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Theorem dstrvprob 24721
Description: The distribution of a random variable is a probability law (TODO: could be shortened using dstrvval 24720) (Contributed by Thierry Arnoux, 10-Feb-2017.)
Hypotheses
Ref Expression
dstrvprob.1  |-  ( ph  ->  P  e. Prob )
dstrvprob.2  |-  ( ph  ->  X  e.  (rRndVar `  P
) )
dstrvprob.3  |-  ( ph  ->  D  =  ( a  e. 𝔅  |->  ( P `  ( XRV/𝑐  _E  a ) ) ) )
Assertion
Ref Expression
dstrvprob  |-  ( ph  ->  D  e. Prob )
Distinct variable groups:    P, a    X, a    D, a    ph, a

Proof of Theorem dstrvprob
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dstrvprob.3 . . . . . 6  |-  ( ph  ->  D  =  ( a  e. 𝔅  |->  ( P `  ( XRV/𝑐  _E  a ) ) ) )
2 dstrvprob.1 . . . . . . . . 9  |-  ( ph  ->  P  e. Prob )
32adantr 452 . . . . . . . 8  |-  ( (
ph  /\  a  e. 𝔅 )  ->  P  e. Prob )
4 dstrvprob.2 . . . . . . . . . 10  |-  ( ph  ->  X  e.  (rRndVar `  P
) )
54adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  a  e. 𝔅 )  ->  X  e.  (rRndVar `  P
) )
6 simpr 448 . . . . . . . . 9  |-  ( (
ph  /\  a  e. 𝔅 )  -> 
a  e. 𝔅 )
73, 5, 6orvcelel 24719 . . . . . . . 8  |-  ( (
ph  /\  a  e. 𝔅 )  -> 
( XRV/𝑐  _E  a )  e.  dom  P )
8 prob01 24663 . . . . . . . 8  |-  ( ( P  e. Prob  /\  ( XRV/𝑐  _E  a )  e.  dom  P )  ->  ( P `  ( XRV/𝑐  _E  a ) )  e.  ( 0 [,] 1
) )
93, 7, 8syl2anc 643 . . . . . . 7  |-  ( (
ph  /\  a  e. 𝔅 )  -> 
( P `  ( XRV/𝑐  _E  a ) )  e.  ( 0 [,] 1
) )
10 elunitrn 24287 . . . . . . . . 9  |-  ( ( P `  ( XRV/𝑐  _E  a ) )  e.  ( 0 [,] 1
)  ->  ( P `  ( XRV/𝑐  _E  a ) )  e.  RR )
1110rexrd 9126 . . . . . . . 8  |-  ( ( P `  ( XRV/𝑐  _E  a ) )  e.  ( 0 [,] 1
)  ->  ( P `  ( XRV/𝑐  _E  a ) )  e. 
RR* )
12 elunitge0 24289 . . . . . . . 8  |-  ( ( P `  ( XRV/𝑐  _E  a ) )  e.  ( 0 [,] 1
)  ->  0  <_  ( P `  ( XRV/𝑐  _E  a ) ) )
13 elxrge0 11000 . . . . . . . 8  |-  ( ( P `  ( XRV/𝑐  _E  a ) )  e.  ( 0 [,]  +oo ) 
<->  ( ( P `  ( XRV/𝑐  _E  a ) )  e. 
