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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 486 | . . . . . 6 ⊢ ((𝐻 ∈ V ∧ 𝑥 ∈ 𝒫 {𝐻, 𝑇}) → 𝑥 ∈ 𝒫 {𝐻, 𝑇}) | |
4 | fvres 6908 | . . . . . 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 9319 | . . . . . . . 8 ⊢ {𝐻, 𝑇} ∈ Fin | |
7 | 3 | elpwid 4611 | . . . . . . . 8 ⊢ ((𝐻 ∈ V ∧ 𝑥 ∈ 𝒫 {𝐻, 𝑇}) → 𝑥 ⊆ {𝐻, 𝑇}) |
8 | ssfi 9170 | . . . . . . . 8 ⊢ (({𝐻, 𝑇} ∈ Fin ∧ 𝑥 ⊆ {𝐻, 𝑇}) → 𝑥 ∈ Fin) | |
9 | 6, 7, 8 | sylancr 588 | . . . . . . 7 ⊢ ((𝐻 ∈ V ∧ 𝑥 ∈ 𝒫 {𝐻, 𝑇}) → 𝑥 ∈ Fin) |
10 | hashcl 14313 | . . . . . . 7 ⊢ (𝑥 ∈ Fin → (♯‘𝑥) ∈ ℕ0) | |
11 | 9, 10 | syl 17 | . . . . . 6 ⊢ ((𝐻 ∈ V ∧ 𝑥 ∈ 𝒫 {𝐻, 𝑇}) → (♯‘𝑥) ∈ ℕ0) |
12 | 11 | nn0red 12530 | . . . . 5 ⊢ ((𝐻 ∈ V ∧ 𝑥 ∈ 𝒫 {𝐻, 𝑇}) → (♯‘𝑥) ∈ ℝ) |
13 | 5, 12 | eqeltrd 2834 | . . . 4 ⊢ ((𝐻 ∈ V ∧ 𝑥 ∈ 𝒫 {𝐻, 𝑇}) → ((♯ ↾ 𝒫 {𝐻, 𝑇})‘𝑥) ∈ ℝ) |
14 | simpr 486 | . . . . 5 ⊢ ((𝐻 ∈ V ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ) | |
15 | 2re 12283 | . . . . . 6 ⊢ 2 ∈ ℝ | |
16 | 15 | a1i 11 | . . . . 5 ⊢ ((𝐻 ∈ V ∧ 𝑦 ∈ ℝ) → 2 ∈ ℝ) |
17 | 2ne0 12313 | . . . . . 6 ⊢ 2 ≠ 0 | |
18 | 17 | a1i 11 | . . . . 5 ⊢ ((𝐻 ∈ V ∧ 𝑦 ∈ ℝ) → 2 ≠ 0) |
19 | rexdiv 32080 | . . . . 5 ⊢ ((𝑦 ∈ ℝ ∧ 2 ∈ ℝ ∧ 2 ≠ 0) → (𝑦 /𝑒 2) = (𝑦 / 2)) | |
20 | 14, 16, 18, 19 | syl3anc 1372 | . . . 4 ⊢ ((𝐻 ∈ V ∧ 𝑦 ∈ ℝ) → (𝑦 /𝑒 2) = (𝑦 / 2)) |
21 | hashresfn 14297 | . . . . 5 ⊢ (♯ ↾ 𝒫 {𝐻, 𝑇}) Fn 𝒫 {𝐻, 𝑇} | |
22 | 21 | a1i 11 | . . . 4 ⊢ (𝐻 ∈ V → (♯ ↾ 𝒫 {𝐻, 𝑇}) Fn 𝒫 {𝐻, 𝑇}) |
23 | pwfi 9175 | . . . . . 6 ⊢ ({𝐻, 𝑇} ∈ Fin ↔ 𝒫 {𝐻, 𝑇} ∈ Fin) | |
24 | 6, 23 | mpbi 229 | . . . . 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 33088 | . . 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 2764 | 1 ⊢ 𝑃 = ((♯ ↾ 𝒫 {𝐻, 𝑇}) ∘f/c /𝑒 2) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 Vcvv 3475 ⊆ wss 3948 𝒫 cpw 4602 {cpr 4630 〈cop 4634 ↾ cres 5678 Fn wfn 6536 ‘cfv 6541 (class class class)co 7406 Fincfn 8936 ℝcr 11106 0cc0 11107 1c1 11108 / cdiv 11868 2c2 12264 ℕ0cn0 12469 ♯chash 14287 /𝑒 cxdiv 32071 ∘f/c cofc 33082 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-card 9931 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-n0 12470 df-xnn0 12542 df-z 12556 df-uz 12820 df-xneg 13089 df-xmul 13091 df-hash 14288 df-xdiv 32072 df-ofc 33083 |
This theorem is referenced by: coinflipprob 33467 |
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