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Theorem measinb 26587
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 26563 . . . . . 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 26530 . . . . 5  |-  ( ( S  e.  U. ran sigAlgebra  /\  x  e.  S  /\  A  e.  S )  ->  ( x  i^i  A
)  e.  S )
73, 4, 5, 6syl3anc 1218 . . . 4  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S )  /\  x  e.  S )  ->  (
x  i^i  A )  e.  S )
8 measvxrge0 26571 . . . 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 2438 . . 3  |-  ( x  e.  S  |->  ( M `
 ( x  i^i 
A ) ) )  =  ( x  e.  S  |->  ( M `  ( x  i^i  A ) ) )
119, 10fmptd 5862 . 2  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S )  ->  (
x  e.  S  |->  ( M `  ( x  i^i  A ) ) ) : S --> ( 0 [,] +oo ) )
12 eqidd 2439 . . 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 3540 . . . . . . 7  |-  ( x  =  (/)  ->  ( x  i^i  A )  =  ( (/)  i^i  A ) )
14 incom 3538 . . . . . . . 8  |-  ( A  i^i  (/) )  =  (
(/)  i^i  A )
15 in0 3658 . . . . . . . 8  |-  ( A  i^i  (/) )  =  (/)
1614, 15eqtr3i 2460 . . . . . . 7  |-  ( (/)  i^i 
A )  =  (/)
1713, 16syl6eq 2486 . . . . . 6  |-  ( x  =  (/)  ->  ( x  i^i  A )  =  (/) )
1817fveq2d 5690 . . . . 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 26572 . . . . 5  |-  ( M  e.  (measures `  S
)  ->  ( M `  (/) )  =  0 )
2120ad2antrr 725 . . . 4  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S )  /\  x  =  (/) )  ->  ( M `  (/) )  =  0 )
2219, 21eqtrd 2470 . . 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 26509 . . . 4  |-  ( S  e.  U. ran sigAlgebra  ->  (/)  e.  S
)
2523, 24syl 16 . . 3  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S )  ->  (/)  e.  S
)
26 0re 9378 . . . 4  |-  0  e.  RR
2726a1i 11 . . 3  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S )  ->  0  e.  RR )
2812, 22, 25, 27fvmptd 5774 . 2  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S )  ->  (
( x  e.  S  |->  ( M `  (
x  i^i  A )
) ) `  (/) )  =  0 )
29 measinblem 26586 . . . . 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 2439 . . . . . 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 3540 . . . . . . . 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 5690 . . . . . 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 26512 . . . . . . 7  |-  ( ( S  e.  U. ran sigAlgebra  /\  z  e.  ~P S  /\  z  ~<_  om )  ->  U. z  e.  S
)
3935, 36, 37, 38syl3anc 1218 . . . . . 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 26530 . . . . . . . 8  |-  ( ( S  e.  U. ran sigAlgebra  /\  U. z  e.  S  /\  A  e.  S )  ->  ( U. z  i^i 
A )  e.  S
)
4235, 39, 40, 41syl3anc 1218 . . . . . . 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 26571 . . . . . . 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 5774 . . . . 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 2439 . . . . . . . 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 3540 . . . . . . . . . 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 5690 . . . . . . . 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 3864 . . . . . . . . . 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 3352 . . . . . . . 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 26530 . . . . . . . . . 10  |-  ( ( S  e.  U. ran sigAlgebra  /\  y  e.  S  /\  A  e.  S )  ->  ( y  i^i  A
)  e.  S )
5855, 53, 56, 57syl3anc 1218 . . . . . . . . 9  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S
)  /\  z  e.  ~P S )  /\  y  e.  z )  ->  (
y  i^i  A )  e.  S )
59 measvxrge0 26571 . . . . . . . . 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 5774 . . . . . . 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 26446 . . . . . 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 2480 . . . 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 2794 . 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 26565 . . 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 1369    e. wcel 1756   A.wral 2710    i^i cin 3322    C_ wss 3323   (/)c0 3632   ~Pcpw 3855   U.cuni 4086  Disj wdisj 4257   class class class wbr 4287    e. cmpt 4345   ran crn 4836   -->wf 5409   ` cfv 5413  (class class class)co 6086   omcom 6471    ~<_ cdom 7300   RRcr 9273   0cc0 9274   +oocpnf 9407   [,]cicc 11295  Σ*cesum 26435  sigAlgebracsiga 26502  measurescmeas 26561
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-inf2 7839  ax-ac2 8624  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  ax-addf 9353  ax-mulf 9354
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-2o 6913  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-ixp 7256  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-acn 8104  df-ac 8278  df-cda 8329  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-fl 11634  df-mod 11701  df-seq 11799  df-exp 11858  df-fac 12044  df-bc 12071  df-hash 12096  df-shft 12548  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-limsup 12941  df-clim 12958  df-rlim 12959  df-sum 13156  df-ef 13345  df-sin 13347  df-cos 13348  df-pi 13350  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-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-pt 14375  df-prds 14378  df-ordt 14431  df-xrs 14432  df-qtop 14437  df-imas 14438  df-xps 14440  df-mre 14516  df-mrc 14517  df-acs 14519  df-ps 15362  df-tsr 15363  df-mnd 15407  df-plusf 15408  df-mhm 15456  df-submnd 15457  df-grp 15536  df-minusg 15537  df-sbg 15538  df-mulg 15539  df-subg 15669  df-cntz 15826  df-cmn 16270  df-abl 16271  df-mgp 16580  df-ur 16592  df-rng 16635  df-cring 16636  df-subrg 16841  df-abv 16880  df-lmod 16928  df-scaf 16929  df-sra 17230  df-rgmod 17231  df-psmet 17784  df-xmet 17785  df-met 17786  df-bl 17787  df-mopn 17788  df-fbas 17789  df-fg 17790  df-cnfld 17794  df-top 18478  df-bases 18480  df-topon 18481  df-topsp 18482  df-cld 18598  df-ntr 18599  df-cls 18600  df-nei 18677  df-lp 18715  df-perf 18716  df-cn 18806  df-cnp 18807  df-haus 18894  df-tx 19110  df-hmeo 19303  df-fil 19394  df-fm 19486  df-flim 19487  df-flf 19488  df-tmd 19618  df-tgp 19619  df-tsms 19672  df-trg 19709  df-xms 19870  df-ms 19871  df-tms 19872  df-nm 20150  df-ngp 20151  df-nrg 20153  df-nlm 20154  df-ii 20428  df-cncf 20429  df-limc 21316  df-dv 21317  df-log 21983  df-esum 26436  df-siga 26503  df-meas 26562
This theorem is referenced by:  measinb2  26589  totprobd  26761  probmeasb  26765
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