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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > coinfliprv | Structured version Visualization version GIF version |
Description: The π we defined for coin-flip is a random variable. (Contributed by Thierry Arnoux, 12-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 |
---|---|
coinfliprv | β’ π β (rRndVarβπ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coinflip.th | . . . . . 6 β’ π» β π | |
2 | coinflip.h | . . . . . . 7 β’ π» β V | |
3 | coinflip.t | . . . . . . 7 β’ π β V | |
4 | 1ex 11192 | . . . . . . 7 β’ 1 β V | |
5 | c0ex 11190 | . . . . . . 7 β’ 0 β V | |
6 | 2, 3, 4, 5 | fpr 7136 | . . . . . 6 β’ (π» β π β {β¨π», 1β©, β¨π, 0β©}:{π», π}βΆ{1, 0}) |
7 | 1, 6 | ax-mp 5 | . . . . 5 β’ {β¨π», 1β©, β¨π, 0β©}:{π», π}βΆ{1, 0} |
8 | coinflip.3 | . . . . . 6 β’ π = {β¨π», 1β©, β¨π, 0β©} | |
9 | 8 | feq1i 6695 | . . . . 5 β’ (π:{π», π}βΆ{1, 0} β {β¨π», 1β©, β¨π, 0β©}:{π», π}βΆ{1, 0}) |
10 | 7, 9 | mpbir 230 | . . . 4 β’ π:{π», π}βΆ{1, 0} |
11 | coinflip.2 | . . . . . 6 β’ π = ((β― βΎ π« {π», π}) βf/c / 2) | |
12 | 2, 3, 1, 11, 8 | coinflipuniv 33311 | . . . . 5 β’ βͺ dom π = {π», π} |
13 | 12 | feq2i 6696 | . . . 4 β’ (π:βͺ dom πβΆ{1, 0} β π:{π», π}βΆ{1, 0}) |
14 | 10, 13 | mpbir 230 | . . 3 β’ π:βͺ dom πβΆ{1, 0} |
15 | 1re 11196 | . . . . 5 β’ 1 β β | |
16 | 0re 11198 | . . . . 5 β’ 0 β β | |
17 | 15, 16 | pm3.2i 471 | . . . 4 β’ (1 β β β§ 0 β β) |
18 | 4, 5 | prss 4816 | . . . 4 β’ ((1 β β β§ 0 β β) β {1, 0} β β) |
19 | 17, 18 | mpbi 229 | . . 3 β’ {1, 0} β β |
20 | fss 6721 | . . 3 β’ ((π:βͺ dom πβΆ{1, 0} β§ {1, 0} β β) β π:βͺ dom πβΆβ) | |
21 | 14, 19, 20 | mp2an 690 | . 2 β’ π:βͺ dom πβΆβ |
22 | imassrn 6060 | . . . . 5 β’ (β‘π β π¦) β ran β‘π | |
23 | dfdm4 5887 | . . . . . 6 β’ dom π = ran β‘π | |
24 | 10 | fdmi 6716 | . . . . . 6 β’ dom π = {π», π} |
25 | 23, 24 | eqtr3i 2761 | . . . . 5 β’ ran β‘π = {π», π} |
26 | 22, 25 | sseqtri 4014 | . . . 4 β’ (β‘π β π¦) β {π», π} |
27 | 2, 3, 1, 11, 8 | coinflipspace 33310 | . . . . . . 7 β’ dom π = π« {π», π} |
28 | 27 | eleq2i 2824 | . . . . . 6 β’ ((β‘π β π¦) β dom π β (β‘π β π¦) β π« {π», π}) |
29 | prex 5425 | . . . . . . . . 9 β’ {β¨π», 1β©, β¨π, 0β©} β V | |
30 | 8, 29 | eqeltri 2828 | . . . . . . . 8 β’ π β V |
31 | cnvexg 7897 | . . . . . . . 8 β’ (π β V β β‘π β V) | |
32 | imaexg 7888 | . . . . . . . 8 β’ (β‘π β V β (β‘π β π¦) β V) | |
33 | 30, 31, 32 | mp2b 10 | . . . . . . 7 β’ (β‘π β π¦) β V |
34 | 33 | elpw 4600 | . . . . . 6 β’ ((β‘π β π¦) β π« {π», π} β (β‘π β π¦) β {π», π}) |
35 | 28, 34 | bitr2i 275 | . . . . 5 β’ ((β‘π β π¦) β {π», π} β (β‘π β π¦) β dom π) |
36 | 35 | biimpi 215 | . . . 4 β’ ((β‘π β π¦) β {π», π} β (β‘π β π¦) β dom π) |
37 | 26, 36 | mp1i 13 | . . 3 β’ (π¦ β π β β (β‘π β π¦) β dom π) |
38 | 37 | rgen 3062 | . 2 β’ βπ¦ β π β (β‘π β π¦) β dom π |
39 | 2, 3, 1, 11, 8 | coinflipprob 33309 | . . . . 5 β’ π β Prob |
40 | 39 | a1i 11 | . . . 4 β’ (π» β V β π β Prob) |
41 | 40 | isrrvv 33273 | . . 3 β’ (π» β V β (π β (rRndVarβπ) β (π:βͺ dom πβΆβ β§ βπ¦ β π β (β‘π β π¦) β dom π))) |
42 | 2, 41 | ax-mp 5 | . 2 β’ (π β (rRndVarβπ) β (π:βͺ dom πβΆβ β§ βπ¦ β π β (β‘π β π¦) β dom π)) |
