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Theorem cndprobval 31813
 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 7138 . 2 (𝐴(cprob‘𝑃)𝐵) = ((cprob‘𝑃)‘⟨𝐴, 𝐵⟩)
2 df-cndprob 31812 . . . . 5 cprob = (𝑝 ∈ Prob ↦ (𝑎 ∈ dom 𝑝, 𝑏 ∈ dom 𝑝 ↦ ((𝑝‘(𝑎𝑏)) / (𝑝𝑏))))
3 dmeq 5736 . . . . . 6 (𝑝 = 𝑃 → dom 𝑝 = dom 𝑃)
4 fveq1 6644 . . . . . . 7 (𝑝 = 𝑃 → (𝑝‘(𝑎𝑏)) = (𝑃‘(𝑎𝑏)))
5 fveq1 6644 . . . . . . 7 (𝑝 = 𝑃 → (𝑝𝑏) = (𝑃𝑏))
64, 5oveq12d 7153 . . . . . 6 (𝑝 = 𝑃 → ((𝑝‘(𝑎𝑏)) / (𝑝𝑏)) = ((𝑃‘(𝑎𝑏)) / (𝑃𝑏)))
73, 3, 6mpoeq123dv 7208 . . . . 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 7596 . . . . . 6 (𝑃 ∈ Prob → dom 𝑃 ∈ V)
10 mpoexga 7760 . . . . . 6 ((dom 𝑃 ∈ V ∧ dom 𝑃 ∈ V) → (𝑎 ∈ dom 𝑃, 𝑏 ∈ dom 𝑃 ↦ ((𝑃‘(𝑎𝑏)) / (𝑃𝑏))) ∈ V)
119, 9, 10syl2anc 587 . . . . 5 (𝑃 ∈ Prob → (𝑎 ∈ dom 𝑃, 𝑏 ∈ dom 𝑃 ↦ ((𝑃‘(𝑎𝑏)) / (𝑃𝑏))) ∈ V)
122, 7, 8, 11fvmptd3 6768 . . . 4 (𝑃 ∈ Prob → (cprob‘𝑃) = (𝑎 ∈ dom 𝑃, 𝑏 ∈ dom 𝑃 ↦ ((𝑃‘(𝑎𝑏)) / (𝑃𝑏))))
13123ad2ant1 1130 . . 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 4140 . . . . 5 (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → (𝑎𝑏) = (𝐴𝐵))
1716fveq2d 6649 . . . 4 (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → (𝑃‘(𝑎𝑏)) = (𝑃‘(𝐴𝐵)))
1815fveq2d 6649 . . . 4 (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → (𝑃𝑏) = (𝑃𝐵))
1917, 18oveq12d 7153 . . 3 (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → ((𝑃‘(𝑎𝑏)) / (𝑃𝑏)) = ((𝑃‘(𝐴𝐵)) / (𝑃𝐵)))
20 simp2 1134 . . 3 ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) → 𝐴 ∈ dom 𝑃)
21 simp3 1135 . . 3 ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) → 𝐵 ∈ dom 𝑃)
22 ovexd 7170 . . 3 ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) → ((𝑃‘(𝐴𝐵)) / (𝑃𝐵)) ∈ V)
2313, 19, 20, 21, 22ovmpod 7282 . 2 ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) → (𝐴(cprob‘𝑃)𝐵) = ((𝑃‘(𝐴𝐵)) / (𝑃𝐵)))
241, 23syl5eqr 2847 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 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  Vcvv 3441   ∩ cin 3880  ⟨cop 4531  dom cdm 5519  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137   / cdiv 11288  Probcprb 31787  cprobccprob 31811 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7673  df-2nd 7674  df-cndprob 31812 This theorem is referenced by:  cndprobin  31814  cndprob01  31815  cndprobtot  31816  cndprobnul  31817  cndprobprob  31818
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