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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 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 11255 | . . . . . . 7 ⊢ 1 ∈ V | |
5 | c0ex 11253 | . . . . . . 7 ⊢ 0 ∈ V | |
6 | 2, 3, 4, 5 | fpr 7174 | . . . . . 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 6728 | . . . . 5 ⊢ (𝑋:{𝐻, 𝑇}⟶{1, 0} ↔ {〈𝐻, 1〉, 〈𝑇, 0〉}:{𝐻, 𝑇}⟶{1, 0}) |
10 | 7, 9 | mpbir 231 | . . . 4 ⊢ 𝑋:{𝐻, 𝑇}⟶{1, 0} |
11 | coinflip.2 | . . . . . 6 ⊢ 𝑃 = ((♯ ↾ 𝒫 {𝐻, 𝑇}) ∘f/c / 2) | |
12 | 2, 3, 1, 11, 8 | coinflipuniv 34463 | . . . . 5 ⊢ ∪ dom 𝑃 = {𝐻, 𝑇} |
13 | 12 | feq2i 6729 | . . . 4 ⊢ (𝑋:∪ dom 𝑃⟶{1, 0} ↔ 𝑋:{𝐻, 𝑇}⟶{1, 0}) |
14 | 10, 13 | mpbir 231 | . . 3 ⊢ 𝑋:∪ dom 𝑃⟶{1, 0} |
15 | 1re 11259 | . . . . 5 ⊢ 1 ∈ ℝ | |
16 | 0re 11261 | . . . . 5 ⊢ 0 ∈ ℝ | |
17 | 15, 16 | pm3.2i 470 | . . . 4 ⊢ (1 ∈ ℝ ∧ 0 ∈ ℝ) |
18 | 4, 5 | prss 4825 | . . . 4 ⊢ ((1 ∈ ℝ ∧ 0 ∈ ℝ) ↔ {1, 0} ⊆ ℝ) |
19 | 17, 18 | mpbi 230 | . . 3 ⊢ {1, 0} ⊆ ℝ |
20 | fss 6753 | . . 3 ⊢ ((𝑋:∪ dom 𝑃⟶{1, 0} ∧ {1, 0} ⊆ ℝ) → 𝑋:∪ dom 𝑃⟶ℝ) | |
21 | 14, 19, 20 | mp2an 692 | . 2 ⊢ 𝑋:∪ dom 𝑃⟶ℝ |
22 | imassrn 6091 | . . . . 5 ⊢ (◡𝑋 “ 𝑦) ⊆ ran ◡𝑋 | |
23 | dfdm4 5909 | . . . . . 6 ⊢ dom 𝑋 = ran ◡𝑋 | |
24 | 10 | fdmi 6748 | . . . . . 6 ⊢ dom 𝑋 = {𝐻, 𝑇} |
25 | 23, 24 | eqtr3i 2765 | . . . . 5 ⊢ ran ◡𝑋 = {𝐻, 𝑇} |
26 | 22, 25 | sseqtri 4032 | . . . 4 ⊢ (◡𝑋 “ 𝑦) ⊆ {𝐻, 𝑇} |
27 | 2, 3, 1, 11, 8 | coinflipspace 34462 | . . . . . . 7 ⊢ dom 𝑃 = 𝒫 {𝐻, 𝑇} |
28 | 27 | eleq2i 2831 | . . . . . 6 ⊢ ((◡𝑋 “ 𝑦) ∈ dom 𝑃 ↔ (◡𝑋 “ 𝑦) ∈ 𝒫 {𝐻, 𝑇}) |
29 | prex 5443 | . . . . . . . . 9 ⊢ {〈𝐻, 1〉, 〈𝑇, 0〉} ∈ V | |
30 | 8, 29 | eqeltri 2835 | . . . . . . . 8 ⊢ 𝑋 ∈ V |
31 | cnvexg 7947 | . . . . . . . 8 ⊢ (𝑋 ∈ V → ◡𝑋 ∈ V) | |
32 | imaexg 7936 | . . . . . . . 8 ⊢ (◡𝑋 ∈ V → (◡𝑋 “ 𝑦) ∈ V) | |
33 | 30, 31, 32 | mp2b 10 | . . . . . . 7 ⊢ (◡𝑋 “ 𝑦) ∈ V |
34 | 33 | elpw 4609 | . . . . . 6 ⊢ ((◡𝑋 “ 𝑦) ∈ 𝒫 {𝐻, 𝑇} ↔ (◡𝑋 “ 𝑦) ⊆ {𝐻, 𝑇}) |
35 | 28, 34 | bitr2i 276 | . . . . 5 ⊢ ((◡𝑋 “ 𝑦) ⊆ {𝐻, 𝑇} ↔ (◡𝑋 “ 𝑦) ∈ dom 𝑃) |
36 | 35 | biimpi 216 | . . . 4 ⊢ ((◡𝑋 “ 𝑦) ⊆ {𝐻, 𝑇} → (◡𝑋 “ 𝑦) ∈ dom 𝑃) |
37 | 26, 36 | mp1i 13 | . . 3 ⊢ (𝑦 ∈ 𝔅ℝ → (◡𝑋 “ 𝑦) ∈ dom 𝑃) |
38 | 37 | rgen 3061 | . 2 ⊢ ∀𝑦 ∈ 𝔅ℝ (◡𝑋 “ 𝑦) ∈ dom 𝑃 |
39 | 2, 3, 1, 11, 8 | coinflipprob 34461 | . . . . 5 ⊢ 𝑃 ∈ Prob |
40 | 39 | a1i 11 | . . . 4 ⊢ (𝐻 ∈ V → 𝑃 ∈ Prob) |
41 | 40 | isrrvv 34425 | . . 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 711 | 1 ⊢ 𝑋 ∈ (rRndVar‘𝑃) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 Vcvv 3478 ⊆ wss 3963 𝒫 cpw 4605 {cpr 4633 〈cop 4637 ∪ cuni 4912 ◡ccnv 5688 dom cdm 5689 ran crn 5690 ↾ cres 5691 “ cima 5692 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ℝcr 11152 0cc0 11153 1c1 11154 / cdiv 11918 2c2 12319 ♯chash 14366 ∘f/c cofc 34076 𝔅ℝcbrsiga 34162 Probcprb 34389 rRndVarcrrv 34422 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 ax-mulf 11233 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-disj 5116 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-dju 9939 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-xnn0 12598 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-fac 14310 df-bc 14339 df-hash 14367 df-shft 15103 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-sum 15720 df-ef 16100 df-sin 16102 df-cos 16103 df-pi 16105 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-ordt 17548 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-ps 18624 df-tsr 18625 df-plusf 18665 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-cntz 19348 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-subrng 20563 df-subrg 20587 df-abv 20827 df-lmod 20877 df-scaf 20878 df-sra 21190 df-rgmod 21191 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-haus 23339 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-tmd 24096 df-tgp 24097 df-tsms 24151 df-trg 24184 df-xms 24346 df-ms 24347 df-tms 24348 df-nm 24611 df-ngp 24612 df-nrg 24614 df-nlm 24615 df-ii 24917 df-cncf 24918 df-limc 25916 df-dv 25917 df-log 26613 df-xdiv 32885 df-esum 34009 df-ofc 34077 df-siga 34090 df-sigagen 34120 df-brsiga 34163 df-meas 34177 df-mbfm 34231 df-prob 34390 df-rrv 34423 |
This theorem is referenced by: (None) |
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