e.  RR  |->  ( abs `  ( ( G `  t )  -  ( F `  t ) ) ) ) )  e.  RR )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   A.wral 2807   _Vcvv 3109   ifcif 3944   class class class wbr 4456    |-> cmpt 4515   dom cdm 5008    o. ccom 5012   -->wf 5590   ` cfv 5594  (class class class)co 6296    oFcof 6537    oRcofr 6538   CCcc 9507   RRcr 9508   0cc0 9509    + caddc 9512   +oocpnf 9642   RR*cxr 9644    <_ cle 9646    - cmin 9824   [,)cico 11556   [,]cicc 11557   abscabs 13079  MblFncmbf 22149   S.1citg1 22150   S.2citg2 22151   L^1cibl 22152
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-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
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-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-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fi 7889  df-sup 7919  df-oi 7953  df-card 8337  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-n0 10817  df-z 10886  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ioo 11558  df-ico 11560  df-icc 11561  df-fz 11698  df-fzo 11822  df-fl 11932  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-clim 13323  df-sum 13521  df-rest 14840  df-topgen 14861  df-psmet 18538  df-xmet 18539  df-met 18540  df-bl 18541  df-mopn 18542  df-top 19526  df-bases 19528  df-topon 19529  df-cmp 20014  df-ovol 22002  df-vol 22003  df-mbf 22154  df-itg1 22155  df-itg2 22156  df-ibl 22157  df-0p 22203
This theorem is referenced by:  ftc1anclem5  30299  ftc1anclem6  30300
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