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Theorem measdivcst 29121
Description: Division of a measure by a positive constant is a measure. (Contributed by Thierry Arnoux, 25-Dec-2016.) (Revised by Thierry Arnoux, 30-Jan-2017.)
Assertion
Ref Expression
measdivcst  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  RR+ )  ->  ( M𝑓/𝑐 /𝑒  A )  e.  (measures `  S ) )

Proof of Theorem measdivcst
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ofcfval3 28997 . . 3  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  RR+ )  ->  ( M𝑓/𝑐 /𝑒  A )  =  ( x  e.  dom  M  |->  ( ( M `  x ) /𝑒  A ) ) )
2 measfrge0 29099 . . . . . 6  |-  ( M  e.  (measures `  S
)  ->  M : S
--> ( 0 [,] +oo ) )
3 fdm 5745 . . . . . 6  |-  ( M : S --> ( 0 [,] +oo )  ->  dom  M  =  S )
42, 3syl 17 . . . . 5  |-  ( M  e.  (measures `  S
)  ->  dom  M  =  S )
54adantr 472 . . . 4  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  RR+ )  ->  dom  M  =  S )
65mpteq1d 4477 . . 3  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  RR+ )  ->  ( x  e.  dom  M  |->  ( ( M `  x ) /𝑒  A ) )  =  ( x  e.  S  |->  ( ( M `  x
) /𝑒 
A ) ) )
71, 6eqtrd 2505 . 2  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  RR+ )  ->  ( M𝑓/𝑐 /𝑒  A )  =  ( x  e.  S  |->  ( ( M `  x
) /𝑒 
A ) ) )
8 measvxrge0 29101 . . . . . 6  |-  ( ( M  e.  (measures `  S
)  /\  x  e.  S )  ->  ( M `  x )  e.  ( 0 [,] +oo ) )
98adantlr 729 . . . . 5  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  x  e.  S )  ->  ( M `  x )  e.  ( 0 [,] +oo ) )
10 simplr 770 . . . . 5  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  x  e.  S )  ->  A  e.  RR+ )
119, 10xrpxdivcld 28479 . . . 4  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  x  e.  S )  ->  (
( M `  x
) /𝑒 
A )  e.  ( 0 [,] +oo )
)
12 eqid 2471 . . . 4  |-  ( x  e.  S  |->  ( ( M `  x ) /𝑒  A ) )  =  ( x  e.  S  |->  ( ( M `  x
) /𝑒 
A ) )
1311, 12fmptd 6061 . . 3  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  RR+ )  ->  ( x  e.  S  |->  ( ( M `  x ) /𝑒  A ) ) : S --> ( 0 [,] +oo ) )
14 measbase 29093 . . . . . . 7  |-  ( M  e.  (measures `  S
)  ->  S  e.  U.
ran sigAlgebra )
15 0elsiga 29010 . . . . . . 7  |-  ( S  e.  U. ran sigAlgebra  ->  (/)  e.  S
)
1614, 15syl 17 . . . . . 6  |-  ( M  e.  (measures `  S
)  ->  (/)  e.  S
)
1716adantr 472 . . . . 5  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  RR+ )  ->  (/)  e.  S
)
18 ovex 6336 . . . . 5  |-  ( ( M `  (/) ) /𝑒  A )  e.  _V
19 fveq2 5879 . . . . . . 7  |-  ( x  =  (/)  ->  ( M `
 x )  =  ( M `  (/) ) )
2019oveq1d 6323 . . . . . 6  |-  ( x  =  (/)  ->  ( ( M `  x ) /𝑒  A )  =  ( ( M `  (/) ) /𝑒  A ) )
2120, 12fvmptg 5961 . . . . 5  |-  ( (
(/)  e.  S  /\  ( ( M `  (/) ) /𝑒  A )  e.  _V )  ->  ( ( x  e.  S  |->  ( ( M `  x ) /𝑒  A ) ) `  (/) )  =  ( ( M `  (/) ) /𝑒  A ) )
