Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  domprobsiga Structured version   Visualization version   GIF version

Theorem domprobsiga 31072
Description: The domain of a probability measure is a sigma-algebra. (Contributed by Thierry Arnoux, 23-Dec-2016.)
Assertion
Ref Expression
domprobsiga (𝑃 ∈ Prob → dom 𝑃 ran sigAlgebra)

Proof of Theorem domprobsiga
StepHypRef Expression
1 domprobmeas 31071 . 2 (𝑃 ∈ Prob → 𝑃 ∈ (measures‘dom 𝑃))
2 measbase 30858 . 2 (𝑃 ∈ (measures‘dom 𝑃) → dom 𝑃 ran sigAlgebra)
31, 2syl 17 1 (𝑃 ∈ Prob → dom 𝑃 ran sigAlgebra)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106   cuni 4671  dom cdm 5355  ran crn 5356  cfv 6135  sigAlgebracsiga 30768  measurescmeas 30856  Probcprb 31068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-fv 6143  df-ov 6925  df-esum 30688  df-meas 30857  df-prob 31069
This theorem is referenced by:  unveldomd  31076  nuleldmp  31078  probdif  31081  totprobd  31087  cndprobin  31095  cndprob01  31096  isrrvv  31104  0rrv  31112  rrvadd  31113  rrvmulc  31114  orrvcval4  31125  orrvcoel  31126  orrvccel  31127
  Copyright terms: Public domain W3C validator