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Theorem probcun 31786
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 Σ construct cannot be used as it can handle infinite indexing set only if they are subsets of , which is not the case here. (Contributed by Thierry Arnoux, 25-Dec-2016.)
Assertion
Ref Expression
probcun ((𝑃 ∈ Prob ∧ 𝐴 ∈ 𝒫 dom 𝑃 ∧ (𝐴 ≼ ω ∧ Disj 𝑥𝐴 𝑥)) → (𝑃 𝐴) = Σ*𝑥𝐴(𝑃𝑥))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑃

Proof of Theorem probcun
StepHypRef Expression
1 domprobmeas 31778 . 2 (𝑃 ∈ Prob → 𝑃 ∈ (measures‘dom 𝑃))
2 measvun 31578 . 2 ((𝑃 ∈ (measures‘dom 𝑃) ∧ 𝐴 ∈ 𝒫 dom 𝑃 ∧ (𝐴 ≼ ω ∧ Disj 𝑥𝐴 𝑥)) → (𝑃 𝐴) = Σ*𝑥𝐴(𝑃𝑥))
31, 2syl3an1 1160 1 ((𝑃 ∈ Prob ∧ 𝐴 ∈ 𝒫 dom 𝑃 ∧ (𝐴 ≼ ω ∧ Disj 𝑥𝐴 𝑥)) → (𝑃 𝐴) = Σ*𝑥𝐴(𝑃𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2111  𝒫 cpw 4497   cuni 4800  Disj wdisj 4995   class class class wbr 5030  dom cdm 5519  cfv 6324  ωcom 7560  cdom 8490  Σ*cesum 31396  measurescmeas 31564  Probcprb 31775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-disj 4996  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-ov 7138  df-esum 31397  df-meas 31565  df-prob 31776
This theorem is referenced by:  probun  31787
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