Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  measinb Unicode version

Theorem measinb 24528
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 731 . . . 4  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S )  /\  x  e.  S )  ->  M  e.  (measures `  S )
)
2 measbase 24504 . . . . . 6  |-  ( M  e.  (measures `  S
)  ->  S  e.  U.
ran sigAlgebra )
32ad2antrr 707 . . . . 5  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S )  /\  x  e.  S )  ->  S  e.  U. ran sigAlgebra )
4 simpr 448 . . . . 5  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S )  /\  x  e.  S )  ->  x  e.  S )
5 simplr 732 . . . . 5  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S )  /\  x  e.  S )  ->  A  e.  S )
6 inelsiga 24471 . . . . 5  |-  ( ( S  e.  U. ran sigAlgebra  /\  x  e.  S  /\  A  e.  S )  ->  ( x  i^i  A
)  e.  S )
73, 4, 5, 6syl3anc 1184 . . . 4  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S )  /\  x  e.  S )  ->  (
x  i^i  A )  e.  S )
8 measvxrge0 24512 . . . 4  |-  ( ( M  e.  (measures `  S
)  /\  ( x  i^i  A )  e.  S
)  ->  ( M `  ( x  i^i  A
) )  e.  ( 0 [,]  +oo )
)
91, 7, 8syl2anc 643 . . 3  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S )  /\  x  e.  S )  ->  ( M `  ( x  i^i  A ) )  e.  ( 0 [,]  +oo ) )
10 eqid 2404 . . 3  |-  ( x  e.  S  |->  ( M `
 ( x  i^i 
A ) ) )  =  ( x  e.  S  |->  ( M `  ( x  i^i  A ) ) )
119, 10fmptd 5852 . 2  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S )  ->  (
x  e.  S  |->  ( M `  ( x  i^i  A ) ) ) : S --> ( 0 [,]  +oo ) )
12 eqidd 2405 . . 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 3495 . . . . . . 7  |-  ( x  =  (/)  ->  ( x  i^i  A )  =  ( (/)  i^i  A ) )
14 incom 3493 . . . . . . . 8  |-  ( A  i^i  (/) )  =  (
(/)  i^i  A )
15 in0 3613 . . . . . . . 8  |-  ( A  i^i  (/) )  =  (/)
1614, 15eqtr3i 2426 . . . . . . 7  |-  ( (/)  i^i 
A )  =  (/)
1713, 16syl6eq 2452 . . . . . 6  |-  ( x  =  (/)  ->  ( x  i^i  A )  =  (/) )
1817fveq2d 5691 . . . . 5  |-  ( x  =  (/)  ->  ( M `
 ( x  i^i 
A ) )  =  ( M `  (/) ) )
1918adantl 453 . . . 4  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S )  /\  x  =  (/) )  ->  ( M `  ( x  i^i  A ) )  =  ( M `  (/) ) )
20 measvnul 24513 . . . . 5  |-  ( M  e.  (measures `  S
)  ->  ( M `  (/) )  =  0 )
2120ad2antrr 707 . . . 4  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S )  /\  x  =  (/) )  ->  ( M `  (/) )  =  0 )
2219, 21eqtrd 2436 . . 3  |-  ( ( ( M  e.  (measures `  S )  /\  A  e.  S )  /\  x  =  (/) )  ->  ( M `  ( x  i^i  A ) )  =  0 )
232adantr 452 . . . 4  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S )  ->  S  e.  U. ran sigAlgebra )
24 0elsiga 24450 . . . 4  |-  ( S  e.  U. ran sigAlgebra  ->  (/)  e.  S
)
2523, 24syl 16 . . 3  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S )  ->  (/)  e.  S
)
26 0re 9047 . . . 4  |-  0  e.  RR
2726a1i 11 . . 3  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S )  ->  0  e.  RR )
2812, 22, 25, 27fvmptd 5769 . 2  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S )  ->  (
( x  e.  S  |->  ( M `  (
x  i^i  A )
) ) `  (/) )  =  0 )
29 measinblem 24527 . . . . 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 2405 . . . . . 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 3495 . . . . . . . 8  |-  ( x  =  U. z  -> 
( x  i^i  A
)  =  ( U. z  i^i  A ) )
3231adantl 453 . . . . . . 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 5691 . . . . . 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 735 . . . . . . . 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 732 . . . . . . 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 733 . . . . . . 7  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S
)  /\  z  e.  ~P S )  /\  (
z  ~<_  om  /\ Disj  y  e.  z y ) )  ->  z  ~<_  om )
38 sigaclcu 24453 . . . . . . 7  |-  ( ( S  e.  U. ran sigAlgebra  /\  z  e.  ~P S  /\  z  ~<_  om )  ->  U. z  e.  S
)
3935, 36, 37, 38syl3anc 1184 . . . . . 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 736 . . . . . . . 8  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S
)  /\  z  e.  ~P S )  /\  (
z  ~<_  om  /\ Disj  y  e.  z y ) )  ->  A  e.  S
)
41 inelsiga 24471 . . . . . . . 8  |-  ( ( S  e.  U. ran sigAlgebra  /\  U. z  e.  S  /\  A  e.  S )  ->  ( U. z  i^i 
A )  e.  S
