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Theorem probmeasb 26727
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 983 . . . . . 6  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  ->  M  e.  (measures `  S ) )
2 simp2 984 . . . . . 6  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  ->  A  e.  S
)
3 measinb 26555 . . . . . 6  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S )  ->  (
y  e.  S  |->  ( M `  ( y  i^i  A ) ) )  e.  (measures `  S
) )
41, 2, 3syl2anc 656 . . . . 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 985 . . . . 5  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  ->  ( M `  A )  e.  RR+ )
6 measdivcstOLD 26558 . . . . 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 656 . . . 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 2442 . . . . . . . 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 458 . . . . . . . . . 10  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A
)  e.  RR+ )  /\  x  e.  S
)  /\  y  =  x )  ->  y  =  x )
109ineq1d 3548 . . . . . . . . 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 5692 . . . . . . . 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 458 . . . . . . . 8  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  x  e.  S )
131adantr 462 . . . . . . . . 9  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  M  e.  (measures `  S )
)
14 measbase 26531 . . . . . . . . . . 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 462 . . . . . . . . . 10  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  A  e.  S )
17 inelsiga 26498 . . . . . . . . . 10  |-  ( ( S  e.  U. ran sigAlgebra  /\  x  e.  S  /\  A  e.  S )  ->  ( x  i^i  A
)  e.  S )
1815, 12, 16, 17syl3anc 1213 . . . . . . . . 9  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  (
x  i^i  A )  e.  S )
19 measvxrge0 26539 . . . . . . . . 9  |-  ( ( M  e.  (measures `  S
)  /\  ( x  i^i  A )  e.  S
)  ->  ( M `  ( x  i^i  A
) )  e.  ( 0 [,] +oo )
)
2013, 18, 19syl2anc 656 . . . . . . . 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 5776 . . . . . . 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 6105 . . . . . 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 11374 . . . . . . . . 9  |-  ( 0 [,] +oo )  C_  RR*
2423, 20sseldi 3351 . . . . . . . 8  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  ( M `  ( x  i^i  A ) )  e. 
RR* )
255adantr 462 . . . . . . . . 9  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  ( M `  A )  e.  RR+ )
2625rpred 11023 . . . . . . . 8  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  ( M `  A )  e.  RR )
27 0xr 9426 . . . . . . . . . 10  |-  0  e.  RR*
28 pnfxr 11088 . . . . . . . . . 10  |- +oo  e.  RR*
29 iccgelb 11348 . . . . . . . . . 10  |-  ( ( 0  e.  RR*  /\ +oo  e.  RR*  /\  ( M `
 ( x  i^i 
A ) )  e.  ( 0 [,] +oo ) )  ->  0  <_  ( M `  (
x  i^i  A )
) )
3027, 28, 29mp3an12 1299 . . . . . . . . 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 3568 . . . . . . . . . 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 26549 . . . . . . . 8  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  ( M `  ( x  i^i  A ) )  <_ 
( M `  A
) )
35 xrrege0 11142 . . . . . . . 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 1214 . . . . . . 7  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  ( M `  ( x  i^i  A ) )  e.  RR )
3725rpne0d 11028 . . . . . . 7  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  ( M `  A )  =/=  0 )
38 rexdiv 26018 . . . . . . 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 1213 . . . . . 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 2473 . . . . 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 4375 . . . 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 11050 . . . . . . . 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 2797 . . . . . . 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 5332 . . . . . . 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 5692 . . . . 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 2446 . . . 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 2521 . . 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 26536 . . 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 4098 . . . 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 5692 . . 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 2442 . . . 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 462 . . . . . 6  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  =  U. S )  -> 