2217, 18, 21sylancl 675 . . . 4  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  RR+ )  ->  ( (
x  e.  S  |->  ( ( M `  x
) /𝑒 
A ) ) `  (/) )  =  ( ( M `  (/) ) /𝑒  A ) )
23 measvnul 29102 . . . . . 6  |-  ( M  e.  (measures `  S
)  ->  ( M `  (/) )  =  0 )
2423oveq1d 6323 . . . . 5  |-  ( M  e.  (measures `  S
)  ->  ( ( M `  (/) ) /𝑒  A )  =  ( 0 /𝑒  A ) )
25 xdiv0rp 28474 . . . . 5  |-  ( A  e.  RR+  ->  ( 0 /𝑒  A )  =  0 )
2624, 25sylan9eq 2525 . . . 4  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  RR+ )  ->  ( ( M `  (/) ) /𝑒  A )  =  0 )
2722, 26eqtrd 2505 . . 3  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  RR+ )  ->  ( (
x  e.  S  |->  ( ( M `  x
) /𝑒 
A ) ) `  (/) )  =  0 )
28 simpll 768 . . . . . 6  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  y  e.  ~P S )  /\  (
y  ~<_  om  /\ Disj  z  e.  y  z ) )  ->  ( M  e.  (measures `  S )  /\  A  e.  RR+ )
)
29 simplr 770 . . . . . 6  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  y  e.  ~P S )  /\  (
y  ~<_  om  /\ Disj  z  e.  y  z ) )  ->  y  e.  ~P S )
30 simprl 772 . . . . . 6  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  y  e.  ~P S )  /\  (
y  ~<_  om  /\ Disj  z  e.  y  z ) )  ->  y  ~<_  om )
31 simprr 774 . . . . . 6  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  y  e.  ~P S )  /\  (
y  ~<_  om  /\ Disj  z  e.  y  z ) )  -> Disj  z  e.  y  z )
32 vex 3034 . . . . . . . . . 10  |-  y  e. 
_V
3332a1i 11 . . . . . . . . 9  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  y  e.  ~P S )  -> 
y  e.  _V )
34 simplll 776 . . . . . . . . . 10  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  y  e.  ~P S )  /\  z  e.  y )  ->  M  e.  (measures `  S )
)
35 selpw 3949 . . . . . . . . . . . 12  |-  ( y  e.  ~P S  <->  y  C_  S )
36 ssel2 3413 . . . . . . . . . . . 12  |-  ( ( y  C_  S  /\  z  e.  y )  ->  z  e.  S )
3735, 36sylanb 480 . . . . . . . . . . 11  |-  ( ( y  e.  ~P S  /\  z  e.  y
)  ->  z  e.  S )
3837adantll 728 . . . . . . . . . 10  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  y  e.  ~P S )  /\  z  e.  y )  ->  z  e.  S )
39 measvxrge0 29101 . . . . . . . . . 10  |-  ( ( M  e.  (measures `  S
)  /\  z  e.  S )  ->  ( M `  z )  e.  ( 0 [,] +oo ) )
4034, 38, 39syl2anc 673 . . . . . . . . 9  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  y  e.  ~P S )  /\  z  e.  y )  ->  ( M `  z )  e.  ( 0 [,] +oo ) )
41 simplr 770 . . . . . . . . 9  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  y  e.  ~P S )  ->  A  e.  RR+ )
4233, 40, 41esumdivc 28978 . . . . . . . 8  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  y  e.  ~P S )  -> 
(Σ* z  e.  y ( M `  z ) /𝑒  A )  = Σ* z  e.  y ( ( M `  z ) /𝑒  A ) )
43423ad2antr1 1195 . . . . . . 7  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  (
