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Theorem measinb 26781
Description: Building a measure restricted to the intersection with a given set. (Contributed by Thierry Arnoux, 25-Dec-2016.)
Assertion
Ref Expression
measinb  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S )  ->  (
x  e.  S  |->  ( M `  ( x  i^i  A ) ) )  e.  (measures `  S
) )
Distinct variable groups:    x, A    x, S    x, M

Proof of Theorem measinb
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll 753 . . . 4  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S )  /\  x  e.  S )  ->  M  e.  (measures `  S )
)
2 measbase 26757 . . . . . 6  |-  ( M  e.  (measures `  S
)  ->  S  e.  U.
ran sigAlgebra )
32ad2antrr 725 . . . . 5  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S )  /\  x  e.  S )  ->  S  e.  U. ran sigAlgebra )
4 simpr 461 . . . . 5  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S )  /\  x  e.  S )  ->  x  e.  S )
5 simplr 754 . . . . 5  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S )  /\  x  e.  S )  ->  A  e.  S )
6 inelsiga 26724 . . . . 5  |-  ( ( S  e.  U. ran sigAlgebra  /\  x  e.  S  /\  A  e.  S )  ->  ( x  i^i  A
)  e.  S )
73, 4, 5, 6syl3anc 1219 . . . 4  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S )  /\  x  e.  S )  ->  (
x  i^i  A )  e.  S )
8 measvxrge0 26765 . . . 4  |-  ( ( M  e.  (measures `  S
)  /\  ( x  i^i  A )  e.  S
)  ->  ( M `  ( x  i^i  A
) )  e.  ( 0 [,] +oo )
)
91, 7, 8syl2anc 661 . . 3  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S )  /\  x  e.  S )  ->  ( M `  ( x  i^i  A ) )  e.  ( 0 [,] +oo ) )
10 eqid 2454 . . 3  |-  ( x  e.  S  |->  ( M `
 ( x  i^i 
A ) ) )  =  ( x  e.  S  |->  ( M `  ( x  i^i  A ) ) )
119, 10fmptd 5977 . 2  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S )  ->  (
x  e.  S  |->  ( M `  ( x  i^i  A ) ) ) : S --> ( 0 [,] +oo ) )
12 eqidd 2455 . . 3  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S )  ->  (
x  e.  S  |->  ( M `  ( x  i^i  A ) ) )  =  ( x  e.  S  |->  ( M `
 ( x  i^i 
A ) ) ) )
13 ineq1 3654 . . . . . . 7  |-  ( x  =  (/)  ->  ( x  i^i  A )  =  ( (/)  i^i  A ) )
14 incom 3652 . . . . . . . 8  |-  ( A  i^i  (/) )  =  (
(/)  i^i  A )
15 in0 3772 . . . . . . . 8  |-  ( A  i^i  (/) )  =  (/)
1614, 15eqtr3i 2485 . . . . . . 7  |-  ( (/)  i^i 
A )  =  (/)
1713, 16syl6eq 2511 . . . . . 6  |-  ( x  =  (/)  ->  ( x  i^i  A )  =  (/) )
1817fveq2d 5804 . . . . 5  |-  ( x  =  (/)  ->  ( M `
 ( x  i^i 
A ) )  =  ( M `  (/) ) )
1918adantl 466 . . . 4  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S )  /\  x  =  (/) )  ->  ( M `  ( x  i^i  A ) )  =  ( M `  (/) ) )
20 measvnul 26766 . . . . 5  |-  ( M  e.  (measures `  S
)  ->  ( M `  (/) )  =  0 )
2120ad2antrr 725 . . . 4  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S )  /\  x  =  (/) )  ->  ( M `  (/) )  =  0 )
2219, 21eqtrd 2495 . . 3  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S )  /\  x  =  (/) )  ->  ( M `  ( x  i^i  A ) )  =  0 )
