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Theorem ftc1anclem4 28395
Description: Lemma for ftc1anc 28400. (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 5838 . . . . . . . . . 10  |-  ( ( G : RR --> RR  /\  t  e.  RR )  ->  ( G `  t
)  e.  RR )
21recnd 9408 . . . . . . . . 9  |-  ( ( G : RR --> RR  /\  t  e.  RR )  ->  ( G `  t
)  e.  CC )
3 i1ff 21113 . . . . . . . . . . 11  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
43ffvelrnda 5840 . . . . . . . . . 10  |-  ( ( F  e.  dom  S.1  /\  t  e.  RR )  ->  ( F `  t )  e.  RR )
54recnd 9408 . . . . . . . . 9  |-  ( ( F  e.  dom  S.1  /\  t  e.  RR )  ->  ( F `  t )  e.  CC )
6 subcl 9605 . . . . . . . . 9  |-  ( ( ( G `  t
)  e.  CC  /\  ( F `  t )  e.  CC )  -> 
( ( G `  t )  -  ( F `  t )
)  e.  CC )
72, 5, 6syl2anr 475 . . . . . . . 8  |-  ( ( ( F  e.  dom  S.1 
/\  t  e.  RR )  /\  ( G : RR
--> RR  /\  t  e.  RR ) )  -> 
( ( G `  t )  -  ( F `  t )
)  e.  CC )
87anandirs 822 . . . . . . 7  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  (
( G `  t
)  -  ( F `
 t ) )  e.  CC )
98abscld 12918 . . . . . 6  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  ( abs `  ( ( G `
 t )  -  ( F `  t ) ) )  e.  RR )
109rexrd 9429 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  ( abs `  ( ( G `
 t )  -  ( F `  t ) ) )  e.  RR* )
118absge0d 12926 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  0  <_  ( abs `  (
( G `  t
)  -  ( F `
 t ) ) ) )
12 elxrge0 11390 . . . . 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 659 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  ( abs `  ( ( G `
 t )  -  ( F `  t ) ) )  e.  ( 0 [,] +oo )
)
14 eqid 2441 . . . 4  |-  ( t  e.  RR  |->  ( abs `  ( ( G `  t )  -  ( F `  t )
) ) )  =  ( t  e.  RR  |->  ( abs `  ( ( G `  t )  -  ( F `  t ) ) ) )
1513, 14fmptd 5864 . . 3  |-  ( ( F  e.  dom  S.1  /\  G : RR --> RR )  ->  ( t  e.  RR  |->  ( abs `  (
( G `  t
)  -  ( F `
 t ) ) ) ) : RR --> ( 0 [,] +oo ) )
16153adant2 1002 . 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 9369 . . . . . . 7  |-  RR  e.  _V
1817a1i 11 . . . . . 6  |-  ( ( F  e.  dom  S.1  /\  G  e.  L^1 
/\  G : RR --> RR )  ->  RR  e.  _V )
19 fvex 5698 . . . . . . 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 5698 . . . . . . 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 2442 . . . . . 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 2442 . . . . . 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 6335 . . . . 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 5692 . . . 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 5741 . . . . . . . . 9  |-  ( G : RR --> RR  ->  G  =  ( t  e.  RR  |->  ( G `  t ) ) )
