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Theorem measdivcst 26783
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 26688 . . 3  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  RR+ )  ->  ( M𝑓/𝑐 /𝑒  A )  =  ( x  e.  dom  M  |->  ( ( M `  x ) /𝑒  A ) ) )
2 measfrge0 26761 . . . . . 6  |-  ( M  e.  (measures `  S
)  ->  M : S
--> ( 0 [,] +oo ) )
3 fdm 5670 . . . . . 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 4480 . . 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 2495 . 2  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  RR+ )  ->  ( M𝑓/𝑐 /𝑒  A )  =  ( x  e.  S  |->  ( ( M `  x
) /𝑒 
A ) ) )
8 measvxrge0 26763 . . . . . 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 26254 . . . 4  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  RR+ )  /\  x  e.  S )  ->  (
( M `  x
) /𝑒 
A )  e.  ( 0 [,] +oo )
)
12 eqid 2454 . . . 4  |-  ( x  e.  S  |->  ( ( M `  x ) /𝑒  A ) )  =  ( x  e.  S  |->  ( ( M `  x
) /𝑒 
A ) )
1311, 12fmptd 5975 . . 3  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  RR+ )  ->  ( x  e.  S  |->  ( ( M `  x ) /𝑒  A ) ) : S --> ( 0 [,] +oo ) )
14 measbase 26755 . . . . . . 7  |-  ( M  e.  (measures `  S
)  ->  S  e.  U.
ran sigAlgebra )
15 0elsiga 26701 . . . . . . 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 6224 . . . . 5  |-  ( ( M `  (/) ) /𝑒  A )  e.  _V
19 fveq2 5798 . . . . . . 7  |-  ( x  =  (/)  ->  ( M `
 x )  =  ( M `  (/) ) )
2019oveq1d 6214 . . . . . 6  |-  ( x  =  (/)  ->  ( ( M `  x ) /𝑒  A )  =  ( ( M `  (/) ) /𝑒  A ) )
2120, 12fvmptg 5880 . . . . 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 26764 . . . . . 6  |-  ( M  e.  (measures `  S
)  ->  ( M `  (/) )  =  0 )
2423oveq1d 6214 . . . . 5  |-  ( M  e.  (measures `  S
)  ->  ( ( M `  (/) ) /𝑒  A )  =  ( 0 /𝑒  A ) )
25 xdiv0rp 26249 . . . . 5  |-  ( A  e.  RR+  ->  ( 0 /𝑒  A )  =  0 )
2624, 25sylan9eq 2515 . . . 4  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  RR+ )  ->  ( ( M `  (/) ) /𝑒  A )  =  0 )
2722, 26eqtrd 2495 . . 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 3079 . . . . . . . . . 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 3974 . . . . . . . . . . . 12  |-  ( y  e.  ~P S  <->  y  C_  S )
37 ssel2 3458 . . . . . . . . . . . 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 26763 . . . . . . . . . 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 26676 . . . . . . . 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 26704 . . . . . . . . . 10  |-  ( ( S  e.  U. ran sigAlgebra  /\  y  e.  ~P S  /\  y  ~<_  om )  ->  U. y  e.  S
)
4945, 46, 47, 48syl3anc 1219 . . . . . . . . 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 5798 . . . . . . . . . . 11  |-  ( x  =  U. y  -> 
( M `  x
)  =  ( M `
 U. y ) )
5150oveq1d 6214 . . . . . . . . . 10  |-  ( x  =  U. y  -> 
( ( M `  x ) /𝑒  A )  =  ( ( M `  U. y ) /𝑒  A ) )
52 ovex 6224 . . . . . . . . . 10  |-  ( ( M `  x ) /𝑒  A )  e.  _V
5351, 12, 52fvmpt3i 5886 . . . . . . . . 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 26767 . . . . . . . . . 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 1223 . . . . . . . . 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 6214 . . . . . . . 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 2495 . . . . . . 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 5798 . . . . . . . . . . . 12  |-  ( x  =  z  ->  ( M `  x )  =  ( M `  z ) )
6261oveq1d 6214 . . . . . . . . . . 11  |-  ( x  =  z  ->  (
( M `  x
) /𝑒 
A )  =  ( ( M `  z
) /𝑒 
A ) )
6362, 12, 52fvmpt3i 5886 . . . . . . . . . 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 26638 . . . . . . . 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 2505 . . . . . 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 2829 . . 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 26757 . . . . . 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 1318 . 2  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  RR+ )  ->  ( x  e.  S  |->  ( ( M `  x ) /𝑒  A ) )  e.  (measures `  S ) )
767, 75eqeltrd 2542 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 1370    e. wcel 1758   A.wral 2798   _Vcvv 3076    C_ wss 3435   (/)c0 3744   ~Pcpw 3967   U.cuni 4198  Disj wdisj 4369   class class class wbr 4399    |-> cmpt 4457   dom cdm 4947   ran crn 4948   -->wf 5521   ` cfv 5525  (class class class)co 6199   omcom 6585    ~<_ cdom 7417   0cc0 9392   +oocpnf 9525   RR+crp 11101   [,]cicc 11413   /𝑒 cxdiv 26236  Σ*cesum 26627  ∘𝑓/𝑐cofc 26681  sigAlgebracsiga 26694  measurescmeas 26753
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469  ax-pre-sup 9470
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-iin 4281  df-disj 4370  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-se 4787  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-of 6429  df-om 6586  df-1st 6686  df-2nd 6687  df-supp 6800  df-recs 6941  df-rdg 6975  df-1o 7029  df-oadd 7033  df-er 7210  df-map 7325  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-fsupp 7731  df-fi 7771  df-sup 7801  df-oi 7834  df-card 8219  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-div 10104  df-nn 10433  df-2 10490  df-3 10491  df-4 10492  df-5 10493  df-6 10494  df-7 10495  df-8 10496  df-9 10497  df-10 10498  df-n0 10690  df-z 10757  df-dec 10866  df-uz 10972  df-q 11064  df-rp 11102  df-xneg 11199  df-xadd 11200  df-xmul 11201  df-ioo 11414  df-ioc 11415  df-ico 11416  df-icc 11417  df-fz 11554  df-fzo 11665  df-seq 11923  df-hash 12220  df-struct 14293  df-ndx 14294  df-slot 14295  df-base 14296  df-sets 14297  df-ress 14298  df-plusg 14369  df-mulr 14370  df-tset 14375  df-ple 14376  df-ds 14378  df-rest 14479  df-topn 14480  df-0g 14498  df-gsum 14499  df-topgen 14500  df-ordt 14557  df-xrs 14558  df-mre 14642  df-mrc 14643  df-acs 14645  df-ps 15488  df-tsr 15489  df-mnd 15533  df-mhm 15582  df-submnd 15583  df-cntz 15953  df-cmn 16399  df-fbas 17938  df-fg 17939  df-top 18634  df-bases 18636  df-topon 18637  df-topsp 18638  df-ntr 18755  df-nei 18833  df-cn 18962  df-cnp 18963  df-haus 19050  df-fil 19550  df-fm 19642  df-flim 19643  df-flf 19644  df-tsms 19828  df-xdiv 26237  df-esum 26628  df-ofc 26682  df-siga 26695  df-meas 26754
This theorem is referenced by:  probfinmeasb  26955
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