Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  probmeasb Structured version   Unicode version

Theorem probmeasb 29258
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 29038 . . . . 5  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S )  ->  (
y  e.  S  |->  ( M `  ( y  i^i  A ) ) )  e.  (measures `  S
) )
2 measdivcstOLD 29041 . . . . 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 1656 . . . 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 2423 . . . . . . . 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 462 . . . . . . . . . 10  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A
)  e.  RR+ )  /\  x  e.  S
)  /\  y  =  x )  ->  y  =  x )
65ineq1d 3663 . . . . . . . . 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 5881 . . . . . . . 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 462 . . . . . . . 8  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  x  e.  S )
9 simp1 1005 . . . . . . . . . 10  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  ->  M  e.  (measures `  S ) )
109adantr 466 . . . . . . . . 9  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  M  e.  (measures `  S )
)
11 measbase 29014 . . . . . . . . . . 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 1006 . . . . . . . . . . 11  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  ->  A  e.  S
)
1413adantr 466 . . . . . . . . . 10  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  A  e.  S )
15 inelsiga 28952 . . . . . . . . . 10  |-  ( ( S  e.  U. ran sigAlgebra  /\  x  e.  S  /\  A  e.  S )  ->  ( x  i^i  A
)  e.  S )
1612, 8, 14, 15syl3anc 1264 . . . . . . . . 9  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  (
x  i^i  A )  e.  S )
17 measvxrge0 29022 . . . . . . . . 9  |-  ( ( M  e.  (measures `  S
)  /\  ( x  i^i  A )  e.  S
)  ->  ( M `  ( x  i^i  A
) )  e.  ( 0 [,] +oo )
)
1810, 16, 17syl2anc 665 . . . . . . . 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 5966 . . . . . . 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 6316 . . . . . 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 11717 . . . . . . . . 9  |-  ( 0 [,] +oo )  C_  RR*
2221, 18sseldi 3462 . . . . . . . 8  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  ( M `  ( x  i^i  A ) )  e. 
RR* )
23 simp3 1007 . . . . . . . . . 10  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  ->  ( M `  A )  e.  RR+ )
2423adantr 466 . . . . . . . . 9  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  ( M `  A )  e.  RR+ )
2524rpred 11341 . . . . . . . 8  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  ( M `  A )  e.  RR )
26 0xr 9687 . . . . . . . . . 10  |-  0  e.  RR*
27 pnfxr 11412 . . . . . . . . . 10  |- +oo  e.  RR*
28 iccgelb 11691 . . . . . . . . . 10  |-  ( ( 0  e.  RR*  /\ +oo  e.  RR*  /\  ( M `
 ( x  i^i 
A ) )  e.  ( 0 [,] +oo ) )  ->  0  <_  ( M `  (
x  i^i  A )
) )
2926, 27, 28mp3an12 1350 . . . . . . . . 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 3683 . . . . . . . . . 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 29032 . . . . . . . 8  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  ( M `  ( x  i^i  A ) )  <_ 
( M `  A
) )
34 xrrege0 11469 . . . . . . . 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 1265 . . . . . . 7  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  ( M `  ( x  i^i  A ) )  e.  RR )
3624rpne0d 11346 . . . . . . 7  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  e.  S )  ->  ( M `  A )  =/=  0 )
37 rexdiv 28389 . . . . . . 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 1264 . . . . . 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 2463 . . . . 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 4507 . . . 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 11369 . . . . . . . 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 2839 . . . . . . 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 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 )
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 5881 . . . . 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 2430 . . . 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 2524 . . 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 29019 . . 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 215 . 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 4226 . . . 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 5881 . . 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 2423 . . . 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 466 . . . . . 6  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  =  U. S )  -> 
( M `  A
)  e.  RR+ )
5453rpcnd 11343 . . . . 5  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  =  U. S )  -> 
( M `  A
)  e.  CC )
5523rpne0d 11346 . . . . . 6  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  ->  ( M `  A )  =/=  0
