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Theorem probmeasb 28162
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 996 . . . . . 6  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  ->  M  e.  (measures `  S ) )
2 simp2 997 . . . . . 6  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  ->  A  e.  S
)
3 measinb 27985 . . . . . 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 998 . . . . 5  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  ->  ( M `  A )  e.  RR+ )
6 measdivcstOLD 27988 . . . . 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 2468 . . . . . . . 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 3704 . . . . . . . . 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 5875 . . . . . . . 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 27961 . . . . . . . . . . 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 27928 . . . . . . . . . 10  |-  ( ( S  e.  U. ran sigAlgebra  /\  x  e.  S  /\  A  e.  S )  ->  ( x  i^i  A
)  e.  S )
1815, 12, 16, 17syl3anc 1228 . . . . . . . . 9  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  (
x  i^i  A )  e.  S )
19 measvxrge0 27969 . . . . . . . . 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 5961 . . . . . . 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 6309 . . . . . 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 11617 . . . . . . . . 9  |-  ( 0 [,] +oo )  C_  RR*
2423, 20sseldi 3507 . . . . . . . 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 11266 . . . . . . . 8  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  ( M `  A )  e.  RR )
27 0xr 9650 . . . . . . . . . 10  |-  0  e.  RR*
28 pnfxr 11331 . . . . . . . . . 10  |- +oo  e.  RR*
29 iccgelb 11591 . . . . . . . . . 10  |-  ( ( 0  e.  RR*  /\ +oo  e.  RR*  /\  ( M `
 ( x  i^i 
A ) )  e.  ( 0 [,] +oo ) )  ->  0  <_  ( M `  (
x  i^i  A )
) )
3027, 28, 29mp3an12 1314 . . . . . . . . 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 3724 . . . . . . . . . 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 27979 . . . . . . . 8  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  ( M `  ( x  i^i  A ) )  <_ 
( M `  A
) )
35 xrrege0 11385 . . . . . . . 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 1229 . . . . . . 7  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  ( M `  ( x  i^i  A ) )  e.  RR )
3725rpne0d 11271 . . . . . . 7  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  ( M `  A )  =/=  0 )
38 rexdiv 27414 . . . . . . 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 1228 . . . . . 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 2508 . . . . 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 4538 . . . 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 11293 . . . . . . . 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 2881 . . . . . . 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 5509 . . . . . . 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 5875 . . . . 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 2475 . . . 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 2569 . . 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 27966 . . 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 4260 . . . 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 5875 . . 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 2468 . . . 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 11268 . . . . 5  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  =  U. S )  -> 
( M `  A
)  e.  CC )
565rpne0d 11271 . . . . . 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 3704 . . . . . . 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 3696 . . . . . . . . . 10  |-  ( U. S  i^i  A )  =  ( A  i^i  U. S )
61 elssuni 4280 . . . . . . . . . . 11  |-  ( A  e.  S  ->  A  C_ 
U. S )
62 df-ss 3495 . . . . . . . . . . 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 2520 . . . . . . . . 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 2508 . . . . . 6  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  =  U. S )  -> 
( x  i^i  A
