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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > coinflippv | Structured version Visualization version GIF version |
Description: The probability of heads is one-half. (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 |
---|---|
coinflippv | ⊢ (𝑃‘{𝐻}) = (1 / 2) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coinflip.2 | . . 3 ⊢ 𝑃 = ((♯ ↾ 𝒫 {𝐻, 𝑇})∘𝑓/𝑐 / 2) | |
2 | 1 | fveq1i 6354 | . 2 ⊢ (𝑃‘{𝐻}) = (((♯ ↾ 𝒫 {𝐻, 𝑇})∘𝑓/𝑐 / 2)‘{𝐻}) |
3 | snsspr1 4490 | . . 3 ⊢ {𝐻} ⊆ {𝐻, 𝑇} | |
4 | prex 5058 | . . . . 5 ⊢ {𝐻, 𝑇} ∈ V | |
5 | 4 | elpw2 4977 | . . . 4 ⊢ ({𝐻} ∈ 𝒫 {𝐻, 𝑇} ↔ {𝐻} ⊆ {𝐻, 𝑇}) |
6 | 5 | biimpri 218 | . . 3 ⊢ ({𝐻} ⊆ {𝐻, 𝑇} → {𝐻} ∈ 𝒫 {𝐻, 𝑇}) |
7 | fveq2 6353 | . . . . . 6 ⊢ (𝑥 = {𝐻} → (♯‘𝑥) = (♯‘{𝐻})) | |
8 | coinflip.h | . . . . . . 7 ⊢ 𝐻 ∈ V | |
9 | hashsng 13371 | . . . . . . 7 ⊢ (𝐻 ∈ V → (♯‘{𝐻}) = 1) | |
10 | 8, 9 | ax-mp 5 | . . . . . 6 ⊢ (♯‘{𝐻}) = 1 |
11 | 7, 10 | syl6eq 2810 | . . . . 5 ⊢ (𝑥 = {𝐻} → (♯‘𝑥) = 1) |
12 | 11 | oveq1d 6829 | . . . 4 ⊢ (𝑥 = {𝐻} → ((♯‘𝑥) / 2) = (1 / 2)) |
13 | 4 | pwex 4997 | . . . . . . 7 ⊢ 𝒫 {𝐻, 𝑇} ∈ V |
14 | 13 | a1i 11 | . . . . . 6 ⊢ (𝐻 ∈ V → 𝒫 {𝐻, 𝑇} ∈ V) |
15 | 2nn0 11521 | . . . . . . 7 ⊢ 2 ∈ ℕ0 | |
16 | 15 | a1i 11 | . . . . . 6 ⊢ (𝐻 ∈ V → 2 ∈ ℕ0) |
17 | prfi 8402 | . . . . . . . . 9 ⊢ {𝐻, 𝑇} ∈ Fin | |
18 | elpwi 4312 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 {𝐻, 𝑇} → 𝑥 ⊆ {𝐻, 𝑇}) | |
19 | ssfi 8347 | . . . . . . . . 9 ⊢ (({𝐻, 𝑇} ∈ Fin ∧ 𝑥 ⊆ {𝐻, 𝑇}) → 𝑥 ∈ Fin) | |
20 | 17, 18, 19 | sylancr 698 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 {𝐻, 𝑇} → 𝑥 ∈ Fin) |
21 | 20 | adantl 473 | . . . . . . 7 ⊢ ((𝐻 ∈ V ∧ 𝑥 ∈ 𝒫 {𝐻, 𝑇}) → 𝑥 ∈ Fin) |
22 | hashcl 13359 | . . . . . . 7 ⊢ (𝑥 ∈ Fin → (♯‘𝑥) ∈ ℕ0) | |
23 | 21, 22 | syl 17 | . . . . . 6 ⊢ ((𝐻 ∈ V ∧ 𝑥 ∈ 𝒫 {𝐻, 𝑇}) → (♯‘𝑥) ∈ ℕ0) |
24 | hashf 13339 | . . . . . . . 8 ⊢ ♯:V⟶(ℕ0 ∪ {+∞}) | |
25 | 24 | a1i 11 | . . . . . . 7 ⊢ (𝐻 ∈ V → ♯:V⟶(ℕ0 ∪ {+∞})) |
26 | ssv 3766 | . . . . . . . 8 ⊢ 𝒫 {𝐻, 𝑇} ⊆ V | |
27 | 26 | a1i 11 | . . . . . . 7 ⊢ (𝐻 ∈ V → 𝒫 {𝐻, 𝑇} ⊆ V) |
28 | 25, 27 | feqresmpt 6413 | . . . . . 6 ⊢ (𝐻 ∈ V → (♯ ↾ 𝒫 {𝐻, 𝑇}) = (𝑥 ∈ 𝒫 {𝐻, 𝑇} ↦ (♯‘𝑥))) |
29 | 14, 16, 23, 28 | ofcfval2 30496 | . . . . 5 ⊢ (𝐻 ∈ V → ((♯ ↾ 𝒫 {𝐻, 𝑇})∘𝑓/𝑐 / 2) = (𝑥 ∈ 𝒫 {𝐻, 𝑇} ↦ ((♯‘𝑥) / 2))) |
30 | 8, 29 | ax-mp 5 | . . . 4 ⊢ ((♯ ↾ 𝒫 {𝐻, 𝑇})∘𝑓/𝑐 / 2) = (𝑥 ∈ 𝒫 {𝐻, 𝑇} ↦ ((♯‘𝑥) / 2)) |
31 | ovex 6842 | . . . 4 ⊢ (1 / 2) ∈ V | |
32 | 12, 30, 31 | fvmpt 6445 | . . 3 ⊢ ({𝐻} ∈ 𝒫 {𝐻, 𝑇} → (((♯ ↾ 𝒫 {𝐻, 𝑇})∘𝑓/𝑐 / 2)‘{𝐻}) = (1 / 2)) |
33 | 3, 6, 32 | mp2b 10 | . 2 ⊢ (((♯ ↾ 𝒫 {𝐻, 𝑇})∘𝑓/𝑐 / 2)‘{𝐻}) = (1 / 2) |
34 | 2, 33 | eqtri 2782 | 1 ⊢ (𝑃‘{𝐻}) = (1 / 2) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1632 ∈ wcel 2139 ≠ wne 2932 Vcvv 3340 ∪ cun 3713 ⊆ wss 3715 𝒫 cpw 4302 {csn 4321 {cpr 4323 〈cop 4327 ↦ cmpt 4881 ↾ cres 5268 ⟶wf 6045 ‘cfv 6049 (class class class)co 6814 Fincfn 8123 0cc0 10148 1c1 10149 +∞cpnf 10283 / cdiv 10896 2c2 11282 ℕ0cn0 11504 ♯chash 13331 ∘𝑓/𝑐cofc 30487 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 ax-pre-mulgt0 10225 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-om 7232 df-1st 7334 df-2nd 7335 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-1o 7730 df-oadd 7734 df-er 7913 df-en 8124 df-dom 8125 df-sdom 8126 df-fin 8127 df-card 8975 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-sub 10480 df-neg 10481 df-nn 11233 df-2 11291 df-n0 11505 df-xnn0 11576 df-z 11590 df-uz 11900 df-fz 12540 df-hash 13332 df-ofc 30488 |
This theorem is referenced by: coinflippvt 30876 |
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