Theorem cndprobval 32112

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 𝑃 ∧ 𝐵 ∈ dom 𝑃) → ((cprob‘𝑃)‘⟨𝐴, 𝐵⟩) = ((𝑃‘(𝐴 ∩ 𝐵)) / (𝑃‘𝐵)))

Proof of Theorem cndprobval

Dummy variables 𝑎 𝑏 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ov 7216	. 2 ⊢ (𝐴(cprob‘𝑃)𝐵) = ((cprob‘𝑃)‘⟨𝐴, 𝐵⟩)
2		df-cndprob 32111	. . . . 5 ⊢ cprob = (𝑝 ∈ Prob ↦ (𝑎 ∈ dom 𝑝, 𝑏 ∈ dom 𝑝 ↦ ((𝑝‘(𝑎 ∩ 𝑏)) / (𝑝‘𝑏))))
3		dmeq 5772	. . . . . 6 ⊢ (𝑝 = 𝑃 → dom 𝑝 = dom 𝑃)
4		fveq1 6716	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (𝑝‘(𝑎 ∩ 𝑏)) = (𝑃‘(𝑎 ∩ 𝑏)))
5		fveq1 6716	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (𝑝‘𝑏) = (𝑃‘𝑏))
6	4, 5	oveq12d 7231	. . . . . 6 ⊢ (𝑝 = 𝑃 → ((𝑝‘(𝑎 ∩ 𝑏)) / (𝑝‘𝑏)) = ((𝑃‘(𝑎 ∩ 𝑏)) / (𝑃‘𝑏)))
7	3, 3, 6	mpoeq123dv 7286	. . . . 5 ⊢ (𝑝 = 𝑃 → (𝑎 ∈ dom 𝑝, 𝑏 ∈ dom 𝑝 ↦ ((𝑝‘(𝑎 ∩ 𝑏)) / (𝑝‘𝑏))) = (𝑎 ∈ dom 𝑃, 𝑏 ∈ dom 𝑃 ↦ ((𝑃‘(𝑎 ∩ 𝑏)) / (𝑃‘𝑏))))
8		id 22	. . . . 5 ⊢ (𝑃 ∈ Prob → 𝑃 ∈ Prob)
9		dmexg 7681	. . . . . 6 ⊢ (𝑃 ∈ Prob → dom 𝑃 ∈ V)
10		mpoexga 7848	. . . . . 6 ⊢ ((dom 𝑃 ∈ V ∧ dom 𝑃 ∈ V) → (𝑎 ∈ dom 𝑃, 𝑏 ∈ dom 𝑃 ↦ ((𝑃‘(𝑎 ∩ 𝑏)) / (𝑃‘𝑏))) ∈ V)
11	9, 9, 10	syl2anc 587	. . . . 5 ⊢ (𝑃 ∈ Prob → (𝑎 ∈ dom 𝑃, 𝑏 ∈ dom 𝑃 ↦ ((𝑃‘(𝑎 ∩ 𝑏)) / (𝑃‘𝑏))) ∈ V)
12	2, 7, 8, 11	fvmptd3 6841	. . . 4 ⊢ (𝑃 ∈ Prob → (cprob‘𝑃) = (𝑎 ∈ dom 𝑃, 𝑏 ∈ dom 𝑃 ↦ ((𝑃‘(𝑎 ∩ 𝑏)) / (𝑃‘𝑏))))
13	12	3ad2ant1 1135	. . 3 ⊢ ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) → (cprob‘𝑃) = (𝑎 ∈ dom 𝑃, 𝑏 ∈ dom 𝑃 ↦ ((𝑃‘(𝑎 ∩ 𝑏)) / (𝑃‘𝑏))))
14		simprl 771	. . . . . 6 ⊢ (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → 𝑎 = 𝐴)
15		simprr 773	. . . . . 6 ⊢ (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → 𝑏 = 𝐵)
16	14, 15	ineq12d 4128	. . . . 5 ⊢ (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → (𝑎 ∩ 𝑏) = (𝐴 ∩ 𝐵))
17	16	fveq2d 6721	. . . 4 ⊢ (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → (𝑃‘(𝑎 ∩ 𝑏)) = (𝑃‘(𝐴 ∩ 𝐵)))
18	15	fveq2d 6721	. . . 4 ⊢ (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → (𝑃‘𝑏) = (𝑃‘𝐵))
19	17, 18	oveq12d 7231	. . 3 ⊢ (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → ((𝑃‘(𝑎 ∩ 𝑏)) / (𝑃‘𝑏)) = ((𝑃‘(𝐴 ∩ 𝐵)) / (𝑃‘𝐵)))
20		simp2 1139	. . 3 ⊢ ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) → 𝐴 ∈ dom 𝑃)
21		simp3 1140	. . 3 ⊢ ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) → 𝐵 ∈ dom 𝑃)
22		ovexd 7248	. . 3 ⊢ ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) → ((𝑃‘(𝐴 ∩ 𝐵)) / (𝑃‘𝐵)) ∈ V)
23	13, 19, 20, 21, 22	ovmpod 7361	. 2 ⊢ ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) → (𝐴(cprob‘𝑃)𝐵) = ((𝑃‘(𝐴 ∩ 𝐵)) / (𝑃‘𝐵)))
24	1, 23	eqtr3id 2792	1 ⊢ ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) → ((cprob‘𝑃)‘⟨𝐴, 𝐵⟩) = ((𝑃‘(𝐴 ∩ 𝐵)) / (𝑃‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110  Vcvv 3408   ∩ cin 3865  ⟨cop 4547  dom cdm 5551  ‘cfv 6380  (class class class)co 7213   ∈ cmpo 7215   / cdiv 11489  Probcprb 32086  cprobccprob 32110

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-1st 7761  df-2nd 7762  df-cndprob 32111

This theorem is referenced by:  cndprobin  32113  cndprob01  32114  cndprobtot  32115  cndprobnul  32116  cndprobprob  32117