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Mathbox for Thierry Arnoux |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > coinflipprob | Structured version Visualization version GIF version | ||
| Description: The π we defined for coin-flip is a probability law. (Contributed by Thierry Arnoux, 15-Jan-2017.) |
| Ref | Expression |
|---|---|
| coinflip.h | β’ π» β V |
| coinflip.t | β’ π β V |
| coinflip.th | β’ π» β π |
| coinflip.2 | β’ π = ((β― βΎ π« {π», π}) βf/c / 2) |
| coinflip.3 | β’ π = {β¨π», 1β©, β¨π, 0β©} |
| Ref | Expression |
|---|---|
| coinflipprob | β’ π β Prob |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | coinflip.h | . . . 4 β’ π» β V | |
| 2 | coinflip.t | . . . 4 β’ π β V | |
| 3 | coinflip.th | . . . 4 β’ π» β π | |
| 4 | coinflip.2 | . . . 4 β’ π = ((β― βΎ π« {π», π}) βf/c / 2) | |
| 5 | coinflip.3 | . . . 4 β’ π = {β¨π», 1β©, β¨π, 0β©} | |
| 6 | 1, 2, 3, 4, 5 | coinfliplem 33775 | . . 3 β’ π = ((β― βΎ π« {π», π}) βf/c /π 2) |
| 7 | unipw 5449 | . . . . . . 7 β’ βͺ π« {π», π} = {π», π} | |
| 8 | prex 5431 | . . . . . . . 8 β’ {π», π} β V | |
| 9 | 8 | pwid 4623 | . . . . . . 7 β’ {π», π} β π« {π», π} |
| 10 | 7, 9 | eqeltri 2827 | . . . . . 6 β’ βͺ π« {π», π} β π« {π», π} |
| 11 | fvres 6909 | . . . . . 6 β’ (βͺ π« {π», π} β π« {π», π} β ((β― βΎ π« {π», π})ββͺ π« {π», π}) = (β―ββͺ π« {π», π})) | |
| 12 | 10, 11 | ax-mp 5 | . . . . 5 β’ ((β― βΎ π« {π», π})ββͺ π« {π», π}) = (β―ββͺ π« {π», π}) |
| 13 | 7 | fveq2i 6893 | . . . . 5 β’ (β―ββͺ π« {π», π}) = (β―β{π», π}) |
| 14 | hashprg 14359 | . . . . . . 7 β’ ((π» β V β§ π β V) β (π» β π β (β―β{π», π}) = 2)) | |
| 15 | 1, 2, 14 | mp2an 688 | . . . . . 6 β’ (π» β π β (β―β{π», π}) = 2) |
| 16 | 3, 15 | mpbi 229 | . . . . 5 β’ (β―β{π», π}) = 2 |
| 17 | 12, 13, 16 | 3eqtri 2762 | . . . 4 β’ ((β― βΎ π« {π», π})ββͺ π« {π», π}) = 2 |
| 18 | 17 | oveq2i 7422 | . . 3 β’ ((β― βΎ π« {π», π}) βf/c /π ((β― βΎ π« {π», π})ββͺ π« {π», π})) = ((β― βΎ π« {π», π}) βf/c /π 2) |
| 19 | 6, 18 | eqtr4i 2761 | . 2 β’ π = ((β― βΎ π« {π», π}) βf/c /π ((β― βΎ π« {π», π})ββͺ π« {π», π})) |
| 20 | pwcntmeas 33523 | . . . 4 β’ ({π», π} β V β (β― βΎ π« {π», π}) β (measuresβπ« {π», π})) | |
| 21 | 8, 20 | ax-mp 5 | . . 3 β’ (β― βΎ π« {π», π}) β (measuresβπ« {π», π}) |
| 22 | 2rp 12983 | . . . 4 β’ 2 β β+ | |
| 23 | 17, 22 | eqeltri 2827 | . . 3 β’ ((β― βΎ π« {π», π})ββͺ π« {π», π}) β β+ |
| 24 | probfinmeasb 33725 | . . 3 β’ (((β― βΎ π« {π», π}) β (measuresβπ« {π», π}) β§ ((β― βΎ π« {π», π})ββͺ π« {π», π}) β β+) β ((β― βΎ π« {π», π}) βf/c /π ((β― βΎ π« {π», π})ββͺ π« {π», π})) β Prob) | |