( M `  A
)  e.  RR+ )
5554rpcnd 11025 . . . . 5  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  =  U. S )  -> 
( M `  A
)  e.  CC )
565rpne0d 11028 . . . . . 6  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  ->  ( M `  A )  =/=  0
)
5756adantr 462 . . . . 5  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  =  U. S )  -> 
( M `  A
)  =/=  0 )
58 simpr 458 . . . . . . . 8  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  =  U. S )  ->  x  =  U. S )
5958ineq1d 3548 . . . . . . 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 3540 . . . . . . . . . 10  |-  ( U. S  i^i  A )  =  ( A  i^i  U. S )
61 elssuni 4118 . . . . . . . . . . 11  |-  ( A  e.  S  ->  A  C_ 
U. S )
62 df-ss 3339 . . . . . . . . . . 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 2485 . . . . . . . . 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 462 . . . . . . 7  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  =  U. S )  -> 
( U. S  i^i  A )  =  A )
6759, 66eqtrd 2473 . . . . . 6  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  =  U. S )  -> 
( x  i^i  A
)  =  A )
6867fveq2d 5692 . . . . 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 10151 . . . 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 26487 . . . . 5  |-  ( S  e.  U. ran sigAlgebra  ->  S  e.  (sigAlgebra `  U. S ) )
71 baselsiga 26478 . . . . 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 9381 . . . . 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 5776 . . 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 2473 . 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 26706 . 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 659 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 960    = wceq 1364    e. wcel 1761    =/= wne 2604   A.wral 2713    i^i cin 3324    C_ wss 3325   U.cuni 4088   class class class wbr 4289    e. cmpt 4347   dom cdm 4836   ran crn 4837   ` cfv 5415  (class class class)co 6090   RRcr 9277   0cc0 9278   1c1 9279   +oocpnf 9411   RR*cxr 9413    <_ cle 9415    / cdiv 9989   RR+crp 10987   [,]cicc 11299   /𝑒 cxdiv 26009  sigAlgebracsiga 26470  measurescmeas 26529  Probcprb 26704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-ac2 8628  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-disj 4260  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-acn 8108  df-ac 8282  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-ioc 11301  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862  df-fac 12048  df-bc 12075  df-hash 12100  df-shft 12552  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-limsup 12945  df-clim 12962  df-rlim 12963  df-sum 13160  df-ef 13349  df-sin 13351  df-cos 13352  df-pi 13354  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-pt 14379  df-prds 14382  df-ordt 14435  df-xrs 14436  df-qtop 14441  df-imas 14442  df-xps 14444  df-mre 14520  df-mrc 14521  df-acs 14523  df-ps 15366  df-tsr 15367  df-mnd 15411  df-plusf 15412  df-mhm 15460  df-submnd 15461  df-grp 15538  df-minusg 15539  df-sbg 15540  df-mulg 15541  df-subg 15671  df-cntz 15828  df-cmn 16272  df-abl 16273  df-mgp 16582  df-ur 16594  df-rng 16637  df-cring 16638  df-subrg 16843  df-abv 16882  df-lmod 16930  df-scaf 16931  df-sra 17231  df-rgmod 17232  df-psmet 17709  df-xmet 17710  df-met 17711  df-bl 17712  df-mopn 17713  df-fbas 17714  df-fg 17715  df-cnfld 17719  df-top 18403  df-bases 18405  df-topon 18406  df-topsp 18407  df-cld 18523  df-ntr 18524  df-cls 18525  df-nei 18602  df-lp 18640  df-perf 18641  df-cn 18731  df-cnp 18732  df-haus 18819  df-tx 19035  df-hmeo 19228  df-fil 19319  df-fm 19411  df-flim 19412  df-flf 19413  df-tmd 19543  df-tgp 19544  df-tsms 19597  df-trg 19634  df-xms 19795  df-ms 19796  df-tms 19797  df-nm 20075  df-ngp 20076  df-nrg 20078  df-nlm 20079  df-ii 20353  df-cncf 20354  df-limc 21241  df-dv 21242  df-log 21951  df-xdiv 26010  df-esum 26404  df-siga 26471  df-meas 26530  df-prob 26705
This theorem is referenced by:  cndprobprob  26735
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