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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > unveldomd | Structured version Visualization version GIF version |
Description: The universe is an element of the domain of the probability, the universe (entire probability space) being βͺ dom π in our construction. (Contributed by Thierry Arnoux, 22-Jan-2017.) |
Ref | Expression |
---|---|
unveldomd.1 | β’ (π β π β Prob) |
Ref | Expression |
---|---|
unveldomd | β’ (π β βͺ dom π β dom π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unveldomd.1 | . 2 β’ (π β π β Prob) | |
2 | domprobsiga 34060 | . 2 β’ (π β Prob β dom π β βͺ ran sigAlgebra) | |
3 | sgon 33772 | . 2 β’ (dom π β βͺ ran sigAlgebra β dom π β (sigAlgebraββͺ dom π)) | |
4 | baselsiga 33763 | . 2 β’ (dom π β (sigAlgebraββͺ dom π) β βͺ dom π β dom π) | |
5 | 1, 2, 3, 4 | 4syl 19 | 1 β’ (π β βͺ dom π β dom π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wcel 2098 βͺ cuni 4901 dom cdm 5670 ran crn 5671 βcfv 6541 sigAlgebracsiga 33756 Probcprb 34056 |
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 2166 ax-ext 2696 ax-sep 5292 ax-nul 5299 ax-pow 5357 ax-pr 5421 ax-un 7736 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4317 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5568 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-fv 6549 df-ov 7417 df-esum 33676 df-siga 33757 df-meas 33844 df-prob 34057 |
This theorem is referenced by: unveldom 34065 probdsb 34071 probtotrnd 34074 cndprobtot 34085 0rrv 34100 rrvadd 34101 dstfrvclim1 34126 |
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