29 absf 12821 . . . . . . . . . . 11  |-  abs : CC
--> RR
3029a1i 11 . . . . . . . . . 10  |-  ( G : RR --> RR  ->  abs
: CC --> RR )
3130feqmptd 5741 . . . . . . . . 9  |-  ( G : RR --> RR  ->  abs  =  ( x  e.  CC  |->  ( abs `  x
) ) )
32 fveq2 5688 . . . . . . . . 9  |-  ( x  =  ( G `  t )  ->  ( abs `  x )  =  ( abs `  ( G `  t )
) )
332, 28, 31, 32fmptco 5873 . . . . . . . 8  |-  ( G : RR --> RR  ->  ( abs  o.  G )  =  ( t  e.  RR  |->  ( abs `  ( G `  t )
) ) )
3433adantl 463 . . . . . . 7  |-  ( ( G  e.  L^1 
/\  G : RR --> RR )  ->  ( abs 
o.  G )  =  ( t  e.  RR  |->  ( abs `  ( G `
 t ) ) ) )
35 iblmbf 21204 . . . . . . . . 9  |-  ( G  e.  L^1  ->  G  e. MblFn )
36 ftc1anclem1 28392 . . . . . . . . 9  |-  ( ( G : RR --> RR  /\  G  e. MblFn )  ->  ( abs  o.  G )  e. MblFn )
3735, 36sylan2 471 . . . . . . . 8  |-  ( ( G : RR --> RR  /\  G  e.  L^1
)  ->  ( abs  o.  G )  e. MblFn )
3837ancoms 450 . . . . . . 7  |-  ( ( G  e.  L^1 
/\  G : RR --> RR )  ->  ( abs 
o.  G )  e. MblFn
)
3934, 38eqeltrrd 2516 . . . . . 6  |-  ( ( G  e.  L^1 
/\  G : RR --> RR )  ->  ( t  e.  RR  |->  ( abs `  ( G `  t
) ) )  e. MblFn
)
40393adant1 1001 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  G  e.  L^1 
/\  G : RR --> RR )  ->  ( t  e.  RR  |->  ( abs `  ( G `  t
) ) )  e. MblFn
)
412abscld 12918 . . . . . . . 8  |-  ( ( G : RR --> RR  /\  t  e.  RR )  ->  ( abs `  ( G `  t )
)  e.  RR )
422absge0d 12926 . . . . . . . 8  |-  ( ( G : RR --> RR  /\  t  e.  RR )  ->  0  <_  ( abs `  ( G `  t
) ) )
43 elrege0 11388 . . . . . . . 8  |-  ( ( abs `  ( G `
 t ) )  e.  ( 0 [,) +oo )  <->  ( ( abs `  ( G `  t
) )  e.  RR  /\  0  <_  ( abs `  ( G `  t
) ) ) )
4441, 42, 43sylanbrc 659 . . . . . . 7  |-  ( ( G : RR --> RR  /\  t  e.  RR )  ->  ( abs `  ( G `  t )
)  e.  ( 0 [,) +oo ) )
45 eqid 2441 . . . . . . 7  |-  ( t  e.  RR  |->  ( abs `  ( G `  t
) ) )  =  ( t  e.  RR  |->  ( abs `  ( G `
 t ) ) )
4644, 45fmptd 5864 . . . . . 6  |-  ( G : RR --> RR  ->  ( t  e.  RR  |->  ( abs `  ( G `
 t ) ) ) : RR --> ( 0 [,) +oo ) )
47463ad2ant3 1006 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  G  e.  L^1 
/\  G : RR --> RR )  ->  ( t  e.  RR  |->  ( abs `  ( G `  t
) ) ) : RR --> ( 0 [,) +oo ) )
48 iftrue 3794 . . . . . . . . 9  |-  ( t  e.  RR  ->  if ( t  e.  RR ,  ( abs `  ( G `  t )
) ,  0 )  =  ( abs `  ( G `  t )
) )
4948mpteq2ia 4371 . . . . . . . 8  |-  ( t  e.  RR  |->  if ( t  e.  RR , 
( abs `  ( G `  t )
) ,  0 ) )  =  ( t  e.  RR  |->  ( abs `  ( G `  t
) ) )
5049fveq2i 5691 . . . . . . 7  |-  ( S.2 `  ( t  e.  RR  |->  if ( t  e.  RR ,  ( abs `  ( G `  t )
) ,  0 ) ) )  =  ( S.2 `  ( t  e.  RR  |->  ( abs `  ( G `  t
