Users' Mathboxes Mathbox for Brendan Leahy < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ftc1anclem4 Structured version   Visualization version   Unicode version

Theorem ftc1anclem4 32013
Description: Lemma for ftc1anc 32018. (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 6018 . . . . . . . . . 10  |-  ( ( G : RR --> RR  /\  t  e.  RR )  ->  ( G `  t
)  e.  RR )
21recnd 9666 . . . . . . . . 9  |-  ( ( G : RR --> RR  /\  t  e.  RR )  ->  ( G `  t
)  e.  CC )
3 i1ff 22627 . . . . . . . . . . 11  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
43ffvelrnda 6020 . . . . . . . . . 10  |-  ( ( F  e.  dom  S.1  /\  t  e.  RR )  ->  ( F `  t )  e.  RR )
54recnd 9666 . . . . . . . . 9  |-  ( ( F  e.  dom  S.1  /\  t  e.  RR )  ->  ( F `  t )  e.  CC )
6 subcl 9871 . . . . . . . . 9  |-  ( ( ( G `  t
)  e.  CC  /\  ( F `  t )  e.  CC )  -> 
( ( G `  t )  -  ( F `  t )
)  e.  CC )
72, 5, 6syl2anr 481 . . . . . . . 8  |-  ( ( ( F  e.  dom  S.1 
/\  t  e.  RR )  /\  ( G : RR
--> RR  /\  t  e.  RR ) )  -> 
( ( G `  t )  -  ( F `  t )
)  e.  CC )
87anandirs 839 . . . . . . 7  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  (
( G `  t
)  -  ( F `
 t ) )  e.  CC )
98abscld 13491 . . . . . 6  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  ( abs `  ( ( G `
 t )  -  ( F `  t ) ) )  e.  RR )
109rexrd 9687 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  ( abs `  ( ( G `
 t )  -  ( F `  t ) ) )  e.  RR* )
118absge0d 13499 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  0  <_  ( abs `  (
( G `  t
)  -  ( F `
 t ) ) ) )
12 elxrge0 11738 . . . . 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 669 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  ( abs `  ( ( G `
 t )  -  ( F `  t ) ) )  e.  ( 0 [,] +oo )
)
14 eqid 2450 . . . 4  |-  ( t  e.  RR  |->  ( abs `  ( ( G `  t )  -  ( F `  t )
) ) )  =  ( t  e.  RR  |->  ( abs `  ( ( G `  t )  -  ( F `  t ) ) ) )
1513, 14fmptd 6044 . . 3  |-  ( ( F  e.  dom  S.1  /\  G : RR --> RR )  ->  ( t  e.  RR  |->  ( abs `  (
( G `  t
)  -  ( F `
 t ) ) ) ) : RR --> ( 0 [,] +oo ) )
16153adant2 1026 . 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 9627 . . . . . . 7  |-  RR  e.  _V
1817a1i 11 . . . . . 6  |-  ( ( F  e.  dom  S.1  /\  G  e.  L^1 
/\  G : RR --> RR )  ->  RR  e.  _V )
19 fvex 5873 . . . . . . 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 5873 . . . . . . 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 2451 . . . . . 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 2451 . . . . . 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 6545 . . . . 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 5867 . . . 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 5916 . . . . . . . . 9  |-  ( G : RR --> RR  ->  G  =  ( t  e.  RR  |->  ( G `  t ) ) )
