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Theorem cndprobval 31098
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 6927 . 2 (𝐴(cprob‘𝑃)𝐵) = ((cprob‘𝑃)‘⟨𝐴, 𝐵⟩)
2 df-cndprob 31097 . . . . 5 cprob = (𝑝 ∈ Prob ↦ (𝑎 ∈ dom 𝑝, 𝑏 ∈ dom 𝑝 ↦ ((𝑝‘(𝑎𝑏)) / (𝑝𝑏))))
3 dmeq 5571 . . . . . 6 (𝑝 = 𝑃 → dom 𝑝 = dom 𝑃)
4 fveq1 6447 . . . . . . 7 (𝑝 = 𝑃 → (𝑝‘(𝑎𝑏)) = (𝑃‘(𝑎𝑏)))
5 fveq1 6447 . . . . . . 7 (𝑝 = 𝑃 → (𝑝𝑏) = (𝑃𝑏))
64, 5oveq12d 6942 . . . . . 6 (𝑝 = 𝑃 → ((𝑝‘(𝑎𝑏)) / (𝑝𝑏)) = ((𝑃‘(𝑎𝑏)) / (𝑃𝑏)))
73, 3, 6mpt2eq123dv 6996 . . . . 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 7377 . . . . . 6 (𝑃 ∈ Prob → dom 𝑃 ∈ V)
10 mpt2exga 7528 . . . . . 6 ((dom 𝑃 ∈ V ∧ dom 𝑃 ∈ V) → (𝑎 ∈ dom 𝑃, 𝑏 ∈ dom 𝑃 ↦ ((𝑃‘(𝑎𝑏)) / (𝑃𝑏))) ∈ V)
119, 9, 10syl2anc 579 . . . . 5 (𝑃 ∈ Prob → (𝑎 ∈ dom 𝑃, 𝑏 ∈ dom 𝑃 ↦ ((𝑃‘(𝑎𝑏)) / (𝑃𝑏))) ∈ V)
122, 7, 8, 11fvmptd3 6566 . . . 4 (𝑃 ∈ Prob → (cprob‘𝑃) = (𝑎 ∈ dom 𝑃, 𝑏 ∈ dom 𝑃 ↦ ((𝑃‘(𝑎𝑏)) / (𝑃𝑏))))
13123ad2ant1 1124 . . 3 ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) → (cprob‘𝑃) = (𝑎 ∈ dom 𝑃, 𝑏 ∈ dom 𝑃 ↦ ((𝑃‘(𝑎𝑏)) / (𝑃𝑏))))
14 simprl 761 . . . . . 6 (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → 𝑎 = 𝐴)
15 simprr 763 . . . . . 6 (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → 𝑏 = 𝐵)
1614, 15ineq12d 4038 . . . . 5 (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → (𝑎𝑏) = (𝐴𝐵))
1716fveq2d 6452 . . . 4 (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → (𝑃‘(𝑎𝑏)) = (𝑃‘(𝐴𝐵)))
1815fveq2d 6452 . . . 4 (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → (𝑃𝑏) = (𝑃𝐵))
1917, 18oveq12d 6942 . . 3 (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → ((𝑃‘(𝑎𝑏)) / (𝑃𝑏)) = ((𝑃‘(𝐴𝐵)) / (𝑃𝐵)))
20 simp2 1128 . . 3 ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) → 𝐴 ∈ dom 𝑃)
21 simp3 1129 . . 3 ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) → 𝐵 ∈ dom 𝑃)
22 ovexd 6958 . . 3 ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) → ((𝑃‘(𝐴𝐵)) / (𝑃𝐵)) ∈ V)
2313, 19, 20, 21, 22ovmpt2d 7067 . 2 ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) → (𝐴(cprob‘𝑃)𝐵) = ((𝑃‘(𝐴𝐵)) / (𝑃𝐵)))
241, 23syl5eqr 2828 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 386  w3a 1071   = wceq 1601  wcel 2107  Vcvv 3398  cin 3791  cop 4404  dom cdm 5357  cfv 6137  (class class class)co 6924  cmpt2 6926   / cdiv 11034  Probcprb 31072  cprobccprob 31096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-1st 7447  df-2nd 7448  df-cndprob 31097
This theorem is referenced by:  cndprobin  31099  cndprob01  31100  cndprobtot  31101  cndprobnul  31102  cndprobprob  31103
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