Theorem ditgpos 23620

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

Step	Hyp	Ref	Expression
1		df-ditg 23611	. 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	2	iftrued 4094	. 2 |- ( ph -> if ( A <_ B , S. ( A (,) B ) C _d x , -u S. ( B (,) A ) C _d x ) = S. ( A (,) B ) C _d x )
4	1, 3	syl5eq 2668	1 |- ( ph -> S__ [ A -> B ] C _d x = S. ( A (,) B ) C _d x )

Syntax hints:   -> wi 4   = wceq 1483   ifcif 4086   class class class wbr 4653  (class class class)co 6650   <_ cle 10075   -ucneg 10267   (,)cioo 12175   S.citg 23387   S__cdit 23610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-if 4087  df-ditg 23611

This theorem is referenced by:  ditgcl  23622  ditgswap  23623  ditgsplitlem  23624  ftc2ditglem  23808  itgsubstlem  23811  itgsubst  23812  ditgeqiooicc  40176  itgiccshift  40196  itgperiod  40197  fourierdlem82  40405