29 absf 13393 . . . . . . . . . . 11  |-  abs : CC
--> RR
3029a1i 11 . . . . . . . . . 10  |-  ( G : RR --> RR  ->  abs
: CC --> RR )
3130feqmptd 5916 . . . . . . . . 9  |-  ( G : RR --> RR  ->  abs  =  ( x  e.  CC  |->  ( abs `  x
) ) )
32 fveq2 5863 . . . . . . . . 9  |-  ( x  =  ( G `  t )  ->  ( abs `  x )  =  ( abs `  ( G `  t )
) )
332, 28, 31, 32fmptco 6054 . . . . . . . 8  |-  ( G : RR --> RR  ->  ( abs  o.  G )  =  ( t  e.  RR  |->  ( abs `  ( G `  t )
) ) )
3433adantl 468 . . . . . . 7  |-  ( ( G  e.  L^1 
/\  G : RR --> RR )  ->  ( abs 
o.  G )  =  ( t  e.  RR  |->  ( abs `  ( G `
 t ) ) ) )
35 iblmbf 22718 . . . . . . . . 9  |-  ( G  e.  L^1  ->  G  e. MblFn )
36 ftc1anclem1 32010 . . . . . . . . 9  |-  ( ( G : RR --> RR  /\  G  e. MblFn )  ->  ( abs  o.  G )  e. MblFn )
3735, 36sylan2 477 . . . . . . . 8  |-  ( ( G : RR --> RR  /\  G  e.  L^1
)  ->  ( abs  o.  G )  e. MblFn )
3837ancoms 455 . . . . . . 7  |-  ( ( G  e.  L^1 
/\  G : RR --> RR )  ->  ( abs 
o.  G )  e. MblFn
)
3934, 38eqeltrrd 2529 . . . . . 6  |-  ( ( G  e.  L^1 
/\  G : RR --> RR )  ->  ( t  e.  RR  |->  ( abs `  ( G `  t
) ) )  e. MblFn
)
40393adant1 1025 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  G  e.  L^1 
/\  G : RR --> RR )  ->  ( t  e.  RR  |->  ( abs `  ( G `  t
) ) )  e. MblFn
)
412abscld 13491 . . . . . . . 8  |-  ( ( G : RR --> RR  /\  t  e.  RR )  ->  ( abs `  ( G `  t )
)  e.  RR )
422absge0d 13499 . . . . . . . 8  |-  ( ( G : RR --> RR  /\  t  e.  RR )  ->  0  <_  ( abs `  ( G `  t
) ) )
43 elrege0 11735 . . . . . . . 8  |-  ( ( abs `  ( G `
 t ) )  e.  ( 0 [,) +oo )  <->  ( ( abs `  ( G `  t
) )  e.  RR  /\  0  <_  ( abs `  ( G `  t
) ) ) )
4441, 42, 43sylanbrc 669 . . . . . . 7  |-  ( ( G : RR --> RR  /\  t  e.  RR )  ->  ( abs `  ( G `  t )
)  e.  ( 0 [,) +oo ) )
45 eqid 2450 . . . . . . 7  |-  ( t  e.  RR  |->  ( abs `  ( G `  t
) ) )  =  ( t  e.  RR  |->  ( abs `  ( G `
 t ) ) )
4644, 45fmptd 6044 . . . . . 6  |-  ( G : RR --> RR  ->  ( t  e.  RR  |->  ( abs `  ( G `
 t ) ) ) : RR --> ( 0 [,) +oo ) )
47463ad2ant3 1030 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  G  e.  L^1 
/\  G : RR --> RR )  ->  ( t  e.  RR  |->  ( abs `  ( G `  t
) ) ) : RR --> ( 0 [,) +oo ) )
48 iftrue 3886 . . . . . . . . 9  |-  ( t  e.  RR  ->  if ( t  e.  RR ,  ( abs `  ( G `  t )
) ,  0 )  =  ( abs `  ( G `  t )
) )
4948mpteq2ia 4484 . . . . . . . 8  |-  ( t  e.  RR  |->  if ( t  e.  RR , 
( abs `  ( G `  t )
) ,  0 ) )  =  ( t  e.  RR  |->  ( abs `  ( G `  t
) ) )
5049fveq2i 5866 . . . . . . 7  |-  ( S.2 `  ( t  e.  RR  |->  if ( t  e.  RR ,  ( abs `  ( G `  t )
) ,  0 ) ) )  =  ( S.2 `  ( t  e.  RR  |->  ( abs `  ( G `  t
) ) ) )
