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Theorem ditgpos 21173
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 21164 . 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 3785 . . 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 2477 1  |-  ( ph  ->  S__ [ A  ->  B ] C  _d x  =  S. ( A (,) B ) C  _d x )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1362   ifcif 3779   class class class wbr 4280  (class class class)co 6080    <_ cle 9407   -ucneg 9584   (,)cioo 11288   S.citg 20940   S__cdit 21163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-clab 2420  df-cleq 2426  df-clel 2429  df-if 3780  df-ditg 21164
This theorem is referenced by:  ditgcl  21175  ditgswap  21176  ditgsplitlem  21177  ftc2ditglem  21359  itgsubstlem  21362  itgsubst  21363
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