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Theorem cndprobval 30273
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 6607 . 2 (𝐴(cprob‘𝑃)𝐵) = ((cprob‘𝑃)‘⟨𝐴, 𝐵⟩)
2 df-cndprob 30272 . . . . . 6 cprob = (𝑝 ∈ Prob ↦ (𝑎 ∈ dom 𝑝, 𝑏 ∈ dom 𝑝 ↦ ((𝑝‘(𝑎𝑏)) / (𝑝𝑏))))
32a1i 11 . . . . 5 (𝑃 ∈ Prob → cprob = (𝑝 ∈ Prob ↦ (𝑎 ∈ dom 𝑝, 𝑏 ∈ dom 𝑝 ↦ ((𝑝‘(𝑎𝑏)) / (𝑝𝑏)))))
4 dmeq 5284 . . . . . . 7 (𝑝 = 𝑃 → dom 𝑝 = dom 𝑃)
5 fveq1 6147 . . . . . . . 8 (𝑝 = 𝑃 → (𝑝‘(𝑎𝑏)) = (𝑃‘(𝑎𝑏)))
6 fveq1 6147 . . . . . . . 8 (𝑝 = 𝑃 → (𝑝𝑏) = (𝑃𝑏))
75, 6oveq12d 6622 . . . . . . 7 (𝑝 = 𝑃 → ((𝑝‘(𝑎𝑏)) / (𝑝𝑏)) = ((𝑃‘(𝑎𝑏)) / (𝑃𝑏)))
84, 4, 7mpt2eq123dv 6670 . . . . . 6 (𝑝 = 𝑃 → (𝑎 ∈ dom 𝑝, 𝑏 ∈ dom 𝑝 ↦ ((𝑝‘(𝑎𝑏)) / (𝑝𝑏))) = (𝑎 ∈ dom 𝑃, 𝑏 ∈ dom 𝑃 ↦ ((𝑃‘(𝑎𝑏)) / (𝑃𝑏))))
98adantl 482 . . . . 5 ((𝑃 ∈ Prob ∧ 𝑝 = 𝑃) → (𝑎 ∈ dom 𝑝, 𝑏 ∈ dom 𝑝 ↦ ((𝑝‘(𝑎𝑏)) / (𝑝𝑏))) = (𝑎 ∈ dom 𝑃, 𝑏 ∈ dom 𝑃 ↦ ((𝑃‘(𝑎𝑏)) / (𝑃𝑏))))
10 id 22 . . . . 5 (𝑃 ∈ Prob → 𝑃 ∈ Prob)
11 dmexg 7044 . . . . . 6 (𝑃 ∈ Prob → dom 𝑃 ∈ V)
12 mpt2exga 7191 . . . . . 6 ((dom 𝑃 ∈ V ∧ dom 𝑃 ∈ V) → (𝑎 ∈ dom 𝑃, 𝑏 ∈ dom 𝑃 ↦ ((𝑃‘(𝑎𝑏)) / (𝑃𝑏))) ∈ V)
1311, 11, 12syl2anc 692 . . . . 5 (𝑃 ∈ Prob → (𝑎 ∈ dom 𝑃, 𝑏 ∈ dom 𝑃 ↦ ((𝑃‘(𝑎𝑏)) / (𝑃𝑏))) ∈ V)
143, 9, 10, 13fvmptd 6245 . . . 4 (𝑃 ∈ Prob → (cprob‘𝑃) = (𝑎 ∈ dom 𝑃, 𝑏 ∈ dom 𝑃 ↦ ((𝑃‘(𝑎𝑏)) / (𝑃𝑏))))
15143ad2ant1 1080 . . 3 ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) → (cprob‘𝑃) = (𝑎 ∈ dom 𝑃, 𝑏 ∈ dom 𝑃 ↦ ((𝑃‘(𝑎𝑏)) / (𝑃𝑏))))
16 simprl 793 . . . . . 6 (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → 𝑎 = 𝐴)
17 simprr 795 . . . . . 6 (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → 𝑏 = 𝐵)
1816, 17ineq12d 3793 . . . . 5 (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → (𝑎𝑏) = (𝐴𝐵))
1918fveq2d 6152 . . . 4 (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → (𝑃‘(𝑎𝑏)) = (𝑃‘(𝐴𝐵)))
2017fveq2d 6152 . . . 4 (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → (𝑃𝑏) = (𝑃𝐵))
2119, 20oveq12d 6622 . . 3 (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → ((𝑃‘(𝑎𝑏)) / (𝑃𝑏)) = ((𝑃‘(𝐴𝐵)) / (𝑃𝐵)))
22 simp2 1060 . . 3 ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) → 𝐴 ∈ dom 𝑃)
23 simp3 1061 . . 3 ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) → 𝐵 ∈ dom 𝑃)
24 ovex 6632 . . . 4 ((𝑃‘(𝐴𝐵)) / (𝑃𝐵)) ∈ V
2524a1i 11 . . 3 ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) → ((𝑃‘(𝐴𝐵)) / (𝑃𝐵)) ∈ V)
2615, 21, 22, 23, 25ovmpt2d 6741 . 2 ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) → (𝐴(cprob‘𝑃)𝐵) = ((𝑃‘(𝐴𝐵)) / (𝑃𝐵)))
271, 26syl5eqr 2669 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 384  w3a 1036   = wceq 1480  wcel 1987  Vcvv 3186  cin 3554  cop 4154  cmpt 4673  dom cdm 5074  cfv 5847  (class class class)co 6604  cmpt2 6606   / cdiv 10628  Probcprb 30247  cprobccprob 30271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-cndprob 30272
This theorem is referenced by:  cndprobin  30274  cndprob01  30275  cndprobtot  30276  cndprobnul  30277  cndprobprob  30278
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