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Theorem probmeasb 26825
Description: Build a probability from a measure and a set with finite measure (Contributed by Thierry Arnoux, 25-Dec-2016.)
Assertion
Ref Expression
probmeasb  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  ->  ( x  e.  S  |->  ( ( M `
 ( x  i^i 
A ) )  / 
( M `  A
) ) )  e. Prob
)
Distinct variable groups:    x, A    x, M    x, S

Proof of Theorem probmeasb
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 simp1 988 . . . . . 6  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  ->  M  e.  (measures `  S ) )
2 simp2 989 . . . . . 6  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  ->  A  e.  S
)
3 measinb 26647 . . . . . 6  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S )  ->  (
y  e.  S  |->  ( M `  ( y  i^i  A ) ) )  e.  (measures `  S
) )
41, 2, 3syl2anc 661 . . . . 5  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  ->  ( y  e.  S  |->  ( M `  ( y  i^i  A
) ) )  e.  (measures `  S )
)
5 simp3 990 . . . . 5  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  ->  ( M `  A )  e.  RR+ )
6 measdivcstOLD 26650 . . . . 5  |-  ( ( ( y  e.  S  |->  ( M `  (
y  i^i  A )
) )  e.  (measures `  S )  /\  ( M `  A )  e.  RR+ )  ->  (
x  e.  S  |->  ( ( ( y  e.  S  |->  ( M `  ( y  i^i  A
) ) ) `  x ) /𝑒  ( M `  A
) ) )  e.  (measures `  S )
)
74, 5, 6syl2anc 661 . . . 4  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  ->  ( x  e.  S  |->  ( ( ( y  e.  S  |->  ( M `  ( y  i^i  A ) ) ) `  x ) /𝑒  ( M `  A ) ) )  e.  (measures `  S ) )
8 eqidd 2444 . . . . . . . 8  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  (
y  e.  S  |->  ( M `  ( y  i^i  A ) ) )  =  ( y  e.  S  |->  ( M `
 ( y  i^i 
A ) ) ) )
9 simpr 461 . . . . . . . . . 10  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A
)  e.  RR+ )  /\  x  e.  S
)  /\  y  =  x )  ->  y  =  x )
109ineq1d 3563 . . . . . . . . 9  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A
)  e.  RR+ )  /\  x  e.  S
)  /\  y  =  x )  ->  (
y  i^i  A )  =  ( x  i^i 
A ) )
1110fveq2d 5707 . . . . . . . 8  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A
)  e.  RR+ )  /\  x  e.  S
)  /\  y  =  x )  ->  ( M `  ( y  i^i  A ) )  =  ( M `  (
x  i^i  A )
) )
12 simpr 461 . . . . . . . 8  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  x  e.  S )
131adantr 465 . . . . . . . . 9  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  M  e.  (measures `  S )
)
14 measbase 26623 . . . . . . . . . . 11  |-  ( M  e.  (measures `  S
)  ->  S  e.  U.
ran sigAlgebra )
1513, 14syl 16 . . . . . . . . . 10  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  S  e.  U. ran sigAlgebra )
162adantr 465 . . . . . . . . . 10  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  A  e.  S )
17 inelsiga 26590 . . . . . . . . . 10  |-  ( ( S  e.  U. ran sigAlgebra  /\  x  e.  S  /\  A  e.  S )  ->  ( x  i^i  A
)  e.  S )
1815, 12, 16, 17syl3anc 1218 . . . . . . . . 9  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  (
x  i^i  A )  e.  S )
19 measvxrge0 26631 . . . . . . . . 9  |-  ( ( M  e.  (measures `  S
)  /\  ( x  i^i  A )  e.  S
)  ->  ( M `  ( x  i^i  A
) )  e.  ( 0 [,] +oo )
)
2013, 18, 19syl2anc 661 . . . . . . . 8  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  ( M `  ( x  i^i  A ) )  e.  ( 0 [,] +oo ) )
218, 11, 12, 20fvmptd 5791 . . . . . . 7  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  (
( y  e.  S  |->  ( M `  (
y  i^i  A )
) ) `  x
)  =  ( M `
 ( x  i^i 
A ) ) )
2221oveq1d 6118 . . . . . 6  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  (
( ( y  e.  S  |->  ( M `  ( y  i^i  A
) ) ) `  x ) /𝑒  ( M `  A
) )  =  ( ( M `  (
x  i^i  A )
) /𝑒 
( M `  A
) ) )
23 iccssxr 11390 . . . . . . . . 9  |-  ( 0 [,] +oo )  C_  RR*
2423, 20sseldi 3366 . . . . . . . 8  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  ( M `  ( x  i^i  A ) )  e. 
