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Theorem measinb 28429
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 751 . . . 4  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S )  /\  x  e.  S )  ->  M  e.  (measures `  S )
)
2 measbase 28405 . . . . . 6  |-  ( M  e.  (measures `  S
)  ->  S  e.  U.
ran sigAlgebra )
32ad2antrr 723 . . . . 5  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S )  /\  x  e.  S )  ->  S  e.  U. ran sigAlgebra )
4 simpr 459 . . . . 5  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S )  /\  x  e.  S )  ->  x  e.  S )
5 simplr 753 . . . . 5  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S )  /\  x  e.  S )  ->  A  e.  S )
6 inelsiga 28365 . . . . 5  |-  ( ( S  e.  U. ran sigAlgebra  /\  x  e.  S  /\  A  e.  S )  ->  ( x  i^i  A
)  e.  S )
73, 4, 5, 6syl3anc 1226 . . . 4  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S )  /\  x  e.  S )  ->  (
x  i^i  A )  e.  S )
8 measvxrge0 28413 . . . 4  |-  ( ( M  e.  (measures `  S
)  /\  ( x  i^i  A )  e.  S
)  ->  ( M `  ( x  i^i  A
) )  e.  ( 0 [,] +oo )
)
91, 7, 8syl2anc 659 . . 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 6031 . 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 3679 . . . . . . 7  |-  ( x  =  (/)  ->  ( x  i^i  A )  =  ( (/)  i^i  A ) )
14 incom 3677 . . . . . . . 8  |-  ( A  i^i  (/) )  =  (
(/)  i^i  A )
15 in0 3810 . . . . . . . 8  |-  ( A  i^i  (/) )  =  (/)
1614, 15eqtr3i 2485 . . . . . . 7  |-  ( (/)  i^i 
A )  =  (/)
1713, 16syl6eq 2511 . . . . . 6  |-  ( x  =  (/)  ->  ( x  i^i  A )  =  (/) )
1817fveq2d 5852 . . . . 5  |-  ( x  =  (/)  ->  ( M `
 ( x  i^i 
A ) )  =  ( M `  (/) ) )
1918adantl 464 . . . 4  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S )  /\  x  =  (/) )  ->  ( M `  ( x  i^i  A ) )  =  ( M `  (/) ) )
20 measvnul 28414 . . . . 5  |-  ( M  e.  (measures `  S
)  ->  ( M `  (/) )  =  0 )
2120ad2antrr 723 . . . 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 463 . . . 4  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S )  ->  S  e.  U. ran sigAlgebra )
24 0elsiga 28344 . . . 4  |-  ( S  e.  U. ran sigAlgebra  ->  (/)  e.  S
)
2523, 24syl 16 . . 3  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S )  ->  (/)  e.  S
)
26 0red 9586 . . 3  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S )  ->  0  e.  RR )
2712, 22, 25, 26fvmptd 5936 . 2  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S )  ->  (
( x  e.  S  |->  ( M `  (
x  i^i  A )
) ) `  (/) )  =  0 )
28 measinblem 28428 . . . . 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
) ) )
29 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 ) ) ) )
30 ineq1 3679 . . . . . . . 8  |-  ( x  =  U. z  -> 
( x  i^i  A
)  =  ( U. z  i^i  A ) )
3130adantl 464 . . . . . . 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 ) )
3231fveq2d 5852 . . . . . 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 ) ) )
33 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 ) )
3433, 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 )
35 simplr 753 . . . . . . 7  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S
)  /\  z  e.  ~P S )  /\  (
z  ~<_  om  /\ Disj  y  e.  z  y ) )  ->  z  e.  ~P S )
36 simprl 754 . . . . . . 7  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S
)  /\  z  e.  ~P S )  /\  (
z  ~<_  om  /\ Disj  y  e.  z  y ) )  ->  z  ~<_  om )
37 sigaclcu 28347 . . . . . . 7  |-  ( ( S  e.  U. ran sigAlgebra  /\  z  e.  ~P S  /\  z  ~<_  om )  ->  U. z  e.  S
)
3834, 35, 36, 37syl3anc 1226 . . . . . 6  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S
)  /\  z  e.  ~P S )  /\  (
z  ~<_  om  /\ Disj  y  e.  z  y ) )  ->  U. z  e.  S
)
39 simpllr 758 . . . . . . . 8  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S
)  /\  z  e.  ~P S )  /\  (
z  ~<_  om  /\ Disj  y  e.  z  y ) )  ->  A  e.  S
)
40 inelsiga 28365 . . . . . . . 8  |-  ( ( S  e.  U. ran sigAlgebra  /\  U. z  e.  S  /\  A  e.  S )  ->  ( U. z  i^i 
A )  e.  S
)
4134, 38, 39, 40syl3anc 1226 . . . . . . 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
)
42 measvxrge0 28413 . . . . . . 7  |-  ( ( M  e.  (measures `  S
)  /\  ( U. z  i^i  A )  e.  S )  ->  ( M `  ( U. z  i^i  A ) )  e.  ( 0 [,] +oo ) )
4333, 41, 42syl2anc 659 . . . . . 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 )
)
4429, 32, 38, 43fvmptd 5936 . . . . 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
) ) )
45 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 ) ) ) )
46 ineq1 3679 . . . . . . . . . 10  |-  ( x  =  y  ->  (
x  i^i  A )  =  ( y  i^i 
A ) )
4746adantl 464 . . . . . . . . 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 ) )
4847fveq2d 5852 . . . . . . . 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 ) ) )
49 elpwi 4008 . . . . . . . . . 10  |-  ( z  e.  ~P S  -> 
z  C_  S )