232adantr 465 . . . 4  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S )  ->  S  e.  U. ran sigAlgebra )
24 0elsiga 26703 . . . 4  |-  ( S  e.  U. ran sigAlgebra  ->  (/)  e.  S
)
2523, 24syl 16 . . 3  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S )  ->  (/)  e.  S
)
26 0re 9498 . . . 4  |-  0  e.  RR
2726a1i 11 . . 3  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S )  ->  0  e.  RR )
2812, 22, 25, 27fvmptd 5889 . 2  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S )  ->  (
( x  e.  S  |->  ( M `  (
x  i^i  A )
) ) `  (/) )  =  0 )
29 measinblem 26780 . . . . 5  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S
)  /\  z  e.  ~P S )  /\  (
z  ~<_  om  /\ Disj  y  e.  z  y ) )  ->  ( M `  ( U. z  i^i  A
) )  = Σ* y  e.  z ( M `  ( y  i^i  A
) ) )
30 eqidd 2455 . . . . . 6  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S
)  /\  z  e.  ~P S )  /\  (
z  ~<_  om  /\ Disj  y  e.  z  y ) )  ->  ( x  e.  S  |->  ( M `  ( x  i^i  A ) ) )  =  ( x  e.  S  |->  ( M `  ( x  i^i  A ) ) ) )
31 ineq1 3654 . . . . . . . 8  |-  ( x  =  U. z  -> 
( x  i^i  A
)  =  ( U. z  i^i  A ) )
3231adantl 466 . . . . . . 7  |-  ( ( ( ( ( M  e.  (measures `  S
)  /\  A  e.  S )  /\  z  e.  ~P S )  /\  ( z  ~<_  om  /\ Disj  y  e.  z  y ) )  /\  x  = 
U. z )  -> 
( x  i^i  A
)  =  ( U. z  i^i  A ) )
3332fveq2d 5804 . . . . . 6  |-  ( ( ( ( ( M  e.  (measures `  S
)  /\  A  e.  S )  /\  z  e.  ~P S )  /\  ( z  ~<_  om  /\ Disj  y  e.  z  y ) )  /\  x  = 
U. z )  -> 
( M `  (
x  i^i  A )
)  =  ( M `
 ( U. z  i^i  A ) ) )
34 simplll 757 . . . . . . . 8  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S
)  /\  z  e.  ~P S )  /\  (
z  ~<_  om  /\ Disj  y  e.  z  y ) )  ->  M  e.  (measures `  S ) )
3534, 2syl 16 . . . . . . 7  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S
)  /\  z  e.  ~P S )  /\  (
z  ~<_  om  /\ Disj  y  e.  z  y ) )  ->  S  e.  U. ran sigAlgebra )
36 simplr 754 . . . . . . 7  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S
)  /\  z  e.  ~P S )  /\  (
z  ~<_  om  /\ Disj  y  e.  z  y ) )  ->  z  e.  ~P S )
37 simprl 755 . . . . . . 7  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S
)  /\  z  e.  ~P S )  /\  (
z  ~<_  om  /\ Disj  y  e.  z  y ) )  ->  z  ~<_  om )
38 sigaclcu 26706 . . . . . . 7  |-  ( ( S  e.  U. ran sigAlgebra  /\  z  e.  ~P S  /\  z  ~<_  om )  ->  U. z  e.  S
)
3935, 36, 37, 38syl3anc 1219 . . . . . 6  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S
)  /\  z  e.  ~P S )  /\  (
z  ~<_  om  /\ Disj  y  e.  z  y ) )  ->  U. z  e.  S
)
40 simpllr 758 . . . . . . . 8  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S
)  /\  z  e.  ~P S )  /\  (
z  ~<_  om  /\ Disj  y  e.  z  y ) )  ->  A  e.  S
)
41 inelsiga 26724 . . . . . . . 8  |-  ( ( S  e.  U. ran sigAlgebra  /\  U. z  e.  S  /\  A  e.  S )  ->  ( U. z  i^i 
A )  e.  S
)
4235, 39, 40, 41syl3anc 1219 . . . . . . 7  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S
)  /\  z  e.  ~P S )  /\  (
z  ~<_  om  /\ Disj  y  e.  z  y ) )  ->  ( U. z  i^i  A )  e.  S
)
