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Theorem probmeasb 29336
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 measinb 29117 . . . . 5  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S )  ->  (
y  e.  S  |->  ( M `  ( y  i^i  A ) ) )  e.  (measures `  S
) )
2 measdivcstOLD 29120 . . . . 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 )
)
31, 2stoic3 1668 . . . 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 ) )
4 eqidd 2472 . . . . . . . 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 ) ) ) )
5 simpr 468 . . . . . . . . . 10  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A
)  e.  RR+ )  /\  x  e.  S
)  /\  y  =  x )  ->  y  =  x )
65ineq1d 3624 . . . . . . . . 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 ) )
76fveq2d 5883 . . . . . . . 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 )
) )
8 simpr 468 . . . . . . . 8  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  x  e.  S )
9 simp1 1030 . . . . . . . . . 10  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  ->  M  e.  (measures `  S ) )
109adantr 472 . . . . . . . . 9  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  M  e.  (measures `  S )
)
11 measbase 29093 . . . . . . . . . . 11  |-  ( M  e.  (measures `  S
)  ->  S  e.  U.
ran sigAlgebra )
1210, 11syl 17 . . . . . . . . . 10  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  S  e.  U. ran sigAlgebra )
13 simp2 1031 . . . . . . . . . . 11  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  ->  A  e.  S
)
1413adantr 472 . . . . . . . . . 10  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  A  e.  S )
15 inelsiga 29031 . . . . . . . . . 10  |-  ( ( S  e.  U. ran sigAlgebra  /\  x  e.  S  /\  A  e.  S )  ->  ( x  i^i  A
)  e.  S )
1612, 8, 14, 15syl3anc 1292 . . . . . . . . 9  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  (
x  i^i  A )  e.  S )
17 measvxrge0 29101 . . . . . . . . 9  |-  ( ( M  e.  (measures `  S
)  /\  ( x  i^i  A )  e.  S
)  ->  ( M `  ( x  i^i  A
) )  e.  ( 0 [,] +oo )
)
1810, 16, 17syl2anc 673 . . . . . . . 8  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  ( M `  ( x  i^i  A ) )  e.  ( 0 [,] +oo ) )
194, 7, 8, 18fvmptd 5969 . . . . . . 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 ) ) )
2019oveq1d 6323 . . . . . 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
) ) )
21 iccssxr 11742 . . . . . . . . 9  |-  ( 0 [,] +oo )  C_  RR*
2221, 18sseldi 3416 . . . . . . . 8  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  ( M `  ( x  i^i  A ) )  e. 
RR* )
23 simp3 1032 . . . . . . . . . 10  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  ->  ( M `  A )  e.  RR+ )
2423adantr 472 . . . . . . . . 9  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  ( M `  A )  e.  RR+ )
2524rpred 11364 . . . . . . . 8  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  ( M `  A )  e.  RR )
26 0xr 9705 . . . . . . . . . 10  |-  0  e.  RR*
27 pnfxr 11435 . . . . . . . . . 10  |- +oo  e.  RR*
28 iccgelb 11716 . . . . . . . . . 10  |-  ( ( 0  e.  RR*  /\ +oo  e.  RR*  /\  ( M `
 ( x  i^i 
A ) )  e.  ( 0 [,] +oo ) )  ->  0  <_  ( M `  (
x  i^i  A )
) )
2926, 27, 28mp3an12 1380 . . . . . . . . 9  |-  ( ( M `  ( x  i^i  A ) )  e.  ( 0 [,] +oo )  ->  0  <_ 
( M `  (
x  i^i  A )
) )
3018, 29syl 17 . . . . . . . 8  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  0  <_  ( M `  (
x  i^i  A )
) )
31 inss2 3644 . . . . . . . . . 10  |-  ( x  i^i  A )  C_  A
3231a1i 11 . . . . . . . . 9  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  (
x  i^i  A )  C_  A )
3310, 16, 14, 32measssd 29111 . . . . . . . 8  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  ( M `  ( x  i^i  A ) )  <_ 
( M `  A
) )
34 xrrege0 11492 . . . . . . . 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 )
3522, 25, 30, 33, 34syl22anc 1293 . . . . . . 7  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  ( M `  ( x  i^i  A ) )  e.  RR )
3624rpne0d 11369 . . . . . . 7  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  ( M `  A )  =/=  0 )
37 rexdiv 28470 . . . . . . 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 ) ) )
3835, 25, 36, 37syl3anc 1292 . . . . . 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 ) ) )
3920, 38eqtrd 2505 . . . . 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 ) ) )
4039mpteq2dva 4482 . . . 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 ) ) ) )
4135, 24rerpdivcld 11392 . . . . . . . 8  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  (
( M `  (
x  i^i  A )
)  /  ( M `
 A ) )  e.  RR )
4241ralrimiva 2809 . . . . . . 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 )
43 dmmptg 5339 . . . . . . 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 )
4442, 43syl 17 . . . . . 6  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  ->  dom  ( x  e.  S  |->  ( ( M `  ( x  i^i  A ) )  /  ( M `  A ) ) )  =  S )
4544fveq2d 5883 . . . . 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
) )
