Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > coinflippvt | Structured version Visualization version GIF version |
Description: The probability of tails is one-half. (Contributed by Thierry Arnoux, 5-Feb-2017.) |
Ref | Expression |
---|---|
coinflip.h | ⊢ 𝐻 ∈ V |
coinflip.t | ⊢ 𝑇 ∈ V |
coinflip.th | ⊢ 𝐻 ≠ 𝑇 |
coinflip.2 | ⊢ 𝑃 = ((♯ ↾ 𝒫 {𝐻, 𝑇}) ∘f/c / 2) |
coinflip.3 | ⊢ 𝑋 = {〈𝐻, 1〉, 〈𝑇, 0〉} |
Ref | Expression |
---|---|
coinflippvt | ⊢ (𝑃‘{𝑇}) = (1 / 2) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coinflip.h | . . . . 5 ⊢ 𝐻 ∈ V | |
2 | coinflip.t | . . . . 5 ⊢ 𝑇 ∈ V | |
3 | coinflip.th | . . . . 5 ⊢ 𝐻 ≠ 𝑇 | |
4 | coinflip.2 | . . . . 5 ⊢ 𝑃 = ((♯ ↾ 𝒫 {𝐻, 𝑇}) ∘f/c / 2) | |
5 | coinflip.3 | . . . . 5 ⊢ 𝑋 = {〈𝐻, 1〉, 〈𝑇, 0〉} | |
6 | 1, 2, 3, 4, 5 | coinflipprob 31744 | . . . 4 ⊢ 𝑃 ∈ Prob |
7 | 1 | prid1 4684 | . . . . . 6 ⊢ 𝐻 ∈ {𝐻, 𝑇} |
8 | snelpwi 5323 | . . . . . 6 ⊢ (𝐻 ∈ {𝐻, 𝑇} → {𝐻} ∈ 𝒫 {𝐻, 𝑇}) | |
9 | 7, 8 | ax-mp 5 | . . . . 5 ⊢ {𝐻} ∈ 𝒫 {𝐻, 𝑇} |
10 | 1, 2, 3, 4, 5 | coinflipspace 31745 | . . . . 5 ⊢ dom 𝑃 = 𝒫 {𝐻, 𝑇} |
11 | 9, 10 | eleqtrri 2912 | . . . 4 ⊢ {𝐻} ∈ dom 𝑃 |
12 | probdsb 31687 | . . . 4 ⊢ ((𝑃 ∈ Prob ∧ {𝐻} ∈ dom 𝑃) → (𝑃‘(∪ dom 𝑃 ∖ {𝐻})) = (1 − (𝑃‘{𝐻}))) | |
13 | 6, 11, 12 | mp2an 690 | . . 3 ⊢ (𝑃‘(∪ dom 𝑃 ∖ {𝐻})) = (1 − (𝑃‘{𝐻})) |
14 | 1, 2, 3, 4, 5 | coinflipuniv 31746 | . . . . . 6 ⊢ ∪ dom 𝑃 = {𝐻, 𝑇} |
15 | 14 | difeq1i 4083 | . . . . 5 ⊢ (∪ dom 𝑃 ∖ {𝐻}) = ({𝐻, 𝑇} ∖ {𝐻}) |
16 | difprsn1 4719 | . . . . . 6 ⊢ (𝐻 ≠ 𝑇 → ({𝐻, 𝑇} ∖ {𝐻}) = {𝑇}) | |
17 | 3, 16 | ax-mp 5 | . . . . 5 ⊢ ({𝐻, 𝑇} ∖ {𝐻}) = {𝑇} |
18 | 15, 17 | eqtri 2844 | . . . 4 ⊢ (∪ dom 𝑃 ∖ {𝐻}) = {𝑇} |
19 | 18 | fveq2i 6659 | . . 3 ⊢ (𝑃‘(∪ dom 𝑃 ∖ {𝐻})) = (𝑃‘{𝑇}) |
20 | 1, 2, 3, 4, 5 | coinflippv 31748 | . . . 4 ⊢ (𝑃‘{𝐻}) = (1 / 2) |
21 | 20 | oveq2i 7153 | . . 3 ⊢ (1 − (𝑃‘{𝐻})) = (1 − (1 / 2)) |
22 | 13, 19, 21 | 3eqtr3i 2852 | . 2 ⊢ (𝑃‘{𝑇}) = (1 − (1 / 2)) |
23 | 1mhlfehlf 11843 | . 2 ⊢ (1 − (1 / 2)) = (1 / 2) | |
24 | 22, 23 | eqtri 2844 | 1 ⊢ (𝑃‘{𝑇}) = (1 / 2) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 ≠ wne 3016 Vcvv 3486 ∖ cdif 3921 𝒫 cpw 4525 {csn 4553 {cpr 4555 〈cop 4559 ∪ cuni 4824 dom cdm 5541 ↾ cres 5543 ‘cfv 6341 (class class class)co 7142 0cc0 10523 1c1 10524 − cmin 10856 / cdiv 11283 2c2 11679 ♯chash 13680 ∘f/c cofc 31361 Probcprb 31672 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-inf2 9090 ax-ac2 9871 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600 ax-pre-sup 10601 ax-addf 10602 ax-mulf 10603 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-int 4863 df-iun 4907 df-iin 4908 df-disj 5018 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-se 5501 