) ) ) )
511adantll 708 . . . . . . . . . 10  |-  ( ( ( G  e.  L^1  /\  G : RR --> RR )  /\  t  e.  RR )  ->  ( G `  t )  e.  RR )
52 simpr 458 . . . . . . . . . . . 12  |-  ( ( G  e.  L^1 
/\  G : RR --> RR )  ->  G : RR
--> RR )
5352feqmptd 5741 . . . . . . . . . . 11  |-  ( ( G  e.  L^1 
/\  G : RR --> RR )  ->  G  =  ( t  e.  RR  |->  ( G `  t ) ) )
54 simpl 454 . . . . . . . . . . 11  |-  ( ( G  e.  L^1 
/\  G : RR --> RR )  ->  G  e.  L^1 )
5553, 54eqeltrrd 2516 . . . . . . . . . 10  |-  ( ( G  e.  L^1 
/\  G : RR --> RR )  ->  ( t  e.  RR  |->  ( G `
 t ) )  e.  L^1 )
5651, 55, 39iblabsnc 28381 . . . . . . . . 9  |-  ( ( G  e.  L^1 
/\  G : RR --> RR )  ->  ( t  e.  RR  |->  ( abs `  ( G `  t
) ) )  e.  L^1 )
5741adantll 708 . . . . . . . . . 10  |-  ( ( ( G  e.  L^1  /\  G : RR --> RR )  /\  t  e.  RR )  ->  ( abs `  ( G `  t ) )  e.  RR )
5842adantll 708 . . . . . . . . . 10  |-  ( ( ( G  e.  L^1  /\  G : RR --> RR )  /\  t  e.  RR )  ->  0  <_  ( abs `  ( G `  t )
) )
5957, 58iblpos 21229 . . . . . . . . 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 460 . . . . . . 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 2526 . . . . . 6  |-  ( ( G  e.  L^1 
/\  G : RR --> RR )  ->  ( S.2 `  ( t  e.  RR  |->  ( abs `  ( G `
 t ) ) ) )  e.  RR )
63623adant1 1001 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  G  e.  L^1 
/\  G : RR --> RR )  ->  ( S.2 `  ( t  e.  RR  |->  ( abs `  ( G `
 t ) ) ) )  e.  RR )
645abscld 12918 . . . . . . . 8  |-  ( ( F  e.  dom  S.1  /\  t  e.  RR )  ->  ( abs `  ( F `  t )
)  e.  RR )
655absge0d 12926 . . . . . . . 8  |-  ( ( F  e.  dom  S.1  /\  t  e.  RR )  ->  0  <_  ( abs `  ( F `  t ) ) )
66 elrege0 11388 . . . . . . . 8  |-  ( ( abs `  ( F `
 t ) )  e.  ( 0 [,) +oo )  <->  ( ( abs `  ( F `  t
) )  e.  RR  /\  0  <_  ( abs `  ( F `  t
) ) ) )
6764, 65, 66sylanbrc 659 . . . . . . 7  |-  ( ( F  e.  dom  S.1  /\  t  e.  RR )  ->  ( abs `  ( F `  t )
)  e.  ( 0 [,) +oo ) )
68 eqid 2441 . . . . . . 7  |-  ( t  e.  RR  |->  ( abs `  ( F `  t
) ) )  =  ( t  e.  RR  |->  ( abs `  ( F `
 t ) ) )
6967, 68fmptd 5864 . . . . . 6  |-  ( F  e.  dom  S.1  ->  ( t  e.  RR  |->  ( abs `  ( F `
 t ) ) ) : RR --> ( 0 [,) +oo ) )
70693ad2ant1 1004 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  G  e.  L^1 
/\  G : RR --> RR )  ->  ( t  e.  RR  |->  ( abs `  ( F `  t
) ) ) : RR --> ( 0 [,) +oo ) )
71 iftrue 3794 . . . . . . . . 9  |-  ( t  e.  RR  ->  if ( t  e.  RR ,  ( abs `  ( F `  t )
) ,  0 )  =  ( abs `  ( F `  t )
) )
7271mpteq2ia 4371 . . . . . . . 8  |-  ( t  e.  RR  |->  if ( t  e.  RR , 
( abs `  ( F `  t )
) ,  0 ) )  =  ( t  e.  RR  |->  ( abs `  ( F `  t
) ) )