43 measvxrge0 26765 . . . . . . 7  |-  ( ( M  e.  (measures `  S
)  /\  ( U. z  i^i  A )  e.  S )  ->  ( M `  ( U. z  i^i  A ) )  e.  ( 0 [,] +oo ) )
4434, 42, 43syl2anc 661 . . . . . 6  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S
)  /\  z  e.  ~P S )  /\  (
z  ~<_  om  /\ Disj  y  e.  z  y ) )  ->  ( M `  ( U. z  i^i  A
) )  e.  ( 0 [,] +oo )
)
4530, 33, 39, 44fvmptd 5889 . . . . 5  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S
)  /\  z  e.  ~P S )  /\  (
z  ~<_  om  /\ Disj  y  e.  z  y ) )  ->  ( ( x  e.  S  |->  ( M `
 ( x  i^i 
A ) ) ) `
 U. z )  =  ( M `  ( U. z  i^i  A
) ) )
46 eqidd 2455 . . . . . . . 8  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S
)  /\  z  e.  ~P S )  /\  y  e.  z )  ->  (
x  e.  S  |->  ( M `  ( x  i^i  A ) ) )  =  ( x  e.  S  |->  ( M `
 ( x  i^i 
A ) ) ) )
47 ineq1 3654 . . . . . . . . . 10  |-  ( x  =  y  ->  (
x  i^i  A )  =  ( y  i^i 
A ) )
4847adantl 466 . . . . . . . . 9  |-  ( ( ( ( ( M  e.  (measures `  S
)  /\  A  e.  S )  /\  z  e.  ~P S )  /\  y  e.  z )  /\  x  =  y
)  ->  ( x  i^i  A )  =  ( y  i^i  A ) )
4948fveq2d 5804 . . . . . . . 8  |-  ( ( ( ( ( M  e.  (measures `  S
)  /\  A  e.  S )  /\  z  e.  ~P S )  /\  y  e.  z )  /\  x  =  y
)  ->  ( M `  ( x  i^i  A
) )  =  ( M `  ( y  i^i  A ) ) )
50 elpwi 3978 . . . . . . . . . 10  |-  ( z  e.  ~P S  -> 
z  C_  S )
5150ad2antlr 726 . . . . . . . . 9  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S
)  /\  z  e.  ~P S )  /\  y  e.  z )  ->  z  C_  S )
52 simpr 461 . . . . . . . . 9  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S
)  /\  z  e.  ~P S )  /\  y  e.  z )  ->  y  e.  z )
5351, 52sseldd 3466 . . . . . . . 8  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S
)  /\  z  e.  ~P S )  /\  y  e.  z )  ->  y  e.  S )
54 simplll 757 . . . . . . . . 9  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S
)  /\  z  e.  ~P S )  /\  y  e.  z )  ->  M  e.  (measures `  S )
)
5554, 2syl 16 . . . . . . . . . 10  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S
)  /\  z  e.  ~P S )  /\  y  e.  z )  ->  S  e.  U. ran sigAlgebra )
56 simpllr 758 . . . . . . . . . 10  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S
)  /\  z  e.  ~P S )  /\  y  e.  z )  ->  A  e.  S )
57 inelsiga 26724 . . . . . . . . . 10  |-  ( ( S  e.  U. ran sigAlgebra  /\  y  e.  S  /\  A  e.  S )  ->  ( y  i^i  A
)  e.  S )
5855, 53, 56, 57syl3anc 1219 . . . . . . . . 9  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S
)  /\  z  e.  ~P S )  /\  y  e.  z )  ->  (
y  i^i  A )  e.  S )
59 measvxrge0 26765 . . . . . . . . 9  |-  ( ( M  e.  (measures `  S
)  /\  ( y  i^i  A )  e.  S
)  ->  ( M `  ( y  i^i  A
) )  e.  ( 0 [,] +oo )
)
6054, 58, 59syl2anc 661 . . . . . . . 8  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S
)  /\  z  e.  ~P S )  /\  y  e.  z )  ->  ( M `  ( y  i^i  A ) )  e.  ( 0 [,] +oo ) )
6146, 49, 53, 60fvmptd 5889 . . . . . . 7  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S
)  /\  z  e.  ~P S )  /\  y  e.  z )  ->  (