y  e.  ~P S  /\  y  ~<_  om  /\ Disj  z  e.  y  z ) )  ->  (Σ* z  e.  y ( M `  z ) /𝑒  A )  = Σ* z  e.  y ( ( M `  z ) /𝑒  A ) )
4414ad2antrr 740 . . . . . . . . . 10  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  (
y  e.  ~P S  /\  y  ~<_  om  /\ Disj  z  e.  y  z ) )  ->  S  e.  U. ran sigAlgebra )
45 simpr1 1036 . . . . . . . . . 10  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  (
y  e.  ~P S  /\  y  ~<_  om  /\ Disj  z  e.  y  z ) )  ->  y  e.  ~P S )
46 simpr2 1037 . . . . . . . . . 10  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  (
y  e.  ~P S  /\  y  ~<_  om  /\ Disj  z  e.  y  z ) )  ->  y  ~<_  om )
47 sigaclcu 29013 . . . . . . . . . 10  |-  ( ( S  e.  U. ran sigAlgebra  /\  y  e.  ~P S  /\  y  ~<_  om )  ->  U. y  e.  S
)
4844, 45, 46, 47syl3anc 1292 . . . . . . . . 9  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  (
y  e.  ~P S  /\  y  ~<_  om  /\ Disj  z  e.  y  z ) )  ->  U. y  e.  S
)
49 fveq2 5879 . . . . . . . . . . 11  |-  ( x  =  U. y  -> 
( M `  x
)  =  ( M `
 U. y ) )
5049oveq1d 6323 . . . . . . . . . 10  |-  ( x  =  U. y  -> 
( ( M `  x ) /𝑒  A )  =  ( ( M `  U. y ) /𝑒  A ) )
51 ovex 6336 . . . . . . . . . 10  |-  ( ( M `  x ) /𝑒  A )  e.  _V
5250, 12, 51fvmpt3i 5968 . . . . . . . . 9  |-  ( U. y  e.  S  ->  ( ( x  e.  S  |->  ( ( M `  x ) /𝑒  A ) ) `  U. y )  =  ( ( M `  U. y ) /𝑒  A ) )
5348, 52syl 17 . . . . . . . 8  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  (
y  e.  ~P S  /\  y  ~<_  om  /\ Disj  z  e.  y  z ) )  ->  ( ( x  e.  S  |->  ( ( M `  x ) /𝑒  A ) ) `  U. y )  =  ( ( M `  U. y ) /𝑒  A ) )
54 simpll 768 . . . . . . . . . 10  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  (
y  e.  ~P S  /\  y  ~<_  om  /\ Disj  z  e.  y  z ) )  ->  M  e.  (measures `  S ) )
55 simpr3 1038 . . . . . . . . . 10  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  (
y  e.  ~P S  /\  y  ~<_  om  /\ Disj  z  e.  y  z ) )  -> Disj  z  e.  y  z )
56 measvun 29105 . . . . . . . . . 10  |-  ( ( M  e.  (measures `  S
)  /\  y  e.  ~P S  /\  (
y  ~<_  om  /\ Disj  z  e.  y  z ) )  ->  ( M `  U. y )  = Σ* z  e.  y ( M `  z ) )
5754, 45, 46, 55, 56syl112anc 1296 . . . . . . . . 9  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  (
y  e.  ~P S  /\  y  ~<_  om  /\ Disj  z  e.  y  z ) )  ->  ( M `  U. y )  = Σ* z  e.  y ( M `  z ) )
5857oveq1d 6323 . . . . . . . 8  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  (
y  e.  ~P S  /\  y  ~<_  om  /\ Disj  z  e.  y  z ) )  ->  ( ( M `
 U. y ) /𝑒  A )  =  (Σ* z  e.  y ( M `  z ) /𝑒  A ) )
5953, 58eqtrd 2505 . . . . . . 7  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  (
y  e.  ~P S  /\  y  ~<_  om  /\ Disj  z  e.  y  z ) )  ->  ( ( x  e.  S  |->  ( ( M `  x ) /𝑒  A ) ) `  U. y )  =  (Σ* z  e.  y ( M `
 z ) /𝑒  A ) )