43 | 21, 38, 42 | mpbir2an 709 | 1 β’ π β (rRndVarβπ) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 396 = wceq 1541 β wcel 2106 β wne 2939 βwral 3060 Vcvv 3473 β wss 3944 π« cpw 4596 {cpr 4624 β¨cop 4628 βͺ cuni 4901 β‘ccnv 5668 dom cdm 5669 ran crn 5670 βΎ cres 5671 β cima 5672 βΆwf 6528 βcfv 6532 (class class class)co 7393 βcr 11091 0cc0 11092 1c1 11093 / cdiv 11853 2c2 12249 β―chash 14272 βf/c cofc 32924 π βcbrsiga 33010 Probcprb 33237 rRndVarcrrv 33270 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-inf2 9618 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 ax-pre-sup 11170 ax-addf 11171 ax-mulf 11172 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-uni 4902 df-int 4944 df-iun 4992 df-iin 4993 df-disj 5107 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 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-isom 6541 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-of 7653 df-om 7839 df-1st 7957 df-2nd 7958 df-supp 8129 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-2o 8449 df-oadd 8452 df-er 8686 df-map 8805 df-pm 8806 df-ixp 8875 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-fsupp 9345 df-fi 9388 df-sup 9419 df-inf 9420 df-oi 9487 df-dju 9878 df-card 9916 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-div 11854 df-nn 12195 df-2 12257 df-3 12258 df-4 12259 df-5 12260 df-6 12261 df-7 12262 df-8 12263 df-9 12264 df-n0 12455 df-xnn0 12527 df-z 12541 df-dec 12660 df-uz 12805 df-q 12915 df-rp 12957 df-xneg 13074 df-xadd 13075 df-xmul 13076 df-ioo 13310 df-ioc 13311 df-ico 13312 df-icc 13313 df-fz 13467 df-fzo 13610 df-fl 13739 df-mod 13817 df-seq 13949 df-exp 14010 df-fac 14216 df-bc 14245 df-hash 14273 df-shft 14996 df-cj 15028 df-re 15029 df-im 15030 df-sqrt 15164 df-abs 15165 df-limsup 15397 df-clim 15414 df-rlim 15415 df-sum 15615 df-ef 15993 df-sin 15995 df-cos 15996 df-pi 15998 df-struct 17062 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17127 df-ress 17156 df-plusg 17192 df-mulr 17193 df-starv 17194 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-hom 17203 df-cco 17204 df-rest 17350 df-topn 17351 df-0g 17369 df-gsum 17370 df-topgen 17371 df-pt 17372 df-prds 17375 df-ordt 17429 df-xrs 17430 df-qtop 17435 df-imas 17436 df-xps 17438 df-mre 17512 df-mrc 17513 df-acs 17515 df-ps 18501 df-tsr 18502 df-plusf 18542 df-mgm 18543 df-sgrp 18592 df-mnd 18603 df-mhm 18647 df-submnd 18648 df-grp 18797 df-minusg 18798 df-sbg 18799 df-mulg 18923 df-subg 18975 df-cntz 19147 df-cmn 19614 df-abl 19615 df-mgp 19947 df-ur 19964 df-ring 20016 df-cring 20017 df-subrg 20310 df-abv 20374 df-lmod 20422 df-scaf 20423 df-sra 20734 df-rgmod 20735 df-psmet 20870 df-xmet 20871 df-met 20872 df-bl 20873 df-mopn 20874 df-fbas 20875 df-fg 20876 df-cnfld 20879 df-top 22325 df-topon 22342 df-topsp 22364 df-bases 22378 df-cld 22452 df-ntr 22453 df-cls 22454 df-nei 22531 df-lp 22569 df-perf 22570 df-cn 22660 df-cnp 22661 df-haus 22748 df-tx 22995 df-hmeo 23188 df-fil 23279 df-fm 23371 df-flim 23372 df-flf 23373 df-tmd 23505 df-tgp 23506 df-tsms 23560 df-trg 23593 df-xms 23755 df-ms 23756 df-tms 23757 df-nm 24020 df-ngp 24021 df-nrg 24023 df-nlm 24024 df-ii 24322 df-cncf 24323 df-limc 25312 df-dv 25313 df-log 25994 df-xdiv 31955 df-esum 32857 df-ofc 32925 df-siga 32938 df-sigagen 32968 df-brsiga 33011 df-meas 33025 df-mbfm 33079 df-prob 33238 df-rrv 33271 |
This theorem is referenced by: (None) |
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