( x  e.  S  |->  ( M `  (
x  i^i  A )
) ) `  y
)  =  ( M `
 ( y  i^i 
A ) ) )
6261esumeq2dv 26640 . . . . . 6  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S )  /\  z  e.  ~P S )  -> Σ* y  e.  z ( ( x  e.  S  |->  ( M `
 ( x  i^i 
A ) ) ) `
 y )  = Σ* y  e.  z ( M `
 ( y  i^i 
A ) ) )
6362adantr 465 . . . . 5  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S
)  /\  z  e.  ~P S )  /\  (
z  ~<_  om  /\ Disj  y  e.  z  y ) )  -> Σ* y  e.  z ( ( x  e.  S  |->  ( M `  (
x  i^i  A )
) ) `  y
)  = Σ* y  e.  z ( M `  (
y  i^i  A )
) )
6429, 45, 633eqtr4d 2505 . . . 4  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S
)  /\  z  e.  ~P S )  /\  (
z  ~<_  om  /\ Disj  y  e.  z  y ) )  ->  ( ( x  e.  S  |->  ( M `
 ( x  i^i 
A ) ) ) `
 U. z )  = Σ* y  e.  z ( ( x  e.  S  |->  ( M `  (
x  i^i  A )
) ) `  y
) )
6564ex 434 . . 3  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S )  /\  z  e.  ~P S )  -> 
( ( z  ~<_  om 
/\ Disj  y  e.  z  y )  ->  ( (
x  e.  S  |->  ( M `  ( x  i^i  A ) ) ) `  U. z
)  = Σ* y  e.  z ( ( x  e.  S  |->  ( M `  ( x  i^i  A ) ) ) `  y
) ) )
6665ralrimiva 2830 . 2  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S )  ->  A. z  e.  ~P  S ( ( z  ~<_  om  /\ Disj  y  e.  z  y )  -> 
( ( x  e.  S  |->  ( M `  ( x  i^i  A ) ) ) `  U. z )  = Σ* y  e.  z ( ( x  e.  S  |->  ( M `
 ( x  i^i 
A ) ) ) `
 y ) ) )
67 ismeas 26759 . . 3  |-  ( S  e.  U. ran sigAlgebra  ->  (
( x  e.  S  |->  ( M `  (
x  i^i  A )
) )  e.  (measures `  S )  <->  ( (
x  e.  S  |->  ( M `  ( x  i^i  A ) ) ) : S --> ( 0 [,] +oo )  /\  ( ( x  e.  S  |->  ( M `  ( x  i^i  A ) ) ) `  (/) )  =  0  /\  A. z  e.  ~P  S ( ( z  ~<_  om  /\ Disj  y  e.  z  y )  -> 
( ( x  e.  S  |->  ( M `  ( x  i^i  A ) ) ) `  U. z )  = Σ* y  e.  z ( ( x  e.  S  |->  ( M `
 ( x  i^i 
A ) ) ) `
 y ) ) ) ) )
6823, 67syl 16 . 2  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S )  ->  (
( x  e.  S  |->  ( M `  (
x  i^i  A )
) )  e.  (measures `  S )  <->  ( (
x  e.  S  |->  ( M `  ( x  i^i  A ) ) ) : S --> ( 0 [,] +oo )  /\  ( ( x  e.  S  |->  ( M `  ( x  i^i  A ) ) ) `  (/) )  =  0  /\  A. z  e.  ~P  S ( ( z  ~<_  om  /\ Disj  y  e.  z  y )  -> 
( ( x  e.  S  |->  ( M `  ( x  i^i  A ) ) ) `  U. z )  = Σ* y  e.  z ( ( x  e.  S  |->  ( M `
 ( x  i^i 
A ) ) ) `
 y ) ) ) ) )
6911, 28, 66, 68mpbir3and 1171 1  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S )  ->  (
x  e.  S  |->  ( M `  ( x  i^i  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 2799    i^i cin 3436    C_ wss 3437   (/)c0 3746   ~Pcpw 3969   U.cuni 4200  Disj wdisj 4371   class class class wbr 4401    |-> cmpt 4459   ran crn 4950   -->wf 5523   ` cfv 5527  (class class class)co 6201   omcom 6587    ~<_ cdom 7419   RRcr 9393   0cc0 9394   +oocpnf 9527   [,]cicc 11415  Σ*cesum 26629  sigAlgebracsiga 26696  measurescmeas 26755
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 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-inf2 7959  ax-ac2 8744  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471  ax-pre-sup 9472  ax-addf 9473  ax-mulf 9474
