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 6673 | . 2 ⊢ (𝑃‘{𝐻}) = (((♯ ↾ 𝒫 {𝐻, 𝑇}) ∘f/c / 2)‘{𝐻}) |
3 | snsspr1 4749 | . . 3 ⊢ {𝐻} ⊆ {𝐻, 𝑇} | |
4 | prex 5335 | . . . . 5 ⊢ {𝐻, 𝑇} ∈ V | |
5 | 4 | elpw2 5250 | . . . 4 ⊢ ({𝐻} ∈ 𝒫 {𝐻, 𝑇} ↔ {𝐻} ⊆ {𝐻, 𝑇}) |
6 | 5 | biimpri 230 | . . 3 ⊢ ({𝐻} ⊆ {𝐻, 𝑇} → {𝐻} ∈ 𝒫 {𝐻, 𝑇}) |
7 | fveq2 6672 | . . . . . 6 ⊢ (𝑥 = {𝐻} → (♯‘𝑥) = (♯‘{𝐻})) | |
8 | coinflip.h | . . . . . . 7 ⊢ 𝐻 ∈ V | |
9 | hashsng 13733 | . . . . . . 7 ⊢ (𝐻 ∈ V → (♯‘{𝐻}) = 1) | |
10 | 8, 9 | ax-mp 5 | . . . . . 6 ⊢ (♯‘{𝐻}) = 1 |
11 | 7, 10 | syl6eq 2874 | . . . . 5 ⊢ (𝑥 = {𝐻} → (♯‘𝑥) = 1) |
12 | 11 | oveq1d 7173 | . . . 4 ⊢ (𝑥 = {𝐻} → ((♯‘𝑥) / 2) = (1 / 2)) |
13 | 4 | pwex 5283 | . . . . . . 7 ⊢ 𝒫 {𝐻, 𝑇} ∈ V |
14 | 13 | a1i 11 | . . . . . 6 ⊢ (𝐻 ∈ V → 𝒫 {𝐻, 𝑇} ∈ V) |
15 | 2nn0 11917 | . . . . . . 7 ⊢ 2 ∈ ℕ0 | |
16 | 15 | a1i 11 | . . . . . 6 ⊢ (𝐻 ∈ V → 2 ∈ ℕ0) |
17 | prfi 8795 | . . . . . . . . 9 ⊢ {𝐻, 𝑇} ∈ Fin | |
18 | elpwi 4550 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 {𝐻, 𝑇} → 𝑥 ⊆ {𝐻, 𝑇}) | |
19 | ssfi 8740 | . . . . . . . . 9 ⊢ (({𝐻, 𝑇} ∈ Fin ∧ 𝑥 ⊆ {𝐻, 𝑇}) → 𝑥 ∈ Fin) | |
20 | 17, 18, 19 | sylancr 589 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 {𝐻, 𝑇} → 𝑥 ∈ Fin) |
21 | 20 | adantl 484 | . . . . . . 7 ⊢ ((𝐻 ∈ V ∧ 𝑥 ∈ 𝒫 {𝐻, 𝑇}) → 𝑥 ∈ Fin) |
22 | hashcl 13720 | . . . . . . 7 ⊢ (𝑥 ∈ Fin → (♯‘𝑥) ∈ ℕ0) | |
23 | 21, 22 | syl 17 | . . . . . 6 ⊢ ((𝐻 ∈ V ∧ 𝑥 ∈ 𝒫 {𝐻, 𝑇}) → (♯‘𝑥) ∈ ℕ0) |
24 | hashf 13701 | . . . . . . . 8 ⊢ ♯:V⟶(ℕ0 ∪ {+∞}) | |
25 | 24 | a1i 11 | . . . . . . 7 ⊢ (𝐻 ∈ V → ♯:V⟶(ℕ0 ∪ {+∞})) |
26 | ssv 3993 | . . . . . . . 8 ⊢ 𝒫 {𝐻, 𝑇} ⊆ V | |
27 | 26 | a1i 11 | . . . . . . 7 ⊢ (𝐻 ∈ V → 𝒫 {𝐻, 𝑇} ⊆ V) |
28 | 25, 27 | feqresmpt 6736 | . . . . . 6 ⊢ (𝐻 ∈ V → (♯ ↾ 𝒫 {𝐻, 𝑇}) = (𝑥 ∈ 𝒫 {𝐻, 𝑇} ↦ (♯‘𝑥))) |
29 | 14, 16, 23, 28 | ofcfval2 31365 | . . . . 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 7191 | . . . 4 ⊢ (1 / 2) ∈ V | |
32 | 12, 30, 31 | fvmpt 6770 | . . 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 2846 | 1 ⊢ (𝑃‘{𝐻}) = (1 / 2) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 Vcvv 3496 ∪ cun 3936 ⊆ wss 3938 𝒫 cpw 4541 {csn 4569 {cpr 4571 〈cop 4575 ↦ cmpt 5148 ↾ cres 5559 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 Fincfn 8511 0cc0 10539 1c1 10540 +∞cpnf 10674 / cdiv 11299 2c2 11695 ℕ0cn0 11900 ♯chash 13693 ∘f/c cofc 31356 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-n0 11901 df-xnn0 11971 df-z 11985 df-uz 12247 df-fz 12896 df-hash 13694 df-ofc 31357 |
This theorem is referenced by: coinflippvt 31744 |
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