Theorem probcun 34420

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 𝑥 ∈ 𝐴 𝑥)) → (𝑃‘∪ 𝐴) = Σ*𝑥 ∈ 𝐴(𝑃‘𝑥))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑃

Proof of Theorem probcun

Step	Hyp	Ref	Expression
1		domprobmeas 34412	. 2 ⊢ (𝑃 ∈ Prob → 𝑃 ∈ (measures‘dom 𝑃))
2		measvun 34210	. 2 ⊢ ((𝑃 ∈ (measures‘dom 𝑃) ∧ 𝐴 ∈ 𝒫 dom 𝑃 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝑥)) → (𝑃‘∪ 𝐴) = Σ*𝑥 ∈ 𝐴(𝑃‘𝑥))
3	1, 2	syl3an1 1164	1 ⊢ ((𝑃 ∈ Prob ∧ 𝐴 ∈ 𝒫 dom 𝑃 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝑥)) → (𝑃‘∪ 𝐴) = Σ*𝑥 ∈ 𝐴(𝑃‘𝑥))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  𝒫 cpw 4600  ∪ cuni 4907  Disj wdisj 5110   class class class wbr 5143  dom cdm 5685  ‘cfv 6561  ωcom 7887   ≼ cdom 8983  Σ^*cesum 34028  measurescmeas 34196  Probcprb 34409

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-disj 5111  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-esum 34029  df-meas 34197  df-prob 34410

This theorem is referenced by:  probun  34421