Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > coinfliprv | Structured version Visualization version GIF version |
Description: The 𝑋 we defined for coin-flip is a random variable. (Contributed by Thierry Arnoux, 12-Jan-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 |
---|---|
coinfliprv | ⊢ 𝑋 ∈ (rRndVar‘𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coinflip.th | . . . . . 6 ⊢ 𝐻 ≠ 𝑇 | |
2 | coinflip.h | . . . . . . 7 ⊢ 𝐻 ∈ V | |
3 | coinflip.t | . . . . . . 7 ⊢ 𝑇 ∈ V | |
4 | 1ex 10625 | . . . . . . 7 ⊢ 1 ∈ V | |
5 | c0ex 10623 | . . . . . . 7 ⊢ 0 ∈ V | |
6 | 2, 3, 4, 5 | fpr 6908 | . . . . . 6 ⊢ (𝐻 ≠ 𝑇 → {〈𝐻, 1〉, 〈𝑇, 0〉}:{𝐻, 𝑇}⟶{1, 0}) |
7 | 1, 6 | ax-mp 5 | . . . . 5 ⊢ {〈𝐻, 1〉, 〈𝑇, 0〉}:{𝐻, 𝑇}⟶{1, 0} |
8 | coinflip.3 | . . . . . 6 ⊢ 𝑋 = {〈𝐻, 1〉, 〈𝑇, 0〉} | |
9 | 8 | feq1i 6498 | . . . . 5 ⊢ (𝑋:{𝐻, 𝑇}⟶{1, 0} ↔ {〈𝐻, 1〉, 〈𝑇, 0〉}:{𝐻, 𝑇}⟶{1, 0}) |
10 | 7, 9 | mpbir 232 | . . . 4 ⊢ 𝑋:{𝐻, 𝑇}⟶{1, 0} |
11 | coinflip.2 | . . . . . 6 ⊢ 𝑃 = ((♯ ↾ 𝒫 {𝐻, 𝑇}) ∘f/c / 2) | |
12 | 2, 3, 1, 11, 8 | coinflipuniv 31638 | . . . . 5 ⊢ ∪ dom 𝑃 = {𝐻, 𝑇} |
13 | 12 | feq2i 6499 | . . . 4 ⊢ (𝑋:∪ dom 𝑃⟶{1, 0} ↔ 𝑋:{𝐻, 𝑇}⟶{1, 0}) |
14 | 10, 13 | mpbir 232 | . . 3 ⊢ 𝑋:∪ dom 𝑃⟶{1, 0} |
15 | 1re 10629 | . . . . 5 ⊢ 1 ∈ ℝ | |
16 | 0re 10631 | . . . . 5 ⊢ 0 ∈ ℝ | |
17 | 15, 16 | pm3.2i 471 | . . . 4 ⊢ (1 ∈ ℝ ∧ 0 ∈ ℝ) |
18 | 4, 5 | prss 4745 | . . . 4 ⊢ ((1 ∈ ℝ ∧ 0 ∈ ℝ) ↔ {1, 0} ⊆ ℝ) |
19 | 17, 18 | mpbi 231 | . . 3 ⊢ {1, 0} ⊆ ℝ |
20 | fss 6520 | . . 3 ⊢ ((𝑋:∪ dom 𝑃⟶{1, 0} ∧ {1, 0} ⊆ ℝ) → 𝑋:∪ dom 𝑃⟶ℝ) | |
21 | 14, 19, 20 | mp2an 688 | . 2 ⊢ 𝑋:∪ dom 𝑃⟶ℝ |
22 | imassrn 5933 | . . . . 5 ⊢ (◡𝑋 “ 𝑦) ⊆ ran ◡𝑋 | |
23 | dfdm4 5757 | . . . . . 6 ⊢ dom 𝑋 = ran ◡𝑋 | |
24 | 10 | fdmi 6517 | . . . . . 6 ⊢ dom 𝑋 = {𝐻, 𝑇} |
25 | 23, 24 | eqtr3i 2843 | . . . . 5 ⊢ ran ◡𝑋 = {𝐻, 𝑇} |
26 | 22, 25 | sseqtri 4000 | . . . 4 ⊢ (◡𝑋 “ 𝑦) ⊆ {𝐻, 𝑇} |
27 | 2, 3, 1, 11, 8 | coinflipspace 31637 | . . . . . . 7 ⊢ dom 𝑃 = 𝒫 {𝐻, 𝑇} |
28 | 27 | eleq2i 2901 | . . . . . 6 ⊢ ((◡𝑋 “ 𝑦) ∈ dom 𝑃 ↔ (◡𝑋 “ 𝑦) ∈ 𝒫 {𝐻, 𝑇}) |
29 | prex 5323 | . . . . . . . . 9 ⊢ {〈𝐻, 1〉, 〈𝑇, 0〉} ∈ V | |
30 | 8, 29 | eqeltri 2906 | . . . . . . . 8 ⊢ 𝑋 ∈ V |
31 | cnvexg 7618 | . . . . . . . 8 ⊢ (𝑋 ∈ V → ◡𝑋 ∈ V) | |
32 | imaexg 7609 | . . . . . . . 8 ⊢ (◡𝑋 ∈ V → (◡𝑋 “ 𝑦) ∈ V) | |
33 | 30, 31, 32 | mp2b 10 | . . . . . . 7 ⊢ (◡𝑋 “ 𝑦) ∈ V |
34 | 33 | elpw 4542 | . . . . . 6 ⊢ ((◡𝑋 “ 𝑦) ∈ 𝒫 {𝐻, 𝑇} ↔ (◡𝑋 “ 𝑦) ⊆ {𝐻, 𝑇}) |
35 | 28, 34 | bitr2i 277 | . . . . 5 ⊢ ((◡𝑋 “ 𝑦) ⊆ {𝐻, 𝑇} ↔ (◡𝑋 “ 𝑦) ∈ dom 𝑃) |
36 | 35 | biimpi 217 | . . . 4 ⊢ ((◡𝑋 “ 𝑦) ⊆ {𝐻, 𝑇} → (◡𝑋 “ 𝑦) ∈ dom 𝑃) |
37 | 26, 36 | mp1i 13 | . . 3 ⊢ (𝑦 ∈ 𝔅ℝ → (◡𝑋 “ 𝑦) ∈ dom 𝑃) |
38 | 37 | rgen 3145 | . 2 ⊢ ∀𝑦 ∈ 𝔅ℝ (◡𝑋 “ 𝑦) ∈ dom 𝑃 |
39 | 2, 3, 1, 11, 8 | coinflipprob 31636 | . . . . 5 ⊢ 𝑃 ∈ Prob |