5049ad2antlr 724 . . . . . . . . 9  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S
)  /\  z  e.  ~P S )  /\  y  e.  z )  ->  z  C_  S )
51 simpr 459 . . . . . . . . 9  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S
)  /\  z  e.  ~P S )  /\  y  e.  z )  ->  y  e.  z )
5250, 51sseldd 3490 . . . . . . . 8  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S
)  /\  z  e.  ~P S )  /\  y  e.  z )  ->  y  e.  S )
53 simplll 757 . . . . . . . . 9  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S
)  /\  z  e.  ~P S )  /\  y  e.  z )  ->  M  e.  (measures `  S )
)
5453, 2syl 16 . . . . . . . . . 10  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S
)  /\  z  e.  ~P S )  /\  y  e.  z )  ->  S  e.  U. ran sigAlgebra )
55 simpllr 758 . . . . . . . . . 10  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S
)  /\  z  e.  ~P S )  /\  y  e.  z )  ->  A  e.  S )
56 inelsiga 28365 . . . . . . . . . 10  |-  ( ( S  e.  U. ran sigAlgebra  /\  y  e.  S  /\  A  e.  S )  ->  ( y  i^i  A
)  e.  S )
5754, 52, 55, 56syl3anc 1226 . . . . . . . . 9  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S
)  /\  z  e.  ~P S )  /\  y  e.  z )  ->  (
y  i^i  A )  e.  S )
58 measvxrge0 28413 . . . . . . . . 9  |-  ( ( M  e.  (measures `  S
)  /\  ( y  i^i  A )  e.  S
)  ->  ( M `  ( y  i^i  A
) )  e.  ( 0 [,] +oo )
)
5953, 57, 58syl2anc 659 . . . . . . . 8  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S
)  /\  z  e.  ~P S )  /\  y  e.  z )  ->  ( M `  ( y  i^i  A ) )  e.  ( 0 [,] +oo ) )
6045, 48, 52, 59fvmptd 5936 . . . . . . 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 ) ) )
6160esumeq2dv 28267 . . . . . 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 ) ) )
6261adantr 463 . . . . 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 )
) )
6328, 44, 623eqtr4d 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
) )
6463ex 432 . . 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
) ) )
6564ralrimiva 2868 . 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 ) ) )
66 ismeas 28407 . . 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 ) ) ) ) )
6723, 66syl 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 ) ) ) ) )
6811, 27, 65, 67mpbir3and 1177 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804    i^i cin 3460    C_ wss 3461   (/)c0 3783   ~Pcpw 3999   U.cuni 4235  Disj wdisj 4410   class class class wbr 4439    |-> cmpt 4497   ran crn 4989   -->wf 5566   ` cfv 5570  (class class class)co 6270   omcom 6673    ~<_ cdom 7507   RRcr 9480   0cc0 9481   +oocpnf 9614   [,]cicc 11535  Σ*cesum 28256  sigAlgebracsiga 28337  measurescmeas 28403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-ac2 8834  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-disj 4411  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-fi 7863  df-sup 7893  df-oi 7927  df-card 8311  df-acn 8314  df-ac 8488  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ioo 11536  df-ioc 11537  df-ico 11538  df-icc 11539  df-fz 11676  df-fzo 11800  df-fl 11910  df-mod 11979  df-seq 12090  df-exp 12149  df-fac 12336  df-bc 12363  df-hash 12388  df-shft 12982  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-limsup 13376  df-clim 13393  df-rlim 13394  df-sum 13591  df-ef 13885  df-sin 13887  df-cos 13888  df-pi 13890  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-starv 14799  df-sca 14800  df-vsca 14801  df-ip 14802  df-tset 14803  df-ple 14804  df-ds 14806  df-unif 14807  df-hom 14808  df-cco 14809  df-rest 14912  df-topn 14913  df-0g 14931  df-gsum 14932  df-topgen 14933  df-pt 14934  df-prds 14937  df-ordt 14990  df-xrs 14991  df-qtop 14996  df-imas 14997  df-xps 14999  df-mre 15075  df-mrc 15076  df-acs 15078  df-ps 16029  df-tsr 16030  df-plusf 16070  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-mhm 16165  df-submnd 16166  df-grp 16256  df-minusg 16257  df-sbg 16258  df-mulg 16259  df-subg 16397  df-cntz 16554  df-cmn 16999  df-abl 17000  df-mgp 17337  df-ur 17349  df-ring 17395  df-cring 17396  df-subrg 17622  df-abv 17661  df-lmod 17709  df-scaf 17710  df-sra 18013  df-rgmod 18014  df-psmet 18606  df-xmet 18607  df-met 18608  df-bl 18609  df-mopn 18610  df-fbas 18611  df-fg 18612  df-cnfld 18616  df-top 19566  df-bases 19568  df-topon 19569  df-topsp 19570  df-cld 19687  df-ntr 19688  df-cls 19689  df-nei 19766  df-lp 19804  df-perf 19805  df-cn 19895  df-cnp 19896  df-haus 19983  df-tx 20229  df-hmeo 20422  df-fil 20513  df-fm 20605  df-flim 20606  df-flf 20607  df-tmd 20737  df-tgp 20738  df-tsms 20791  df-trg 20828  df-xms 20989  df-ms 20990  df-tms 20991  df-nm 21269  df-ngp 21270  df-nrg 21272  df-nlm 21273  df-ii 21547  df-cncf 21548  df-limc 22436  df-dv 22437  df-log 23110  df-esum 28257  df-siga 28338  df-meas 28404
This theorem is referenced by:  measinb2  28431  totprobd  28629  probmeasb  28633
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