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Theorem ditgex 22794
Description: A directed integral is a set. (Contributed by Mario Carneiro, 7-Sep-2014.)
Assertion
Ref Expression
ditgex  |-  S__ [ A  ->  B ] C  _d x  e.  _V

Proof of Theorem ditgex
StepHypRef Expression
1 df-ditg 22789 . 2  |-  S__ [ A  ->  B ] C  _d x  =  if ( A  <_  B ,  S. ( A (,) B
) C  _d x ,  -u S. ( B (,) A ) C  _d x )
2 itgex 22715 . . 3  |-  S. ( A (,) B ) C  _d x  e. 
_V
3 negex 9874 . . 3  |-  -u S. ( B (,) A ) C  _d x  e. 
_V
42, 3ifex 3977 . 2  |-  if ( A  <_  B ,  S. ( A (,) B
) C  _d x ,  -u S. ( B (,) A ) C  _d x )  e. 
_V
51, 4eqeltri 2506 1  |-  S__ [ A  ->  B ] C  _d x  e.  _V
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1868   _Vcvv 3081   ifcif 3909   class class class wbr 4420  (class class class)co 6302    <_ cle 9677   -ucneg 9862   (,)cioo 11636   S.citg 22563   S__cdit 22788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-nul 4552
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-v 3083  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-uni 4217  df-iota 5562  df-fv 5606  df-ov 6305  df-neg 9864  df-sum 13741  df-itg 22568  df-ditg 22789
This theorem is referenced by:  itgsubstlem  22987
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