RR* )
255adantr 465 . . . . . . . . 9  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  ( M `  A )  e.  RR+ )
2625rpred 11039 . . . . . . . 8  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  ( M `  A )  e.  RR )
27 0xr 9442 . . . . . . . . . 10  |-  0  e.  RR*
28 pnfxr 11104 . . . . . . . . . 10  |- +oo  e.  RR*
29 iccgelb 11364 . . . . . . . . . 10  |-  ( ( 0  e.  RR*  /\ +oo  e.  RR*  /\  ( M `
 ( x  i^i 
A ) )  e.  ( 0 [,] +oo ) )  ->  0  <_  ( M `  (
x  i^i  A )
) )
3027, 28, 29mp3an12 1304 . . . . . . . . 9  |-  ( ( M `  ( x  i^i  A ) )  e.  ( 0 [,] +oo )  ->  0  <_ 
( M `  (
x  i^i  A )
) )
3120, 30syl 16 . . . . . . . 8  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  0  <_  ( M `  (
x  i^i  A )
) )
32 inss2 3583 . . . . . . . . . 10  |-  ( x  i^i  A )  C_  A
3332a1i 11 . . . . . . . . 9  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  (
x  i^i  A )  C_  A )
3413, 18, 16, 33measssd 26641 . . . . . . . 8  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  ( M `  ( x  i^i  A ) )  <_ 
( M `  A
) )
35 xrrege0 11158 . . . . . . . 8  |-  ( ( ( ( M `  ( x  i^i  A ) )  e.  RR*  /\  ( M `  A )  e.  RR )  /\  (
0  <_  ( M `  ( x  i^i  A
) )  /\  ( M `  ( x  i^i  A ) )  <_ 
( M `  A
) ) )  -> 
( M `  (
x  i^i  A )
)  e.  RR )
3624, 26, 31, 34, 35syl22anc 1219 . . . . . . 7  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  ( M `  ( x  i^i  A ) )  e.  RR )
3725rpne0d 11044 . . . . . . 7  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  ( M `  A )  =/=  0 )
38 rexdiv 26113 . . . . . . 7  |-  ( ( ( M `  (
x  i^i  A )
)  e.  RR  /\  ( M `  A )  e.  RR  /\  ( M `  A )  =/=  0 )  ->  (
( M `  (
x  i^i  A )
) /𝑒 
( M `  A
) )  =  ( ( M `  (
x  i^i  A )
)  /  ( M `
 A ) ) )
3936, 26, 37, 38syl3anc 1218 . . . . . 6  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  (
( M `  (
x  i^i  A )
) /𝑒 
( M `  A
) )  =  ( ( M `  (
x  i^i  A )
)  /  ( M `
 A ) ) )
4022, 39eqtrd 2475 . . . . 5  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  (
( ( y  e.  S  |->  ( M `  ( y  i^i  A
) ) ) `  x ) /𝑒  ( M `  A
) )  =  ( ( M `  (
x  i^i  A )
)  /  ( M `
 A ) ) )
4140mpteq2dva 4390 . . . 4  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  ->  ( x  e.  S  |->  ( ( ( y  e.  S  |->  ( M `  ( y  i^i  A ) ) ) `  x ) /𝑒  ( M `  A ) ) )  =  ( x  e.  S  |->  ( ( M `  (
x  i^i  A )
)  /  ( M `
 A ) ) ) )
4236, 25rerpdivcld 11066 . . . . . . . 8  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  (
( M `  (
x  i^i  A )
)  /  ( M `
 A ) )  e.  RR )
4342ralrimiva 2811 . . . . . . 7  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  ->  A. x  e.  S  ( ( M `  ( x  i^i  A ) )  /  ( M `
 A ) )  e.  RR )
44 dmmptg 5347 . . . . . . 7  |-  ( A. x  e.  S  (
( M `  (
x  i^i  A )
)  /  ( M `
 A ) )  e.  RR  ->  dom  ( x  e.  S  |->  ( ( M `  ( x  i^i  A ) )  /  ( M `
