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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rrvadd | Structured version Visualization version GIF version |
Description: The sum of two random variables is a random variable. (Contributed by Thierry Arnoux, 4-Jun-2017.) |
Ref | Expression |
---|---|
rrvadd.1 | β’ (π β π β Prob) |
rrvadd.2 | β’ (π β π β (rRndVarβπ)) |
rrvadd.3 | β’ (π β π β (rRndVarβπ)) |
Ref | Expression |
---|---|
rrvadd | β’ (π β (π βf + π) β (rRndVarβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfmpt1 5249 | . . . 4 β’ β²π(π β βͺ dom π β¦ β¨(πβπ), (πβπ)β©) | |
2 | rrvadd.1 | . . . . 5 β’ (π β π β Prob) | |
3 | rrvadd.2 | . . . . 5 β’ (π β π β (rRndVarβπ)) | |
4 | 2, 3 | rrvvf 33973 | . . . 4 β’ (π β π:βͺ dom πβΆβ) |
5 | rrvadd.3 | . . . . 5 β’ (π β π β (rRndVarβπ)) | |
6 | 2, 5 | rrvvf 33973 | . . . 4 β’ (π β π:βͺ dom πβΆβ) |
7 | 2 | unveldomd 33944 | . . . 4 β’ (π β βͺ dom π β dom π) |
8 | eqidd 2727 | . . . 4 β’ (π β (π β βͺ dom π β¦ β¨(πβπ), (πβπ)β©) = (π β βͺ dom π β¦ β¨(πβπ), (πβπ)β©)) | |
9 | eqidd 2727 | . . . 4 β’ (π β (π₯ β β, π¦ β β β¦ (π₯ + π¦)) = (π₯ β β, π¦ β β β¦ (π₯ + π¦))) | |
10 | 1, 4, 6, 7, 8, 9 | ofoprabco 32394 | . . 3 β’ (π β (π βf + π) = ((π₯ β β, π¦ β β β¦ (π₯ + π¦)) β (π β βͺ dom π β¦ β¨(πβπ), (πβπ)β©))) |
11 | domprobsiga 33940 | . . . . 5 β’ (π β Prob β dom π β βͺ ran sigAlgebra) | |
12 | 2, 11 | syl 17 | . . . 4 β’ (π β dom π β βͺ ran sigAlgebra) |
13 | brsigarn 33712 | . . . . . 6 β’ π β β (sigAlgebraββ) | |
14 | elrnsiga 33654 | . . . . . 6 β’ (π β β (sigAlgebraββ) β π β β βͺ ran sigAlgebra) | |
15 | 13, 14 | mp1i 13 | . . . . 5 β’ (π β π β β βͺ ran sigAlgebra) |
16 | sxsiga 33719 | . . . . 5 β’ ((π β β βͺ ran sigAlgebra β§ π β β βͺ ran sigAlgebra) β (π β Γs π β) β βͺ ran sigAlgebra) | |
17 | 15, 15, 16 | syl2anc 583 | . . . 4 β’ (π β (π β Γs π β) β βͺ ran sigAlgebra) |
18 | 2 | rrvmbfm 33971 | . . . . . 6 β’ (π β (π β (rRndVarβπ) β π β (dom πMblFnMπ β))) |
19 | 3, 18 | mpbid 231 | . . . . 5 β’ (π β π β (dom πMblFnMπ β)) |
20 | 2 | rrvmbfm 33971 | . . . . . 6 β’ (π β (π β (rRndVarβπ) β π β (dom πMblFnMπ β))) |
21 | 5, 20 | mpbid 231 | . . . . 5 β’ (π β π β (dom πMblFnMπ β)) |
22 | fveq2 6884 | . . . . . . 7 β’ (π = π β (πβπ) = (πβπ)) | |
23 | fveq2 6884 | . . . . . . 7 β’ (π = π β (πβπ) = (πβπ)) | |
24 | 22, 23 | opeq12d 4876 | . . . . . 6 β’ (π = π β β¨(πβπ), (πβπ)β© = β¨(πβπ), (πβπ)β©) |
25 | 24 | cbvmptv 5254 | . . . . 5 β’ (π β βͺ dom π β¦ β¨(πβπ), (πβπ)β©) = (π β βͺ dom π β¦ β¨(πβπ), (πβπ)β©) |
26 | 12, 15, 15, 19, 21, 25 | mbfmco2 33794 | . . . 4 β’ (π β (π β βͺ dom π β¦ β¨(πβπ), (πβπ)β©) β (dom πMblFnM(π β Γs π β))) |
27 | eqid 2726 | . . . . . . 7 β’ (topGenβran (,)) = (topGenβran (,)) | |
28 | 27 | raddcn 33439 | . . . . . 6 β’ (π₯ β β, π¦ β β β¦ (π₯ + π¦)) β (((topGenβran (,)) Γt (topGenβran (,))) Cn (topGenβran (,))) |
29 | 28 | a1i 11 | . . . . 5 β’ (π β (π₯ β β, π¦ β β β¦ (π₯ + π¦)) β (((topGenβran (,)) Γt (topGenβran (,))) Cn (topGenβran (,)))) |
30 | 27 | sxbrsiga 33819 | . . . . . 6 β’ (π β Γs π β) = (sigaGenβ((topGenβran (,)) Γt (topGenβran (,)))) |
31 | 30 | a1i 11 | . . . . 5 β’ (π β (π β Γs π β) = (sigaGenβ((topGenβran (,)) Γt (topGenβran (,))))) |
32 | df-brsiga 33710 | . . . . . 6 β’ π β = (sigaGenβ(topGenβran (,))) | |
