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Theorem cndprobval 28555
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 6199 . 2  |-  ( A (cprob `  P ) B )  =  ( (cprob `  P ) `  <. A ,  B >. )
2 df-cndprob 28554 . . . . . 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 5116 . . . . . . 7  |-  ( p  =  P  ->  dom  p  =  dom  P )
5 fveq1 5773 . . . . . . . 8  |-  ( p  =  P  ->  (
p `  ( a  i^i  b ) )  =  ( P `  (
a  i^i  b )
) )
6 fveq1 5773 . . . . . . . 8  |-  ( p  =  P  ->  (
p `  b )  =  ( P `  b ) )
75, 6oveq12d 6214 . . . . . . 7  |-  ( p  =  P  ->  (
( p `  (
a  i^i  b )
)  /  ( p `
 b ) )  =  ( ( P `
 ( a  i^i  b ) )  / 
( P `  b
) ) )
84, 4, 7mpt2eq123dv 6258 . . . . . 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 464 . . . . 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 6630 . . . . . 6  |-  ( P  e. Prob  ->  dom  P  e.  _V )
12 mpt2exga 6775 . . . . . 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 659 . . . . 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 5862 . . . 4  |-  ( P  e. Prob  ->  (cprob `  P
)  =  ( a  e.  dom  P , 
b  e.  dom  P  |->  ( ( P `  ( a  i^i  b
) )  /  ( P `  b )
) ) )
15143ad2ant1 1015 . . 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 754 . . . . . 6  |-  ( ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  ( a  =  A  /\  b  =  B ) )  -> 
a  =  A )
17 simprr 755 . . . . . 6  |-  ( ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  ( a  =  A  /\  b  =  B ) )  -> 
b  =  B )
1816, 17ineq12d 3615 . . . . 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 5778 . . . 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 5778 . . . 4  |-  ( ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  ( a  =  A  /\  b  =  B ) )  -> 
( P `  b
)  =  ( P `
 B ) )
2119, 20oveq12d 6214 . . 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 995 . . 3  |-  ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  ->  A  e.  dom  P )
23 simp3 996 . . 3  |-  ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  ->  B  e.  dom  P )
24 ovex 6224 . . . 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 6329 . 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 2437 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 367    /\ w3a 971    = wceq 1399    e. wcel 1826   _Vcvv 3034    i^i cin 3388   <.cop 3950    |-> cmpt 4425   dom cdm 4913   ` cfv 5496  (class class class)co 6196    |-> cmpt2 6198    / cdiv 10123  Probcprb 28529  cprobccprob 28553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-1st 6699  df-2nd 6700  df-cndprob 28554
This theorem is referenced by:  cndprobin  28556  cndprob01  28557  cndprobtot  28558  cndprobnul  28559  cndprobprob  28560
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