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Theorem ftc1anclem4 28613
Description: Lemma for ftc1anc 28618. (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 5945 . . . . . . . . . 10  |-  ( ( G : RR --> RR  /\  t  e.  RR )  ->  ( G `  t
)  e.  RR )
21recnd 9518 . . . . . . . . 9  |-  ( ( G : RR --> RR  /\  t  e.  RR )  ->  ( G `  t
)  e.  CC )
3 i1ff 21282 . . . . . . . . . . 11  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
43ffvelrnda 5947 . . . . . . . . . 10  |-  ( ( F  e.  dom  S.1  /\  t  e.  RR )  ->  ( F `  t )  e.  RR )
54recnd 9518 . . . . . . . . 9  |-  ( ( F  e.  dom  S.1  /\  t  e.  RR )  ->  ( F `  t )  e.  CC )
6 subcl 9715 . . . . . . . . 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 827 . . . . . . 7  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  (
( G `  t
)  -  ( F `
 t ) )  e.  CC )
98abscld 13035 . . . . . 6  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  ( abs `  ( ( G `
 t )  -  ( F `  t ) ) )  e.  RR )
109rexrd 9539 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  ( abs `  ( ( G `
 t )  -  ( F `  t ) ) )  e.  RR* )
118absge0d 13043 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  0  <_  ( abs `  (
( G `  t
)  -  ( F `
 t ) ) ) )
12 elxrge0 11506 . . . . 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 2452 . . . 4  |-  ( t  e.  RR  |->  ( abs `  ( ( G `  t )  -  ( F `  t )
) ) )  =  ( t  e.  RR  |->  ( abs `  ( ( G `  t )  -  ( F `  t ) ) ) )
1513, 14fmptd 5971 . . 3  |-  ( ( F  e.  dom  S.1  /\  G : RR --> RR )  ->  ( t  e.  RR  |->  ( abs `  (
( G `  t
)  -  ( F `
 t ) ) ) ) : RR --> ( 0 [,] +oo ) )
16153adant2 1007 . 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 9479 . . . . . . 7  |-  RR  e.  _V
1817a1i 11 . . . . . 6  |-  ( ( F  e.  dom  S.1  /\  G  e.  L^1 
/\  G : RR --> RR )  ->  RR  e.  _V )
19 fvex 5804 . . . . . . 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 5804 . . . . . . 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 2453 . . . . . 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 2453 . . . . . 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 6441 . . . . 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 5798 . . . 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 5848 . . . . . . . . 9  |-  ( G : RR --> RR  ->  G  =  ( t  e.  RR  |->  ( G `  t ) ) )
29 absf 12938 . . . . . . . . . . 11  |-  abs : CC
--> RR
3029a1i 11 . . . . . . . . . 10  |-  ( G : RR --> RR  ->  abs
: CC --> RR )
3130feqmptd 5848 . . . . . . . . 9  |-  ( G : RR --> RR  ->  abs  =  ( x  e.  CC  |->  ( abs `  x
) ) )
32 fveq2 5794 . . . . . . . . 9  |-  ( x  =  ( G `  t )  ->  ( abs `  x )  =  ( abs `  ( G `  t )
) )
332, 28, 31, 32fmptco 5980 . . . . . . . 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 21373 . . . . . . . . 9  |-  ( G  e.  L^1  ->  G  e. MblFn )
36 ftc1anclem1 28610 . . . . . . . . 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 2541 . . . . . 6  |-  ( ( G  e.  L^1 
/\  G : RR --> RR )  ->  ( t  e.  RR  |->  ( abs `  ( G `  t
) ) )  e. MblFn
)
40393adant1 1006 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  G  e.  L^1 
/\  G : RR --> RR )  ->  ( t  e.  RR  |->  ( abs `  ( G `  t
) ) )  e. MblFn
)
412abscld 13035 . . . . . . . 8  |-  ( ( G : RR --> RR  /\  t  e.  RR )  ->  ( abs `  ( G `  t )
)  e.  RR )
422absge0d 13043 . . . . . . . 8  |-  ( ( G : RR --> RR  /\  t  e.  RR )  ->  0  <_  ( abs `  ( G `  t
) ) )
43 elrege0 11504 . . . . . . . 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 2452 . . . . . . 7  |-  ( t  e.  RR  |->  ( abs `  ( G `  t
) ) )  =  ( t  e.  RR  |->  ( abs `  ( G `
 t ) ) )
4644, 45fmptd 5971 . . . . . 6  |-  ( G : RR --> RR  ->  ( t  e.  RR  |->  ( abs `  ( G `
 t ) ) ) : RR --> ( 0 [,) +oo ) )