60 fveq2 5879 . . . . . . . . . . . 12  |-  ( x  =  z  ->  ( M `  x )  =  ( M `  z ) )
6160oveq1d 6323 . . . . . . . . . . 11  |-  ( x  =  z  ->  (
( M `  x
) /𝑒 
A )  =  ( ( M `  z
) /𝑒 
A ) )
6261, 12, 51fvmpt3i 5968 . . . . . . . . . 10  |-  ( z  e.  S  ->  (
( x  e.  S  |->  ( ( M `  x ) /𝑒  A ) ) `  z )  =  ( ( M `  z
) /𝑒 
A ) )
6337, 62syl 17 . . . . . . . . 9  |-  ( ( y  e.  ~P S  /\  z  e.  y
)  ->  ( (
x  e.  S  |->  ( ( M `  x
) /𝑒 
A ) ) `  z )  =  ( ( M `  z
) /𝑒 
A ) )
6463esumeq2dv 28933 . . . . . . . 8  |-  ( y  e.  ~P S  -> Σ* z  e.  y ( ( x  e.  S  |->  ( ( M `  x ) /𝑒  A ) ) `  z
)  = Σ* z  e.  y ( ( M `  z ) /𝑒  A ) )
6545, 64syl 17 . . . . . . 7  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  (
y  e.  ~P S  /\  y  ~<_  om  /\ Disj  z  e.  y  z ) )  -> Σ* z  e.  y ( ( x  e.  S  |->  ( ( M `  x ) /𝑒  A ) ) `  z )  = Σ* z  e.  y ( ( M `
 z ) /𝑒  A ) )
6643, 59, 653eqtr4d 2515 . . . . . 6  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  (
y  e.  ~P S  /\  y  ~<_  om  /\ Disj  z  e.  y  z ) )  ->  ( ( x  e.  S  |->  ( ( M `  x ) /𝑒  A ) ) `  U. y )  = Σ* z  e.  y ( ( x  e.  S  |->  ( ( M `  x ) /𝑒  A ) ) `  z
) )
6728, 29, 30, 31, 66syl13anc 1294 . . . . 5  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  y  e.  ~P S )  /\  (
y  ~<_  om  /\ Disj  z  e.  y  z ) )  ->  ( ( x  e.  S  |->  ( ( M `  x ) /𝑒  A ) ) `  U. y )  = Σ* z  e.  y ( ( x  e.  S  |->  ( ( M `  x ) /𝑒  A ) ) `  z
) )
6867ex 441 . . . 4  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  y  e.  ~P S )  -> 
( ( y  ~<_  om 
/\ Disj  z  e.  y  z )  ->  ( (
x  e.  S  |->  ( ( M `  x
) /𝑒 
A ) ) `  U. y )  = Σ* z  e.  y ( ( x  e.  S  |->  ( ( M `  x ) /𝑒  A ) ) `  z
) ) )
6968ralrimiva 2809 . . 3  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  RR+ )  ->  A. y  e.  ~P  S ( ( y  ~<_  om  /\ Disj  z  e.  y  z )  -> 
( ( x  e.  S  |->  ( ( M `
 x ) /𝑒  A ) ) `  U. y
)  = Σ* z  e.  y ( ( x  e.  S  |->  ( ( M `
 x ) /𝑒  A ) ) `  z ) ) )
70 ismeas 29095 . . . . . 6  |-  ( S  e.  U. ran sigAlgebra  ->  (
( x  e.  S  |->  ( ( M `  x ) /𝑒  A ) )  e.  (measures `  S )  <->  ( ( x  e.  S  |->  ( ( M `  x ) /𝑒  A ) ) : S --> ( 0 [,] +oo )  /\  (
( x  e.  S  |->  ( ( M `  x ) /𝑒  A ) ) `  (/) )  =  0  /\ 
A. y  e.  ~P  S ( ( y  ~<_  om  /\ Disj  z  e.  y  z )  ->  (
( x  e.  S  |->  ( ( M `  x ) /𝑒  A ) ) `  U. y )  = Σ* z  e.  y ( ( x  e.  S  |->  ( ( M `  x ) /𝑒  A ) ) `  z
) ) ) ) )
7114, 70syl 17 . . . . 5  |-  ( M  e.  (measures `  S
)  ->  ( (
x  e.  S  |->  ( ( M `  x
) /𝑒 
A ) )  e.  (measures `  S )  <->  ( ( x  e.  S  |->  ( ( M `  x ) /𝑒  A ) ) : S --> ( 0 [,] +oo )  /\  (
( x  e.  S  |->  ( ( M `  x ) /𝑒  A ) ) `  (/) )  =  0  /\ 
A. y  e.  ~P  S ( ( y  ~<_  om  /\ Disj  z  e.  y  z )  ->  (
( x  e.  S  |->  ( ( M `  x ) /𝑒  A ) ) `  U. y )  = Σ* z  e.  y ( ( x  e.  S  |->  ( ( M `  x ) /𝑒  A ) ) `  z