511adantll 719 . . . . . . . . . 10  |-  ( ( ( G  e.  L^1  /\  G : RR --> RR )  /\  t  e.  RR )  ->  ( G `  t )  e.  RR )
52 simpr 463 . . . . . . . . . . . 12  |-  ( ( G  e.  L^1 
/\  G : RR --> RR )  ->  G : RR
--> RR )
5352feqmptd 5916 . . . . . . . . . . 11  |-  ( ( G  e.  L^1 
/\  G : RR --> RR )  ->  G  =  ( t  e.  RR  |->  ( G `  t ) ) )
54 simpl 459 . . . . . . . . . . 11  |-  ( ( G  e.  L^1 
/\  G : RR --> RR )  ->  G  e.  L^1 )
5553, 54eqeltrrd 2529 . . . . . . . . . 10  |-  ( ( G  e.  L^1 
/\  G : RR --> RR )  ->  ( t  e.  RR  |->  ( G `
 t ) )  e.  L^1 )
5651, 55, 39iblabsnc 31999 . . . . . . . . 9  |-  ( ( G  e.  L^1 
/\  G : RR --> RR )  ->  ( t  e.  RR  |->  ( abs `  ( G `  t
) ) )  e.  L^1 )
5741adantll 719 . . . . . . . . . 10  |-  ( ( ( G  e.  L^1  /\  G : RR --> RR )  /\  t  e.  RR )  ->  ( abs `  ( G `  t ) )  e.  RR )
5842adantll 719 . . . . . . . . . 10  |-  ( ( ( G  e.  L^1  /\  G : RR --> RR )  /\  t  e.  RR )  ->  0  <_  ( abs `  ( G `  t )
) )
5957, 58iblpos 22743 . . . . . . . . 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 214 . . . . . . . 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 465 . . . . . . 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 2533 . . . . . 6  |-  ( ( G  e.  L^1 
/\  G : RR --> RR )  ->  ( S.2 `  ( t  e.  RR  |->  ( abs `  ( G `
 t ) ) ) )  e.  RR )
63623adant1 1025 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  G  e.  L^1 
/\  G : RR --> RR )  ->  ( S.2 `  ( t  e.  RR  |->  ( abs `  ( G `
 t ) ) ) )  e.  RR )
645abscld 13491 . . . . . . . 8  |-  ( ( F  e.  dom  S.1  /\  t  e.  RR )  ->  ( abs `  ( F `  t )
)  e.  RR )
655absge0d 13499 . . . . . . . 8  |-  ( ( F  e.  dom  S.1  /\  t  e.  RR )  ->  0  <_  ( abs `  ( F `  t ) ) )
66 elrege0 11735 . . . . . . . 8  |-  ( ( abs `  ( F `
 t ) )  e.  ( 0 [,) +oo )  <->  ( ( abs `  ( F `  t
) )  e.  RR  /\  0  <_  ( abs `  ( F `  t
) ) ) )
6764, 65, 66sylanbrc 669 . . . . . . 7  |-  ( ( F  e.  dom  S.1  /\  t  e.  RR )  ->  ( abs `  ( F `  t )
)  e.  ( 0 [,) +oo ) )
68 eqid 2450 . . . . . . 7  |-  ( t  e.  RR  |->  ( abs `  ( F `  t
) ) )  =  ( t  e.  RR  |->  ( abs `  ( F `
 t ) ) )
6967, 68fmptd 6044 . . . . . 6  |-  ( F  e.  dom  S.1  ->  ( t  e.  RR  |->  ( abs `  ( F `
 t ) ) ) : RR --> ( 0 [,) +oo ) )
70693ad2ant1 1028 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  G  e.  L^1 
/\  G : RR --> RR )  ->  ( t  e.  RR  |->  ( abs `  ( F `  t
) ) ) : RR --> ( 0 [,) +oo ) )
71 iftrue 3886 . . . . . . . . 9  |-  ( t  e.  RR  ->  if ( t  e.  RR ,  ( abs `  ( F `  t )
) ,  0 )  =  ( abs `  ( F `  t )
) )
7271mpteq2ia 4484 . . . . . . . 8  |-  ( t  e.  RR  |->  if ( t  e.  RR , 
( abs `  ( F `  t )
) ,  0 ) )  =  ( t  e.  RR  |->  ( abs `  ( F `  t
) ) )