RR*  /\  0  <_  ( P `  ( XRV/𝑐  _E  a ) ) ) )
1411, 12, 13sylanbrc 646 . . . . . . 7  |-  ( ( P `  ( XRV/𝑐  _E  a ) )  e.  ( 0 [,] 1
)  ->  ( P `  ( XRV/𝑐  _E  a ) )  e.  ( 0 [,]  +oo ) )
159, 14syl 16 . . . . . 6  |-  ( (
ph  /\  a  e. 𝔅 )  -> 
( P `  ( XRV/𝑐  _E  a ) )  e.  ( 0 [,]  +oo ) )
161, 15fmpt3d 24062 . . . . 5  |-  ( ph  ->  D :𝔅 --> ( 0 [,]  +oo ) )
17 simpr 448 . . . . . . . . 9  |-  ( (
ph  /\  a  =  (/) )  ->  a  =  (/) )
1817oveq2d 6089 . . . . . . . 8  |-  ( (
ph  /\  a  =  (/) )  ->  ( XRV/𝑐  _E  a
)  =  ( XRV/𝑐  _E  (/) ) )
1918fveq2d 5724 . . . . . . 7  |-  ( (
ph  /\  a  =  (/) )  ->  ( P `  ( XRV/𝑐  _E  a ) )  =  ( P `  ( XRV/𝑐  _E  (/) ) ) )
20 brsigarn 24530 . . . . . . . . 9  |- 𝔅  e.  (sigAlgebra `  RR )
21 elrnsiga 24501 . . . . . . . . 9  |-  (𝔅  e.  (sigAlgebra `  RR )  -> 𝔅  e.  U. ran sigAlgebra )
22 0elsiga 24489 . . . . . . . . 9  |-  (𝔅  e.  U. ran sigAlgebra  ->  (/)  e. 𝔅 )
2320, 21, 22mp2b 10 . . . . . . . 8  |-  (/)  e. 𝔅
2423a1i 11 . . . . . . 7  |-  ( ph  -> 
(/)  e. 𝔅 )
252, 4, 24orvcelel 24719 . . . . . . . 8  |-  ( ph  ->  ( XRV/𝑐  _E  (/) )  e.  dom  P )
262, 25probvalrnd 24674 . . . . . . 7  |-  ( ph  ->  ( P `  ( XRV/𝑐  _E  (/) ) )  e.  RR )
271, 19, 24, 26fvmptd 5802 . . . . . 6  |-  ( ph  ->  ( D `  (/) )  =  ( P `  ( XRV/𝑐  _E  (/) ) ) )
282, 4, 24orvcelval 24718 . . . . . . 7  |-  ( ph  ->  ( XRV/𝑐  _E  (/) )  =  ( `' X " (/) ) )
2928fveq2d 5724 . . . . . 6  |-  ( ph  ->  ( P `  ( XRV/𝑐  _E  (/) ) )  =  ( P `  ( `' X " (/) ) ) )
30 ima0 5213 . . . . . . . 8  |-  ( `' X " (/) )  =  (/)
3130fveq2i 5723 . . . . . . 7  |-  ( P `
 ( `' X "
(/) ) )  =  ( P `  (/) )
32 probnul 24664 . . . . . . . 8  |-  ( P  e. Prob  ->  ( P `  (/) )  =  0 )
332, 32syl 16 . . . . . . 7  |-  ( ph  ->  ( P `  (/) )  =  0 )
3431, 33syl5eq 2479 . . . . . 6  |-  ( ph  ->  ( P `  ( `' X " (/) ) )  =  0 )
3527, 29, 343eqtrd 2471 . . . . 5  |-  ( ph  ->  ( D `  (/) )  =  0 )
362, 4rrvvf 24694 . . . . . . . . . . . 12  |-  ( ph  ->  X : U. dom  P --> RR )
3736ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  ~P𝔅
)  /\  ( x  ~<_  om  /\ Disj  a  e.  x
a ) )  ->  X : U. dom  P --> RR )
38 ffun 5585 . . . . . . . . . . 11  |-  ( X : U. dom  P --> RR  ->  Fun  X )
3937, 38syl 16 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  ~P𝔅
)  /\  ( x  ~<_  om  /\ Disj  a  e.  x
a ) )  ->  Fun  X )
40 unipreima 24048 . . . . . . . . . . 11  |-  ( Fun 
X  ->  ( `' X " U. x )  =  U_ a  e.  x  ( `' X " a ) )
4140fveq2d 5724 . . . . . . . . . 10  |-  ( Fun 
X  ->  ( P `  ( `' X " U. x ) )  =  ( P `  U_ a  e.  x  ( `' X " a ) ) )
4239, 41syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  ~P𝔅
)  /\  ( x  ~<_  om  /\ Disj  a  e.  x
a ) )  -> 
( P `  ( `' X " U. x
) )  =  ( P `  U_ a  e.  x  ( `' X " a ) ) )
432ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  ~P𝔅