7372fveq2i 5691 . . . . . . 7  |-  ( S.2 `  ( t  e.  RR  |->  if ( t  e.  RR ,  ( abs `  ( F `  t )
) ,  0 ) ) )  =  ( S.2 `  ( t  e.  RR  |->  ( abs `  ( F `  t
) ) ) )
743feqmptd 5741 . . . . . . . . . . 11  |-  ( F  e.  dom  S.1  ->  F  =  ( t  e.  RR  |->  ( F `  t ) ) )
75 i1fibl 21244 . . . . . . . . . . 11  |-  ( F  e.  dom  S.1  ->  F  e.  L^1 )
7674, 75eqeltrrd 2516 . . . . . . . . . 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 5741 . . . . . . . . . . . 12  |-  ( F  e.  dom  S.1  ->  abs  =  ( x  e.  CC  |->  ( abs `  x
) ) )
79 fveq2 5688 . . . . . . . . . . . 12  |-  ( x  =  ( F `  t )  ->  ( abs `  x )  =  ( abs `  ( F `  t )
) )
805, 74, 78, 79fmptco 5873 . . . . . . . . . . 11  |-  ( F  e.  dom  S.1  ->  ( abs  o.  F )  =  ( t  e.  RR  |->  ( abs `  ( F `  t )
) ) )
81 i1fmbf 21112 . . . . . . . . . . . 12  |-  ( F  e.  dom  S.1  ->  F  e. MblFn )
82 ftc1anclem1 28392 . . . . . . . . . . . 12  |-  ( ( F : RR --> RR  /\  F  e. MblFn )  ->  ( abs  o.  F )  e. MblFn )
833, 81, 82syl2anc 656 . . . . . . . . . . 11  |-  ( F  e.  dom  S.1  ->  ( abs  o.  F )  e. MblFn )
8480, 83eqeltrrd 2516 . . . . . . . . . 10  |-  ( F  e.  dom  S.1  ->  ( t  e.  RR  |->  ( abs `  ( F `
 t ) ) )  e. MblFn )
854, 76, 84iblabsnc 28381 . . . . . . . . 9  |-  ( F  e.  dom  S.1  ->  ( t  e.  RR  |->  ( abs `  ( F `
 t ) ) )  e.  L^1 )
8664, 65iblpos 21229 . . . . . . . . 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 460 . . . . . . 7  |-  ( F  e.  dom  S.1  ->  ( S.2 `  ( t  e.  RR  |->  if ( t  e.  RR , 
( abs `  ( F `  t )
) ,  0 ) ) )  e.  RR )
8973, 88syl5eqelr 2526 . . . . . 6  |-  ( F  e.  dom  S.1  ->  ( S.2 `  ( t  e.  RR  |->  ( abs `  ( F `  t
) ) ) )  e.  RR )
90893ad2ant1 1004 . . . . 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 28371 . . . 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 2475 . . 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 9409 . . 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 2515 . 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 9361 . . . . . . . . 9  |-  ( ( ( abs `  ( G `  t )
)  e.  RR  /\  ( abs `  ( F `
 t ) )  e.  RR )  -> 
( ( abs `  ( G `  t )
)  +  ( abs `  ( F `  t
) ) )  e.  RR )
9641, 64, 95syl2anr 475 . . . . . . . 8  |-  ( ( ( F  e.  dom  S.1 
/\  t  e.  RR )  /\  ( G : RR
--> RR  /\  t  e.  RR ) )  -> 
( ( abs `  ( G `  t )
)  +  ( abs `  ( F `  t
) ) )  e.  RR )
9796anandirs 822 . . . . . . 7  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  (
( abs `  ( G `  t )
)  +  ( abs `  ( F `  t
) ) )  e.  RR )
9897rexrd 9429 . . . . . 6  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  (
( abs `  ( G `  t )
)  +  ( abs `  ( F `  t
) ) )  e. 
