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Theorem dstrvprob 24682
Description: The distribution of a random variable is a probability law (TODO: could be shortened using dstrvval 24681) (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 24680 . . . . . . . 8  |-  ( (
ph  /\  a  e. 𝔅 )  -> 
( XRV/𝑐  _E  a )  e.  dom  P )
8 prob01 24624 . . . . . . . 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 24248 . . . . . . . . 9  |-  ( ( P `  ( XRV/𝑐  _E  a ) )  e.  ( 0 [,] 1
)  ->  ( P `  ( XRV/𝑐  _E  a ) )  e.  RR )
1110rexrd 9090 . . . . . . . 8  |-  ( ( P `  ( XRV/𝑐  _E  a ) )  e.  ( 0 [,] 1
)  ->  ( P `  ( XRV/𝑐  _E  a ) )  e. 
RR* )
12 elunitge0 24250 . . . . . . . 8  |-  ( ( P `  ( XRV/𝑐  _E  a ) )  e.  ( 0 [,] 1
)  ->  0  <_  ( P `  ( XRV/𝑐  _E  a ) ) )
13 elxrge0 10964 . . . . . . . 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 24023 . . . . 5  |-  ( ph  ->  D :𝔅 --> ( 0 [,]  +oo ) )
17 simpr 448 . . . . . . . . 9  |-  ( (
ph  /\  a  =  (/) )  ->  a  =  (/) )
1817oveq2d 6056 . . . . . . . 8  |-  ( (
ph  /\  a  =  (/) )  ->  ( XRV/𝑐  _E  a
)  =  ( XRV/𝑐  _E  (/) ) )
1918fveq2d 5691 . . . . . . 7  |-  ( (
ph  /\  a  =  (/) )  ->  ( P `  ( XRV/𝑐  _E  a ) )  =  ( P `  ( XRV/𝑐  _E  (/) ) ) )
20 brsigarn 24491 . . . . . . . . 9  |- 𝔅  e.  (sigAlgebra `  RR )
21 elrnsiga 24462 . . . . . . . . 9  |-  (𝔅  e.  (sigAlgebra `  RR )  -> 𝔅  e.  U. ran sigAlgebra )
22 0elsiga 24450 . . . . . . . . 9  |-  (𝔅  e.  U. ran sigAlgebra  ->  (/)  e. 𝔅 )
2320, 21, 22mp2b 10 . . . . . . . 8  |-  (/)  e. 𝔅
2423a1i 11 . . . . . . 7  |-  ( ph  -> 
(/)  e. 𝔅 )
252, 4, 24orvcelel 24680 . . . . . . . 8  |-  ( ph  ->  ( XRV/𝑐  _E  (/) )  e.  dom  P )
262, 25probvalrnd 24635 . . . . . . 7  |-  ( ph  ->  ( P `  ( XRV/𝑐  _E  (/) ) )  e.  RR )
271, 19, 24, 26fvmptd 5769 . . . . . 6  |-  ( ph  ->  ( D `  (/) )  =  ( P `  ( XRV/𝑐  _E  (/) ) ) )
282, 4, 24orvcelval 24679 . . . . . . 7  |-  ( ph  ->  ( XRV/𝑐  _E  (/) )  =  ( `' X " (/) ) )
2928fveq2d 5691 . . . . . 6  |-  ( ph  ->  ( P `  ( XRV/𝑐  _E  (/) ) )  =  ( P `  ( `' X " (/) ) ) )
30 ima0 5180 . . . . . . . 8  |-  ( `' X " (/) )  =  (/)
3130fveq2i 5690 . . . . . . 7  |-  ( P `
 ( `' X "
(/) ) )  =  ( P `  (/) )
32 probnul 24625 . . . . . . . 8  |-  ( P  e. Prob  ->  ( P `  (/) )  =  0 )
332, 32syl 16 . . . . . . 7  |-  ( ph  ->  ( P `  (/) )  =  0 )
3431, 33syl5eq 2448 . . . . . 6  |-  ( ph  ->  ( P `  ( `' X " (/) ) )  =  0 )
3527, 29, 343eqtrd 2440 . . . . 5  |-  ( ph  ->  ( D `  (/) )  =  0 )
362, 4rrvvf 24655 . . . . . . . . . . . 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 5552 . . . . . . . . . . 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 24009 . . . . . . . . . . 11  |-  ( Fun 
X  ->  ( `' X " U. x )  =  U_ a  e.  x  ( `' X " a ) )
4140fveq2d 5691 . . . . . . . . . 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 24621 . . . . . . . . . . 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 1626 . . . . . . . . . . . 12  |-  F/ a ( ph  /\  x  e.  ~P𝔅
)
47 nfv 1626 . . . . . . . . . . . . 13  |-  F/ a  x  ~<_  om
48 nfdisj1 4155 . . . . . . . . . . . . 13  |-  F/ aDisj  a  e.  x a
