Nearby theorems
|
|
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
| 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 33477 | . . 3 β’ π = ((β― βΎ π« {π», π}) βf/c /π 2) |
| 7 | unipw 5451 | . . . . . . 7 β’ βͺ π« {π», π} = {π», π} | |
| 8 | prex 5433 | . . . . . . . 8 β’ {π», π} β V | |
| 9 | 8 | pwid 4625 | . . . . . . 7 β’ {π», π} β π« {π», π} |
| 10 | 7, 9 | eqeltri 2830 | . . . . . 6 β’ βͺ π« {π», π} β π« {π», π} |
| 11 | fvres 6911 | . . . . . 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 6895 | . . . . 5 β’ (β―ββͺ π« {π», π}) = (β―β{π», π}) |
| 14 | hashprg 14355 | . . . . . . 7 β’ ((π» β V β§ π β V) β (π» β π β (β―β{π», π}) = 2)) | |
| 15 | 1, 2, 14 | mp2an 691 | . . . . . 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 2765 | . . . 4 β’ ((β― βΎ π« {π», π})ββͺ π« {π», π}) = 2 |
| 18 | 17 | oveq2i 7420 | . . 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 2764 | . 2 β’ π = ((β― βΎ π« {π», π}) βf/c /π ((β― βΎ π« {π», π})ββͺ π« {π», π})) |
| 20 | pwcntmeas 33225 | . . . 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 12979 | . . . 4 β’ 2 β β+ | |
| 23 | 17, 22 | eqeltri 2830 | . . 3 β’ ((β― βΎ π« {π», π})ββͺ π« {π», π}) β β+ |
| 24 | probfinmeasb 33427 | . . 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 691 | . 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 2830 | 1 β’ π β Prob |
| Colors of variables: wff setvar class |
| Syntax hints: β wb 205 = wceq 1542 β wcel 2107 β wne 2941 Vcvv 3475 π« cpw 4603 {cpr 4631 β¨cop 4635 βͺ cuni 4909 βΎ cres 5679 βcfv 6544 (class class class)co 7409 0cc0 11110 1c1 11111 / cdiv 11871 2c2 12267 β+crp 12974 β―chash 14290 /π cxdiv 32083 βf/c cofc 33093 measurescmeas 33193 Probcprb 33406 |
| 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-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189 ax-mulf 11190 |
| 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-rmo 3377 df-reu 3378 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-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-disj 5115 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 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-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 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-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-oadd 8470 df-er 8703 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9505 df-dju 9896 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-xnn0 12545 df-z 12559 df-dec 12678 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ioo 13328 df-ioc 13329 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-fl 13757 df-mod 13835 df-seq 13967 df-exp 14028 df-fac 14234 df-bc 14263 df-hash 14291 df-shft 15014 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-limsup 15415 df-clim 15432 df-rlim 15433 df-sum 15633 df-ef 16011 df-sin 16013 df-cos 16014 df-pi 16016 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-hom 17221 df-cco 17222 df-rest 17368 df-topn 17369 df-0g 17387 df-gsum 17388 df-topgen 17389 df-pt 17390 df-prds 17393 df-ordt 17447 df-xrs 17448 df-qtop 17453 df-imas 17454 df-xps 17456 df-mre 17530 df-mrc 17531 df-acs 17533 df-ps 18519 df-tsr 18520 df-plusf 18560 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-mhm 18671 df-submnd 18672 df-grp 18822 df-minusg 18823 df-sbg 18824 df-mulg 18951 df-subg 19003 df-cntz 19181 df-cmn 19650 df-abl 19651 df-mgp 19988 df-ur 20005 df-ring 20058 df-cring 20059 df-subrg 20317 df-abv 20425 df-lmod 20473 df-scaf 20474 df-sra 20785 df-rgmod 20786 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-fbas 20941 df-fg 20942 df-cnfld 20945 df-top 22396 df-topon 22413 df-topsp 22435 df-bases 22449 df-cld 22523 df-ntr 22524 df-cls 22525 df-nei 22602 df-lp 22640 df-perf 22641 df-cn 22731 df-cnp 22732 df-haus 22819 df-tx 23066 df-hmeo 23259 df-fil 23350 df-fm 23442 df-flim 23443 df-flf 23444 df-tmd 23576 df-tgp 23577 df-tsms 23631 df-trg 23664 df-xms 23826 df-ms 23827 df-tms 23828 df-nm 24091 df-ngp 24092 df-nrg 24094 df-nlm 24095 df-ii 24393 df-cncf 24394 df-limc 25383 df-dv 25384 df-log 26065 df-xdiv 32084 df-esum 33026 df-ofc 33094 df-siga 33107 df-meas 33194 df-prob 33407 |
| This theorem is referenced by: coinfliprv 33481 coinflippvt 33483 |
| Copyright terms: Public domain | W3C validator |