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Mirrors > Home > MPE Home > Th. List > Mathboxes > isrrvv | Structured version Visualization version GIF version |
Description: Elementhood to the set of real-valued random variables with respect to the probability π. (Contributed by Thierry Arnoux, 25-Jan-2017.) |
Ref | Expression |
---|---|
isrrvv.1 | β’ (π β π β Prob) |
Ref | Expression |
---|---|
isrrvv | β’ (π β (π β (rRndVarβπ) β (π:βͺ dom πβΆβ β§ βπ¦ β π β (β‘π β π¦) β dom π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isrrvv.1 | . . 3 β’ (π β π β Prob) | |
2 | 1 | rrvmbfm 33441 | . 2 β’ (π β (π β (rRndVarβπ) β π β (dom πMblFnMπ β))) |
3 | domprobsiga 33410 | . . . 4 β’ (π β Prob β dom π β βͺ ran sigAlgebra) | |
4 | 1, 3 | syl 17 | . . 3 β’ (π β dom π β βͺ ran sigAlgebra) |
5 | brsigarn 33182 | . . . 4 β’ π β β (sigAlgebraββ) | |
6 | elrnsiga 33124 | . . . 4 β’ (π β β (sigAlgebraββ) β π β β βͺ ran sigAlgebra) | |
7 | 5, 6 | mp1i 13 | . . 3 β’ (π β π β β βͺ ran sigAlgebra) |
8 | 4, 7 | ismbfm 33249 | . 2 β’ (π β (π β (dom πMblFnMπ β) β (π β (βͺ π β βm βͺ dom π) β§ βπ¦ β π β (β‘π β π¦) β dom π))) |
9 | unibrsiga 33184 | . . . . . 6 β’ βͺ π β = β | |
10 | 9 | oveq1i 7419 | . . . . 5 β’ (βͺ π β βm βͺ dom π) = (β βm βͺ dom π) |
11 | 10 | eleq2i 2826 | . . . 4 β’ (π β (βͺ π β βm βͺ dom π) β π β (β βm βͺ dom π)) |
12 | reex 11201 | . . . . 5 β’ β β V | |
13 | 4 | uniexd 7732 | . . . . 5 β’ (π β βͺ dom π β V) |
14 | elmapg 8833 | . . . . 5 β’ ((β β V β§ βͺ dom π β V) β (π β (β βm βͺ dom π) β π:βͺ dom πβΆβ)) | |
15 | 12, 13, 14 | sylancr 588 | . . . 4 β’ (π β (π β (β βm βͺ dom π) β π:βͺ dom πβΆβ)) |
16 | 11, 15 | bitrid 283 | . . 3 β’ (π β (π β (βͺ π β βm βͺ dom π) β π:βͺ dom πβΆβ)) |
17 | 16 | anbi1d 631 | . 2 β’ (π β ((π β (βͺ π β βm βͺ dom π) β§ βπ¦ β π β (β‘π β π¦) β dom π) β (π:βͺ dom πβΆβ β§ βπ¦ β π β (β‘π β π¦) β dom π))) |
18 | 2, 8, 17 | 3bitrd 305 | 1 β’ (π β (π β (rRndVarβπ) β (π:βͺ dom πβΆβ β§ βπ¦ β π β (β‘π β π¦) β dom π))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 β wcel 2107 βwral 3062 Vcvv 3475 βͺ cuni 4909 β‘ccnv 5676 dom cdm 5677 ran crn 5678 β cima 5680 βΆwf 6540 βcfv 6544 (class class class)co 7409 βm cmap 8820 βcr 11109 sigAlgebracsiga 33106 π βcbrsiga 33179 MblFnMcmbfm 33247 Probcprb 33406 rRndVarcrrv 33439 |
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 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-pre-lttri 11184 ax-pre-lttrn 11185 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-ioo 13328 df-topgen 17389 df-top 22396 df-bases 22449 df-esum 33026 df-siga 33107 df-sigagen 33137 df-brsiga 33180 df-meas 33194 df-mbfm 33248 df-prob 33407 df-rrv 33440 |
This theorem is referenced by: rrvvf 33443 rrvfinvima 33449 0rrv 33450 coinfliprv 33481 |
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