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Theorem measdivcst 26608
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 26513 . . 3  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  RR+ )  ->  ( M𝑓/𝑐 /𝑒  A )  =  ( x  e.  dom  M  |->  ( ( M `  x ) /𝑒  A ) ) )
2 measfrge0 26586 . . . . . 6  |-  ( M  e.  (measures `  S
)  ->  M : S
--> ( 0 [,] +oo ) )
3 fdm 5558 . . . . . 6  |-  ( M : S --> ( 0 [,] +oo )  ->  dom  M  =  S )
42, 3syl 16 . . . . 5  |-  ( M  e.  (measures `  S
)  ->  dom  M  =  S )
54adantr 465 . . . 4  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  RR+ )  ->  dom  M  =  S )
65mpteq1d 4368 . . 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 2470 . 2  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  RR+ )  ->  ( M𝑓/𝑐 /𝑒  A )  =  ( x  e.  S  |->  ( ( M `  x
) /𝑒 
A ) ) )
8 measvxrge0 26588 . . . . . 6  |-  ( ( M  e.  (measures `  S
)  /\  x  e.  S )  ->  ( M `  x )  e.  ( 0 [,] +oo ) )
98adantlr 714 . . . . 5  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  x  e.  S )  ->  ( M `  x )  e.  ( 0 [,] +oo ) )
10 simplr 754 . . . . 5  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  x  e.  S )  ->  A  e.  RR+ )
119, 10xrpxdivcld 26078 . . . 4  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  x  e.  S )  ->  (
( M `  x
) /𝑒 
A )  e.  ( 0 [,] +oo )
)
12 eqid 2438 . . . 4  |-  ( x  e.  S  |->  ( ( M `  x ) /𝑒  A ) )  =  ( x  e.  S  |->  ( ( M `  x
) /𝑒 
A ) )
1311, 12fmptd 5862 . . 3  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  RR+ )  ->  ( x  e.  S  |->  ( ( M `  x ) /𝑒  A ) ) : S --> ( 0 [,] +oo ) )
14 measbase 26580 . . . . . . 7  |-  ( M  e.  (measures `  S
)  ->  S  e.  U.
ran sigAlgebra )
15 0elsiga 26526 . . . . . . 7  |-  ( S  e.  U. ran sigAlgebra  ->  (/)  e.  S
)
1614, 15syl 16 . . . . . 6  |-  ( M  e.  (measures `  S
)  ->  (/)  e.  S
)
1716adantr 465 . . . . 5  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  RR+ )  ->  (/)  e.  S
)
18 ovex 6111 . . . . 5  |-  ( ( M `  (/) ) /𝑒  A )  e.  _V
19 fveq2 5686 . . . . . . 7  |-  ( x  =  (/)  ->  ( M `
 x )  =  ( M `  (/) ) )
2019oveq1d 6101 . . . . . 6  |-  ( x  =  (/)  ->  ( ( M `  x ) /𝑒  A )  =  ( ( M `  (/) ) /𝑒  A ) )
2120, 12fvmptg 5767 . . . . 5  |-  ( (
(/)  e.  S  /\  ( ( M `  (/) ) /𝑒  A )  e.  _V )  ->  ( ( x  e.  S  |->  ( ( M `  x ) /𝑒  A ) ) `  (/) )  =  ( ( M `  (/) ) /𝑒  A ) )
2217, 18, 21sylancl 662 . . . 4  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  RR+ )  ->  ( (
x  e.  S  |->  ( ( M `  x
) /𝑒 
A ) ) `  (/) )  =  ( ( M `  (/) ) /𝑒  A ) )
23 measvnul 26589 . . . . . 6  |-  ( M  e.  (measures `  S
)  ->  ( M `  (/) )  =  0 )
2423oveq1d 6101 . . . . 5  |-  ( M  e.  (measures `  S
)  ->  ( ( M `  (/) ) /𝑒  A )  =  ( 0 /𝑒  A ) )
25 xdiv0rp 26073 . . . . 5  |-  ( A  e.  RR+  ->  ( 0 /𝑒  A )  =  0 )
2624, 25sylan9eq 2490 . . . 4  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  RR+ )  ->  ( ( M `  (/) ) /𝑒  A )  =  0 )
2722, 26eqtrd 2470 . . 3  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  RR+ )  ->  ( (
x  e.  S  |->  ( ( M `  x
) /𝑒 
A ) ) `  (/) )  =  0 )
28 simpll 753 . . . . . 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 754 . . . . . . 7  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  y  e.  ~P S )  /\  (
y  ~<_  om  /\ Disj  z  e.  y  z ) )  ->  y  e.  ~P S )
30 simprl 755 . . . . . . 7  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  y  e.  ~P S )  /\  (
y  ~<_  om  /\ Disj  z  e.  y  z ) )  ->  y  ~<_  om )
31 simprr 756 . . . . . . 7  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  y  e.  ~P S )  /\  (
y  ~<_  om  /\ Disj  z  e.  y  z ) )  -> Disj  z  e.  y  z )
3229, 30, 313jca 1168 . . . . . 6  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  y  e.  ~P S )  /\  (
y  ~<_  om  /\ Disj  z  e.  y  z ) )  ->  ( y  e. 
