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Theorem ditgpos 19745
Description: Value of the directed integral in the forward direction. (Contributed by Mario Carneiro, 13-Aug-2014.)
Hypothesis
Ref Expression
ditgpos.1  |-  ( ph  ->  A  <_  B )
Assertion
Ref Expression
ditgpos  |-  ( ph  ->  S__ [ A  ->  B ] C  _d x  =  S. ( A (,) B ) C  _d x )
Distinct variable groups:    x, A    x, B    ph, x
Allowed substitution hint:    C( x)

Proof of Theorem ditgpos
StepHypRef Expression
1 df-ditg 19519 . 2  |-  S__ [ A  ->  B ] C  _d x  =  if ( A  <_  B ,  S. ( A (,) B
) C  _d x ,  -u S. ( B (,) A ) C  _d x )
2 ditgpos.1 . . 3  |-  ( ph  ->  A  <_  B )
3 iftrue 3747 . . 3  |-  ( A  <_  B  ->  if ( A  <_  B ,  S. ( A (,) B
) C  _d x ,  -u S. ( B (,) A ) C  _d x )  =  S. ( A (,) B ) C  _d x )
42, 3syl 16 . 2  |-  ( ph  ->  if ( A  <_  B ,  S. ( A (,) B ) C  _d x ,  -u S. ( B (,) A
) C  _d x )  =  S. ( A (,) B ) C  _d x )
51, 4syl5eq 2482 1  |-  ( ph  ->  S__ [ A  ->  B ] C  _d x  =  S. ( A (,) B ) C  _d x )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653   ifcif 3741   class class class wbr 4214  (class class class)co 6083    <_ cle 9123   -ucneg 9294   (,)cioo 10918   S.citg 19512   S__cdit 19513
This theorem is referenced by:  ditgcl  19747  ditgswap  19748  ditgsplitlem  19749  ftc2ditglem  19931  itgsubstlem  19934  itgsubst  19935
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-if 3742  df-ditg 19519
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