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Mirrors > Home > MPE Home > Th. List > Mathboxes > prob01 | Structured version Visualization version GIF version |
Description: A probability is an element of [ 0 , 1 ]. First axiom of Kolmogorov. (Contributed by Thierry Arnoux, 25-Dec-2016.) |
Ref | Expression |
---|---|
prob01 | β’ ((π β Prob β§ π΄ β dom π) β (πβπ΄) β (0[,]1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | domprobmeas 32771 | . . . . 5 β’ (π β Prob β π β (measuresβdom π)) | |
2 | measvxrge0 32565 | . . . . 5 β’ ((π β (measuresβdom π) β§ π΄ β dom π) β (πβπ΄) β (0[,]+β)) | |
3 | 1, 2 | sylan 581 | . . . 4 β’ ((π β Prob β§ π΄ β dom π) β (πβπ΄) β (0[,]+β)) |
4 | elxrge0 13303 | . . . 4 β’ ((πβπ΄) β (0[,]+β) β ((πβπ΄) β β* β§ 0 β€ (πβπ΄))) | |
5 | 3, 4 | sylib 217 | . . 3 β’ ((π β Prob β§ π΄ β dom π) β ((πβπ΄) β β* β§ 0 β€ (πβπ΄))) |
6 | 1 | adantr 482 | . . . . 5 β’ ((π β Prob β§ π΄ β dom π) β π β (measuresβdom π)) |
7 | simpr 486 | . . . . 5 β’ ((π β Prob β§ π΄ β dom π) β π΄ β dom π) | |
8 | measbase 32557 | . . . . . 6 β’ (π β (measuresβdom π) β dom π β βͺ ran sigAlgebra) | |
9 | unielsiga 32488 | . . . . . 6 β’ (dom π β βͺ ran sigAlgebra β βͺ dom π β dom π) | |
10 | 6, 8, 9 | 3syl 18 | . . . . 5 β’ ((π β Prob β§ π΄ β dom π) β βͺ dom π β dom π) |
11 | elssuni 4897 | . . . . . 6 β’ (π΄ β dom π β π΄ β βͺ dom π) | |
12 | 11 | adantl 483 | . . . . 5 β’ ((π β Prob β§ π΄ β dom π) β π΄ β βͺ dom π) |
13 | 6, 7, 10, 12 | measssd 32575 | . . . 4 β’ ((π β Prob β§ π΄ β dom π) β (πβπ΄) β€ (πββͺ dom π)) |
14 | probtot 32773 | . . . . . 6 β’ (π β Prob β (πββͺ dom π) = 1) | |
15 | 14 | breq2d 5116 | . . . . 5 β’ (π β Prob β ((πβπ΄) β€ (πββͺ dom π) β (πβπ΄) β€ 1)) |
16 | 15 | adantr 482 | . . . 4 β’ ((π β Prob β§ π΄ β dom π) β ((πβπ΄) β€ (πββͺ dom π) β (πβπ΄) β€ 1)) |
17 | 13, 16 | mpbid 231 | . . 3 β’ ((π β Prob β§ π΄ β dom π) β (πβπ΄) β€ 1) |
18 | df-3an 1090 | . . 3 β’ (((πβπ΄) β β* β§ 0 β€ (πβπ΄) β§ (πβπ΄) β€ 1) β (((πβπ΄) β β* β§ 0 β€ (πβπ΄)) β§ (πβπ΄) β€ 1)) | |
19 | 5, 17, 18 | sylanbrc 584 | . 2 β’ ((π β Prob β§ π΄ β dom π) β ((πβπ΄) β β* β§ 0 β€ (πβπ΄) β§ (πβπ΄) β€ 1)) |
20 | 0xr 11136 | . . 3 β’ 0 β β* | |
21 | 1xr 11148 | . . 3 β’ 1 β β* | |
22 | elicc1 13237 | . . 3 β’ ((0 β β* β§ 1 β β*) β ((πβπ΄) β (0[,]1) β ((πβπ΄) β β* β§ 0 β€ (πβπ΄) β§ (πβπ΄) β€ 1))) | |
23 | 20, 21, 22 | mp2an 691 | . 2 β’ ((πβπ΄) β (0[,]1) β ((πβπ΄) β β* β§ 0 β€ (πβπ΄) β§ (πβπ΄) β€ 1)) |
24 | 19, 23 | sylibr 233 | 1 β’ ((π β Prob β§ π΄ β dom π) β (πβπ΄) β (0[,]1)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 β§ w3a 1088 β wcel 2107 β wss 3909 βͺ cuni 4864 class class class wbr 5104 dom cdm 5631 ran crn 5632 βcfv 6492 (class class class)co 7350 0cc0 10985 1c1 10986 +βcpnf 11120 β*cxr 11122 β€ cle 11124 [,]cicc 13196 sigAlgebracsiga 32468 measurescmeas 32555 Probcprb 32768 |
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 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-inf2 9511 ax-ac2 10333 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 ax-pre-sup 11063 ax-addf 11064 ax-mulf 11065 |
