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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 | ā¢ š» ā V |
coinflip.t | ā¢ š ā V |
coinflip.th | ā¢ š» ā š |
coinflip.2 | ā¢ š = ((āÆ ā¾ š« {š», š}) āf/c / 2) |
coinflip.3 | ā¢ š = {āØš», 1ā©, āØš, 0ā©} |
Ref | Expression |
---|---|
coinflippv | ā¢ (šā{š»}) = (1 / 2) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coinflip.2 | . . 3 ā¢ š = ((āÆ ā¾ š« {š», š}) āf/c / 2) | |
2 | 1 | fveq1i 6844 | . 2 ā¢ (šā{š»}) = (((āÆ ā¾ š« {š», š}) āf/c / 2)ā{š»}) |
3 | snsspr1 4775 | . . 3 ā¢ {š»} ā {š», š} | |
4 | prex 5390 | . . . . 5 ā¢ {š», š} ā V | |
5 | 4 | elpw2 5303 | . . . 4 ā¢ ({š»} ā š« {š», š} ā {š»} ā {š», š}) |
6 | 5 | biimpri 227 | . . 3 ā¢ ({š»} ā {š», š} ā {š»} ā š« {š», š}) |
7 | fveq2 6843 | . . . . . 6 ā¢ (š„ = {š»} ā (āÆāš„) = (āÆā{š»})) | |
8 | coinflip.h | . . . . . . 7 ā¢ š» ā V | |
9 | hashsng 14275 | . . . . . . 7 ā¢ (š» ā V ā (āÆā{š»}) = 1) | |
10 | 8, 9 | ax-mp 5 | . . . . . 6 ā¢ (āÆā{š»}) = 1 |
11 | 7, 10 | eqtrdi 2789 | . . . . 5 ā¢ (š„ = {š»} ā (āÆāš„) = 1) |
12 | 11 | oveq1d 7373 | . . . 4 ā¢ (š„ = {š»} ā ((āÆāš„) / 2) = (1 / 2)) |
13 | 4 | pwex 5336 | . . . . . . 7 ā¢ š« {š», š} ā V |
14 | 13 | a1i 11 | . . . . . 6 ā¢ (š» ā V ā š« {š», š} ā V) |
15 | 2nn0 12435 | . . . . . . 7 ā¢ 2 ā ā0 | |
16 | 15 | a1i 11 | . . . . . 6 ā¢ (š» ā V ā 2 ā ā0) |
17 | prfi 9269 | . . . . . . . . 9 ā¢ {š», š} ā Fin | |
18 | elpwi 4568 | . . . . . . . . 9 ā¢ (š„ ā š« {š», š} ā š„ ā {š», š}) | |
19 | ssfi 9120 | . . . . . . . . 9 ā¢ (({š», š} ā Fin ā§ š„ ā {š», š}) ā š„ ā Fin) | |
20 | 17, 18, 19 | sylancr 588 | . . . . . . . 8 ā¢ (š„ ā š« {š», š} ā š„ ā Fin) |
21 | 20 | adantl 483 | . . . . . . 7 ā¢ ((š» ā V ā§ š„ ā š« {š», š}) ā š„ ā Fin) |
22 | hashcl 14262 | . . . . . . 7 ā¢ (š„ ā Fin ā (āÆāš„) ā ā0) | |
23 | 21, 22 | syl 17 | . . . . . 6 ā¢ ((š» ā V ā§ š„ ā š« {š», š}) ā (āÆāš„) ā ā0) |
24 | hashf 14244 | . . . . . . . 8 ā¢ āÆ:Vā¶(ā0 āŖ {+ā}) | |
25 | 24 | a1i 11 | . . . . . . 7 ā¢ (š» ā V ā āÆ:Vā¶(ā0 āŖ {+ā})) |
26 | ssv 3969 | . . . . . . . 8 ā¢ š« {š», š} ā V | |
27 | 26 | a1i 11 | . . . . . . 7 ā¢ (š» ā V ā š« {š», š} ā V) |
28 | 25, 27 | feqresmpt 6912 | . . . . . 6 ā¢ (š» ā V ā (āÆ ā¾ š« {š», š}) = (š„ ā š« {š», š} ā¦ (āÆāš„))) |
29 | 14, 16, 23, 28 | ofcfval2 32760 | . . . . 5 ā¢ (š» ā V ā ((āÆ ā¾ š« {š», š}) āf/c / 2) = (š„ ā š« {š», š} ā¦ ((āÆāš„) / 2))) |
30 | 8, 29 | ax-mp 5 | . . . 4 ā¢ ((āÆ ā¾ š« {š», š}) āf/c / 2) = (š„ ā š« {š», š} ā¦ ((āÆāš„) / 2)) |
31 | ovex 7391 | . . . 4 ā¢ (1 / 2) ā V | |
32 | 12, 30, 31 | fvmpt 6949 | . . 3 ā¢ ({š»} ā š« {š», š} ā (((āÆ ā¾ š« {š», š}) āf/c / 2)ā{š»}) = (1 / 2)) |
33 | 3, 6, 32 | mp2b 10 | . 2 ā¢ (((āÆ ā¾ š« {š», š}) āf/c / 2)ā{š»}) = (1 / 2) |
34 | 2, 33 | eqtri 2761 | 1 ā¢ (šā{š»}) = (1 / 2) |
Colors of variables: wff setvar class |
Syntax hints: ā§ wa 397 = wceq 1542 ā wcel 2107 ā wne 2940 Vcvv 3444 āŖ cun 3909 ā wss 3911 š« cpw 4561 {csn 4587 {cpr 4589 āØcop 4593 ā¦ cmpt 5189 ā¾ cres 5636 ā¶wf 6493 ācfv 6497 (class class class)co 7358 Fincfn 8886 0cc0 11056 1c1 11057 +ācpnf 11191 / cdiv 11817 2c2 12213 ā0cn0 12418 āÆchash 14236 āf/c cofc 32751 |
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 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
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 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-n0 12419 df-xnn0 12491 df-z 12505 df-uz 12769 df-fz 13431 df-hash 14237 df-ofc 32752 |
This theorem is referenced by: coinflippvt 33141 |
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