Theorem probcun 34556

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 34548	. 2 ⊢ (𝑃 ∈ Prob → 𝑃 ∈ (measures‘dom 𝑃))
2		measvun 34347	. 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 1542   ∈ wcel 2114  𝒫 cpw 4555  ∪ cuni 4864  Disj wdisj 5066   class class class wbr 5099  dom cdm 5625  ‘cfv 6493  ωcom 7810   ≼ cdom 8885  Σ^*cesum 34165  measurescmeas 34333  Probcprb 34545

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3351  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-disj 5067  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-fv 6501  df-ov 7363  df-esum 34166  df-meas 34334  df-prob 34546

This theorem is referenced by:  probun  34557