Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > coinfliplem | Structured version Visualization version GIF version |
Description: Division in the extended real numbers can be used for the coin-flip example. (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 |
---|---|
coinfliplem | ⊢ 𝑃 = ((♯ ↾ 𝒫 {𝐻, 𝑇}) ∘f/c /𝑒 2) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coinflip.2 | . 2 ⊢ 𝑃 = ((♯ ↾ 𝒫 {𝐻, 𝑇}) ∘f/c / 2) | |
2 | coinflip.h | . . 3 ⊢ 𝐻 ∈ V | |
3 | simpr 487 | . . . . . 6 ⊢ ((𝐻 ∈ V ∧ 𝑥 ∈ 𝒫 {𝐻, 𝑇}) → 𝑥 ∈ 𝒫 {𝐻, 𝑇}) | |
4 | fvres 6691 | . . . . . 6 ⊢ (𝑥 ∈ 𝒫 {𝐻, 𝑇} → ((♯ ↾ 𝒫 {𝐻, 𝑇})‘𝑥) = (♯‘𝑥)) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ ((𝐻 ∈ V ∧ 𝑥 ∈ 𝒫 {𝐻, 𝑇}) → ((♯ ↾ 𝒫 {𝐻, 𝑇})‘𝑥) = (♯‘𝑥)) |
6 | prfi 8795 | . . . . . . . 8 ⊢ {𝐻, 𝑇} ∈ Fin | |
7 | 3 | elpwid 4552 | . . . . . . . 8 ⊢ ((𝐻 ∈ V ∧ 𝑥 ∈ 𝒫 {𝐻, 𝑇}) → 𝑥 ⊆ {𝐻, 𝑇}) |
8 | ssfi 8740 | . . . . . . . 8 ⊢ (({𝐻, 𝑇} ∈ Fin ∧ 𝑥 ⊆ {𝐻, 𝑇}) → 𝑥 ∈ Fin) | |
9 | 6, 7, 8 | sylancr 589 | . . . . . . 7 ⊢ ((𝐻 ∈ V ∧ 𝑥 ∈ 𝒫 {𝐻, 𝑇}) → 𝑥 ∈ Fin) |
10 | hashcl 13720 | . . . . . . 7 ⊢ (𝑥 ∈ Fin → (♯‘𝑥) ∈ ℕ0) | |
11 | 9, 10 | syl 17 | . . . . . 6 ⊢ ((𝐻 ∈ V ∧ 𝑥 ∈ 𝒫 {𝐻, 𝑇}) → (♯‘𝑥) ∈ ℕ0) |
12 | 11 | nn0red 11959 | . . . . 5 ⊢ ((𝐻 ∈ V ∧ 𝑥 ∈ 𝒫 {𝐻, 𝑇}) → (♯‘𝑥) ∈ ℝ) |
13 | 5, 12 | eqeltrd 2915 | . . . 4 ⊢ ((𝐻 ∈ V ∧ 𝑥 ∈ 𝒫 {𝐻, 𝑇}) → ((♯ ↾ 𝒫 {𝐻, 𝑇})‘𝑥) ∈ ℝ) |
14 | simpr 487 | . . . . 5 ⊢ ((𝐻 ∈ V ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ) | |
15 | 2re 11714 | . . . . . 6 ⊢ 2 ∈ ℝ | |
16 | 15 | a1i 11 | . . . . 5 ⊢ ((𝐻 ∈ V ∧ 𝑦 ∈ ℝ) → 2 ∈ ℝ) |
17 | 2ne0 11744 | . . . . . 6 ⊢ 2 ≠ 0 | |
18 | 17 | a1i 11 | . . . . 5 ⊢ ((𝐻 ∈ V ∧ 𝑦 ∈ ℝ) → 2 ≠ 0) |
19 | rexdiv 30604 | . . . . 5 ⊢ ((𝑦 ∈ ℝ ∧ 2 ∈ ℝ ∧ 2 ≠ 0) → (𝑦 /𝑒 2) = (𝑦 / 2)) | |
20 | 14, 16, 18, 19 | syl3anc 1367 | . . . 4 ⊢ ((𝐻 ∈ V ∧ 𝑦 ∈ ℝ) → (𝑦 /𝑒 2) = (𝑦 / 2)) |
21 | hashresfn 13703 | . . . . 5 ⊢ (♯ ↾ 𝒫 {𝐻, 𝑇}) Fn 𝒫 {𝐻, 𝑇} | |
22 | 21 | a1i 11 | . . . 4 ⊢ (𝐻 ∈ V → (♯ ↾ 𝒫 {𝐻, 𝑇}) Fn 𝒫 {𝐻, 𝑇}) |
23 | pwfi 8821 | . . . . . 6 ⊢ ({𝐻, 𝑇} ∈ Fin ↔ 𝒫 {𝐻, 𝑇} ∈ Fin) | |
24 | 6, 23 | mpbi 232 | . . . . 5 ⊢ 𝒫 {𝐻, 𝑇} ∈ Fin |
25 | 24 | a1i 11 | . . . 4 ⊢ (𝐻 ∈ V → 𝒫 {𝐻, 𝑇} ∈ Fin) |
26 | 15 | a1i 11 | . . . 4 ⊢ (𝐻 ∈ V → 2 ∈ ℝ) |
27 | 13, 20, 22, 25, 26 | ofcfeqd2 31362 | . . 3 ⊢ (𝐻 ∈ V → ((♯ ↾ 𝒫 {𝐻, 𝑇}) ∘f/c /𝑒 2) = ((♯ ↾ 𝒫 {𝐻, 𝑇}) ∘f/c / 2)) |
28 | 2, 27 | ax-mp 5 | . 2 ⊢ ((♯ ↾ 𝒫 {𝐻, 𝑇}) ∘f/c /𝑒 2) = ((♯ ↾ 𝒫 {𝐻, 𝑇}) ∘f/c / 2) |
29 | 1, 28 | eqtr4i 2849 | 1 ⊢ 𝑃 = ((♯ ↾ 𝒫 {𝐻, 𝑇}) ∘f/c /𝑒 2) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 Vcvv 3496 ⊆ wss 3938 𝒫 cpw 4541 {cpr 4571 〈cop 4575 ↾ cres 5559 Fn wfn 6352 ‘cfv 6357 (class class class)co 7158 Fincfn 8511 ℝcr 10538 0cc0 10539 1c1 10540 / cdiv 11299 2c2 11695 ℕ0cn0 11900 ♯chash 13693 /𝑒 cxdiv 30595 ∘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-rmo 3148 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-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 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-div 11300 df-nn 11641 df-2 11703 df-n0 11901 df-xnn0 11971 df-z 11985 df-uz 12247 df-xneg 12510 df-xmul 12512 df-hash 13694 df-xdiv 30596 df-ofc 31357 |
This theorem is referenced by: coinflipprob 31739 |
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