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Theorem ditgpos 19259
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 19033 . 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 3605 . . 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 15 . 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 2360 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 1633   ifcif 3599   class class class wbr 4060  (class class class)co 5900    <_ cle 8913   -ucneg 9083   (,)cioo 10703   S.citg 19026   S__cdit 19027
This theorem is referenced by:  ditgcl  19261  ditgswap  19262  ditgsplitlem  19263  ftc2ditglem  19445  itgsubstlem  19448  itgsubst  19449
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-if 3600  df-ditg 19033
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