4947, 48nfan 1842 . . . . . . . . . . . 12  |-  F/ a ( x  ~<_  om  /\ Disj  a  e.  x a )
5046, 49nfan 1842 . . . . . . . . . . 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 3769 . . . . . . . . . . . . . 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 24661 . . . . . . . . . . . . . 14  |-  ( ph  ->  A. a  e. 𝔅  ( `' X "
a )  e.  dom  P )
5756r19.21bi 2764 . . . . . . . . . . . . 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 2747 . . . . . . . . . 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 23979 . . . . . . . . . . 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 24521 . . . . . . . . . 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 2436 . . . . . . . 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 24453 . . . . . . . . . 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 24681 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ~P𝔅
)  /\  ( x  ~<_  om  /\ Disj  a  e.  x
a ) )  -> 
( D `  U. x )  =  ( P `  ( `' X " U. x
) ) )
751, 9fvmpt2d 5773 . . . . . . . . . . . . 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 24679 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  x  e.  ~P𝔅
)  /\  ( x  ~<_  om  /\ Disj  a  e.  x
a ) )  /\  a  e.  x )  ->  ( XRV/𝑐  _E  a )  =  ( `' X " a ) )
8079fveq2d 5691 . . . . . . . . . . . 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 2436 . . . . . . . . . . 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 2747 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  ~P𝔅
)  /\  ( x  ~<_  om  /\ Disj  a  e.  x
a ) )  ->  A. a  e.  x  ( D `  a )  =  ( P `  ( `' X " a ) ) )
8450, 83esumeq2d 24387 . . . . . . . 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 2446 . . . . . . 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 2749 . . . . 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 24506 . . . . . 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 5031 . . . . . 6  |-  ( ph  ->  dom  D  =  dom  ( a  e. 𝔅 
|->  ( P `  ( XRV/𝑐  _E  a ) ) ) )
9215ralrimiva 2749 . . . . . . 7  |-  ( ph  ->  A. a  e. 𝔅  ( P `  ( XRV/𝑐  _E  a ) )  e.  ( 0 [,]  +oo ) )
93 dmmptg 5326 . . . . . . 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 2436 . . . . 5  |-  ( ph  ->  dom  D  = 𝔅 )
9695fveq2d 5691 . . . 4  |-  ( ph  ->  (measures `  dom  D )  =  (measures ` 𝔅 ) )
9790, 96eleqtrrd 2481 . . 3  |-  ( ph  ->  D  e.  (measures `  dom  D ) )
98 measbasedom 24509 . . 3  |-  ( D  e.  U. ran measures  <->  D  e.  (measures `  dom  D ) )
9997, 98sylibr 204 . 2  |-  ( ph  ->  D  e.  U. ran measures )
10095unieqd 3986 . . . . 5  |-  ( ph  ->  U. dom  D  = 
U.𝔅 )
101 unibrsiga 24493 . . . . 5  |-  U.𝔅  =  RR
102100, 101syl6eq 2452 . . . 4  |-  ( ph  ->  U. dom  D  =  RR )
103102fveq2d 5691 . . 3  |-  ( ph  ->  ( D `  U. dom  D )  =  ( D `  RR ) )
104 simpr 448 . . . . . . . 8  |-  ( (
ph  /\  a  =  RR )  ->  a  =  RR )
105104oveq2d 6056 . . . . . . 7  |-  ( (
ph  /\  a  =  RR )  ->  ( XRV/𝑐  _E  a )  =  ( XRV/𝑐  _E  RR ) )
106 baselsiga 24451 . . . . . . . . . 10  |-  (𝔅  e.  (sigAlgebra `  RR )  ->  RR  e. 𝔅 )
10720, 106mp1i 12 . . . . . . . . 9  |-  ( ph  ->  RR  e. 𝔅 )
1082, 4, 107orvcelval 24679 . . . . . . . 8  |-  ( ph  ->  ( XRV/𝑐  _E  RR )  =  ( `' X " RR ) )
