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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 33050 | . 2 β’ (π β Prob β π β (measuresβdom π)) | |
2 | measvun 32848 | . 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 1164 | 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 397 β§ w3a 1088 = wceq 1542 β wcel 2107 π« cpw 4565 βͺ cuni 4870 Disj wdisj 5075 class class class wbr 5110 dom cdm 5638 βcfv 6501 Οcom 7807 βΌ cdom 8888 Ξ£*cesum 32666 measurescmeas 32834 Probcprb 33047 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3356 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-disj 5076 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-fv 6509 df-ov 7365 df-esum 32667 df-meas 32835 df-prob 33048 |
This theorem is referenced by: probun 33059 |
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