33 | 32 | a1i 11 | . . . . 5 β’ (π β π β = (sigaGenβ(topGenβran (,)))) |
34 | 29, 31, 33 | cnmbfm 33792 | . . . 4 β’ (π β (π₯ β β, π¦ β β β¦ (π₯ + π¦)) β ((π β Γs π β)MblFnMπ β)) |
35 | 12, 17, 15, 26, 34 | mbfmco 33793 | . . 3 β’ (π β ((π₯ β β, π¦ β β β¦ (π₯ + π¦)) β (π β βͺ dom π β¦ β¨(πβπ), (πβπ)β©)) β (dom πMblFnMπ β)) |
36 | 10, 35 | eqeltrd 2827 | . 2 β’ (π β (π βf + π) β (dom πMblFnMπ β)) |
37 | 2 | rrvmbfm 33971 | . 2 β’ (π β ((π βf + π) β (rRndVarβπ) β (π βf + π) β (dom πMblFnMπ β))) |
38 | 36, 37 | mpbird 257 | 1 β’ (π β (π βf + π) β (rRndVarβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 β¨cop 4629 βͺ cuni 4902 β¦ cmpt 5224 dom cdm 5669 ran crn 5670 β ccom 5673 βcfv 6536 (class class class)co 7404 β cmpo 7406 βf cof 7664 βcr 11108 + caddc 11112 (,)cioo 13327 topGenctg 17390 Cn ccn 23079 Γt ctx 23415 sigAlgebracsiga 33636 sigaGencsigagen 33666 π βcbrsiga 33709 Γs csx 33716 MblFnMcmbfm 33777 Probcprb 33936 rRndVarcrrv 33969 |
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-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-inf2 9635 ax-ac2 10457 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 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-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 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-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 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-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8144 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-2o 8465 df-oadd 8468 df-omul 8469 df-er 8702 df-map 8821 df-pm 8822 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-fi 9405 df-sup 9436 df-inf 9437 df-oi 9504 df-dju 9895 df-card 9933 df-acn 9936 df-ac 10110 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-q 12934 df-rp 12978 df-xneg 13095 df-xadd 13096 df-xmul 13097 df-ioo 13331 df-ioc 13332 df-ico 13333 df-icc 13334 df-fz 13488 df-fzo 13631 df-fl 13760 df-mod 13838 df-seq 13970 df-exp 14031 df-fac 14237 df-bc 14266 df-hash 14294 df-shft 15018 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-limsup 15419 df-clim 15436 df-rlim 15437 df-sum 15637 df-ef 16015 df-sin 16017 df-cos 16018 df-pi 16020 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-hom 17228 df-cco 17229 df-rest 17375 df-topn 17376 df-0g 17394 df-gsum 17395 df-topgen 17396 df-pt 17397 df-prds 17400 df-xrs 17455 df-qtop 17460 df-imas 17461 df-xps 17463 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 df-mulg 18994 df-cntz 19231 df-cmn 19700 df-psmet 21228 df-xmet 21229 df-met 21230 df-bl 21231 df-mopn 21232 df-fbas 21233 df-fg 21234 df-cnfld 21237 df-refld 21494 df-top 22747 df-topon 22764 df-topsp 22786 df-bases 22800 df-cld 22874 df-ntr 22875 df-cls 22876 df-nei 22953 df-lp 22991 df-perf 22992 df-cn 23082 df-cnp 23083 df-haus 23170 df-cmp 23242 df-tx 23417 df-hmeo 23610 df-fil 23701 df-fm 23793 df-flim 23794 df-flf 23795 df-fcls 23796 df-xms 24177 df-ms 24178 df-tms 24179 df-cncf 24749 df-cfil 25134 df-cmet 25136 df-cms 25214 df-limc 25746 df-dv 25747 df-log 26441 df-cxp 26442 df-logb 26648 df-esum 33556 df-siga 33637 df-sigagen 33667 df-brsiga 33710 df-sx 33717 df-meas 33724 df-mbfm 33778 df-prob 33937 df-rrv 33970 |
This theorem is referenced by: rrvsum 33983 |
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