RR* )
9941adantll 708 . . . . . . 7  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  ( abs `  ( G `  t ) )  e.  RR )
10064adantlr 709 . . . . . . 7  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  ( abs `  ( F `  t ) )  e.  RR )
10142adantll 708 . . . . . . 7  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  0  <_  ( abs `  ( G `  t )
) )
10265adantlr 709 . . . . . . 7  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  0  <_  ( abs `  ( F `  t )
) )
10399, 100, 101, 102addge0d 9911 . . . . . 6  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  0  <_  ( ( abs `  ( G `  t )
)  +  ( abs `  ( F `  t
) ) ) )
104 elxrge0 11390 . . . . . 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 659 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  (
( abs `  ( G `  t )
)  +  ( abs `  ( F `  t
) ) )  e.  ( 0 [,] +oo ) )
106 eqid 2441 . . . . 5  |-  ( t  e.  RR  |->  ( ( abs `  ( G `
 t ) )  +  ( abs `  ( F `  t )
) ) )  =  ( t  e.  RR  |->  ( ( abs `  ( G `  t )
)  +  ( abs `  ( F `  t
) ) ) )
107105, 106fmptd 5864 . . . 4  |-  ( ( F  e.  dom  S.1  /\  G : RR --> RR )  ->  ( t  e.  RR  |->  ( ( abs `  ( G `  t
) )  +  ( abs `  ( F `
 t ) ) ) ) : RR --> ( 0 [,] +oo ) )
1081073adant2 1002 . . 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 12817 . . . . . . . 8  |-  ( ( ( G `  t
)  e.  CC  /\  ( F `  t )  e.  CC )  -> 
( abs `  (
( G `  t
)  -  ( F `
 t ) ) )  <_  ( ( abs `  ( G `  t ) )  +  ( abs `  ( F `  t )
) ) )
1102, 5, 109syl2anr 475 . . . . . . 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 822 . . . . . 6  |-  ( ( ( F  e.  dom  S.1 
/\  G : RR --> RR )  /\  t  e.  RR )  ->  ( abs `  ( ( G `
 t )  -  ( F `  t ) ) )  <_  (
( abs `  ( G `  t )
)  +  ( abs `  ( F `  t
) ) ) )
112111ralrimiva 2797 . . . . 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 2442 . . . . . 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 2442 . . . . . 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 6336 . . . . 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 1002 . . 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 21176 . . 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 1213 . 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 21175 . 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 1213 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 960    = wceq 1364    e. wcel 1761   A.wral 2713   _Vcvv 2970   ifcif 3788   class class class wbr 4289    e. cmpt 4347   dom cdm 4836    o. ccom 4840   -->wf 5411   ` cfv 5415  (class class class)co 6090    oFcof 6317    oRcofr 6318   CCcc 9276   RRcr 9277   0cc0 9278    + caddc 9281   +oocpnf 9411   RR*cxr 9413    <_ cle 9415    - cmin 9591   [,)cico 11298   [,]cicc 11299   abscabs 12719  MblFncmbf 21053   S.1citg1 21054   S.2citg2 21055   L^1cibl 21056
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-disj 4260  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-ofr 6320  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-fl 11638  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-clim 12962  df-sum 13160  df-rest 14357  df-topgen 14378  df-psmet 17768  df-xmet 17769  df-met 17770  df-bl 17771  df-mopn 17772  df-top 18462  df-bases 18464  df-topon 18465  df-cmp 18949  df-ovol 20907  df-vol 20908  df-mbf 21058  df-itg1 21059  df-itg2 21060  df-ibl 21061  df-0p 21107
This theorem is referenced by:  ftc1anclem5  28396  ftc1anclem6  28397
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