) ) ) ) )
7271biimprd 231 . . . 4  |-  ( M  e.  (measures `  S
)  ->  ( (
( x  e.  S  |->  ( ( M `  x ) /𝑒  A ) ) : S --> ( 0 [,] +oo )  /\  (
( x  e.  S  |->  ( ( M `  x ) /𝑒  A ) ) `  (/) )  =  0  /\ 
A. y  e.  ~P  S ( ( y  ~<_  om  /\ Disj  z  e.  y  z )  ->  (
( x  e.  S  |->  ( ( M `  x ) /𝑒  A ) ) `  U. y )  = Σ* z  e.  y ( ( x  e.  S  |->  ( ( M `  x ) /𝑒  A ) ) `  z
) ) )  -> 
( x  e.  S  |->  ( ( M `  x ) /𝑒  A ) )  e.  (measures `  S )
) )
7372adantr 472 . . 3  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  RR+ )  ->  ( (
( x  e.  S  |->  ( ( M `  x ) /𝑒  A ) ) : S --> ( 0 [,] +oo )  /\  (
( x  e.  S  |->  ( ( M `  x ) /𝑒  A ) ) `  (/) )  =  0  /\ 
A. y  e.  ~P  S ( ( y  ~<_  om  /\ Disj  z  e.  y  z )  ->  (
( x  e.  S  |->  ( ( M `  x ) /𝑒  A ) ) `  U. y )  = Σ* z  e.  y ( ( x  e.  S  |->  ( ( M `  x ) /𝑒  A ) ) `  z
) ) )  -> 
( x  e.  S  |->  ( ( M `  x ) /𝑒  A ) )  e.  (measures `  S )
) )
7413, 27, 69, 73mp3and 1393 . 2  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  RR+ )  ->  ( x  e.  S  |->  ( ( M `  x ) /𝑒  A ) )  e.  (measures `  S ) )
757, 74eqeltrd 2549 1  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  RR+ )  ->  ( M𝑓/𝑐 /𝑒  A )  e.  (measures `  S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   A.wral 2756   _Vcvv 3031    C_ wss 3390   (/)c0 3722   ~Pcpw 3942   U.cuni 4190  Disj wdisj 4366   class class class wbr 4395    |-> cmpt 4454   dom cdm 4839   ran crn 4840   -->wf 5585   ` cfv 5589  (class class class)co 6308   omcom 6711    ~<_ cdom 7585   0cc0 9557   +oocpnf 9690   RR+crp 11325   [,]cicc 11663   /𝑒 cxdiv 28461  Σ*cesum 28922  ∘𝑓/𝑐cofc 28990  sigAlgebracsiga 29003  measurescmeas 29091
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-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
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-oadd 7204  df-er 7381  df-map 7492  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-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-seq 12252  df-hash 12554  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-tset 15287  df-ple 15288  df-ds 15290  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-ordt 15477  df-xrs 15478  df-mre 15570  df-mrc 15571  df-acs 15573  df-ps 16524  df-tsr 16525  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-mhm 16660  df-submnd 16661  df-cntz 17049  df-cmn 17510  df-fbas 19044  df-fg 19045  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-ntr 20112  df-nei 20191  df-cn 20320  df-cnp 20321  df-haus 20408  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-tsms 21219  df-xdiv 28462  df-esum 28923  df-ofc 28991  df-siga 29004  df-meas 29092
This theorem is referenced by:  probfinmeasb  29335
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