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 6707 | . 2 ⊢ (𝑃‘{𝐻}) = (((♯ ↾ 𝒫 {𝐻, 𝑇}) ∘f/c / 2)‘{𝐻}) |
3 | snsspr1 4717 | . . 3 ⊢ {𝐻} ⊆ {𝐻, 𝑇} | |
4 | prex 5314 | . . . . 5 ⊢ {𝐻, 𝑇} ∈ V | |
5 | 4 | elpw2 5227 | . . . 4 ⊢ ({𝐻} ∈ 𝒫 {𝐻, 𝑇} ↔ {𝐻} ⊆ {𝐻, 𝑇}) |
6 | 5 | biimpri 231 | . . 3 ⊢ ({𝐻} ⊆ {𝐻, 𝑇} → {𝐻} ∈ 𝒫 {𝐻, 𝑇}) |
7 | fveq2 6706 | . . . . . 6 ⊢ (𝑥 = {𝐻} → (♯‘𝑥) = (♯‘{𝐻})) | |
8 | coinflip.h | . . . . . . 7 ⊢ 𝐻 ∈ V | |
9 | hashsng 13919 | . . . . . . 7 ⊢ (𝐻 ∈ V → (♯‘{𝐻}) = 1) | |
10 | 8, 9 | ax-mp 5 | . . . . . 6 ⊢ (♯‘{𝐻}) = 1 |
11 | 7, 10 | eqtrdi 2790 | . . . . 5 ⊢ (𝑥 = {𝐻} → (♯‘𝑥) = 1) |
12 | 11 | oveq1d 7217 | . . . 4 ⊢ (𝑥 = {𝐻} → ((♯‘𝑥) / 2) = (1 / 2)) |
13 | 4 | pwex 5262 | . . . . . . 7 ⊢ 𝒫 {𝐻, 𝑇} ∈ V |
14 | 13 | a1i 11 | . . . . . 6 ⊢ (𝐻 ∈ V → 𝒫 {𝐻, 𝑇} ∈ V) |
15 | 2nn0 12090 | . . . . . . 7 ⊢ 2 ∈ ℕ0 | |
16 | 15 | a1i 11 | . . . . . 6 ⊢ (𝐻 ∈ V → 2 ∈ ℕ0) |
17 | prfi 8935 | . . . . . . . . 9 ⊢ {𝐻, 𝑇} ∈ Fin | |
18 | elpwi 4512 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 {𝐻, 𝑇} → 𝑥 ⊆ {𝐻, 𝑇}) | |
19 | ssfi 8840 | . . . . . . . . 9 ⊢ (({𝐻, 𝑇} ∈ Fin ∧ 𝑥 ⊆ {𝐻, 𝑇}) → 𝑥 ∈ Fin) | |
20 | 17, 18, 19 | sylancr 590 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 {𝐻, 𝑇} → 𝑥 ∈ Fin) |
21 | 20 | adantl 485 | . . . . . . 7 ⊢ ((𝐻 ∈ V ∧ 𝑥 ∈ 𝒫 {𝐻, 𝑇}) → 𝑥 ∈ Fin) |
22 | hashcl 13906 | . . . . . . 7 ⊢ (𝑥 ∈ Fin → (♯‘𝑥) ∈ ℕ0) | |
23 | 21, 22 | syl 17 | . . . . . 6 ⊢ ((𝐻 ∈ V ∧ 𝑥 ∈ 𝒫 {𝐻, 𝑇}) → (♯‘𝑥) ∈ ℕ0) |
24 | hashf 13887 | . . . . . . . 8 ⊢ ♯:V⟶(ℕ0 ∪ {+∞}) | |
25 | 24 | a1i 11 | . . . . . . 7 ⊢ (𝐻 ∈ V → ♯:V⟶(ℕ0 ∪ {+∞})) |
26 | ssv 3915 | . . . . . . . 8 ⊢ 𝒫 {𝐻, 𝑇} ⊆ V | |
27 | 26 | a1i 11 | . . . . . . 7 ⊢ (𝐻 ∈ V → 𝒫 {𝐻, 𝑇} ⊆ V) |
28 | 25, 27 | feqresmpt 6770 | . . . . . 6 ⊢ (𝐻 ∈ V → (♯ ↾ 𝒫 {𝐻, 𝑇}) = (𝑥 ∈ 𝒫 {𝐻, 𝑇} ↦ (♯‘𝑥))) |
29 | 14, 16, 23, 28 | ofcfval2 31756 | . . . . 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 7235 | . . . 4 ⊢ (1 / 2) ∈ V | |
32 | 12, 30, 31 | fvmpt 6807 | . . 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 2762 | 1 ⊢ (𝑃‘{𝐻}) = (1 / 2) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1543 ∈ wcel 2110 ≠ wne 2935 Vcvv 3401 ∪ cun 3855 ⊆ wss 3857 𝒫 cpw 4503 {csn 4531 {cpr 4533 〈cop 4537 ↦ cmpt 5124 ↾ cres 5542 ⟶wf 6365 ‘cfv 6369 (class class class)co 7202 Fincfn 8615 0cc0 10712 1c1 10713 +∞cpnf 10847 / cdiv 11472 2c2 11868 ℕ0cn0 12073 ♯chash 13879 ∘f/c cofc 31747 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-card 9538 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-2 11876 df-n0 12074 df-xnn0 12146 df-z 12160 df-uz 12422 df-fz 13079 df-hash 13880 df-ofc 31748 |
This theorem is referenced by: coinflippvt 32135 |
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