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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > probcun | Structured version Visualization version GIF version |
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 Ξ£ construct cannot be used as it can handle infinite indexing set only if they are subsets of β€, which is not the case here. (Contributed by Thierry Arnoux, 25-Dec-2016.) |
Ref | Expression |
---|---|
probcun | β’ ((π β Prob β§ π΄ β π« dom π β§ (π΄ βΌ Ο β§ Disj π₯ β π΄ π₯)) β (πββͺ π΄) = Ξ£*π₯ β π΄(πβπ₯)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | domprobmeas 33938 | . 2 β’ (π β Prob β π β (measuresβdom π)) | |
2 | measvun 33736 | . 2 β’ ((π β (measuresβdom π) β§ π΄ β π« dom π β§ (π΄ βΌ Ο β§ Disj π₯ β π΄ π₯)) β (πββͺ π΄) = Ξ£*π₯ β π΄(πβπ₯)) | |
3 | 1, 2 | syl3an1 1160 | 1 β’ ((π β Prob β§ π΄ β π« dom π β§ (π΄ βΌ Ο β§ Disj π₯ β π΄ π₯)) β (πββͺ π΄) = Ξ£*π₯ β π΄(πβπ₯)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 π« cpw 4597 βͺ cuni 4902 Disj wdisj 5106 class class class wbr 5141 dom cdm 5669 βcfv 6536 Οcom 7851 βΌ cdom 8936 Ξ£*cesum 33554 measurescmeas 33722 Probcprb 33935 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-disj 5107 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-fv 6544 df-ov 7407 df-esum 33555 df-meas 33723 df-prob 33936 |
This theorem is referenced by: probun 33947 |
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