7372fveq2i 5866 . . . . . . 7  |-  ( S.2 `  ( t  e.  RR  |->  if ( t  e.  RR ,  ( abs `  ( F `  t )
) ,  0 ) ) )  =  ( S.2 `  ( t  e.  RR  |->  ( abs `  ( F `  t
) ) ) )
743feqmptd 5916 . . . . . . . . . . 11  |-  ( F  e.  dom  S.1  ->  F  =  ( t  e.  RR  |->  ( F `  t ) ) )
75 i1fibl 22758 . . . . . . . . . . 11  |-  ( F  e.  dom  S.1  ->  F  e.  L^1 )
7674, 75eqeltrrd 2529 . . . . . . . . . 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 5916 . . . . . . . . . . . 12  |-  ( F  e.  dom  S.1  ->  abs  =  ( x  e.  CC  |->  ( abs `  x
) ) )
79 fveq2 5863 . . . . . . . . . . . 12  |-  ( x  =  ( F `  t )  ->  ( abs `  x )  =  ( abs `  ( F `  t )
) )
805, 74, 78, 79fmptco 6054 . . . . . . . . . . 11  |-  ( F  e.  dom  S.1  ->  ( abs  o.  F )  =  ( t  e.  RR  |->  ( abs `  ( F `  t )
) ) )
81 i1fmbf 22626 . . . . . . . . . . . 12  |-  ( F  e.  dom  S.1  ->  F  e. MblFn )
82 ftc1anclem1 32010 . . . . . . . . . . . 12  |-  ( ( F : RR --> RR  /\  F  e. MblFn )  ->  ( abs  o.  F )  e. MblFn )
833, 81, 82syl2anc 666 . . . . . . . . . . 11  |-  ( F  e.  dom  S.1  ->  ( abs  o.  F )  e. MblFn )
8480, 83eqeltrrd 2529 . . . . . . . . . 10  |-  ( F  e.  dom  S.1  ->  ( t  e.  RR  |->  ( abs `  ( F `
 t ) ) )  e. MblFn )
854, 76, 84iblabsnc 31999 . . . . . . . . 9  |-  ( F  e.  dom  S.1  ->  ( t  e.  RR  |->  ( abs `  ( F `
 t ) ) )  e.  L^1 )
8664, 65iblpos 22743 . . . . . . . . 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 214 . . . . . . . 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 465 . . . . . . 7  |-  ( F  e.  dom  S.1  ->  ( S.2 `  ( t  e.  RR  |->  if ( t  e.  RR , 
( abs `  ( F `  t )
) ,  0 ) ) )  e.  RR )
8973, 88syl5eqelr 2533 . . . . . 6  |-  ( F  e.  dom  S.1  ->  ( S.2 `  ( t  e.  RR  |->  ( abs `  ( F `  t
) ) ) )  e.  RR )
90893ad2ant1 1028 . . . . 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 31989 . . . 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 2486 . . 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 9667 . . 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 2528 . 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 9619 . . . . . . . . 9  |-  ( ( ( abs `  ( G `  t )
)  e.  RR  /\  ( abs `  ( F `
 t ) )  e.  RR )  -> 
( ( abs `  ( G `  t )
)  +  ( abs `  ( F `  t
) ) )  e.  RR )
9641, 64, 95syl2anr 481 . . . . . . . 8  |-  ( ( ( F  e.  dom  S.1 
/\  t  e.  RR )  /\  ( G : RR
--> RR  /\  t  e.  RR ) )  -> 
( ( abs `  ( G `  t )
)  +  ( abs `  ( F `  t
) ) )  e.  RR )
9796anandirs 839 . . . . . . 7  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  (
( abs `  ( G `  t )
)  +  ( abs `  ( F `  t
) ) )  e.  RR )
9897rexrd 9687 . . . . . 6  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  (
( abs `  ( G `  t )
)  +  ( abs `  ( F `  t
) ) )  e. 
