MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ditgpos Structured version   Unicode version

Theorem ditgpos 22128
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 22119 . 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 3951 . . 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 2520 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 1379   ifcif 3945   class class class wbr 4453  (class class class)co 6295    <_ cle 9641   -ucneg 9818   (,)cioo 11541   S.citg 21895   S__cdit 22118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-if 3946  df-ditg 22119
This theorem is referenced by:  ditgcl  22130  ditgswap  22131  ditgsplitlem  22132  ftc2ditglem  22314  itgsubstlem  22317  itgsubst  22318  ditgeqiooicc  31601  itgiccshift  31621  itgperiod  31622  fourierdlem82  31812
  Copyright terms: Public domain W3C validator