Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cndprobval Structured version   Visualization version   GIF version

Theorem cndprobval 34436
Description: The value of the conditional probability , i.e. the probability for the event 𝐴, given 𝐵, under the probability law 𝑃. (Contributed by Thierry Arnoux, 21-Jan-2017.)
Assertion
Ref Expression
cndprobval ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) → ((cprob‘𝑃)‘⟨𝐴, 𝐵⟩) = ((𝑃‘(𝐴𝐵)) / (𝑃𝐵)))

Proof of Theorem cndprobval
Dummy variables 𝑎 𝑏 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ov 7435 . 2 (𝐴(cprob‘𝑃)𝐵) = ((cprob‘𝑃)‘⟨𝐴, 𝐵⟩)
2 df-cndprob 34435 . . . . 5 cprob = (𝑝 ∈ Prob ↦ (𝑎 ∈ dom 𝑝, 𝑏 ∈ dom 𝑝 ↦ ((𝑝‘(𝑎𝑏)) / (𝑝𝑏))))
3 dmeq 5913 . . . . . 6 (𝑝 = 𝑃 → dom 𝑝 = dom 𝑃)
4 fveq1 6904 . . . . . . 7 (𝑝 = 𝑃 → (𝑝‘(𝑎𝑏)) = (𝑃‘(𝑎𝑏)))
5 fveq1 6904 . . . . . . 7 (𝑝 = 𝑃 → (𝑝𝑏) = (𝑃𝑏))
64, 5oveq12d 7450 . . . . . 6 (𝑝 = 𝑃 → ((𝑝‘(𝑎𝑏)) / (𝑝𝑏)) = ((𝑃‘(𝑎𝑏)) / (𝑃𝑏)))
73, 3, 6mpoeq123dv 7509 . . . . 5 (𝑝 = 𝑃 → (𝑎 ∈ dom 𝑝, 𝑏 ∈ dom 𝑝 ↦ ((𝑝‘(𝑎𝑏)) / (𝑝𝑏))) = (𝑎 ∈ dom 𝑃, 𝑏 ∈ dom 𝑃 ↦ ((𝑃‘(𝑎𝑏)) / (𝑃𝑏))))
8 id 22 . . . . 5 (𝑃 ∈ Prob → 𝑃 ∈ Prob)
9 dmexg 7924 . . . . . 6 (𝑃 ∈ Prob → dom 𝑃 ∈ V)
10 mpoexga 8103 . . . . . 6 ((dom 𝑃 ∈ V ∧ dom 𝑃 ∈ V) → (𝑎 ∈ dom 𝑃, 𝑏 ∈ dom 𝑃 ↦ ((𝑃‘(𝑎𝑏)) / (𝑃𝑏))) ∈ V)
119, 9, 10syl2anc 584 . . . . 5 (𝑃 ∈ Prob → (𝑎 ∈ dom 𝑃, 𝑏 ∈ dom 𝑃 ↦ ((𝑃‘(𝑎𝑏)) / (𝑃𝑏))) ∈ V)
122, 7, 8, 11fvmptd3 7038 . . . 4 (𝑃 ∈ Prob → (cprob‘𝑃) = (𝑎 ∈ dom 𝑃, 𝑏 ∈ dom 𝑃 ↦ ((𝑃‘(𝑎𝑏)) / (𝑃𝑏))))
13123ad2ant1 1133 . . 3 ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) → (cprob‘𝑃) = (𝑎 ∈ dom 𝑃, 𝑏 ∈ dom 𝑃 ↦ ((𝑃‘(𝑎𝑏)) / (𝑃𝑏))))
14 simprl 770 . . . . . 6 (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → 𝑎 = 𝐴)
15 simprr 772 . . . . . 6 (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → 𝑏 = 𝐵)
1614, 15ineq12d 4220 . . . . 5 (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → (𝑎𝑏) = (𝐴𝐵))
1716fveq2d 6909 . . . 4 (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → (𝑃‘(𝑎𝑏)) = (𝑃‘(𝐴𝐵)))
1815fveq2d 6909 . . . 4 (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → (𝑃𝑏) = (𝑃𝐵))
1917, 18oveq12d 7450 . . 3 (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → ((𝑃‘(𝑎𝑏)) / (𝑃𝑏)) = ((𝑃‘(𝐴𝐵)) / (𝑃𝐵)))
20 simp2 1137 . . 3 ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) → 𝐴 ∈ dom 𝑃)
21 simp3 1138 . . 3 ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) → 𝐵 ∈ dom 𝑃)
22 ovexd 7467 . . 3 ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) → ((𝑃‘(𝐴𝐵)) / (𝑃𝐵)) ∈ V)
2313, 19, 20, 21, 22ovmpod 7586 . 2 ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) → (𝐴(cprob‘𝑃)𝐵) = ((𝑃‘(𝐴𝐵)) / (𝑃𝐵)))
241, 23eqtr3id 2790 1 ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) → ((cprob‘𝑃)‘⟨𝐴, 𝐵⟩) = ((𝑃‘(𝐴𝐵)) / (𝑃𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1539  wcel 2107  Vcvv 3479  cin 3949  cop 4631  dom cdm 5684  cfv 6560  (class class class)co 7432  cmpo 7434   / cdiv 11921  Probcprb 34410  cprobccprob 34434
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-cndprob 34435
This theorem is referenced by:  cndprobin  34437  cndprob01  34438  cndprobtot  34439  cndprobnul  34440  cndprobprob  34441
  Copyright terms: Public domain W3C validator