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Theorem probcun 30814
 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 ℤ, 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 30806 . 2 (𝑃 ∈ Prob → 𝑃 ∈ (measures‘dom 𝑃))
2 measvun 30606 . 2 ((𝑃 ∈ (measures‘dom 𝑃) ∧ 𝐴 ∈ 𝒫 dom 𝑃 ∧ (𝐴 ≼ ω ∧ Disj 𝑥𝐴 𝑥)) → (𝑃 𝐴) = Σ*𝑥𝐴(𝑃𝑥))
31, 2syl3an1 1165 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 382   ∧ w3a 1070   = wceq 1630   ∈ wcel 2144  𝒫 cpw 4295  ∪ cuni 4572  Disj wdisj 4752   class class class wbr 4784  dom cdm 5249  ‘cfv 6031  ωcom 7211   ≼ cdom 8106  Σ*cesum 30423  measurescmeas 30592  Probcprb 30803 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-disj 4753  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6795  df-esum 30424  df-meas 30593  df-prob 30804 This theorem is referenced by:  probun  30815
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