RR* )
9941adantll 719 . . . . . . 7  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  ( abs `  ( G `  t ) )  e.  RR )
10064adantlr 720 . . . . . . 7  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  ( abs `  ( F `  t ) )  e.  RR )
10142adantll 719 . . . . . . 7  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  0  <_  ( abs `  ( G `  t )
) )
10265adantlr 720 . . . . . . 7  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  0  <_  ( abs `  ( F `  t )
) )
10399, 100, 101, 102addge0d 10186 . . . . . 6  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  0  <_  ( ( abs `  ( G `  t )
)  +  ( abs `  ( F `  t
) ) ) )
104 elxrge0 11738 . . . . . 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 669 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  (
( abs `  ( G `  t )
)  +  ( abs `  ( F `  t
) ) )  e.  ( 0 [,] +oo ) )
106 eqid 2450 . . . . 5  |-  ( t  e.  RR  |->  ( ( abs `  ( G `
 t ) )  +  ( abs `  ( F `  t )
) ) )  =  ( t  e.  RR  |->  ( ( abs `  ( G `  t )
)  +  ( abs `  ( F `  t
) ) ) )
107105, 106fmptd 6044 . . . 4  |-  ( ( F  e.  dom  S.1  /\  G : RR --> RR )  ->  ( t  e.  RR  |->  ( ( abs `  ( G `  t
) )  +  ( abs `  ( F `
 t ) ) ) ) : RR --> ( 0 [,] +oo ) )
1081073adant2 1026 . . 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 13389 . . . . . . . 8  |-  ( ( ( G `  t
)  e.  CC  /\  ( F `  t )  e.  CC )  -> 
( abs `  (
( G `  t
)  -  ( F `
 t ) ) )  <_  ( ( abs `  ( G `  t ) )  +  ( abs `  ( F `  t )
) ) )
1102, 5, 109syl2anr 481 . . . . . . 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 839 . . . . . 6  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  ( abs `  ( ( G `
 t )  -  ( F `  t ) ) )  <_  (
( abs `  ( G `  t )
)  +  ( abs `  ( F `  t
) ) ) )
112111ralrimiva 2801 . . . . 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 2451 . . . . . 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 2451 . . . . . 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 6546 . . . . 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 236 . . . 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 1026 . . 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 22690 . . 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 1267 . 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 22689 . 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 1267 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 371    /\ w3a 984    = wceq 1443    e. wcel 1886   A.wral 2736   _Vcvv 3044   ifcif 3880   class class class wbr 4401    |-> cmpt 4460   dom cdm 4833    o. ccom 4837   -->wf 5577   ` cfv 5581  (class class class)co 6288    oFcof 6526    oRcofr 6527   CCcc 9534   RRcr 9535   0cc0 9536    + caddc 9539   +oocpnf 9669   RR*cxr 9671    <_ cle 9673    - cmin 9857   [,)cico 11634   [,]cicc 11635   abscabs 13290  MblFncmbf 22565   S.1citg1 22566   S.2citg2 22567   L^1cibl 22568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614  ax-addf 9615
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-disj 4373  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-of 6528  df-ofr 6529  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-2o 7180  df-oadd 7183  df-er 7360  df-map 7471  df-pm 7472  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-fi 7922  df-sup 7953  df-inf 7954  df-oi 8022  df-card 8370  df-cda 8595  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-n0 10867  df-z 10935  df-uz 11157  df-q 11262  df-rp 11300  df-xneg 11406  df-xadd 11407  df-xmul 11408  df-ioo 11636  df-ico 11638  df-icc 11639  df-fz 11782  df-fzo 11913  df-fl 12025  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-clim 13545  df-sum 13746  df-rest 15314  df-topgen 15335  df-psmet 18955  df-xmet 18956  df-met 18957  df-bl 18958  df-mopn 18959  df-top 19914  df-bases 19915  df-topon 19916  df-cmp 20395  df-ovol 22409  df-vol 22411  df-mbf 22570  df-itg1 22571  df-itg2 22572  df-ibl 22573  df-0p 22621
This theorem is referenced by:  ftc1anclem5  32014  ftc1anclem6  32015
  Copyright terms: Public domain W3C validator