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 477 | . . . . . 6 ⊢ ((𝐻 ∈ V ∧ 𝑥 ∈ 𝒫 {𝐻, 𝑇}) → 𝑥 ∈ 𝒫 {𝐻, 𝑇}) | |
4 | fvres 6207 | . . . . . 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 8235 | . . . . . . . 8 ⊢ {𝐻, 𝑇} ∈ Fin | |
7 | 3 | elpwid 4170 | . . . . . . . 8 ⊢ ((𝐻 ∈ V ∧ 𝑥 ∈ 𝒫 {𝐻, 𝑇}) → 𝑥 ⊆ {𝐻, 𝑇}) |
8 | ssfi 8180 | . . . . . . . 8 ⊢ (({𝐻, 𝑇} ∈ Fin ∧ 𝑥 ⊆ {𝐻, 𝑇}) → 𝑥 ∈ Fin) | |
9 | 6, 7, 8 | sylancr 695 | . . . . . . 7 ⊢ ((𝐻 ∈ V ∧ 𝑥 ∈ 𝒫 {𝐻, 𝑇}) → 𝑥 ∈ Fin) |
10 | hashcl 13147 | . . . . . . 7 ⊢ (𝑥 ∈ Fin → (#‘𝑥) ∈ ℕ0) | |
11 | 9, 10 | syl 17 | . . . . . 6 ⊢ ((𝐻 ∈ V ∧ 𝑥 ∈ 𝒫 {𝐻, 𝑇}) → (#‘𝑥) ∈ ℕ0) |
12 | 11 | nn0red 11352 | . . . . 5 ⊢ ((𝐻 ∈ V ∧ 𝑥 ∈ 𝒫 {𝐻, 𝑇}) → (#‘𝑥) ∈ ℝ) |
13 | 5, 12 | eqeltrd 2701 | . . . 4 ⊢ ((𝐻 ∈ V ∧ 𝑥 ∈ 𝒫 {𝐻, 𝑇}) → ((# ↾ 𝒫 {𝐻, 𝑇})‘𝑥) ∈ ℝ) |
14 | simpr 477 | . . . . 5 ⊢ ((𝐻 ∈ V ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ) | |
15 | 2re 11090 | . . . . . 6 ⊢ 2 ∈ ℝ | |
16 | 15 | a1i 11 | . . . . 5 ⊢ ((𝐻 ∈ V ∧ 𝑦 ∈ ℝ) → 2 ∈ ℝ) |
17 | 2ne0 11113 | . . . . . 6 ⊢ 2 ≠ 0 | |
18 | 17 | a1i 11 | . . . . 5 ⊢ ((𝐻 ∈ V ∧ 𝑦 ∈ ℝ) → 2 ≠ 0) |
19 | rexdiv 29634 | . . . . 5 ⊢ ((𝑦 ∈ ℝ ∧ 2 ∈ ℝ ∧ 2 ≠ 0) → (𝑦 /𝑒 2) = (𝑦 / 2)) | |
20 | 14, 16, 18, 19 | syl3anc 1326 | . . . 4 ⊢ ((𝐻 ∈ V ∧ 𝑦 ∈ ℝ) → (𝑦 /𝑒 2) = (𝑦 / 2)) |
21 | hashresfn 13128 | . . . . 5 ⊢ (# ↾ 𝒫 {𝐻, 𝑇}) Fn 𝒫 {𝐻, 𝑇} | |
22 | 21 | a1i 11 | . . . 4 ⊢ (𝐻 ∈ V → (# ↾ 𝒫 {𝐻, 𝑇}) Fn 𝒫 {𝐻, 𝑇}) |
23 | pwfi 8261 | . . . . . 6 ⊢ ({𝐻, 𝑇} ∈ Fin ↔ 𝒫 {𝐻, 𝑇} ∈ Fin) | |
24 | 6, 23 | mpbi 220 | . . . . 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 30163 | . . 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 2647 | 1 ⊢ 𝑃 = ((# ↾ 𝒫 {𝐻, 𝑇})∘𝑓/𝑐 /𝑒 2) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 ⊆ wss 3574 𝒫 cpw 4158 {cpr 4179 〈cop 4183 ↾ cres 5116 Fn wfn 5883 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 ℝcr 9935 0cc0 9936 1c1 9937 / cdiv 10684 2c2 11070 ℕ0cn0 11292 #chash 13117 /𝑒 cxdiv 29625 ∘𝑓/𝑐cofc 30157 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-xneg 11946 df-xmul 11948 df-hash 13118 df-xdiv 29626 df-ofc 30158 |
This theorem is referenced by: coinflipprob 30541 |
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