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 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-iin 4283  df-disj 4372  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-se 4789  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-isom 5536  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-of 6431  df-om 6588  df-1st 6688  df-2nd 6689  df-supp 6802  df-recs 6943  df-rdg 6977  df-1o 7031  df-2o 7032  df-oadd 7035  df-er 7212  df-map 7327  df-pm 7328  df-ixp 7375  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-fsupp 7733  df-fi 7773  df-sup 7803  df-oi 7836  df-card 8221  df-acn 8224  df-ac 8398  df-cda 8449  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-div 10106  df-nn 10435  df-2 10492  df-3 10493  df-4 10494  df-5 10495  df-6 10496  df-7 10497  df-8 10498  df-9 10499  df-10 10500  df-n0 10692  df-z 10759  df-dec 10868  df-uz 10974  df-q 11066  df-rp 11104  df-xneg 11201  df-xadd 11202  df-xmul 11203  df-ioo 11416  df-ioc 11417  df-ico 11418  df-icc 11419  df-fz 11556  df-fzo 11667  df-fl 11760  df-mod 11827  df-seq 11925  df-exp 11984  df-fac 12170  df-bc 12197  df-hash 12222  df-shft 12675  df-cj 12707  df-re 12708  df-im 12709  df-sqr 12843  df-abs 12844  df-limsup 13068  df-clim 13085  df-rlim 13086  df-sum 13283  df-ef 13472  df-sin 13474  df-cos 13475  df-pi 13477  df-struct 14295  df-ndx 14296  df-slot 14297  df-base 14298  df-sets 14299  df-ress 14300  df-plusg 14371  df-mulr 14372  df-starv 14373  df-sca 14374  df-vsca 14375  df-ip 14376  df-tset 14377  df-ple 14378  df-ds 14380  df-unif 14381  df-hom 14382  df-cco 14383  df-rest 14481  df-topn 14482  df-0g 14500  df-gsum 14501  df-topgen 14502  df-pt 14503  df-prds 14506  df-ordt 14559  df-xrs 14560  df-qtop 14565  df-imas 14566  df-xps 14568  df-mre 14644  df-mrc 14645  df-acs 14647  df-ps 15490  df-tsr 15491  df-mnd 15535  df-plusf 15536  df-mhm 15584  df-submnd 15585  df-grp 15665  df-minusg 15666  df-sbg 15667  df-mulg 15668  df-subg 15798  df-cntz 15955  df-cmn 16401  df-abl 16402  df-mgp 16715  df-ur 16727  df-rng 16771  df-cring 16772  df-subrg 16987  df-abv 17026  df-lmod 17074  df-scaf 17075  df-sra 17377  df-rgmod 17378  df-psmet 17935  df-xmet 17936  df-met 17937  df-bl 17938  df-mopn 17939  df-fbas 17940  df-fg 17941  df-cnfld 17945  df-top 18636  df-bases 18638  df-topon 18639  df-topsp 18640  df-cld 18756  df-ntr 18757  df-cls 18758  df-nei 18835  df-lp 18873  df-perf 18874  df-cn 18964  df-cnp 18965  df-haus 19052  df-tx 19268  df-hmeo 19461  df-fil 19552  df-fm 19644  df-flim 19645  df-flf 19646  df-tmd 19776  df-tgp 19777  df-tsms 19830  df-trg 19867  df-xms 20028  df-ms 20029  df-tms 20030  df-nm 20308  df-ngp 20309  df-nrg 20311  df-nlm 20312  df-ii 20586  df-cncf 20587  df-limc 21475  df-dv 21476  df-log 22142  df-esum 26630  df-siga 26697  df-meas 26756
This theorem is referenced by:  measinb2  26783  totprobd  26954  probmeasb  26958
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