~P S  /\  y  ~<_  om  /\ Disj  z  e.  y 
z ) )
33 vex 2970 . . . . . . . . . 10  |-  y  e. 
_V
3433a1i 11 . . . . . . . . 9  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  y  e.  ~P S )  -> 
y  e.  _V )
35 simplll 757 . . . . . . . . . 10  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  y  e.  ~P S )  /\  z  e.  y )  ->  M  e.  (measures `  S )
)
36 selpw 3862 . . . . . . . . . . . 12  |-  ( y  e.  ~P S  <->  y  C_  S )
37 ssel2 3346 . . . . . . . . . . . 12  |-  ( ( y  C_  S  /\  z  e.  y )  ->  z  e.  S )
3836, 37sylanb 472 . . . . . . . . . . 11  |-  ( ( y  e.  ~P S  /\  z  e.  y
)  ->  z  e.  S )
3938adantll 713 . . . . . . . . . 10  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  y  e.  ~P S )  /\  z  e.  y )  ->  z  e.  S )
40 measvxrge0 26588 . . . . . . . . . 10  |-  ( ( M  e.  (measures `  S
)  /\  z  e.  S )  ->  ( M `  z )  e.  ( 0 [,] +oo ) )
4135, 39, 40syl2anc 661 . . . . . . . . 9  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  y  e.  ~P S )  /\  z  e.  y )  ->  ( M `  z )  e.  ( 0 [,] +oo ) )
42 simplr 754 . . . . . . . . 9  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  y  e.  ~P S )  ->  A  e.  RR+ )
4334, 41, 42esumdivc 26501 . . . . . . . 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 ) )
44433ad2antr1 1153 . . . . . . 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 ) )
4514ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  (
y  e.  ~P S  /\  y  ~<_  om  /\ Disj  z  e.  y  z ) )  ->  S  e.  U. ran sigAlgebra )
46 simpr1 994 . . . . . . . . . 10  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  (
y  e.  ~P S  /\  y  ~<_  om  /\ Disj  z  e.  y  z ) )  ->  y  e.  ~P S )
47 simpr2 995 . . . . . . . . . 10  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  (
y  e.  ~P S  /\  y  ~<_  om  /\ Disj  z  e.  y  z ) )  ->  y  ~<_  om )
48 sigaclcu 26529 . . . . . . . . . 10  |-  ( ( S  e.  U. ran sigAlgebra  /\  y  e.  ~P S  /\  y  ~<_  om )  ->  U. y  e.  S
)
4945, 46, 47, 48syl3anc 1218 . . . . . . . . 9  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  (
y  e.  ~P S  /\  y  ~<_  om  /\ Disj  z  e.  y  z ) )  ->  U. y  e.  S
)
50 fveq2 5686 . . . . . . . . . . 11  |-  ( x  =  U. y  -> 
( M `  x
)  =  ( M `
 U. y ) )
5150oveq1d 6101 . . . . . . . . . 10  |-  ( x  =  U. y  -> 
( ( M `  x ) /𝑒  A )  =  ( ( M `  U. y ) /𝑒  A ) )
52 ovex 6111 . . . . . . . . . 10  |-  ( ( M `  x ) /𝑒  A )  e.  _V
5351, 12, 52fvmpt3i 5773 . . . . . . . . 9  |-  ( U. y  e.  S  ->  ( ( x  e.  S  |->  ( ( M `  x ) /𝑒  A ) ) `  U. y )  =  ( ( M `  U. y ) /𝑒  A ) )
5449, 53syl 16 . . . . . . . 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 ) )
55 simpll 753 . . . . . . . . . 10  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  (
y  e.  ~P S  /\  y  ~<_  om  /\ Disj  z  e.  y  z ) )  ->  M  e.  (measures `  S ) )
56 simpr3 996 . . . . . . . . . 10  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  (
y  e.  ~P S  /\  y  ~<_  om  /\ Disj  z  e.  y  z ) )  -> Disj  z  e.  y  z )
57 measvun 26592 . . . . . . . . . 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 ) )
5855, 46, 47, 56, 57syl112anc 1222 . . . . . . . . 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 ) )