109108adantr 452 . . . . . . 7  |-  ( (
ph  /\  a  =  RR )  ->  ( XRV/𝑐  _E  RR )  =  ( `' X " RR ) )
110105, 109eqtrd 2436 . . . . . 6  |-  ( (
ph  /\  a  =  RR )  ->  ( XRV/𝑐  _E  a )  =  ( `' X " RR ) )
111110fveq2d 5691 . . . . 5  |-  ( (
ph  /\  a  =  RR )  ->  ( P `
 ( XRV/𝑐  _E  a ) )  =  ( P `  ( `' X " RR ) ) )
112 fimacnv 5821 . . . . . . . . 9  |-  ( X : U. dom  P --> RR  ->  ( `' X " RR )  =  U. dom  P )
11336, 112syl 16 . . . . . . . 8  |-  ( ph  ->  ( `' X " RR )  =  U. dom  P )
114113fveq2d 5691 . . . . . . 7  |-  ( ph  ->  ( P `  ( `' X " RR ) )  =  ( P `
 U. dom  P
) )
115 probtot 24623 . . . . . . . 8  |-  ( P  e. Prob  ->  ( P `  U. dom  P )  =  1 )
1162, 115syl 16 . . . . . . 7  |-  ( ph  ->  ( P `  U. dom  P )  =  1 )
117114, 116eqtrd 2436 . . . . . 6  |-  ( ph  ->  ( P `  ( `' X " RR ) )  =  1 )
118117adantr 452 . . . . 5  |-  ( (
ph  /\  a  =  RR )  ->  ( P `
 ( `' X " RR ) )  =  1 )
119111, 118eqtrd 2436 . . . 4  |-  ( (
ph  /\  a  =  RR )  ->  ( P `
 ( XRV/𝑐  _E  a ) )  =  1 )
120 1re 9046 . . . . 5  |-  1  e.  RR
121120a1i 11 . . . 4  |-  ( ph  ->  1  e.  RR )
1221, 119, 107, 121fvmptd 5769 . . 3  |-  ( ph  ->  ( D `  RR )  =  1 )
123103, 122eqtrd 2436 . 2  |-  ( ph  ->  ( D `  U. dom  D )  =  1 )
124 elprob 24620 . 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 1649    e. wcel 1721   A.wral 2666   (/)c0 3588   ~Pcpw 3759   U.cuni 3975   U_ciun 4053  Disj wdisj 4142   class class class wbr 4172    e. cmpt 4226    _E cep 4452   omcom 4804   `'ccnv 4836   dom cdm 4837   ran crn 4838   "cima 4840   Fun wfun 5407   -->wf 5409   ` cfv 5413  (class class class)co 6040    ~<_ cdom 7066   RRcr 8945   0cc0 8946   1c1 8947    +oocpnf 9073   RR*cxr 9075    <_ cle 9077   [,]cicc 10875  Σ*cesum 24377  sigAlgebracsiga 24443  𝔅cbrsiga 24488  measurescmeas 24502  Probcprb 24618  rRndVarcrrv 24651  ∘RV/𝑐corvc 24666
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-ac2 8299  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-disj 4143  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-acn 7785  df-ac 7953  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435  df-ef 12625  df-sin 12627  df-cos 12628  df-pi 12630  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-ordt 13680  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-ps 14584  df-tsr 14585  df-mnd 14645  df-plusf 14646  df-mhm 14693  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-mulg 14770  df-subg 14896  df-cntz 15071  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-cring 15619  df-ur 15620  df-subrg 15821  df-abv 15860  df-lmod 15907  df-scaf 15908  df-sra 16199  df-rgmod 16200  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-tmd 18055  df-tgp 18056  df-tsms 18109  df-trg 18142  df-xms 18303  df-ms 18304  df-tms 18305  df-nm 18583  df-ngp 18584  df-nrg 18586  df-nlm 18587  df-ii 18860  df-cncf 18861  df-limc 19706  df-dv 19707  df-log 20407  df-esum 24378  df-siga 24444  df-sigagen 24475  df-brsiga 24489  df-meas 24503  df-mbfm 24554  df-prob 24619  df-rrv 24652  df-orvc 24667
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