MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ditgex Structured version   Visualization version   GIF version

Theorem ditgex 23836
Description: A directed integral is a set. (Contributed by Mario Carneiro, 7-Sep-2014.)
Assertion
Ref Expression
ditgex ⨜[𝐴𝐵]𝐶 d𝑥 ∈ V

Proof of Theorem ditgex
StepHypRef Expression
1 df-ditg 23831 . 2 ⨜[𝐴𝐵]𝐶 d𝑥 = if(𝐴𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥)
2 itgex 23757 . . 3 ∫(𝐴(,)𝐵)𝐶 d𝑥 ∈ V
3 negex 10485 . . 3 -∫(𝐵(,)𝐴)𝐶 d𝑥 ∈ V
42, 3ifex 4296 . 2 if(𝐴𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥) ∈ V
51, 4eqeltri 2846 1 ⨜[𝐴𝐵]𝐶 d𝑥 ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2145  Vcvv 3351  ifcif 4226   class class class wbr 4787  (class class class)co 6796  cle 10281  -cneg 10473  (,)cioo 12380  citg 23606  cdit 23830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-nul 4924
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-uni 4576  df-iota 5993  df-fv 6038  df-ov 6799  df-neg 10475  df-sum 14625  df-itg 23611  df-ditg 23831
This theorem is referenced by:  itgsubstlem  24031
  Copyright terms: Public domain W3C validator