)  =  A )
6867fveq2d 5875 . . . . 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 10378 . . . 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 27917 . . . . 5  |-  ( S  e.  U. ran sigAlgebra  ->  S  e.  (sigAlgebra `  U. S ) )
71 baselsiga 27908 . . . . 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 9605 . . . . 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 5961 . . 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 2508 . 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 28141 . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817    i^i cin 3480    C_ wss 3481   U.cuni 4250   class class class wbr 4452    |-> cmpt 4510   dom cdm 5004   ran crn 5005   ` cfv 5593  (class class class)co 6294   RRcr 9501   0cc0 9502   1c1 9503   +oocpnf 9635   RR*cxr 9637    <_ cle 9639    / cdiv 10216   RR+crp 11230   [,]cicc 11542   /𝑒 cxdiv 27405  sigAlgebracsiga 27900  measurescmeas 27959  Probcprb 28139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-inf2 8068  ax-ac2 8853  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579  ax-pre-sup 9580  ax-addf 9581  ax-mulf 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-int 4288  df-iun 4332  df-iin 4333  df-disj 4423  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-se 4844  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-isom 5602  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-of 6534  df-om 6695  df-1st 6794  df-2nd 6795  df-supp 6912  df-recs 7052  df-rdg 7086  df-1o 7140  df-2o 7141  df-oadd 7144  df-er 7321  df-map 7432  df-pm 7433  df-ixp 7480  df-en 7527  df-dom 7528  df-sdom 7529  df-fin 7530  df-fsupp 7840  df-fi 7881  df-sup 7911  df-oi 7945  df-card 8330  df-acn 8333  df-ac 8507  df-cda 8558  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-div 10217  df-nn 10547  df-2 10604  df-3 10605  df-4 10606  df-5 10607  df-6 10608  df-7 10609  df-8 10610  df-9 10611  df-10 10612  df-n0 10806  df-z 10875  df-dec 10987  df-uz 11093  df-q 11193  df-rp 11231  df-xneg 11328  df-xadd 11329  df-xmul 11330  df-ioo 11543  df-ioc 11544  df-ico 11545  df-icc 11546  df-fz 11683  df-fzo 11803  df-fl 11907  df-mod 11975  df-seq 12086  df-exp 12145  df-fac 12332  df-bc 12359  df-hash 12384  df-shft 12875  df-cj 12907  df-re 12908  df-im 12909  df-sqrt 13043  df-abs 13044  df-limsup 13269  df-clim 13286  df-rlim 13287  df-sum 13484  df-ef 13677  df-sin 13679  df-cos 13680  df-pi 13682  df-struct 14504  df-ndx 14505  df-slot 14506  df-base 14507  df-sets 14508  df-ress 14509  df-plusg 14580  df-mulr 14581  df-starv 14582  df-sca 14583  df-vsca 14584  df-ip 14585  df-tset 14586  df-ple 14587  df-ds 14589  df-unif 14590  df-hom 14591  df-cco 14592  df-rest 14690  df-topn 14691  df-0g 14709  df-gsum 14710  df-topgen 14711  df-pt 14712  df-prds 14715  df-ordt 14768  df-xrs 14769  df-qtop 14774  df-imas 14775  df-xps 14777  df-mre 14853  df-mrc 14854  df-acs 14856  df-ps 15699  df-tsr 15700  df-plusf 15740  df-mgm 15741  df-sgrp 15764  df-mnd 15774  df-mhm 15819  df-submnd 15820  df-grp 15906  df-minusg 15907  df-sbg 15908  df-mulg 15909  df-subg 16047  df-cntz 16204  df-cmn 16650  df-abl 16651  df-mgp 16991  df-ur 17003  df-ring 17049  df-cring 17050  df-subrg 17275  df-abv 17314  df-lmod 17362  df-scaf 17363  df-sra 17666  df-rgmod 17667  df-psmet 18258  df-xmet 18259  df-met 18260  df-bl 18261  df-mopn 18262  df-fbas 18263  df-fg 18264  df-cnfld 18268  df-top 19245  df-bases 19247  df-topon 19248  df-topsp 19249  df-cld 19365  df-ntr 19366  df-cls 19367  df-nei 19444  df-lp 19482  df-perf 19483  df-cn 19573  df-cnp 19574  df-haus 19661  df-tx 19908  df-hmeo 20101  df-fil 20192  df-fm 20284  df-flim 20285  df-flf 20286  df-tmd 20416  df-tgp 20417  df-tsms 20470  df-trg 20507  df-xms 20668  df-ms 20669  df-tms 20670  df-nm 20948  df-ngp 20949  df-nrg 20951  df-nlm 20952  df-ii 21226  df-cncf 21227  df-limc 22115  df-dv 22116  df-log 22787  df-xdiv 27406  df-esum 27834  df-siga 27901  df-meas 27960  df-prob 28140
This theorem is referenced by:  cndprobprob  28170
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