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 34415
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 7434 . 2 (𝐴(cprob‘𝑃)𝐵) = ((cprob‘𝑃)‘⟨𝐴, 𝐵⟩)
2 df-cndprob 34414 . . . . 5 cprob = (𝑝 ∈ Prob ↦ (𝑎 ∈ dom 𝑝, 𝑏 ∈ dom 𝑝 ↦ ((𝑝‘(𝑎𝑏)) / (𝑝𝑏))))
3 dmeq 5917 . . . . . 6 (𝑝 = 𝑃 → dom 𝑝 = dom 𝑃)
4 fveq1 6906 . . . . . . 7 (𝑝 = 𝑃 → (𝑝‘(𝑎𝑏)) = (𝑃‘(𝑎𝑏)))
5 fveq1 6906 . . . . . . 7 (𝑝 = 𝑃 → (𝑝𝑏) = (𝑃𝑏))
64, 5oveq12d 7449 . . . . . 6 (𝑝 = 𝑃 → ((𝑝‘(𝑎𝑏)) / (𝑝𝑏)) = ((𝑃‘(𝑎𝑏)) / (𝑃𝑏)))
73, 3, 6mpoeq123dv 7508 . . . . 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 8101 . . . . . 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 7039 . . . 4 (𝑃 ∈ Prob → (cprob‘𝑃) = (𝑎 ∈ dom 𝑃, 𝑏 ∈ dom 𝑃 ↦ ((𝑃‘(𝑎𝑏)) / (𝑃𝑏))))
13123ad2ant1 1132 . . 3 ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) → (cprob‘𝑃) = (𝑎 ∈ dom 𝑃, 𝑏 ∈ dom 𝑃 ↦ ((𝑃‘(𝑎𝑏)) / (𝑃𝑏))))
14 simprl 771 . . . . . 6 (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → 𝑎 = 𝐴)
15 simprr 773 . . . . . 6 (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → 𝑏 = 𝐵)
1614, 15ineq12d 4229 . . . . 5 (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → (𝑎𝑏) = (𝐴𝐵))
1716fveq2d 6911 . . . 4 (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → (𝑃‘(𝑎𝑏)) = (𝑃‘(𝐴𝐵)))
1815fveq2d 6911 . . . 4 (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → (𝑃𝑏) = (𝑃𝐵))
1917, 18oveq12d 7449 . . 3 (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → ((𝑃‘(𝑎𝑏)) / (𝑃𝑏)) = ((𝑃‘(𝐴𝐵)) / (𝑃𝐵)))
20 simp2 1136 . . 3 ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) → 𝐴 ∈ dom 𝑃)
21 simp3 1137 . . 3 ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) → 𝐵 ∈ dom 𝑃)
22 ovexd 7466 . . 3 ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) → ((𝑃‘(𝐴𝐵)) / (𝑃𝐵)) ∈ V)
2313, 19, 20, 21, 22ovmpod 7585 . 2 ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) → (𝐴(cprob‘𝑃)𝐵) = ((𝑃‘(𝐴𝐵)) / (𝑃𝐵)))
241, 23eqtr3id 2789 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 1537  wcel 2106  Vcvv 3478  cin 3962  cop 4637  dom cdm 5689  cfv 6563  (class class class)co 7431  cmpo 7433   / cdiv 11918  Probcprb 34389  cprobccprob 34413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-cndprob 34414
This theorem is referenced by:  cndprobin  34416  cndprob01  34417  cndprobtot  34418  cndprobnul  34419  cndprobprob  34420
  Copyright terms: Public domain W3C validator