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-isom 6350 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-of 7395 df-om 7567 df-1st 7675 df-2nd 7676 df-supp 7817 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-1o 8088 df-2o 8089 df-oadd 8092 df-er 8275 df-map 8394 df-pm 8395 df-ixp 8448 df-en 8496 df-dom 8497 df-sdom 8498 df-fin 8499 df-fsupp 8820 df-fi 8861 df-sup 8892 df-inf 8893 df-oi 8960 df-dju 9316 df-card 9354 df-acn 9357 df-ac 9528 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-div 11284 df-nn 11625 df-2 11687 df-3 11688 df-4 11689 df-5 11690 df-6 11691 df-7 11692 df-8 11693 df-9 11694 df-n0 11885 df-xnn0 11955 df-z 11969 df-dec 12086 df-uz 12231 df-q 12336 df-rp 12377 df-xneg 12494 df-xadd 12495 df-xmul 12496 df-ioo 12729 df-ioc 12730 df-ico 12731 df-icc 12732 df-fz 12883 df-fzo 13024 df-fl 13152 df-mod 13228 df-seq 13360 df-exp 13420 df-fac 13624 df-bc 13653 df-hash 13681 df-shft 14411 df-cj 14443 df-re 14444 df-im 14445 df-sqrt 14579 df-abs 14580 df-limsup 14813 df-clim 14830 df-rlim 14831 df-sum 15028 df-ef 15406 df-sin 15408 df-cos 15409 df-pi 15411 df-struct 16468 df-ndx 16469 df-slot 16470 df-base 16472 df-sets 16473 df-ress 16474 df-plusg 16561 df-mulr 16562 df-starv 16563 df-sca 16564 df-vsca 16565 df-ip 16566 df-tset 16567 df-ple 16568 df-ds 16570 df-unif 16571 df-hom 16572 df-cco 16573 df-rest 16679 df-topn 16680 df-0g 16698 df-gsum 16699 df-topgen 16700 df-pt 16701 df-prds 16704 df-ordt 16757 df-xrs 16758 df-qtop 16763 df-imas 16764 df-xps 16766 df-mre 16840 df-mrc 16841 df-acs 16843 df-ps 17793 df-tsr 17794 df-plusf 17834 df-mgm 17835 df-sgrp 17884 df-mnd 17895 df-mhm 17939 df-submnd 17940 df-grp 18089 df-minusg 18090 df-sbg 18091 df-mulg 18208 df-subg 18259 df-cntz 18430 df-cmn 18891 df-abl 18892 df-mgp 19223 df-ur 19235 df-ring 19282 df-cring 19283 df-subrg 19516 df-abv 19571 df-lmod 19619 df-scaf 19620 df-sra 19927 df-rgmod 19928 df-psmet 20520 df-xmet 20521 df-met 20522 df-bl 20523 df-mopn 20524 df-fbas 20525 df-fg 20526 df-cnfld 20529 df-top 21485 df-topon 21502 df-topsp 21524 df-bases 21537 df-cld 21610 df-ntr 21611 df-cls 21612 df-nei 21689 df-lp 21727 df-perf 21728 df-cn 21818 df-cnp 21819 df-haus 21906 df-tx 22153 df-hmeo 22346 df-fil 22437 df-fm 22529 df-flim 22530 df-flf 22531 df-tmd 22663 df-tgp 22664 df-tsms 22718 df-trg 22751 df-xms 22913 df-ms 22914 df-tms 22915 df-nm 23175 df-ngp 23176 df-nrg 23178 df-nlm 23179 df-ii 23468 df-cncf 23469 df-limc 24449 df-dv 24450 df-log 25126 df-xdiv 30580 df-esum 31294 df-ofc 31362 df-siga 31375 df-meas 31462 df-prob 31673 |
This theorem is referenced by: (None) |
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