)
5655adantr 466 . . . . 5  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  =  U. S )  -> 
( M `  A
)  =/=  0 )
57 simpr 462 . . . . . . . 8  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  =  U. S )  ->  x  =  U. S )
5857ineq1d 3663 . . . . . . 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 3655 . . . . . . . . . 10  |-  ( U. S  i^i  A )  =  ( A  i^i  U. S )
60 elssuni 4245 . . . . . . . . . . 11  |-  ( A  e.  S  ->  A  C_ 
U. S )
61 df-ss 3450 . . . . . . . . . . 11  |-  ( A 
C_  U. S  <->  ( A  i^i  U. S )  =  A )
6260, 61sylib 199 . . . . . . . . . 10  |-  ( A  e.  S  ->  ( A  i^i  U. S )  =  A )
6359, 62syl5eq 2475 . . . . . . . . 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 466 . . . . . . 7  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  =  U. S )  -> 
( U. S  i^i  A )  =  A )
6658, 65eqtrd 2463 . . . . . 6  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  /\  x  =  U. S )  -> 
( x  i^i  A
)  =  A )
6766fveq2d 5881 . . . . 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 10431 . . . 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 28941 . . . . 5  |-  ( S  e.  U. ran sigAlgebra  ->  S  e.  (sigAlgebra `  U. S ) )
70 baselsiga 28932 . . . . 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 9658 . . . 4  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S  /\  ( M `  A )  e.  RR+ )  ->  1  e.  RR )
7352, 68, 71, 72fvmptd 5966 . . 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 2463 . 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 29237 . 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 668 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 370    /\ w3a 982    = wceq 1437    e. wcel 1868    =/= wne 2618   A.wral 2775    i^i cin 3435    C_ wss 3436   U.cuni 4216   class class class wbr 4420    |-> cmpt 4479   dom cdm 4849   ran crn 4850   ` cfv 5597  (class class class)co 6301   RRcr 9538   0cc0 9539   1c1 9540   +oocpnf 9672   RR*cxr 9674    <_ cle 9676    / cdiv 10269   RR+crp 11302   [,]cicc 11638   /𝑒 cxdiv 28380  sigAlgebracsiga 28924  measurescmeas 29012  Probcprb 29235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-inf2 8148  ax-ac2 8893  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-iin 4299  df-disj 4392  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-se 4809  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-isom 5606  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-of 6541  df-om 6703  df-1st 6803  df-2nd 6804  df-supp 6922  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-1o 7186  df-2o 7187  df-oadd 7190  df-er 7367  df-map 7478  df-pm 7479  df-ixp 7527  df-en 7574  df-dom 7575  df-sdom 7576  df-fin 7577  df-fsupp 7886  df-fi 7927  df-sup 7958  df-inf 7959  df-oi 8027  df-card 8374  df-acn 8377  df-ac 8547  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ioc 11640  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-fl 12027  df-mod 12096  df-seq 12213  df-exp 12272  df-fac 12459  df-bc 12487  df-hash 12515  df-shft 13118  df-cj 13150  df-re 13151  df-im 13152  df-sqrt 13286  df-abs 13287  df-limsup 13513  df-clim 13539  df-rlim 13540  df-sum 13740  df-ef 14108  df-sin 14110  df-cos 14111  df-pi 14113  df-struct 15110  df-ndx 15111  df-slot 15112  df-base 15113  df-sets 15114  df-ress 15115  df-plusg 15190  df-mulr 15191  df-starv 15192  df-sca 15193  df-vsca 15194  df-ip 15195  df-tset 15196  df-ple 15197  df-ds 15199  df-unif 15200  df-hom 15201  df-cco 15202  df-rest 15308  df-topn 15309  df-0g 15327  df-gsum 15328  df-topgen 15329  df-pt 15330  df-prds 15333  df-ordt 15386  df-xrs 15387  df-qtop 15393  df-imas 15394  df-xps 15397  df-mre 15479  df-mrc 15480  df-acs 15482  df-ps 16433  df-tsr 16434  df-plusf 16474  df-mgm 16475  df-sgrp 16514  df-mnd 16524  df-mhm 16569  df-submnd 16570  df-grp 16660  df-minusg 16661  df-sbg 16662  df-mulg 16663  df-subg 16801  df-cntz 16958  df-cmn 17419  df-abl 17420  df-mgp 17711  df-ur 17723  df-ring 17769  df-cring 17770  df-subrg 17993  df-abv 18032  df-lmod 18080  df-scaf 18081  df-sra 18382  df-rgmod 18383  df-psmet 18949  df-xmet 18950  df-met 18951  df-bl 18952  df-mopn 18953  df-fbas 18954  df-fg 18955  df-cnfld 18958  df-top 19907  df-bases 19908  df-topon 19909  df-topsp 19910  df-cld 20020  df-ntr 20021  df-cls 20022  df-nei 20100  df-lp 20138  df-perf 20139  df-cn 20229  df-cnp 20230  df-haus 20317  df-tx 20563  df-hmeo 20756  df-fil 20847  df-fm 20939  df-flim 20940  df-flf 20941  df-tmd 21073  df-tgp 21074  df-tsms 21127  df-trg 21160  df-xms 21321  df-ms 21322  df-tms 21323  df-nm 21583  df-ngp 21584  df-nrg 21586  df-nlm 21587  df-ii 21895  df-cncf 21896  df-limc 22807  df-dv 22808  df-log 23492  df-xdiv 28381  df-esum 28844  df-siga 28925  df-meas 29013  df-prob 29236
This theorem is referenced by:  cndprobprob  29266
  Copyright terms: Public domain W3C validator