 A ) ) )  =  S )
4543, 44syl 16 . . . . . 6  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  ->  dom  ( x  e.  S  |->  ( ( M `  ( x  i^i  A ) )  /  ( M `  A ) ) )  =  S )
4645fveq2d 5707 . . . . 5  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  ->  (measures `  dom  ( x  e.  S  |->  ( ( M `  ( x  i^i  A ) )  /  ( M `  A ) ) ) )  =  (measures `  S
) )
4746eqcomd 2448 . . . 4  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  ->  (measures `  S )  =  (measures `  dom  ( x  e.  S  |->  ( ( M `  ( x  i^i  A ) )  /  ( M `  A ) ) ) ) )
487, 41, 473eltr3d 2523 . . 3  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  ->  ( x  e.  S  |->  ( ( M `
 ( x  i^i 
A ) )  / 
( M `  A
) ) )  e.  (measures `  dom  ( x  e.  S  |->  ( ( M `  ( x  i^i  A ) )  /  ( M `  A ) ) ) ) )
49 measbasedom 26628 . . 3  |-  ( ( x  e.  S  |->  ( ( M `  (
x  i^i  A )
)  /  ( M `
 A ) ) )  e.  U. ran measures  <->  ( x  e.  S  |->  ( ( M `  ( x  i^i  A ) )  /  ( M `  A ) ) )  e.  (measures `  dom  ( x  e.  S  |->  ( ( M `  ( x  i^i  A ) )  /  ( M `
 A ) ) ) ) )
5048, 49sylibr 212 . 2  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  ->  ( x  e.  S  |->  ( ( M `
 ( x  i^i 
A ) )  / 
( M `  A
) ) )  e. 
U. ran measures )
5145unieqd 4113 . . . 4  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  ->  U. dom  ( x  e.  S  |->  ( ( M `  ( x  i^i  A ) )  /  ( M `  A ) ) )  =  U. S )
5251fveq2d 5707 . . 3  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  ->  ( ( x  e.  S  |->  ( ( M `  ( x  i^i  A ) )  /  ( M `  A ) ) ) `
 U. dom  (
x  e.  S  |->  ( ( M `  (
x  i^i  A )
)  /  ( M `
 A ) ) ) )  =  ( ( x  e.  S  |->  ( ( M `  ( x  i^i  A ) )  /  ( M `
 A ) ) ) `  U. S
) )
53 eqidd 2444 . . . 4  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  ->  ( x  e.  S  |->  ( ( M `
 ( x  i^i 
A ) )  / 
( M `  A
) ) )  =  ( x  e.  S  |->  ( ( M `  ( x  i^i  A ) )  /  ( M `
 A ) ) ) )
545adantr 465 . . . . . 6  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  =  U. S )  -> 
( M `  A
)  e.  RR+ )
5554rpcnd 11041 . . . . 5  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  =  U. S )  -> 
( M `  A
)  e.  CC )
565rpne0d 11044 . . . . . 6  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  ->  ( M `  A )  =/=  0
)
5756adantr 465 . . . . 5  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  =  U. S )  -> 
( M `  A
)  =/=  0 )
58 simpr 461 . . . . . . . 8  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  =  U. S )  ->  x  =  U. S )
5958ineq1d 3563 . . . . . . 7  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  =  U. S )  -> 
( x  i^i  A
)  =  ( U. S  i^i  A ) )
60 incom 3555 . . . . . . . . . 10  |-  ( U. S  i^i  A )  =  ( A  i^i  U. S )
61 elssuni 4133 . . . . . . . . . . 11  |-  ( A  e.  S  ->  A  C_ 
U. S )
62 df-ss 3354 . . . . . . . . . . 11  |-  ( A 
