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Theorem ditgpos 22126
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 22117 . 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 )
32iftrued 3930 . 2  |-  ( ph  ->  if ( A  <_  B ,  S. ( A (,) B ) C  _d x ,  -u S. ( B (,) A
) C  _d x )  =  S. ( A (,) B ) C  _d x )
41, 3syl5eq 2494 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 1381   ifcif 3922   class class class wbr 4433  (class class class)co 6277    <_ cle 9627   -ucneg 9806   (,)cioo 11533   S.citg 21893   S__cdit 22116
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-clab 2427  df-cleq 2433  df-clel 2436  df-if 3923  df-ditg 22117
This theorem is referenced by:  ditgcl  22128  ditgswap  22129  ditgsplitlem  22130  ftc2ditglem  22312  itgsubstlem  22315  itgsubst  22316  ditgeqiooicc  31645  itgiccshift  31665  itgperiod  31666  fourierdlem82  31856
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