)  /\  ( x  ~<_  om  /\ Disj  a  e.  x
a ) )  ->  P  e. Prob )
44 domprobmeas 24660 . . . . . . . . . . 11  |-  ( P  e. Prob  ->  P  e.  (measures `  dom  P ) )
4543, 44syl 16 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  ~P𝔅
)  /\  ( x  ~<_  om  /\ Disj  a  e.  x
a ) )  ->  P  e.  (measures `  dom  P ) )
46 nfv 1629 . . . . . . . . . . . 12  |-  F/ a ( ph  /\  x  e.  ~P𝔅
)
47 nfv 1629 . . . . . . . . . . . . 13  |-  F/ a  x  ~<_  om
48 nfdisj1 4187 . . . . . . . . . . . . 13  |-  F/ aDisj  a  e.  x a
4947, 48nfan 1846 . . . . . . . . . . . 12  |-  F/ a ( x  ~<_  om  /\ Disj  a  e.  x a )
5046, 49nfan 1846 . . . . . . . . . . 11  |-  F/ a ( ( ph  /\  x  e.  ~P𝔅
)  /\  ( x  ~<_  om  /\ Disj  a  e.  x
a ) )
51 simplll 735 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  x  e.  ~P𝔅
)  /\  ( x  ~<_  om  /\ Disj  a  e.  x
a ) )  /\  a  e.  x )  ->  ph )
52 simpr 448 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  x  e.  ~P𝔅
)  /\  ( x  ~<_  om  /\ Disj  a  e.  x
a ) )  /\  a  e.  x )  ->  a  e.  x )
53 simpllr 736 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  x  e.  ~P𝔅
)  /\  ( x  ~<_  om  /\ Disj  a  e.  x
a ) )  /\  a  e.  x )  ->  x  e.  ~P𝔅
)
54 elelpwi 3801 . . . . . . . . . . . . . 14  |-  ( ( a  e.  x  /\  x  e.  ~P𝔅
)  ->  a  e. 𝔅 )
5552, 53, 54syl2anc 643 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  x  e.  ~P𝔅
)  /\  ( x  ~<_  om  /\ Disj  a  e.  x
a ) )  /\  a  e.  x )  ->  a  e. 𝔅 )
562, 4rrvfinvima 24700 . . . . . . . . . . . . . 14  |-  ( ph  ->  A. a  e. 𝔅  ( `' X "
a )  e.  dom  P )
5756r19.21bi 2796 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e. 𝔅 )  -> 
( `' X "
a )  e.  dom  P )
5851, 55, 57syl2anc 643 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  x  e.  ~P𝔅
)  /\  ( x  ~<_  om  /\ Disj  a  e.  x
a ) )  /\  a  e.  x )  ->  ( `' X "
a )  e.  dom  P )
5958ex 424 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  ~P𝔅
)  /\  ( x  ~<_  om  /\ Disj  a  e.  x
a ) )  -> 
( a  e.  x  ->  ( `' X "
a )  e.  dom  P ) )
6050, 59ralrimi 2779 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  ~P𝔅
)  /\  ( x  ~<_  om  /\ Disj  a  e.  x
a ) )  ->  A. a  e.  x  ( `' X " a )  e.  dom  P )
61 simprl 733 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  ~P𝔅
)  /\  ( x  ~<_  om  /\ Disj  a  e.  x
a ) )  ->  x  ~<_  om )
62 simprr 734 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  ~P𝔅
)  /\  ( x  ~<_  om  /\ Disj  a  e.  x
a ) )  -> Disj  a  e.  x a )
63 disjpreima 24018 . . . . . . . . . . 11  |-  ( ( Fun  X  /\ Disj  a  e.  x a )  -> Disj  a  e.  x ( `' X " a ) )
6439, 62, 63syl2anc 643 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  ~P𝔅
)  /\  ( x  ~<_  om  /\ Disj  a  e.  x
a ) )  -> Disj  a  e.  x ( `' X " a ) )
65 measvuni 24560 . . . . . . . . . 10  |-  ( ( P  e.  (measures `  dom  P )  /\  A. a  e.  x  ( `' X " a )  e. 
