Theorem probcun 35161

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 35153	. 2 ⊢ (𝑃 ∈ Prob → 𝑃 ∈ (measures‘dom 𝑃))
2		measvun 34953	. 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 1181	1 ⊢ ((𝑃 ∈ Prob ∧ 𝐴 ∈ 𝒫 dom 𝑃 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝑥)) → (𝑃‘∪ 𝐴) = Σ*𝑥 ∈ 𝐴(𝑃‘𝑥))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  𝒫 cpw 4560  ∪ cuni 4872  Disj wdisj 5075   class class class wbr 5108  dom cdm 5658  ‘cfv 6566  ωcom 7911   ≼ cdom 9009  Σ^*cesum 34770  measurescmeas 34939  Probcprb 35150

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-disj 5076  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-fv 6574  df-ov 7454  df-esum 34771  df-meas 34940  df-prob 35151

This theorem is used by:  probun  35162