Theorem probcun 30480

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 sum_ construct cannot be used as it can handle infinite indexing set only if they are subsets of ZZ, which is not the case here. (Contributed by Thierry Arnoux, 25-Dec-2016.)

Assertion

Ref	Expression
probcun	|- ( ( P e. Prob /\ A e. ~P dom P /\ ( A ~<_ om /\ Disj x e. A x ) ) -> ( P ` U. A ) = Σ* x e. A ( P ` x ) )

Distinct variable groups:   x, A   x, P

Proof of Theorem probcun

Step	Hyp	Ref	Expression
1		domprobmeas 30472	. 2 |- ( P e. Prob -> P e. (measures ` dom P ) )
2		measvun 30272	. 2 |- ( ( P e. (measures ` dom P ) /\ A e. ~P dom P /\ ( A ~<_ om /\ Disj x e. A x ) ) -> ( P ` U. A ) = Σ* x e. A ( P ` x ) )
3	1, 2	syl3an1 1359	1 |- ( ( P e. Prob /\ A e. ~P dom P /\ ( A ~<_ om /\ Disj x e. A x ) ) -> ( P ` U. A ) = Σ* x e. A ( P ` x ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   ~Pcpw 4158   U.cuni 4436  Disj wdisj 4620   class class class wbr 4653   dom cdm 5114   ` cfv 5888   omcom 7065   ~<_ cdom 7953  Σ^*cesum 30089  measurescmeas 30258  Probcprb 30469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-esum 30090  df-meas 30259  df-prob 30470

This theorem is referenced by:  probun  30481