dom  P  /\  (
x  ~<_  om  /\ Disj  a  e.  x ( `' X " a ) ) )  ->  ( P `  U_ a  e.  x  ( `' X " a ) )  = Σ* a  e.  x
( P `  ( `' X " a ) ) )
6645, 60, 61, 64, 65syl112anc 1188 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  ~P𝔅
)  /\  ( x  ~<_  om  /\ Disj  a  e.  x
a ) )  -> 
( P `  U_ a  e.  x  ( `' X " a ) )  = Σ* a  e.  x ( P `  ( `' X " a ) ) )
6742, 66eqtrd 2467 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ~P𝔅
)  /\  ( x  ~<_  om  /\ Disj  a  e.  x
a ) )  -> 
( P `  ( `' X " U. x
) )  = Σ* a  e.  x ( P `  ( `' X " a ) ) )
684ad2antrr 707 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  ~P𝔅
)  /\  ( x  ~<_  om  /\ Disj  a  e.  x
a ) )  ->  X  e.  (rRndVar `  P
) )
691ad2antrr 707 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  ~P𝔅
)  /\  ( x  ~<_  om  /\ Disj  a  e.  x
a ) )  ->  D  =  ( a  e. 𝔅  |->  ( P `  ( XRV/𝑐  _E  a ) ) ) )
7020, 21mp1i 12 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  ~P𝔅
)  /\  ( x  ~<_  om  /\ Disj  a  e.  x
a ) )  -> 𝔅  e.  U. ran sigAlgebra )
71 simplr 732 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  ~P𝔅
)  /\  ( x  ~<_  om  /\ Disj  a  e.  x
a ) )  ->  x  e.  ~P𝔅
)
72 sigaclcu 24492 . . . . . . . . . 10  |-  ( (𝔅  e.  U.
ran sigAlgebra  /\  x  e.  ~P𝔅  /\  x  ~<_  om )  ->  U. x  e. 𝔅 )
7370, 71, 61, 72syl3anc 1184 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  ~P𝔅
)  /\  ( x  ~<_  om  /\ Disj  a  e.  x
a ) )  ->  U. x  e. 𝔅 )
7443, 68, 69, 73dstrvval 24720 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ~P𝔅
)  /\  ( x  ~<_  om  /\ Disj  a  e.  x
a ) )  -> 
( D `  U. x )  =  ( P `  ( `' X " U. x
) ) )
751, 9fvmpt2d 5806 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e. 𝔅 )  -> 
( D `  a
)  =  ( P `
 ( XRV/𝑐  _E  a ) ) )
7651, 55, 75syl2anc 643 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  x  e.  ~P𝔅
)  /\  ( x  ~<_  om  /\ Disj  a  e.  x
a ) )  /\  a  e.  x )  ->  ( D `  a
)  =  ( P `
 ( XRV/𝑐  _E  a ) ) )
7743adantr 452 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  x  e.  ~P𝔅
)  /\  ( x  ~<_  om  /\ Disj  a  e.  x
a ) )  /\  a  e.  x )  ->  P  e. Prob )
7868adantr 452 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  x  e.  ~P𝔅
)  /\  ( x  ~<_  om  /\ Disj  a  e.  x
a ) )  /\  a  e.  x )  ->  X  e.  (rRndVar `  P
) )
7977, 78, 55orvcelval 24718 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  x  e.  ~P𝔅
)  /\  ( x  ~<_  om  /\ Disj  a  e.  x
a ) )  /\  a  e.  x )  ->  ( XRV/𝑐  _E  a )  =  ( `' X " a ) )
8079fveq2d 5724 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  x  e.  ~P𝔅