47463ad2ant3 1011 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  G  e.  L^1 
/\  G : RR --> RR )  ->  ( t  e.  RR  |->  ( abs `  ( G `  t
) ) ) : RR --> ( 0 [,) +oo ) )
48 iftrue 3900 . . . . . . . . 9  |-  ( t  e.  RR  ->  if ( t  e.  RR ,  ( abs `  ( G `  t )
) ,  0 )  =  ( abs `  ( G `  t )
) )
4948mpteq2ia 4477 . . . . . . . 8  |-  ( t  e.  RR  |->  if ( t  e.  RR , 
( abs `  ( G `  t )
) ,  0 ) )  =  ( t  e.  RR  |->  ( abs `  ( G `  t
) ) )
5049fveq2i 5797 . . . . . . 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 5848 . . . . . . . . . . 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 2541 . . . . . . . . . 10  |-  ( ( G  e.  L^1 
/\  G : RR --> RR )  ->  ( t  e.  RR  |->  ( G `
 t ) )  e.  L^1 )
5651, 55, 39iblabsnc 28599 . . . . . . . . 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 21398 . . . . . . . . 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 2545 . . . . . 6  |-  ( ( G  e.  L^1 
/\  G : RR --> RR )  ->  ( S.2 `  ( t  e.  RR  |->  ( abs `  ( G `
 t ) ) ) )  e.  RR )
63623adant1 1006 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  G  e.  L^1 
/\  G : RR --> RR )  ->  ( S.2 `  ( t  e.  RR  |->  ( abs `  ( G `
 t ) ) ) )  e.  RR )
645abscld 13035 . . . . . . . 8  |-  ( ( F  e.  dom  S.1  /\  t  e.  RR )  ->  ( abs `  ( F `  t )
)  e.  RR )
655absge0d 13043 . . . . . . . 8  |-  ( ( F  e.  dom  S.1  /\  t  e.  RR )  ->  0  <_  ( abs `  ( F `  t ) ) )
66 elrege0 11504 . . . . . . . 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 2452 . . . . . . 7  |-  ( t  e.  RR  |->  ( abs `  ( F `  t
) ) )  =  ( t  e.  RR  |->  ( abs `  ( F `
 t ) ) )
6967, 68fmptd 5971 . . . . . 6  |-  ( F  e.  dom  S.1  ->  ( t  e.  RR  |->  ( abs `  ( F `
 t ) ) ) : RR --> ( 0 [,) +oo ) )
70693ad2ant1 1009 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  G  e.  L^1 
/\  G : RR --> RR )  ->  ( t  e.  RR  |->  ( abs `  ( F `  t
) ) ) : RR --> ( 0 [,) +oo ) )
71 iftrue 3900 . . . . . . . . 9  |-  ( t  e.  RR  ->  if ( t  e.  RR ,  ( abs `  ( F `  t )
) ,  0 )  =  ( abs `  ( F `  t )
) )
7271mpteq2ia 4477 . . . . . . . 8  |-  ( t  e.  RR  |->  if ( t  e.  RR , 
( abs `  ( F `  t )
) ,  0 ) )  =  ( t  e.  RR  |->  ( abs `  ( F `  t
) ) )
7372fveq2i 5797 . . . . . . 7  |-  ( S.2 `  ( t  e.  RR  |->  if ( t  e.  RR ,  ( abs `  ( F `  t )
) ,  0 ) ) )  =  ( S.2 `  ( t  e.  RR  |->  ( abs `  ( F `  t
) ) ) )
743feqmptd 5848 . . . . . . . . . . 11  |-  ( F  e.  dom  S.1  ->  F  =  ( t  e.  RR  |->  ( F `  t ) ) )
75 i1fibl 21413 . . . . . . . . . . 11  |-  ( F  e.  dom  S.1  ->  F  e.  L^1 )
7674, 75eqeltrrd 2541 . . . . . . . . . 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 5848 . . . . . . . . . . . 12  |-  ( F  e.  dom  S.1  ->  abs  =  ( x  e.  CC  |->  ( abs `  x
) ) )
79 fveq2 5794 . . . . . . . . . . . 12  |-  ( x  =  ( F `  t )  ->  ( abs `  x )  =  ( abs `  ( F `  t )
) )
805, 74, 78, 79fmptco 5980 . . . . . . . . . . 11  |-  ( F  e.  dom  S.1  ->  ( abs  o.  F )  =  ( t  e.  RR  |->  ( abs `  ( F `  t )
) ) )
81 i1fmbf 21281 . . . . . . . . . . . 12  |-  ( F  e.  dom  S.1  ->  F  e. MblFn )
82 ftc1anclem1 28610 . . . . . . . . . . . 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 2541 . . . . . . . . . 10  |-  ( F  e.  dom  S.1  ->  ( t  e.  RR  |->  ( abs `  ( F `
 t ) ) )  e. MblFn )
854, 76, 84iblabsnc 28599 . . . . . . . . 9  |-  ( F  e.  dom  S.1  ->  ( t  e.  RR  |->  ( abs `  ( F `
 t ) ) )  e.  L^1 )
8664, 65iblpos 21398 . . . . . . . . 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 2545 . . . . . 6  |-  ( F  e.  dom  S.1  ->  ( S.2 `  ( t  e.  RR  |->  ( abs `  ( F `  t
) ) ) )  e.  RR )
