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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 11257 | . . . . . . 7 ⊢ 1 ∈ V | |
| 5 | c0ex 11255 | . . . . . . 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 6727 | . . . . 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 34484 | . . . . 5 ⊢ ∪ dom 𝑃 = {𝐻, 𝑇} |
| 13 | 12 | feq2i 6728 | . . . 4 ⊢ (𝑋:∪ dom 𝑃⟶{1, 0} ↔ 𝑋:{𝐻, 𝑇}⟶{1, 0}) |
| 14 | 10, 13 | mpbir 231 | . . 3 ⊢ 𝑋:∪ dom 𝑃⟶{1, 0} |
| 15 | 1re 11261 | . . . . 5 ⊢ 1 ∈ ℝ | |
| 16 | 0re 11263 | . . . . 5 ⊢ 0 ∈ ℝ | |
| 17 | 15, 16 | pm3.2i 470 | . . . 4 ⊢ (1 ∈ ℝ ∧ 0 ∈ ℝ) |
| 18 | 4, 5 | prss 4820 | . . . 4 ⊢ ((1 ∈ ℝ ∧ 0 ∈ ℝ) ↔ {1, 0} ⊆ ℝ) |
| 19 | 17, 18 | mpbi 230 | . . 3 ⊢ {1, 0} ⊆ ℝ |
| 20 | fss 6752 | . . 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 6089 | . . . . 5 ⊢ (◡𝑋 “ 𝑦) ⊆ ran ◡𝑋 | |
| 23 | dfdm4 5906 | . . . . . 6 ⊢ dom 𝑋 = ran ◡𝑋 | |
| 24 | 10 | fdmi 6747 | . . . . . 6 ⊢ dom 𝑋 = {𝐻, 𝑇} |
| 25 | 23, 24 | eqtr3i 2767 | . . . . 5 ⊢ ran ◡𝑋 = {𝐻, 𝑇} |
| 26 | 22, 25 | sseqtri 4032 | . . . 4 ⊢ (◡𝑋 “ 𝑦) ⊆ {𝐻, 𝑇} |
| 27 | 2, 3, 1, 11, 8 | coinflipspace 34483 | . . . . . . 7 ⊢ dom 𝑃 = 𝒫 {𝐻, 𝑇} |
| 28 | 27 | eleq2i 2833 | . . . . . 6 ⊢ ((◡𝑋 “ 𝑦) ∈ dom 𝑃 ↔ (◡𝑋 “ 𝑦) ∈ 𝒫 {𝐻, 𝑇}) |
| 29 | prex 5437 | . . . . . . . . 9 ⊢ {〈𝐻, 1〉, 〈𝑇, 0〉} ∈ V | |
| 30 | 8, 29 | eqeltri 2837 | . . . . . . . 8 ⊢ 𝑋 ∈ V |
| 31 | cnvexg 7946 | . . . . . . . 8 ⊢ (𝑋 ∈ V → ◡𝑋 ∈ V) | |
| 32 | imaexg 7935 | . . . . . . . 8 ⊢ (◡𝑋 ∈ V → (◡𝑋 “ 𝑦) ∈ V) | |
| 33 | 30, 31, 32 | mp2b 10 | . . . . . . 7 ⊢ (◡𝑋 “ 𝑦) ∈ V |
| 34 | 33 | elpw 4604 | . . . . . 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 3063 | . 2 ⊢ ∀𝑦 ∈ 𝔅ℝ (◡𝑋 “ 𝑦) ∈ dom 𝑃 |
| 39 | 2, 3, 1, 11, 8 | coinflipprob 34482 | . . . . 5 ⊢ 𝑃 ∈ Prob |
| 40 | 39 | a1i 11 | . . . 4 ⊢ (𝐻 ∈ V → 𝑃 ∈ Prob) |
| 41 | 40 | isrrvv 34445 | . . 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 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 Vcvv 3480 ⊆ wss 3951 𝒫 cpw 4600 {cpr 4628 〈cop 4632 ∪ cuni 4907 ◡ccnv 5684 dom cdm 5685 ran crn 5686 ↾ cres 5687 “ cima 5688 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 0cc0 11155 1c1 11156 / cdiv 11920 2c2 12321 ♯chash 14369 ∘f/c cofc 34096 𝔅ℝcbrsiga 34182 Probcprb 34409 rRndVarcrrv 34442 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 ax-mulf 11235 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-disj 5111 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-xnn0 12600 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-shft 15106 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-sum 15723 df-ef 16103 df-sin 16105 df-cos 16106 df-pi 16108 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-ordt 17546 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-ps 18611 df-tsr 18612 df-plusf 18652 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-subrng 20546 df-subrg 20570 df-abv 20810 df-lmod 20860 df-scaf 20861 df-sra 21172 df-rgmod 21173 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-perf 23145 df-cn 23235 df-cnp 23236 df-haus 23323 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-tmd 24080 df-tgp 24081 df-tsms 24135 df-trg 24168 df-xms 24330 df-ms 24331 df-tms 24332 df-nm 24595 df-ngp 24596 df-nrg 24598 df-nlm 24599 df-ii 24903 df-cncf 24904 df-limc 25901 df-dv 25902 df-log 26598 df-xdiv 32900 df-esum 34029 df-ofc 34097 df-siga 34110 df-sigagen 34140 df-brsiga 34183 df-meas 34197 df-mbfm 34251 df-prob 34410 df-rrv 34443 |
| This theorem is referenced by: (None) |
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