Theorem cndprobval 30495

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.

Step	Hyp	Ref	Expression
1		df-ov 6653	. 2 |- ( A (cprob ` P ) B ) = ( (cprob ` P ) ` <. A , B >. )
2		df-cndprob 30494	. . . . . 6 |- cprob = ( p e. Prob |-> ( a e. dom p , b e. dom p |-> ( ( p ` ( a i^i b ) ) / ( p ` b ) ) ) )
3	2	a1i 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 5324	. . . . . . 7 |- ( p = P -> dom p = dom P )
5		fveq1 6190	. . . . . . . 8 |- ( p = P -> ( p ` ( a i^i b ) ) = ( P ` ( a i^i b ) ) )
6		fveq1 6190	. . . . . . . 8 |- ( p = P -> ( p ` b ) = ( P ` b ) )
7	5, 6	oveq12d 6668	. . . . . . 7 |- ( p = P -> ( ( p ` ( a i^i b ) ) / ( p ` b ) ) = ( ( P ` ( a i^i b ) ) / ( P ` b ) ) )
8	4, 4, 7	mpt2eq123dv 6717	. . . . . 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 ) ) ) )
9	8	adantl 482	. . . . 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 7097	. . . . . 6 |- ( P e. Prob -> dom P e. _V )
12		mpt2exga 7246	. . . . . 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 )
13	11, 11, 12	syl2anc 693	. . . . 5 |- ( P e. Prob -> ( a e. dom P , b e. dom P |-> ( ( P ` ( a i^i b ) ) / ( P ` b ) ) ) e. _V )
14	3, 9, 10, 13	fvmptd 6288	. . . 4 |- ( P e. Prob -> (cprob ` P ) = ( a e. dom P , b e. dom P |-> ( ( P ` ( a i^i b ) ) / ( P ` b ) ) ) )
15	14	3ad2ant1 1082	. . 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 794	. . . . . 6 |- ( ( ( P e. Prob /\ A e. dom P /\ B e. dom P ) /\ ( a = A /\ b = B ) ) -> a = A )
17		simprr 796	. . . . . 6 |- ( ( ( P e. Prob /\ A e. dom P /\ B e. dom P ) /\ ( a = A /\ b = B ) ) -> b = B )
18	16, 17	ineq12d 3815	. . . . 5 |- ( ( ( P e. Prob /\ A e. dom P /\ B e. dom P ) /\ ( a = A /\ b = B ) ) -> ( a i^i b ) = ( A i^i B ) )
19	18	fveq2d 6195	. . . 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 ) ) )
20	17	fveq2d 6195	. . . 4 |- ( ( ( P e. Prob /\ A e. dom P /\ B e. dom P ) /\ ( a = A /\ b = B ) ) -> ( P ` b ) = ( P ` B ) )
21	19, 20	oveq12d 6668	. . 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 1062	. . 3 |- ( ( P e. Prob /\ A e. dom P /\ B e. dom P ) -> A e. dom P )
23		simp3 1063	. . 3 |- ( ( P e. Prob /\ A e. dom P /\ B e. dom P ) -> B e. dom P )
24		ovexd 6680	. . 3 |- ( ( P e. Prob /\ A e. dom P /\ B e. dom P ) -> ( ( P ` ( A i^i B ) ) / ( P ` B ) ) e. _V )
25	15, 21, 22, 23, 24	ovmpt2d 6788	. 2 |- ( ( P e. Prob /\ A e. dom P /\ B e. dom P ) -> ( A (cprob ` P ) B ) = ( ( P ` ( A i^i B ) ) / ( P ` B ) ) )
26	1, 25	syl5eqr 2670	1 |- ( ( P e. Prob /\ A e. dom P /\ B e. dom P ) -> ( (cprob ` P ) ` <. A , B >. ) = ( ( P ` ( A i^i B ) ) / ( P ` B ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   <.cop 4183   |-> cmpt 4729   dom cdm 5114   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   / cdiv 10684  Probcprb 30469  cprobccprob 30493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-cndprob 30494

This theorem is referenced by:  cndprobin  30496  cndprob01  30497  cndprobtot  30498  cndprobnul  30499  cndprobprob  30500