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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 | ⊢ 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 6497 | . 2 ⊢ (𝑃‘{𝐻}) = (((♯ ↾ 𝒫 {𝐻, 𝑇})∘𝑓/𝑐 / 2)‘{𝐻}) |
3 | snsspr1 4617 | . . 3 ⊢ {𝐻} ⊆ {𝐻, 𝑇} | |
4 | prex 5185 | . . . . 5 ⊢ {𝐻, 𝑇} ∈ V | |
5 | 4 | elpw2 5100 | . . . 4 ⊢ ({𝐻} ∈ 𝒫 {𝐻, 𝑇} ↔ {𝐻} ⊆ {𝐻, 𝑇}) |
6 | 5 | biimpri 220 | . . 3 ⊢ ({𝐻} ⊆ {𝐻, 𝑇} → {𝐻} ∈ 𝒫 {𝐻, 𝑇}) |
7 | fveq2 6496 | . . . . . 6 ⊢ (𝑥 = {𝐻} → (♯‘𝑥) = (♯‘{𝐻})) | |
8 | coinflip.h | . . . . . . 7 ⊢ 𝐻 ∈ V | |
9 | hashsng 13542 | . . . . . . 7 ⊢ (𝐻 ∈ V → (♯‘{𝐻}) = 1) | |
10 | 8, 9 | ax-mp 5 | . . . . . 6 ⊢ (♯‘{𝐻}) = 1 |
11 | 7, 10 | syl6eq 2824 | . . . . 5 ⊢ (𝑥 = {𝐻} → (♯‘𝑥) = 1) |
12 | 11 | oveq1d 6989 | . . . 4 ⊢ (𝑥 = {𝐻} → ((♯‘𝑥) / 2) = (1 / 2)) |
13 | 4 | pwex 5130 | . . . . . . 7 ⊢ 𝒫 {𝐻, 𝑇} ∈ V |
14 | 13 | a1i 11 | . . . . . 6 ⊢ (𝐻 ∈ V → 𝒫 {𝐻, 𝑇} ∈ V) |
15 | 2nn0 11724 | . . . . . . 7 ⊢ 2 ∈ ℕ0 | |
16 | 15 | a1i 11 | . . . . . 6 ⊢ (𝐻 ∈ V → 2 ∈ ℕ0) |
17 | prfi 8586 | . . . . . . . . 9 ⊢ {𝐻, 𝑇} ∈ Fin | |
18 | elpwi 4426 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 {𝐻, 𝑇} → 𝑥 ⊆ {𝐻, 𝑇}) | |
19 | ssfi 8531 | . . . . . . . . 9 ⊢ (({𝐻, 𝑇} ∈ Fin ∧ 𝑥 ⊆ {𝐻, 𝑇}) → 𝑥 ∈ Fin) | |
20 | 17, 18, 19 | sylancr 578 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 {𝐻, 𝑇} → 𝑥 ∈ Fin) |
21 | 20 | adantl 474 | . . . . . . 7 ⊢ ((𝐻 ∈ V ∧ 𝑥 ∈ 𝒫 {𝐻, 𝑇}) → 𝑥 ∈ Fin) |
22 | hashcl 13530 | . . . . . . 7 ⊢ (𝑥 ∈ Fin → (♯‘𝑥) ∈ ℕ0) | |
23 | 21, 22 | syl 17 | . . . . . 6 ⊢ ((𝐻 ∈ V ∧ 𝑥 ∈ 𝒫 {𝐻, 𝑇}) → (♯‘𝑥) ∈ ℕ0) |
24 | hashf 13511 | . . . . . . . 8 ⊢ ♯:V⟶(ℕ0 ∪ {+∞}) | |
25 | 24 | a1i 11 | . . . . . . 7 ⊢ (𝐻 ∈ V → ♯:V⟶(ℕ0 ∪ {+∞})) |
26 | ssv 3875 | . . . . . . . 8 ⊢ 𝒫 {𝐻, 𝑇} ⊆ V | |
27 | 26 | a1i 11 | . . . . . . 7 ⊢ (𝐻 ∈ V → 𝒫 {𝐻, 𝑇} ⊆ V) |
28 | 25, 27 | feqresmpt 6561 | . . . . . 6 ⊢ (𝐻 ∈ V → (♯ ↾ 𝒫 {𝐻, 𝑇}) = (𝑥 ∈ 𝒫 {𝐻, 𝑇} ↦ (♯‘𝑥))) |
29 | 14, 16, 23, 28 | ofcfval2 31036 | . . . . 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 7006 | . . . 4 ⊢ (1 / 2) ∈ V | |
32 | 12, 30, 31 | fvmpt 6593 | . . 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 2796 | 1 ⊢ (𝑃‘{𝐻}) = (1 / 2) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 387 = wceq 1507 ∈ wcel 2050 ≠ wne 2961 Vcvv 3409 ∪ cun 3821 ⊆ wss 3823 𝒫 cpw 4416 {csn 4435 {cpr 4437 〈cop 4441 ↦ cmpt 5004 ↾ cres 5405 ⟶wf 6181 ‘cfv 6185 (class class class)co 6974 Fincfn 8304 0cc0 10333 1c1 10334 +∞cpnf 10469 / cdiv 11096 2c2 11493 ℕ0cn0 11705 ♯chash 13503 ∘𝑓/𝑐cofc 31027 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-1st 7499 df-2nd 7500 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-oadd 7907 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-card 9160 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-nn 11438 df-2 11501 df-n0 11706 df-xnn0 11778 df-z 11792 df-uz 12057 df-fz 12707 df-hash 13504 df-ofc 31028 |
This theorem is referenced by: coinflippvt 31417 |
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