C_  U. S  <->  ( A  i^i  U. S )  =  A )
6361, 62sylib 196 . . . . . . . . . 10  |-  ( A  e.  S  ->  ( A  i^i  U. S )  =  A )
6460, 63syl5eq 2487 . . . . . . . . 9  |-  ( A  e.  S  ->  ( U. S  i^i  A )  =  A )
652, 64syl 16 . . . . . . . 8  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  ->  ( U. S  i^i  A )  =  A )
6665adantr 465 . . . . . . 7  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  =  U. S )  -> 
( U. S  i^i  A )  =  A )
6759, 66eqtrd 2475 . . . . . 6  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  =  U. S )  -> 
( x  i^i  A
)  =  A )
6867fveq2d 5707 . . . . 5  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  =  U. S )  -> 
( M `  (
x  i^i  A )
)  =  ( M `
 A ) )
6955, 57, 68diveq1bd 10167 . . . 4  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  =  U. S )  -> 
( ( M `  ( x  i^i  A ) )  /  ( M `
 A ) )  =  1 )
70 sgon 26579 . . . . 5  |-  ( S  e.  U. ran sigAlgebra  ->  S  e.  (sigAlgebra `  U. S ) )
71 baselsiga 26570 . . . . 5  |-  ( S  e.  (sigAlgebra `  U. S )  ->  U. S  e.  S
)
721, 14, 70, 714syl 21 . . . 4  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  ->  U. S  e.  S
)
73 1re 9397 . . . . 5  |-  1  e.  RR
7473a1i 11 . . . 4  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  ->  1  e.  RR )
7553, 69, 72, 74fvmptd 5791 . . 3  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  ->  ( ( x  e.  S  |->  ( ( M `  ( x  i^i  A ) )  /  ( M `  A ) ) ) `
 U. S )  =  1 )
7652, 75eqtrd 2475 . 2  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  ->  ( ( x  e.  S  |->  ( ( M `  ( x  i^i  A ) )  /  ( M `  A ) ) ) `
 U. dom  (
x  e.  S  |->  ( ( M `  (
x  i^i  A )
)  /  ( M `
 A ) ) ) )  =  1 )
77 elprob 26804 . 2  |-  ( ( x  e.  S  |->  ( ( M `  (
x  i^i  A )
)  /  ( M `
 A ) ) )  e. Prob  <->  ( (
x  e.  S  |->  ( ( M `  (
x  i^i  A )
)  /  ( M `
 A ) ) )  e.  U. ran measures  /\  ( ( x  e.  S  |->  ( ( M `
 ( x  i^i 
A ) )  / 
( M `  A
) ) ) `  U. dom  ( x  e.  S  |->  ( ( M `
 ( x  i^i 
A ) )  / 
( M `  A
) ) ) )  =  1 ) )
7850, 76, 77sylanbrc 664 1  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  ->  ( x  e.  S  |->  ( ( M `
 ( x  i^i 
A ) )  / 
( M `  A
) ) )  e. Prob
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2618   A.wral 2727    i^i cin 3339    C_ wss 3340   U.cuni 4103   class class class wbr 4304    e. cmpt 4362   dom cdm 4852   ran crn 4853   ` cfv 5430  (class class class)co 6103   RRcr 9293   0cc0 9294   1c1 9295   +oocpnf 9427   RR*cxr 9429    <_ cle 9431    / cdiv 10005   RR+crp 11003   [,]cicc 11315   /𝑒 cxdiv 26104  sigAlgebracsiga 26562  measurescmeas 26621  Probcprb 26802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-inf2 7859  ax-ac2 8644  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-pre-sup 9372  ax-addf 9373  ax-mulf 9374