)  /\  ( x  ~<_  om  /\ Disj  a  e.  x
a ) )  /\  a  e.  x )  ->  ( P `  ( XRV/𝑐  _E  a ) )  =  ( P `  ( `' X " a ) ) )
8176, 80eqtrd 2467 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  x  e.  ~P𝔅
)  /\  ( x  ~<_  om  /\ Disj  a  e.  x
a ) )  /\  a  e.  x )  ->  ( D `  a
)  =  ( P `
 ( `' X " a ) ) )
8281ex 424 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  ~P𝔅
)  /\  ( x  ~<_  om  /\ Disj  a  e.  x
a ) )  -> 
( a  e.  x  ->  ( D `  a
)  =  ( P `
 ( `' X " a ) ) ) )
8350, 82ralrimi 2779 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  ~P𝔅
)  /\  ( x  ~<_  om  /\ Disj  a  e.  x
a ) )  ->  A. a  e.  x  ( D `  a )  =  ( P `  ( `' X " a ) ) )
8450, 83esumeq2d 24426 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ~P𝔅
)  /\  ( x  ~<_  om  /\ Disj  a  e.  x
a ) )  -> Σ* a  e.  x ( D `  a )  = Σ* a  e.  x ( P `  ( `' X " a ) ) )
8567, 74, 843eqtr4d 2477 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ~P𝔅
)  /\  ( x  ~<_  om  /\ Disj  a  e.  x
a ) )  -> 
( D `  U. x )  = Σ* a  e.  x ( D `  a ) )
8685ex 424 . . . . . 6  |-  ( (
ph  /\  x  e.  ~P𝔅 )  ->  ( ( x  ~<_  om  /\ Disj  a  e.  x
a )  ->  ( D `  U. x )  = Σ* a  e.  x ( D `  a ) ) )
8786ralrimiva 2781 . . . . 5  |-  ( ph  ->  A. x  e.  ~P 𝔅 ( ( x  ~<_  om  /\ Disj  a  e.  x a )  -> 
( D `  U. x )  = Σ* a  e.  x ( D `  a ) ) )
88 ismeas 24545 . . . . . 6  |-  (𝔅  e.  U. ran sigAlgebra  ->  ( D  e.  (measures ` 𝔅 )  <->  ( D :𝔅 --> ( 0 [,]  +oo )  /\  ( D `  (/) )  =  0  /\  A. x  e.  ~P 𝔅 ( ( x  ~<_  om 
/\ Disj  a  e.  x a )  ->  ( D `  U. x )  = Σ* a  e.  x ( D `
 a ) ) ) ) )
8920, 21, 88mp2b 10 . . . . 5  |-  ( D  e.  (measures ` 𝔅 )  <->  ( D :𝔅 --> ( 0 [,]  +oo )  /\  ( D `  (/) )  =  0  /\  A. x  e.  ~P 𝔅 ( ( x  ~<_  om 
/\ Disj  a  e.  x a )  ->  ( D `  U. x )  = Σ* a  e.  x ( D `
 a ) ) ) )
9016, 35, 87, 89syl3anbrc 1138 . . . 4  |-  ( ph  ->  D  e.  (measures ` 𝔅 ) )
911dmeqd 5064 . . . . . 6  |-  ( ph  ->  dom  D  =  dom  ( a  e. 𝔅 
|->  ( P `  ( XRV/𝑐  _E  a ) ) ) )
9215ralrimiva 2781 . . . . . . 7  |-  ( ph  ->  A. a  e. 𝔅  ( P `  ( XRV/𝑐  _E  a ) )  e.  ( 0 [,]  +oo ) )
93 dmmptg 5359 . . . . . . 7  |-  ( A. a  e. 𝔅  ( P `  ( XRV/𝑐  _E  a ) )  e.  ( 0 [,]  +oo )  ->  dom  ( a  e. 𝔅  |->  ( P `  ( XRV/𝑐  _E  a ) ) )  = 𝔅
)
9492, 93syl 16 . . . . . 6  |-  ( ph  ->  dom  ( a  e. 𝔅  |->  ( P `
 ( XRV/𝑐  _E  a ) ) )  = 𝔅
)
9591, 94eqtrd 2467 . . . . 5  |-  ( ph  ->  dom  D  = 𝔅 )
9695fveq2d 5724 . . . 4  |-  ( ph  ->  (measures `  dom  D )  =  (measures ` 𝔅 ) )
9790, 96eleqtrrd 2512 . . 3  |-  ( ph  ->  D  e.  (measures `  dom  D ) )
