Theorem cndprobval 34617

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 7359	. 2 ⊢ (𝐴(cprob‘𝑃)𝐵) = ((cprob‘𝑃)‘⟨𝐴, 𝐵⟩)
2		df-cndprob 34616	. . . . 5 ⊢ cprob = (𝑝 ∈ Prob ↦ (𝑎 ∈ dom 𝑝, 𝑏 ∈ dom 𝑝 ↦ ((𝑝‘(𝑎 ∩ 𝑏)) / (𝑝‘𝑏))))
3		dmeq 5845	. . . . . 6 ⊢ (𝑝 = 𝑃 → dom 𝑝 = dom 𝑃)
4		fveq1 6826	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (𝑝‘(𝑎 ∩ 𝑏)) = (𝑃‘(𝑎 ∩ 𝑏)))
5		fveq1 6826	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (𝑝‘𝑏) = (𝑃‘𝑏))
6	4, 5	oveq12d 7374	. . . . . 6 ⊢ (𝑝 = 𝑃 → ((𝑝‘(𝑎 ∩ 𝑏)) / (𝑝‘𝑏)) = ((𝑃‘(𝑎 ∩ 𝑏)) / (𝑃‘𝑏)))
7	3, 3, 6	mpoeq123dv 7431	. . . . 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 7841	. . . . . 6 ⊢ (𝑃 ∈ Prob → dom 𝑃 ∈ V)
10		mpoexga 8019	. . . . . 6 ⊢ ((dom 𝑃 ∈ V ∧ dom 𝑃 ∈ V) → (𝑎 ∈ dom 𝑃, 𝑏 ∈ dom 𝑃 ↦ ((𝑃‘(𝑎 ∩ 𝑏)) / (𝑃‘𝑏))) ∈ V)
11	9, 9, 10	syl2anc 590	. . . . 5 ⊢ (𝑃 ∈ Prob → (𝑎 ∈ dom 𝑃, 𝑏 ∈ dom 𝑃 ↦ ((𝑃‘(𝑎 ∩ 𝑏)) / (𝑃‘𝑏))) ∈ V)
12	2, 7, 8, 11	fvmptd3 6959	. . . 4 ⊢ (𝑃 ∈ Prob → (cprob‘𝑃) = (𝑎 ∈ dom 𝑃, 𝑏 ∈ dom 𝑃 ↦ ((𝑃‘(𝑎 ∩ 𝑏)) / (𝑃‘𝑏))))
13	12	3ad2ant1 1139	. . 3 ⊢ ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) → (cprob‘𝑃) = (𝑎 ∈ dom 𝑃, 𝑏 ∈ dom 𝑃 ↦ ((𝑃‘(𝑎 ∩ 𝑏)) / (𝑃‘𝑏))))
14		simprl 776	. . . . . 6 ⊢ (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → 𝑎 = 𝐴)
15		simprr 778	. . . . . 6 ⊢ (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → 𝑏 = 𝐵)
16	14, 15	ineq12d 4150	. . . . 5 ⊢ (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → (𝑎 ∩ 𝑏) = (𝐴 ∩ 𝐵))
17	16	fveq2d 6831	. . . 4 ⊢ (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → (𝑃‘(𝑎 ∩ 𝑏)) = (𝑃‘(𝐴 ∩ 𝐵)))
18	15	fveq2d 6831	. . . 4 ⊢ (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → (𝑃‘𝑏) = (𝑃‘𝐵))
19	17, 18	oveq12d 7374	. . 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 1143	. . 3 ⊢ ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) → 𝐴 ∈ dom 𝑃)
21		simp3 1144	. . 3 ⊢ ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) → 𝐵 ∈ dom 𝑃)
22		ovexd 7391	. . 3 ⊢ ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) → ((𝑃‘(𝐴 ∩ 𝐵)) / (𝑃‘𝐵)) ∈ V)
23	13, 19, 20, 21, 22	ovmpod 7508	. 2 ⊢ ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) → (𝐴(cprob‘𝑃)𝐵) = ((𝑃‘(𝐴 ∩ 𝐵)) / (𝑃‘𝐵)))
24	1, 23	eqtr3id 2788	1 ⊢ ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) → ((cprob‘𝑃)‘⟨𝐴, 𝐵⟩) = ((𝑃‘(𝐴 ∩ 𝐵)) / (𝑃‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  Vcvv 3431   ∩ cin 3882  ⟨cop 4561  dom cdm 5618  ‘cfv 6485  (class class class)co 7356   ∈ cmpo 7358   / cdiv 11798  Probcprb 34591  cprobccprob 34615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-cndprob 34616

This theorem is referenced by:  cndprobin  34618  cndprob01  34619  cndprobtot  34620  cndprobnul  34621  cndprobprob  34622