5958oveq1d 6101 . . . . . . . 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 ) )
6054, 59eqtrd 2470 . . . . . . 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 ) )
61 fveq2 5686 . . . . . . . . . . . 12  |-  ( x  =  z  ->  ( M `  x )  =  ( M `  z ) )
6261oveq1d 6101 . . . . . . . . . . 11  |-  ( x  =  z  ->  (
( M `  x
) /𝑒 
A )  =  ( ( M `  z
) /𝑒 
A ) )
6362, 12, 52fvmpt3i 5773 . . . . . . . . . 10  |-  ( z  e.  S  ->  (
( x  e.  S  |->  ( ( M `  x ) /𝑒  A ) ) `  z )  =  ( ( M `  z
) /𝑒 
A ) )
6438, 63syl 16 . . . . . . . . 9  |-  ( ( y  e.  ~P S  /\  z  e.  y
)  ->  ( (
x  e.  S  |->  ( ( M `  x
) /𝑒 
A ) ) `  z )  =  ( ( M `  z
) /𝑒 
A ) )
6564esumeq2dv 26463 . . . . . . . 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 ) )
6646, 65syl 16 . . . . . . 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 ) )
6744, 60, 663eqtr4d 2480 . . . . . 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
) )
6828, 32, 67syl2anc 661 . . . . 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
) )
6968ex 434 . . . 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
) ) )
7069ralrimiva 2794 . . 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 ) ) )
71 ismeas 26582 . . . . . 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
) ) ) ) )
7214, 71syl 16 . . . . 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
) ) ) ) )
7372biimprd 223 . . . 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 )
) )
7473adantr 465 . . 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 )
) )
7513, 27, 70, 74mp3and 1317 . 2  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  RR+ )  ->  ( x  e.  S  |->  ( ( M `  x ) /𝑒  A ) )  e.  (measures `  S ) )
767, 75eqeltrd 2512 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 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2710   _Vcvv 2967    C_ wss 3323   (/)c0 3632   ~Pcpw 3855   U.cuni 4086  Disj wdisj 4257   class class class wbr 4287    e. cmpt 4345   dom cdm 4835   ran crn 4836   -->wf 5409   ` cfv 5413  (class class class)co 6086   omcom 6471    ~<_ cdom 7300   0cc0 9274   +oocpnf 9407   RR+crp 10983   [,]cicc 11295   /𝑒 cxdiv 26060  Σ*cesum 26452  ∘𝑓/𝑐cofc 26506  sigAlgebracsiga 26519  measurescmeas 26578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-disj 4258  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-ioc 11297  df-ico 11298  df-icc 11299  df-fz 11430  df-fzo 11541  df-seq 11799  df-hash 12096  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-tset 14249  df-ple 14250  df-ds 14252  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-ordt 14431  df-xrs 14432  df-mre 14516  df-mrc 14517  df-acs 14519  df-ps 15362  df-tsr 15363  df-mnd 15407  df-mhm 15456  df-submnd 15457  df-cntz 15826  df-cmn 16270  df-fbas 17794  df-fg 17795  df-top 18483  df-bases 18485  df-topon 18486  df-topsp 18487  df-ntr 18604  df-nei 18682  df-cn 18811  df-cnp 18812  df-haus 18899  df-fil 19399  df-fm 19491  df-flim 19492  df-flf 19493  df-tsms 19677  df-xdiv 26061  df-esum 26453  df-ofc 26507  df-siga 26520  df-meas 26579
This theorem is referenced by:  probfinmeasb  26781
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