40 | 39 | a1i 11 | . . . 4 ⊢ (𝐻 ∈ V → 𝑃 ∈ Prob) |
41 | 40 | isrrvv 31600 | . . 3 ⊢ (𝐻 ∈ V → (𝑋 ∈ (rRndVar‘𝑃) ↔ (𝑋:∪ dom 𝑃⟶ℝ ∧ ∀𝑦 ∈ 𝔅ℝ (◡𝑋 “ 𝑦) ∈ dom 𝑃))) |
42 | 2, 41 | ax-mp 5 | . 2 ⊢ (𝑋 ∈ (rRndVar‘𝑃) ↔ (𝑋:∪ dom 𝑃⟶ℝ ∧ ∀𝑦 ∈ 𝔅ℝ (◡𝑋 “ 𝑦) ∈ dom 𝑃)) |
43 | 21, 38, 42 | mpbir2an 707 | 1 ⊢ 𝑋 ∈ (rRndVar‘𝑃) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∀wral 3135 Vcvv 3492 ⊆ wss 3933 𝒫 cpw 4535 {cpr 4559 〈cop 4563 ∪ cuni 4830 ◡ccnv 5547 dom cdm 5548 ran crn 5549 ↾ cres 5550 “ cima 5551 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ℝcr 10524 0cc0 10525 1c1 10526 / cdiv 11285 2c2 11680 ♯chash 13678 ∘f/c cofc 31253 𝔅ℝcbrsiga 31339 Probcprb 31564 rRndVarcrrv 31597 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-disj 5023 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-fi 8863 df-sup 8894 df-inf 8895 df-oi 8962 df-dju 9318 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-xnn0 11956 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ioc 12731 df-ico 12732 df-icc 12733 df-fz 12881 df-fzo 13022 df-fl 13150 df-mod 13226 df-seq 13358 df-exp 13418 df-fac 13622 df-bc 13651 df-hash 13679 df-shft 14414 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-limsup 14816 df-clim 14833 df-rlim 14834 df-sum 15031 df-ef 15409 df-sin 15411 df-cos 15412 df-pi 15414 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-hom 16577 df-cco 16578 df-rest 16684 df-topn 16685 df-0g 16703 df-gsum 16704 df-topgen 16705 df-pt 16706 df-prds 16709 df-ordt 16762 df-xrs 16763 df-qtop 16768 df-imas 16769 df-xps 16771 df-mre 16845 df-mrc 16846 df-acs 16848 df-ps 17798 df-tsr 17799 df-plusf 17839 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mhm 17944 df-submnd 17945 df-grp 18044 df-minusg 18045 df-sbg 18046 df-mulg 18163 df-subg 18214 df-cntz 18385 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-cring 19229 df-subrg 19462 df-abv 19517 df-lmod 19565 df-scaf 19566 df-sra 19873 df-rgmod 19874 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-fbas 20470 df-fg 20471 df-cnfld 20474 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-cld 21555 df-ntr 21556 df-cls 21557 df-nei 21634 df-lp 21672 df-perf 21673 df-cn 21763 df-cnp 21764 df-haus 21851 df-tx 22098 df-hmeo 22291 df-fil 22382 df-fm 22474 df-flim 22475 df-flf 22476 df-tmd 22608 df-tgp 22609 df-tsms 22662 df-trg 22695 df-xms 22857 df-ms 22858 df-tms 22859 df-nm 23119 df-ngp 23120 df-nrg 23122 df-nlm 23123 df-ii 23412 df-cncf 23413 df-limc 24391 df-dv 24392 df-log 25067 df-xdiv 30521 df-esum 31186 df-ofc 31254 df-siga 31267 df-sigagen 31297 df-brsiga 31340 df-meas 31354 df-mbfm 31408 df-prob 31565 df-rrv 31598 |
This theorem is referenced by: (None) |
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