90893ad2ant1 1009 . . . . 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 28589 . . . 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 2495 . . 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 9519 . . 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 2540 . 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 9471 . . . . . . . . 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 827 . . . . . . 7  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  (
( abs `  ( G `  t )
)  +  ( abs `  ( F `  t
) ) )  e.  RR )
9897rexrd 9539 . . . . . 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 10021 . . . . . 6  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  0  <_  ( ( abs `  ( G `  t )
)  +  ( abs `  ( F `  t
) ) ) )
104 elxrge0 11506 . . . . . 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 2452 . . . . 5  |-  ( t  e.  RR  |->  ( ( abs `  ( G `
 t ) )  +  ( abs `  ( F `  t )
) ) )  =  ( t  e.  RR  |->  ( ( abs `  ( G `  t )
)  +  ( abs `  ( F `  t
) ) ) )
107105, 106fmptd 5971 . . . 4  |-  ( ( F  e.  dom  S.1  /\  G : RR --> RR )  ->  ( t  e.  RR  |->  ( ( abs `  ( G `  t
) )  +  ( abs `  ( F `
 t ) ) ) ) : RR --> ( 0 [,] +oo ) )
1081073adant2 1007 . . 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 12934 . . . . . . . 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 827 . . . . . 6  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  ( abs `  ( ( G `
 t )  -  ( F `  t ) ) )  <_  (
( abs `  ( G `  t )
)  +  ( abs `  ( F `  t
) ) ) )
112111ralrimiva 2827 . . . . 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 2453 . . . . . 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 2453 . . . . . 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 6442 . . . . 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 1007 . . 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 21345 . . 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 1219 . 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 21344 . 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 1219 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 965    = wceq 1370    e. wcel 1758   A.wral 2796   _Vcvv 3072   ifcif 3894   class class class wbr 4395    |-> cmpt 4453   dom cdm 4943    o. ccom 4947   -->wf 5517   ` cfv 5521  (class class class)co 6195    oFcof 6423    oRcofr 6424   CCcc 9386   RRcr 9387   0cc0 9388    + caddc 9391   +oocpnf 9521   RR*cxr 9523    <_ cle 9525    - cmin 9701   [,)cico 11408   [,]cicc 11409   abscabs 12836  MblFncmbf 21222   S.1citg1 21223   S.2citg2 21224   L^1cibl 21225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-inf2 7953  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465  ax-pre-sup 9466  ax-addf 9467
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-disj 4366  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-isom 5530  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-of 6425  df-ofr 6426  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-1o 7025  df-2o 7026  df-oadd 7029  df-er 7206  df-map 7321  df-pm 7322  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-fi 7767  df-sup 7797  df-oi 7830  df-card 8215  df-cda 8443  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-div 10100  df-nn 10429  df-2 10486  df-3 10487  df-n0 10686  df-z 10753  df-uz 10968  df-q 11060  df-rp 11098  df-xneg 11195  df-xadd 11196  df-xmul 11197  df-ioo 11410  df-ico 11412  df-icc 11413  df-fz 11550  df-fzo 11661  df-fl 11754  df-seq 11919  df-exp 11978  df-hash 12216  df-cj 12701  df-re 12702  df-im 12703  df-sqr 12837  df-abs 12838  df-clim 13079  df-sum 13277  df-rest 14475  df-topgen 14496  df-psmet 17929  df-xmet 17930  df-met 17931  df-bl 17932  df-mopn 17933  df-top 18630  df-bases 18632  df-topon 18633  df-cmp 19117  df-ovol 21075  df-vol 21076  df-mbf 21227  df-itg1 21228  df-itg2 21229  df-ibl 21230  df-0p 21276
This theorem is referenced by:  ftc1anclem5  28614  ftc1anclem6  28615
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