98 measbasedom 24548 . . 3  |-  ( D  e.  U. ran measures  <->  D  e.  (measures `  dom  D ) )
9997, 98sylibr 204 . 2  |-  ( ph  ->  D  e.  U. ran measures )
10095unieqd 4018 . . . . 5  |-  ( ph  ->  U. dom  D  = 
U.𝔅 )
101 unibrsiga 24532 . . . . 5  |-  U.𝔅  =  RR
102100, 101syl6eq 2483 . . . 4  |-  ( ph  ->  U. dom  D  =  RR )
103102fveq2d 5724 . . 3  |-  ( ph  ->  ( D `  U. dom  D )  =  ( D `  RR ) )
104 simpr 448 . . . . . . . 8  |-  ( (
ph  /\  a  =  RR )  ->  a  =  RR )
105104oveq2d 6089 . . . . . . 7  |-  ( (
ph  /\  a  =  RR )  ->  ( XRV/𝑐  _E  a )  =  ( XRV/𝑐  _E  RR ) )
106 baselsiga 24490 . . . . . . . . . 10  |-  (𝔅  e.  (sigAlgebra `  RR )  ->  RR  e. 𝔅 )
10720, 106mp1i 12 . . . . . . . . 9  |-  ( ph  ->  RR  e. 𝔅 )
1082, 4, 107orvcelval 24718 . . . . . . . 8  |-  ( ph  ->  ( XRV/𝑐  _E  RR )  =  ( `' X " RR ) )
109108adantr 452 . . . . . . 7  |-  ( (
ph  /\  a  =  RR )  ->  ( XRV/𝑐  _E  RR )  =  ( `' X " RR ) )
110105, 109eqtrd 2467 . . . . . 6  |-  ( (
ph  /\  a  =  RR )  ->  ( XRV/𝑐  _E  a )  =  ( `' X " RR ) )
111110fveq2d 5724 . . . . 5  |-  ( (
ph  /\  a  =  RR )  ->  ( P `
 ( XRV/𝑐  _E  a ) )  =  ( P `  ( `' X " RR ) ) )
112 fimacnv 5854 . . . . . . . . 9  |-  ( X : U. dom  P --> RR  ->  ( `' X " RR )  =  U. dom  P )
11336, 112syl 16 . . . . . . . 8  |-  ( ph  ->  ( `' X " RR )  =  U. dom  P )
114113fveq2d 5724 . . . . . . 7  |-  ( ph  ->  ( P `  ( `' X " RR ) )  =  ( P `
 U. dom  P
) )
115 probtot 24662 . . . . . . . 8  |-  ( P  e. Prob  ->  ( P `  U. dom  P )  =  1 )
1162, 115syl 16 . . . . . . 7  |-  ( ph  ->  ( P `  U. dom  P )  =  1 )
117114, 116eqtrd 2467 . . . . . 6  |-  ( ph  ->  ( P `  ( `' X " RR ) )  =  1 )
118117adantr 452 . . . . 5  |-  ( (
ph  /\  a  =  RR )  ->  ( P `
 ( `' X " RR ) )  =  1 )
119111, 118eqtrd 2467 . . . 4  |-  ( (
ph  /\  a  =  RR )  ->  ( P `
 ( XRV/𝑐  _E  a ) )  =  1 )
120 1re 9082 . . . . 5  |-  1  e.  RR
121120a1i 11 . . . 4  |-  ( ph  ->  1  e.  RR )
1221, 119, 107, 121fvmptd 5802 . . 3  |-  ( ph  ->  ( D `  RR )  =  1 )
123103, 122eqtrd 2467 . 2  |-  ( ph  ->  ( D `  U. dom  D )  =  1 )
124 elprob 24659 . 2  |-  ( D  e. Prob 
<->  ( D  e.  U. ran measures 
/\  ( D `  U. dom  D )  =  1 ) )
12599, 123, 124sylanbrc 646 1  |-  ( ph  ->  D  e. Prob )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   (/)c0 3620   ~Pcpw 3791   U.cuni 4007   U_ciun 4085  Disj wdisj 4174   class class class wbr 4204    e. cmpt 4258    _E cep 4484   omcom 4837   `'ccnv 4869   dom cdm 4870   ran crn 4871   "cima 4873   Fun wfun 5440   -->wf 5442   ` cfv 5446  (class class class)co 6073    ~<_ cdom 7099   RRcr 8981   0cc0 8982   1c1 8983    +oocpnf 9109   RR*cxr 9111    <_ cle 9113   [,]cicc 10911  Σ*cesum 24416  sigAlgebracsiga 24482  𝔅cbrsiga 24527  measurescmeas 24541  Probcprb 24657  rRndVarcrrv 24690  ∘RV/𝑐corvc 24705