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-iin 4186  df-disj 4275  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-se 4692  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-isom 5439  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-of 6332  df-om 6489  df-1st 6589  df-2nd 6590  df-supp 6703  df-recs 6844  df-rdg 6878  df-1o 6932  df-2o 6933  df-oadd 6936  df-er 7113  df-map 7228  df-pm 7229  df-ixp 7276  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-fsupp 7633  df-fi 7673  df-sup 7703  df-oi 7736  df-card 8121  df-acn 8124  df-ac 8298  df-cda 8349  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-div 10006  df-nn 10335  df-2 10392  df-3 10393  df-4 10394  df-5 10395  df-6 10396  df-7 10397  df-8 10398  df-9 10399  df-10 10400  df-n0 10592  df-z 10659  df-dec 10768  df-uz 10874  df-q 10966  df-rp 11004  df-xneg 11101  df-xadd 11102  df-xmul 11103  df-ioo 11316  df-ioc 11317  df-ico 11318  df-icc 11319  df-fz 11450  df-fzo 11561  df-fl 11654  df-mod 11721  df-seq 11819  df-exp 11878  df-fac 12064  df-bc 12091  df-hash 12116  df-shft 12568  df-cj 12600  df-re 12601  df-im 12602  df-sqr 12736  df-abs 12737  df-limsup 12961  df-clim 12978  df-rlim 12979  df-sum 13176  df-ef 13365  df-sin 13367  df-cos 13368  df-pi 13370  df-struct 14188  df-ndx 14189  df-slot 14190  df-base 14191  df-sets 14192  df-ress 14193  df-plusg 14263  df-mulr 14264  df-starv 14265  df-sca 14266  df-vsca 14267  df-ip 14268  df-tset 14269  df-ple 14270  df-ds 14272  df-unif 14273  df-hom 14274  df-cco 14275  df-rest 14373  df-topn 14374  df-0g 14392  df-gsum 14393  df-topgen 14394  df-pt 14395  df-prds 14398  df-ordt 14451  df-xrs 14452  df-qtop 14457  df-imas 14458  df-xps 14460  df-mre 14536  df-mrc 14537  df-acs 14539  df-ps 15382  df-tsr 15383  df-mnd 15427  df-plusf 15428  df-mhm 15476  df-submnd 15477  df-grp 15557  df-minusg 15558  df-sbg 15559  df-mulg 15560  df-subg 15690  df-cntz 15847  df-cmn 16291  df-abl 16292  df-mgp 16604  df-ur 16616  df-rng 16659  df-cring 16660  df-subrg 16875  df-abv 16914  df-lmod 16962  df-scaf 16963  df-sra 17265  df-rgmod 17266  df-psmet 17821  df-xmet 17822  df-met 17823  df-bl 17824  df-mopn 17825  df-fbas 17826  df-fg 17827  df-cnfld 17831  df-top 18515  df-bases 18517  df-topon 18518  df-topsp 18519  df-cld 18635  df-ntr 18636  df-cls 18637  df-nei 18714  df-lp 18752  df-perf 18753  df-cn 18843  df-cnp 18844  df-haus 18931  df-tx 19147  df-hmeo 19340  df-fil 19431  df-fm 19523  df-flim 19524  df-flf 19525  df-tmd 19655  df-tgp 19656  df-tsms 19709  df-trg 19746  df-xms 19907  df-ms 19908  df-tms 19909  df-nm 20187  df-ngp 20188  df-nrg 20190  df-nlm 20191  df-ii 20465  df-cncf 20466  df-limc 21353  df-dv 21354  df-log 22020  df-xdiv 26105  df-esum 26496  df-siga 26563  df-meas 26622  df-prob 26803
This theorem is referenced by:  cndprobprob  26833
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