)
4235, 39, 40, 41syl3anc 1184 . . . . . . 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 24512 . . . . . . 7  |-  ( ( M  e.  (measures `  S
)  /\  ( U. z  i^i  A )  e.  S )  ->  ( M `  ( U. z  i^i  A ) )  e.  ( 0 [,] 
+oo ) )
4434, 42, 43syl2anc 643 . . . . . 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 5769 . . . . 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 2405 . . . . . . . 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 3495 . . . . . . . . . 10  |-  ( x  =  y  ->  (
x  i^i  A )  =  ( y  i^i 
A ) )
4847adantl 453 . . . . . . . . 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 5691 . . . . . . . 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 3767 . . . . . . . . . 10  |-  ( z  e.  ~P S  -> 
z  C_  S )
5150ad2antlr 708 . . . . . . . . 9  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S
)  /\  z  e.  ~P S )  /\  y  e.  z )  ->  z  C_  S )
52 simpr 448 . . . . . . . . 9  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S
)  /\  z  e.  ~P S )  /\  y  e.  z )  ->  y  e.  z )
5351, 52sseldd 3309 . . . . . . . 8  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S
)  /\  z  e.  ~P S )  /\  y  e.  z )  ->  y  e.  S )
54 simplll 735 . . . . . . . . 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 736 . . . . . . . . . 10  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S
)  /\  z  e.  ~P S )  /\  y  e.  z )  ->  A  e.  S )
57 inelsiga 24471 . . . . . . . . . 10  |-  ( ( S  e.  U. ran sigAlgebra  /\  y  e.  S  /\  A  e.  S )  ->  ( y  i^i  A
)  e.  S )
5855, 53, 56, 57syl3anc 1184 . . . . . . . . 9  |-  ( ( ( ( M  e.  (measures `  S )  /\  A  e.  S
)  /\  z  e.  ~P S )  /\  y  e.  z )  ->  (
y  i^i  A )  e.  S )
59 measvxrge0 24512 . . . . . . . . 9  |-  ( ( M  e.  (measures `  S
)  /\  ( y  i^i  A )  e.  S
)  ->  ( M `  ( y  i^i  A
) )  e.  ( 0 [,]  +oo )
)
6054, 58, 59syl2anc 643 . . . . . . . 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 5769 . . . . . . 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 24388 . . . . . 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 452 . . . . 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 2446 . . . 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 424 . . 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 2749 . 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 24506 . . 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 1137 1  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S )  ->  (
x  e.  S  |->  ( M `  ( x  i^i  A ) ) )  e.  (measures `  S
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666    i^i cin 3279    C_ wss 3280   (/)c0 3588   ~Pcpw 3759   U.cuni 3975  Disj wdisj 4142   class class class wbr 4172    e. cmpt 4226   omcom 4804   ran crn 4838   -->wf 5409   ` cfv 5413  (class class class)co 6040    ~<_ cdom 7066   RRcr 8945   0cc0 8946    +oocpnf 9073   [,]cicc 10875  Σ*cesum 24377  sigAlgebracsiga 24443  measurescmeas 24502
This theorem is referenced by:  measinb2  24530  totprobd  24637  probmeasb  24641
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-ac2 8299  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-disj 4143  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  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-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-acn 7785  df-ac 7953  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435  df-ef 12625  df-sin 12627  df-cos 12628  df-pi 12630  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-ordt 13680  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-ps 14584  df-tsr 14585  df-mnd 14645  df-plusf 14646  df-mhm 14693  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-mulg 14770  df-subg 14896  df-cntz 15071  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-cring 15619  df-ur 15620  df-subrg 15821  df-abv 15860  df-lmod 15907  df-scaf 15908  df-sra 16199  df-rgmod 16200  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-tmd 18055  df-tgp 18056  df-tsms 18109  df-trg 18142  df-xms 18303  df-ms 18304  df-tms 18305  df-nm 18583  df-ngp 18584  df-nrg 18586  df-nlm 18587  df-ii 18860  df-cncf 18861  df-limc 19706  df-dv 19707  df-log 20407  df-esum 24378  df-siga 24444  df-meas 24503
  Copyright terms: Public domain W3C validator