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Theorem probcun 34071
Description: The probability of the union of a countable disjoint set of events is the sum of their probabilities. (Third axiom of Kolmogorov) Here, the Ξ£ construct cannot be used as it can handle infinite indexing set only if they are subsets of β„€, which is not the case here. (Contributed by Thierry Arnoux, 25-Dec-2016.)
Assertion
Ref Expression
probcun ((𝑃 ∈ Prob ∧ 𝐴 ∈ 𝒫 dom 𝑃 ∧ (𝐴 β‰Ό Ο‰ ∧ Disj π‘₯ ∈ 𝐴 π‘₯)) β†’ (π‘ƒβ€˜βˆͺ 𝐴) = Ξ£*π‘₯ ∈ 𝐴(π‘ƒβ€˜π‘₯))
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝑃

Proof of Theorem probcun
StepHypRef Expression
1 domprobmeas 34063 . 2 (𝑃 ∈ Prob β†’ 𝑃 ∈ (measuresβ€˜dom 𝑃))
2 measvun 33861 . 2 ((𝑃 ∈ (measuresβ€˜dom 𝑃) ∧ 𝐴 ∈ 𝒫 dom 𝑃 ∧ (𝐴 β‰Ό Ο‰ ∧ Disj π‘₯ ∈ 𝐴 π‘₯)) β†’ (π‘ƒβ€˜βˆͺ 𝐴) = Ξ£*π‘₯ ∈ 𝐴(π‘ƒβ€˜π‘₯))
31, 2syl3an1 1160 1 ((𝑃 ∈ Prob ∧ 𝐴 ∈ 𝒫 dom 𝑃 ∧ (𝐴 β‰Ό Ο‰ ∧ Disj π‘₯ ∈ 𝐴 π‘₯)) β†’ (π‘ƒβ€˜βˆͺ 𝐴) = Ξ£*π‘₯ ∈ 𝐴(π‘ƒβ€˜π‘₯))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  π’« cpw 4606  βˆͺ cuni 4912  Disj wdisj 5117   class class class wbr 5152  dom cdm 5682  β€˜cfv 6553  Ο‰com 7876   β‰Ό cdom 8968  Ξ£*cesum 33679  measurescmeas 33847  Probcprb 34060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-disj 5118  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-ov 7429  df-esum 33680  df-meas 33848  df-prob 34061
This theorem is referenced by:  probun  34072
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