| 25 | 21, 23, 24 | mp2an 688 | . 2 β’ ((β― βΎ π« {π», π}) βf/c /π ((β― βΎ π« {π», π})ββͺ π« {π», π})) β Prob |
| 26 | 19, 25 | eqeltri 2827 | 1 β’ π β Prob |
| Colors of variables: wff setvar class |
| Syntax hints: β wb 205 = wceq 1539 β wcel 2104 β wne 2938 Vcvv 3472 π« cpw 4601 {cpr 4629 β¨cop 4633 βͺ cuni 4907 βΎ cres 5677 βcfv 6542 (class class class)co 7411 0cc0 11112 1c1 11113 / cdiv 11875 2c2 12271 β+crp 12978 β―chash 14294 /π cxdiv 32350 βf/c cofc 33391 measurescmeas 33491 Probcprb 33704 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-disj 5113 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-oadd 8472 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-dju 9898 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-xnn0 12549 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12979 df-xneg 13096 df-xadd 13097 df-xmul 13098 df-ioo 13332 df-ioc 13333 df-ico 13334 df-icc 13335 df-fz 13489 df-fzo 13632 df-fl 13761 df-mod 13839 df-seq 13971 df-exp 14032 df-fac 14238 df-bc 14267 df-hash 14295 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 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-starv 17216 df-sca 17217 df-vsca 17218 df-ip 17219 df-tset 17220 df-ple 17221 df-ds 17223 df-unif 17224 df-hom 17225 df-cco 17226 df-rest 17372 df-topn 17373 df-0g 17391 df-gsum 17392 df-topgen 17393 df-pt 17394 df-prds 17397 df-ordt 17451 df-xrs 17452 df-qtop 17457 df-imas 17458 df-xps 17460 df-mre 17534 df-mrc 17535 df-acs 17537 df-ps 18523 df-tsr 18524 df-plusf 18564 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18705 df-submnd 18706 df-grp 18858 df-minusg 18859 df-sbg 18860 df-mulg 18987 df-subg 19039 df-cntz 19222 df-cmn 19691 df-abl 19692 df-mgp 20029 df-rng 20047 df-ur 20076 df-ring 20129 df-cring 20130 df-subrng 20434 df-subrg 20459 df-abv 20568 df-lmod 20616 df-scaf 20617 df-sra 20930 df-rgmod 20931 df-psmet 21136 df-xmet 21137 df-met 21138 df-bl 21139 df-mopn 21140 df-fbas 21141 df-fg 21142 df-cnfld 21145 df-top 22616 df-topon 22633 df-topsp 22655 df-bases 22669 df-cld 22743 df-ntr 22744 df-cls 22745 df-nei 22822 df-lp 22860 df-perf 22861 df-cn 22951 df-cnp 22952 df-haus 23039 df-tx 23286 df-hmeo 23479 df-fil 23570 df-fm 23662 df-flim 23663 df-flf 23664 df-tmd 23796 df-tgp 23797 df-tsms 23851 df-trg 23884 df-xms 24046 df-ms 24047 df-tms 24048 df-nm 24311 df-ngp 24312 df-nrg 24314 df-nlm 24315 df-ii 24617 df-cncf 24618 df-limc 25615 df-dv 25616 df-log 26301 df-xdiv 32351 df-esum 33324 df-ofc 33392 df-siga 33405 df-meas 33492 df-prob 33705 |
| This theorem is referenced by: coinfliprv 33779 coinflippvt 33781 |
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