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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > coinflipspace | Structured version Visualization version GIF version |
Description: The space of our coin-flip probability. (Contributed by Thierry Arnoux, 15-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 |
---|---|
coinflipspace | ⊢ dom 𝑃 = 𝒫 {𝐻, 𝑇} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coinflip.2 | . . 3 ⊢ 𝑃 = ((♯ ↾ 𝒫 {𝐻, 𝑇})∘𝑓/𝑐 / 2) | |
2 | 1 | dmeqi 5572 | . 2 ⊢ dom 𝑃 = dom ((♯ ↾ 𝒫 {𝐻, 𝑇})∘𝑓/𝑐 / 2) |
3 | coinflip.h | . . 3 ⊢ 𝐻 ∈ V | |
4 | hashresfn 13449 | . . . . 5 ⊢ (♯ ↾ 𝒫 {𝐻, 𝑇}) Fn 𝒫 {𝐻, 𝑇} | |
5 | 4 | a1i 11 | . . . 4 ⊢ (𝐻 ∈ V → (♯ ↾ 𝒫 {𝐻, 𝑇}) Fn 𝒫 {𝐻, 𝑇}) |
6 | prex 5143 | . . . . 5 ⊢ {𝐻, 𝑇} ∈ V | |
7 | pwexg 5092 | . . . . 5 ⊢ ({𝐻, 𝑇} ∈ V → 𝒫 {𝐻, 𝑇} ∈ V) | |
8 | 6, 7 | mp1i 13 | . . . 4 ⊢ (𝐻 ∈ V → 𝒫 {𝐻, 𝑇} ∈ V) |
9 | 2re 11453 | . . . . 5 ⊢ 2 ∈ ℝ | |
10 | 9 | a1i 11 | . . . 4 ⊢ (𝐻 ∈ V → 2 ∈ ℝ) |
11 | 5, 8, 10 | ofcfn 30764 | . . 3 ⊢ (𝐻 ∈ V → ((♯ ↾ 𝒫 {𝐻, 𝑇})∘𝑓/𝑐 / 2) Fn 𝒫 {𝐻, 𝑇}) |
12 | fndm 6237 | . . 3 ⊢ (((♯ ↾ 𝒫 {𝐻, 𝑇})∘𝑓/𝑐 / 2) Fn 𝒫 {𝐻, 𝑇} → dom ((♯ ↾ 𝒫 {𝐻, 𝑇})∘𝑓/𝑐 / 2) = 𝒫 {𝐻, 𝑇}) | |
13 | 3, 11, 12 | mp2b 10 | . 2 ⊢ dom ((♯ ↾ 𝒫 {𝐻, 𝑇})∘𝑓/𝑐 / 2) = 𝒫 {𝐻, 𝑇} |
14 | 2, 13 | eqtri 2802 | 1 ⊢ dom 𝑃 = 𝒫 {𝐻, 𝑇} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1601 ∈ wcel 2107 ≠ wne 2969 Vcvv 3398 𝒫 cpw 4379 {cpr 4400 〈cop 4404 dom cdm 5357 ↾ cres 5359 Fn wfn 6132 (class class class)co 6924 ℝcr 10273 0cc0 10274 1c1 10275 / cdiv 11034 2c2 11434 ♯chash 13439 ∘𝑓/𝑐cofc 30759 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-card 9100 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11379 df-2 11442 df-n0 11647 df-xnn0 11719 df-z 11733 df-uz 11997 df-hash 13440 df-ofc 30760 |
This theorem is referenced by: coinflipuniv 31146 coinfliprv 31147 coinflippvt 31149 |
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