Theorem cndprobval 35112

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 7424	. 2 ⊢ (𝐴(cprob‘𝑃)𝐵) = ((cprob‘𝑃)‘⟨𝐴, 𝐵⟩)
2		df-cndprob 35111	. . . . 5 ⊢ cprob = (𝑝 ∈ Prob ↦ (𝑎 ∈ dom 𝑝, 𝑏 ∈ dom 𝑝 ↦ ((𝑝‘(𝑎 ∩ 𝑏)) / (𝑝‘𝑏))))
3		dmeq 5884	. . . . . 6 ⊢ (𝑝 = 𝑃 → dom 𝑝 = dom 𝑃)
4		fveq1 6885	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (𝑝‘(𝑎 ∩ 𝑏)) = (𝑃‘(𝑎 ∩ 𝑏)))
5		fveq1 6885	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (𝑝‘𝑏) = (𝑃‘𝑏))
6	4, 5	oveq12d 7439	. . . . . 6 ⊢ (𝑝 = 𝑃 → ((𝑝‘(𝑎 ∩ 𝑏)) / (𝑝‘𝑏)) = ((𝑃‘(𝑎 ∩ 𝑏)) / (𝑃‘𝑏)))
7	3, 3, 6	mpoeq123dv 7496	. . . . 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 23	. . . . 5 ⊢ (𝑃 ∈ Prob → 𝑃 ∈ Prob)
9		dmexg 7914	. . . . . 6 ⊢ (𝑃 ∈ Prob → dom 𝑃 ∈ V)
10		mpoexga 8091	. . . . . 6 ⊢ ((dom 𝑃 ∈ V ∧ dom 𝑃 ∈ V) → (𝑎 ∈ dom 𝑃, 𝑏 ∈ dom 𝑃 ↦ ((𝑃‘(𝑎 ∩ 𝑏)) / (𝑃‘𝑏))) ∈ V)
11	9, 9, 10	syl2anc 596	. . . . 5 ⊢ (𝑃 ∈ Prob → (𝑎 ∈ dom 𝑃, 𝑏 ∈ dom 𝑃 ↦ ((𝑃‘(𝑎 ∩ 𝑏)) / (𝑃‘𝑏))) ∈ V)
12	2, 7, 8, 11	fvmptd3 7018	. . . 4 ⊢ (𝑃 ∈ Prob → (cprob‘𝑃) = (𝑎 ∈ dom 𝑃, 𝑏 ∈ dom 𝑃 ↦ ((𝑃‘(𝑎 ∩ 𝑏)) / (𝑃‘𝑏))))
13	12	3ad2ant1 1151	. . 3 ⊢ ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) → (cprob‘𝑃) = (𝑎 ∈ dom 𝑃, 𝑏 ∈ dom 𝑃 ↦ ((𝑃‘(𝑎 ∩ 𝑏)) / (𝑃‘𝑏))))
14		simprl 783	. . . . . 6 ⊢ (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → 𝑎 = 𝐴)
15		simprr 785	. . . . . 6 ⊢ (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → 𝑏 = 𝐵)
16	14, 15	ineq12d 4118	. . . . 5 ⊢ (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → (𝑎 ∩ 𝑏) = (𝐴 ∩ 𝐵))
17	16	fveq2d 6890	. . . 4 ⊢ (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → (𝑃‘(𝑎 ∩ 𝑏)) = (𝑃‘(𝐴 ∩ 𝐵)))
18	15	fveq2d 6890	. . . 4 ⊢ (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → (𝑃‘𝑏) = (𝑃‘𝐵))
19	17, 18	oveq12d 7439	. . 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 1155	. . 3 ⊢ ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) → 𝐴 ∈ dom 𝑃)
21		simp3 1156	. . 3 ⊢ ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) → 𝐵 ∈ dom 𝑃)
22		ovexd 7456	. . 3 ⊢ ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) → ((𝑃‘(𝐴 ∩ 𝐵)) / (𝑃‘𝐵)) ∈ V)
23	13, 19, 20, 21, 22	ovmpod 7573	. 2 ⊢ ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) → (𝐴(cprob‘𝑃)𝐵) = ((𝑃‘(𝐴 ∩ 𝐵)) / (𝑃‘𝐵)))
24	1, 23	eqtr3id 2809	1 ⊢ ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) → ((cprob‘𝑃)‘⟨𝐴, 𝐵⟩) = ((𝑃‘(𝐴 ∩ 𝐵)) / (𝑃‘𝐵)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  Vcvv 3450   ∩ cin 4089  ⟨cop 4834  dom cdm 5650  ‘cfv 6539  (class class class)co 7421   ∈ cmpo 7423   / cdiv 11977  Probcprb 35086  cprobccprob 35110

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7424  df-oprab 7425  df-mpo 7426  df-1st 8002  df-2nd 8003  df-cndprob 35111

This theorem is used by:  cndprobin  35113  cndprob01  35114  cndprobtot  35115  cndprobnul  35116  cndprobprob  35117