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 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-iin 4956 df-disj 5070 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-of 7608 df-om 7794 df-1st 7912 df-2nd 7913 df-supp 8061 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-1o 8380 df-2o 8381 df-er 8582 df-map 8701 df-pm 8702 df-ixp 8770 df-en 8818 df-dom 8819 df-sdom 8820 df-fin 8821 df-fsupp 9240 df-fi 9281 df-sup 9312 df-inf 9313 df-oi 9380 df-dju 9771 df-card 9809 df-acn 9812 df-ac 9986 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-div 11747 df-nn 12088 df-2 12150 df-3 12151 df-4 12152 df-5 12153 df-6 12154 df-7 12155 df-8 12156 df-9 12157 df-n0 12348 df-z 12434 df-dec 12552 df-uz 12697 df-q 12803 df-rp 12845 df-xneg 12962 df-xadd 12963 df-xmul 12964 df-ioo 13197 df-ioc 13198 df-ico 13199 df-icc 13200 df-fz 13354 df-fzo 13497 df-fl 13626 df-mod 13704 df-seq 13836 df-exp 13897 df-fac 14102 df-bc 14131 df-hash 14159 df-shft 14886 df-cj 14918 df-re 14919 df-im 14920 df-sqrt 15054 df-abs 15055 df-limsup 15288 df-clim 15305 df-rlim 15306 df-sum 15506 df-ef 15885 df-sin 15887 df-cos 15888 df-pi 15890 df-struct 16954 df-sets 16971 df-slot 16989 df-ndx 17001 df-base 17019 df-ress 17048 df-plusg 17081 df-mulr 17082 df-starv 17083 df-sca 17084 df-vsca 17085 df-ip 17086 df-tset 17087 df-ple 17088 df-ds 17090 df-unif 17091 df-hom 17092 df-cco 17093 df-rest 17239 df-topn 17240 df-0g 17258 df-gsum 17259 df-topgen 17260 df-pt 17261 df-prds 17264 df-ordt 17318 df-xrs 17319 df-qtop 17324 df-imas 17325 df-xps 17327 df-mre 17401 df-mrc 17402 df-acs 17404 df-ps 18390 df-tsr 18391 df-plusf 18431 df-mgm 18432 df-sgrp 18481 df-mnd 18492 df-mhm 18536 df-submnd 18537 df-grp 18686 df-minusg 18687 df-sbg 18688 df-mulg 18807 df-subg 18858 df-cntz 19029 df-cmn 19493 df-abl 19494 df-mgp 19826 df-ur 19843 df-ring 19890 df-cring 19891 df-subrg 20143 df-abv 20199 df-lmod 20247 df-scaf 20248 df-sra 20556 df-rgmod 20557 df-psmet 20711 df-xmet 20712 df-met 20713 df-bl 20714 df-mopn 20715 df-fbas 20716 df-fg 20717 df-cnfld 20720 df-top 22165 df-topon 22182 df-topsp 22204 df-bases 22218 df-cld 22292 df-ntr 22293 df-cls 22294 df-nei 22371 df-lp 22409 df-perf 22410 df-cn 22500 df-cnp 22501 df-haus 22588 df-tx 22835 df-hmeo 23028 df-fil 23119 df-fm 23211 df-flim 23212 df-flf 23213 df-tmd 23345 df-tgp 23346 df-tsms 23400 df-trg 23433 df-xms 23595 df-ms 23596 df-tms 23597 df-nm 23860 df-ngp 23861 df-nrg 23863 df-nlm 23864 df-ii 24162 df-cncf 24163 df-limc 25152 df-dv 25153 df-log 25834 df-esum 32388 df-siga 32469 df-meas 32556 df-prob 32769 |
This theorem is referenced by: probun 32780 probdif 32781 probvalrnd 32785 totprobd 32787 cndprobin 32795 cndprob01 32796 cndprobtot 32797 cndprobnul 32798 cndprobprob 32799 bayesth 32800 dstrvprob 32832 dstfrvclim1 32838 |
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