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Theorem ftc1anclem4 30011
Description: Lemma for ftc1anc 30016. (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 9634 . . . . . . . . 9  |-  ( ( G : RR --> RR  /\  t  e.  RR )  ->  ( G `  t
)  e.  CC )
3 i1ff 21951 . . . . . . . . . . 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 9634 . . . . . . . . 9  |-  ( ( F  e.  dom  S.1  /\  t  e.  RR )  ->  ( F `  t )  e.  CC )
6 subcl 9831 . . . . . . . . 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 829 . . . . . . 7  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  (
( G `  t
)  -  ( F `
 t ) )  e.  CC )
98abscld 13247 . . . . . 6  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  ( abs `  ( ( G `
 t )  -  ( F `  t ) ) )  e.  RR )
109rexrd 9655 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  ( abs `  ( ( G `
 t )  -  ( F `  t ) ) )  e.  RR* )
118absge0d 13255 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  0  <_  ( abs `  (
( G `  t
)  -  ( F `
 t ) ) ) )
12 elxrge0 11641 . . . . 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 2467 . . . 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 9595 . . . . . . 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 2468 . . . . . 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 2468 . . . . . 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 6551 . . . . 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 5927 . . . . . . . . 9  |-  ( G : RR --> RR  ->  G  =  ( t  e.  RR  |->  ( G `  t ) ) )
29 absf 13150 . . . . . . . . . . 11  |-  abs : CC
--> RR
3029a1i 11 . . . . . . . . . 10  |-  ( G : RR --> RR  ->  abs
: CC --> RR )
3130feqmptd 5927 . . . . . . . . 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 22042 . . . . . . . . 9  |-  ( G  e.  L^1  ->  G  e. MblFn )
36 ftc1anclem1 30008 . . . . . . . . 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 2556 . . . . . 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 13247 . . . . . . . 8  |-  ( ( G : RR --> RR  /\  t  e.  RR )  ->  ( abs `  ( G `  t )
)  e.  RR )
422absge0d 13255 . . . . . . . 8  |-  ( ( G : RR --> RR  /\  t  e.  RR )  ->  0  <_  ( abs `  ( G `  t
) ) )
43 elrege0 11639 . . . . . . . 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 2467 . . . . . . 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 3951 . . . . . . . . 9  |-  ( t  e.  RR  ->  if ( t  e.  RR ,  ( abs `  ( G `  t )
) ,  0 )  =  ( abs `  ( G `  t )
) )
4948mpteq2ia 4535 . . . . . . . 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 5927 . . . . . . . . . . 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 2556 . . . . . . . . . 10  |-  ( ( G  e.  L^1 
/\  G : RR --> RR )  ->  ( t  e.  RR  |->  ( G `
 t ) )  e.  L^1 )
5651, 55, 39iblabsnc 29997 . . . . . . . . 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 22067 . . . . . . . . 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 2560 . . . . . 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 13247 . . . . . . . 8  |-  ( ( F  e.  dom  S.1  /\  t  e.  RR )  ->  ( abs `  ( F `  t )
)  e.  RR )
655absge0d 13255 . . . . . . . 8  |-  ( ( F  e.  dom  S.1  /\  t  e.  RR )  ->  0  <_  ( abs `  ( F `  t ) ) )
66 elrege0 11639 . . . . . . . 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 2467 . . . . . . 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 3951 . . . . . . . . 9  |-  ( t  e.  RR  ->  if ( t  e.  RR ,  ( abs `  ( F `  t )
) ,  0 )  =  ( abs `  ( F `  t )
) )
7271mpteq2ia 4535 . . . . . . . 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 5927 . . . . . . . . . . 11  |-  ( F  e.  dom  S.1  ->  F  =  ( t  e.  RR  |->  ( F `  t ) ) )
75 i1fibl 22082 . . . . . . . . . . 11  |-  ( F  e.  dom  S.1  ->  F  e.  L^1 )
7674, 75eqeltrrd 2556 . . . . . . . . . 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 5927 . . . . . . . . . . . 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 21950 . . . . . . . . . . . 12  |-  ( F  e.  dom  S.1  ->  F  e. MblFn )
82 ftc1anclem1 30008 . . . . . . . . . . . 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 2556 . . . . . . . . . 10  |-  ( F  e.  dom  S.1  ->  ( t  e.  RR  |->  ( abs `  ( F `
 t ) ) )  e. MblFn )
854, 76, 84iblabsnc 29997 . . . . . . . . 9  |-  ( F  e.  dom  S.1  ->  ( t  e.  RR  |->  ( abs `  ( F `
 t ) ) )  e.  L^1 )
8664, 65iblpos 22067 . . . . . . . . 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 2560 . . . . . 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 29987 . . . 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 2510 . . 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 9635 . . 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 2555 . 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 9587 . . . . . . . . 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 829 . . . . . . 7  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  (
( abs `  ( G `  t )
)  +  ( abs `  ( F `  t
) ) )  e.  RR )
9897rexrd 9655 . . . . . 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 10140 . . . . . 6  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  0  <_  ( ( abs `  ( G `  t )
)  +  ( abs `  ( F `  t
) ) ) )
104 elxrge0 11641 . . . . . 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 2467 . . . . 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 13146 . . . . . . . 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 829 . . . . . 6  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  ( abs `  ( ( G `
 t )  -  ( F `  t ) ) )  <_  (
( abs `  ( G `  t )
)  +  ( abs `  ( F `  t
) ) ) )
112111ralrimiva 2881 . . . . 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 2468 . . . . . 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 2468 . . . . . 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 6552 . . . . 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 22014 . . 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 22013 . 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 1379    e. wcel 1767   A.wral 2817   _Vcvv 3118   ifcif 3945   class class class wbr 4453    |-> cmpt 4511   dom cdm 5005    o. ccom 5009   -->wf 5590   ` cfv 5594  (class class class)co 6295    oFcof 6533    oRcofr 6534   CCcc 9502   RRcr 9503   0cc0 9504    + caddc 9507   +oocpnf 9637   RR*cxr 9639    <_ cle 9641    - cmin 9817   [,)cico 11543   [,]cicc 11544   abscabs 13047  MblFncmbf 21891   S.1citg1 21892   S.2citg2 21893   L^1cibl 21894
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582  ax-addf 9583
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-disj 4424  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-ofr 6536  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fi 7883  df-sup 7913  df-oi 7947  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-ioo 11545  df-ico 11547  df-icc 11548  df-fz 11685  df-fzo 11805  df-fl 11909  df-seq 12088  df-exp 12147  df-hash 12386  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-clim 13291  df-sum 13489  df-rest 14695  df-topgen 14716  df-psmet 18281  df-xmet 18282  df-met 18283  df-bl 18284  df-mopn 18285  df-top 19268  df-bases 19270  df-topon 19271  df-cmp 19755  df-ovol 21744  df-vol 21745  df-mbf 21896  df-itg1 21897  df-itg2 21898  df-ibl 21899  df-0p 21945
This theorem is referenced by:  ftc1anclem5  30012  ftc1anclem6  30013
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