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Theorem cndprobval 26955
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  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  -> 
( (cprob `  P
) `  <. A ,  B >. )  =  ( ( P `  ( A  i^i  B ) )  /  ( P `  B ) ) )

Proof of Theorem cndprobval
Dummy variables  a 
b  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ov 6198 . 2  |-  ( A (cprob `  P ) B )  =  ( (cprob `  P ) `  <. A ,  B >. )
2 df-cndprob 26954 . . . . . 6  |- cprob  =  ( p  e. Prob  |->  ( a  e.  dom  p ,  b  e.  dom  p  |->  ( ( p `  ( a  i^i  b
) )  /  (
p `  b )
) ) )
32a1i 11 . . . . 5  |-  ( P  e. Prob  -> cprob  =  ( p  e. Prob 
|->  ( a  e.  dom  p ,  b  e.  dom  p  |->  ( ( p `
 ( a  i^i  b ) )  / 
( p `  b
) ) ) ) )
4 dmeq 5143 . . . . . . 7  |-  ( p  =  P  ->  dom  p  =  dom  P )
5 fveq1 5793 . . . . . . . 8  |-  ( p  =  P  ->  (
p `  ( a  i^i  b ) )  =  ( P `  (
a  i^i  b )
) )
6 fveq1 5793 . . . . . . . 8  |-  ( p  =  P  ->  (
p `  b )  =  ( P `  b ) )
75, 6oveq12d 6213 . . . . . . 7  |-  ( p  =  P  ->  (
( p `  (
a  i^i  b )
)  /  ( p `
 b ) )  =  ( ( P `
 ( a  i^i  b ) )  / 
( P `  b
) ) )
84, 4, 7mpt2eq123dv 6252 . . . . . 6  |-  ( p  =  P  ->  (
a  e.  dom  p ,  b  e.  dom  p  |->  ( ( p `
 ( a  i^i  b ) )  / 
( p `  b
) ) )  =  ( a  e.  dom  P ,  b  e.  dom  P 
|->  ( ( P `  ( a  i^i  b
) )  /  ( P `  b )
) ) )
98adantl 466 . . . . 5  |-  ( ( P  e. Prob  /\  p  =  P )  ->  (
a  e.  dom  p ,  b  e.  dom  p  |->  ( ( p `
 ( a  i^i  b ) )  / 
( p `  b
) ) )  =  ( a  e.  dom  P ,  b  e.  dom  P 
|->  ( ( P `  ( a  i^i  b
) )  /  ( P `  b )
) ) )
10 id 22 . . . . 5  |-  ( P  e. Prob  ->  P  e. Prob )
11 dmexg 6614 . . . . . 6  |-  ( P  e. Prob  ->  dom  P  e.  _V )
12 mpt2exga 6754 . . . . . 6  |-  ( ( dom  P  e.  _V  /\ 
dom  P  e.  _V )  ->  ( a  e. 
dom  P ,  b  e.  dom  P  |->  ( ( P `  (
a  i^i  b )
)  /  ( P `
 b ) ) )  e.  _V )
1311, 11, 12syl2anc 661 . . . . 5  |-  ( P  e. Prob  ->  ( a  e. 
dom  P ,  b  e.  dom  P  |->  ( ( P `  (
a  i^i  b )
)  /  ( P `
 b ) ) )  e.  _V )
143, 9, 10, 13fvmptd 5883 . . . 4  |-  ( P  e. Prob  ->  (cprob `  P
)  =  ( a  e.  dom  P , 
b  e.  dom  P  |->  ( ( P `  ( a  i^i  b
) )  /  ( P `  b )
) ) )
15143ad2ant1 1009 . . 3  |-  ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  -> 
(cprob `  P )  =  ( a  e. 
dom  P ,  b  e.  dom  P  |->  ( ( P `  (
a  i^i  b )
)  /  ( P `
 b ) ) ) )
16 simprl 755 . . . . . 6  |-  ( ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  ( a  =  A  /\  b  =  B ) )  -> 
a  =  A )
17 simprr 756 . . . . . 6  |-  ( ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  ( a  =  A  /\  b  =  B ) )  -> 
b  =  B )
1816, 17ineq12d 3656 . . . . 5  |-  ( ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  ( a  =  A  /\  b  =  B ) )  -> 
( a  i^i  b
)  =  ( A  i^i  B ) )
1918fveq2d 5798 . . . 4  |-  ( ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  ( a  =  A  /\  b  =  B ) )  -> 
( P `  (
a  i^i  b )
)  =  ( P `
 ( A  i^i  B ) ) )
2017fveq2d 5798 . . . 4  |-  ( ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  ( a  =  A  /\  b  =  B ) )  -> 
( P `  b
)  =  ( P `
 B ) )
2119, 20oveq12d 6213 . . 3  |-  ( ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  ( a  =  A  /\  b  =  B ) )  -> 
( ( P `  ( a  i^i  b
) )  /  ( P `  b )
)  =  ( ( P `  ( A  i^i  B ) )  /  ( P `  B ) ) )
22 simp2 989 . . 3  |-  ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  ->  A  e.  dom  P )
23 simp3 990 . . 3  |-  ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  ->  B  e.  dom  P )
24 ovex 6220 . . . 4  |-  ( ( P `  ( A  i^i  B ) )  /  ( P `  B ) )  e. 
_V
2524a1i 11 . . 3  |-  ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  -> 
( ( P `  ( A  i^i  B ) )  /  ( P `
 B ) )  e.  _V )
2615, 21, 22, 23, 25ovmpt2d 6323 . 2  |-  ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  -> 
( A (cprob `  P ) B )  =  ( ( P `
 ( A  i^i  B ) )  /  ( P `  B )
) )
271, 26syl5eqr 2507 1  |-  ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  -> 
( (cprob `  P
) `  <. A ,  B >. )  =  ( ( P `  ( A  i^i  B ) )  /  ( P `  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   _Vcvv 3072    i^i cin 3430   <.cop 3986    |-> cmpt 4453   dom cdm 4943   ` cfv 5521  (class class class)co 6195    |-> cmpt2 6197    / cdiv 10099  Probcprb 26929  cprobccprob 26953
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-1st 6682  df-2nd 6683  df-cndprob 26954
This theorem is referenced by:  cndprobin  26956  cndprob01  26957  cndprobtot  26958  cndprobnul  26959  cndprobprob  26960
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