4645eqcomd 2477 . . . 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 ) ) ) ) )
473, 40, 463eltr3d 2563 . . 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 ) ) ) ) )
48 measbasedom 29098 . . 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 ) ) ) ) )
4947, 48sylibr 217 . 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 )
5044unieqd 4200 . . . 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 )
5150fveq2d 5883 . . 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
) )
52 eqidd 2472 . . . 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 ) ) ) )
5323adantr 472 . . . . . 6  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  =  U. S )  -> 
( M `  A
)  e.  RR+ )
5453rpcnd 11366 . . . . 5  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  =  U. S )  -> 
( M `  A
)  e.  CC )
5523rpne0d 11369 . . . . . 6  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  ->  ( M `  A )  =/=  0
)
5655adantr 472 . . . . 5  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  =  U. S )  -> 
( M `  A
)  =/=  0 )
57 simpr 468 . . . . . . . 8  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  =  U. S )  ->  x  =  U. S )
5857ineq1d 3624 . . . . . . 7  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  =  U. S )  -> 
( x  i^i  A
)  =  ( U. S  i^i  A ) )
59 incom 3616 . . . . . . . . . 10  |-  ( U. S  i^i  A )  =  ( A  i^i  U. S )
60 elssuni 4219 . . . . . . . . . . 11  |-  ( A  e.  S  ->  A  C_ 
U. S )
61 df-ss 3404 . . . . . . . . . . 11  |-  ( A 
C_  U. S  <->  ( A  i^i  U. S )  =  A )
6260, 61sylib 201 . . . . . . . . . 10  |-  ( A  e.  S  ->  ( A  i^i  U. S )  =  A )
6359, 62syl5eq 2517 . . . . . . . . 9  |-  ( A  e.  S  ->  ( U. S  i^i  A )  =  A )
6413, 63syl 17 . . . . . . . 8  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  ->  ( U. S  i^i  A )  =  A )
6564adantr 472 . . . . . . 7  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  =  U. S )  -> 
( U. S  i^i  A )  =  A )
6658, 65eqtrd 2505 . . . . . 6  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  =  U. S )  -> 
( x  i^i  A
)  =  A )
6766fveq2d 5883 . . . . 5  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  =  U. S )  -> 
( M `  (
x  i^i  A )
)  =  ( M `
 A ) )
6854, 56, 67diveq1bd 10453 . . . 4  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  =  U. S )  -> 
( ( M `  ( x  i^i  A ) )  /  ( M `
 A ) )  =  1 )
69 sgon 29020 . . . . 5  |-  ( S  e.  U. ran sigAlgebra  ->  S  e.  (sigAlgebra `  U. S ) )
70 baselsiga 29011 . . . . 5  |-  ( S  e.  (sigAlgebra `  U. S )  ->  U. S  e.  S
)
719, 11, 69, 704syl 19 . . . 4  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  ->  U. S  e.  S
)
72 1red 9676 . . . 4  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  ->  1  e.  RR )
7352, 68, 71, 72fvmptd 5969 . . 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 )
7451, 73eqtrd 2505 . 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 )
75 elprob 29315 . 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 ) )
7649, 74, 75sylanbrc 677 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 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756    i^i cin 3389    C_ wss 3390   U.cuni 4190   class class class wbr 4395    |-> cmpt 4454   dom cdm 4839   ran crn 4840   ` cfv 5589  (class class class)co 6308   RRcr 9556   0cc0 9557   1c1 9558   +oocpnf 9690   RR*cxr 9692    <_ cle 9694    / cdiv 10291   RR+crp 11325   [,]cicc 11663   /𝑒 cxdiv 28461  sigAlgebracsiga 29003  measurescmeas 29091  Probcprb 29313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-ac2 8911  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-disj 4367  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-acn 8394  df-ac 8565  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ioc 11665  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526  df-hash 12554  df-shft 13207  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-limsup 13603  df-clim 13629  df-rlim 13630  df-sum 13830  df-ef 14198  df-sin 14200  df-cos 14201  df-pi 14203  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-ordt 15477  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-ps 16524  df-tsr 16525  df-plusf 16565  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-mhm 16660  df-submnd 16661  df-grp 16751  df-minusg 16752  df-sbg 16753  df-mulg 16754  df-subg 16892  df-cntz 17049  df-cmn 17510  df-abl 17511  df-mgp 17802  df-ur 17814  df-ring 17860  df-cring 17861  df-subrg 18084  df-abv 18123  df-lmod 18171  df-scaf 18172  df-sra 18473  df-rgmod 18474  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-tmd 21165  df-tgp 21166  df-tsms 21219  df-trg 21252  df-xms 21413  df-ms 21414  df-tms 21415  df-nm 21675  df-ngp 21676  df-nrg 21678  df-nlm 21679  df-ii 21987  df-cncf 21988  df-limc 22900  df-dv 22901  df-log 23585  df-xdiv 28462  df-esum 28923  df-siga 29004  df-meas 29092  df-prob 29314
This theorem is referenced by:  cndprobprob  29344
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