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-ac2 8335  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061  ax-mulf 9062
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-disj 4175  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-sup 7438  df-oi 7471  df-card 7818  df-acn 7821  df-ac 7989  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-q 10567  df-rp 10605  df-xneg 10702  df-xadd 10703  df-xmul 10704  df-ioo 10912  df-ioc 10913  df-ico 10914  df-icc 10915  df-fz 11036  df-fzo 11128  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375  df-fac 11559  df-bc 11586  df-hash 11611  df-shft 11874  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-limsup 12257  df-clim 12274  df-rlim 12275  df-sum 12472  df-ef 12662  df-sin 12664  df-cos 12665  df-pi 12667  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-starv 13536  df-sca 13537  df-vsca 13538  df-tset 13540  df-ple 13541  df-ds 13543  df-unif 13544  df-hom 13545  df-cco 13546  df-rest 13642  df-topn 13643  df-topgen 13659  df-pt 13660  df-prds 13663  df-ordt 13717  df-xrs 13718  df-0g 13719  df-gsum 13720  df-qtop 13725  df-imas 13726  df-xps 13728  df-mre 13803  df-mrc 13804  df-acs 13806  df-ps 14621  df-tsr 14622  df-mnd 14682  df-plusf 14683  df-mhm 14730  df-submnd 14731  df-grp 14804  df-minusg 14805  df-sbg 14806  df-mulg 14807  df-subg 14933  df-cntz 15108  df-cmn 15406  df-abl 15407  df-mgp 15641  df-rng 15655  df-cring 15656  df-ur 15657  df-subrg 15858  df-abv 15897  df-lmod 15944  df-scaf 15945  df-sra 16236  df-rgmod 16237  df-psmet 16686  df-xmet 16687  df-met 16688  df-bl 16689  df-mopn 16690  df-fbas 16691  df-fg 16692  df-cnfld 16696  df-top 16955  df-bases 16957  df-topon 16958  df-topsp 16959  df-cld 17075  df-ntr 17076  df-cls 17077  df-nei 17154  df-lp 17192  df-perf 17193  df-cn 17283  df-cnp 17284  df-haus 17371  df-tx 17586  df-hmeo 17779  df-fil 17870  df-fm 17962  df-flim 17963  df-flf 17964  df-tmd 18094  df-tgp 18095  df-tsms 18148  df-trg 18181  df-xms 18342  df-ms 18343  df-tms 18344  df-nm 18622  df-ngp 18623  df-nrg 18625  df-nlm 18626  df-ii 18899  df-cncf 18900  df-limc 19745  df-dv 19746  df-log 20446  df-esum 24417  df-siga 24483  df-sigagen 24514  df-brsiga 24528  df-meas 24542  df-mbfm 24593  df-prob 24658  df-rrv 24691  df-orvc 24706
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