HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  occllem Unicode version

Theorem occllem 21882
Description: Lemma for occl 21883. (Contributed by NM, 7-Aug-2000.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
occl.1  |-  ( ph  ->  A  C_  ~H )
occl.2  |-  ( ph  ->  F  e.  Cauchy )
occl.3  |-  ( ph  ->  F : NN --> ( _|_ `  A ) )
occl.4  |-  ( ph  ->  B  e.  A )
Assertion
Ref Expression
occllem  |-  ( ph  ->  ( (  ~~>v  `  F
)  .ih  B )  =  0 )

Proof of Theorem occllem
Dummy variables  x  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . . 4  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
21cnfldhaus 18294 . . 3  |-  ( TopOpen ` fld )  e.  Haus
32a1i 10 . 2  |-  ( ph  ->  ( TopOpen ` fld )  e.  Haus )
4 occl.2 . . . . . . 7  |-  ( ph  ->  F  e.  Cauchy )
5 ax-hcompl 21781 . . . . . . 7  |-  ( F  e.  Cauchy  ->  E. x  e.  ~H  F  ~~>v  x )
6 hlimf 21817 . . . . . . . . . 10  |-  ~~>v  : dom  ~~>v  --> ~H
7 ffn 5389 . . . . . . . . . 10  |-  (  ~~>v  : dom  ~~>v  --> ~H  ->  ~~>v  Fn  dom  ~~>v  )
86, 7ax-mp 8 . . . . . . . . 9  |-  ~~>v  Fn  dom  ~~>v
9 fnbr 5346 . . . . . . . . 9  |-  ( ( 
~~>v  Fn  dom  ~~>v  /\  F  ~~>v  x )  ->  F  e.  dom  ~~>v  )
108, 9mpan 651 . . . . . . . 8  |-  ( F 
~~>v  x  ->  F  e.  dom 
~~>v  )
1110rexlimivw 2663 . . . . . . 7  |-  ( E. x  e.  ~H  F  ~~>v  x  ->  F  e.  dom  ~~>v  )
124, 5, 113syl 18 . . . . . 6  |-  ( ph  ->  F  e.  dom  ~~>v  )
13 ffun 5391 . . . . . . 7  |-  (  ~~>v  : dom  ~~>v  --> ~H  ->  Fun  ~~>v  )
14 funfvbrb 5638 . . . . . . 7  |-  ( Fun  ~~>v 
->  ( F  e.  dom  ~~>v  <->  F  ~~>v  (  ~~>v  `  F )
) )
156, 13, 14mp2b 9 . . . . . 6  |-  ( F  e.  dom  ~~>v  <->  F  ~~>v  (  ~~>v  `  F ) )
1612, 15sylib 188 . . . . 5  |-  ( ph  ->  F  ~~>v  (  ~~>v  `  F
) )
17 eqid 2283 . . . . . . . 8  |-  <. <.  +h  ,  .h  >. ,  normh >.  =  <. <.  +h  ,  .h  >. ,  normh >.
18 eqid 2283 . . . . . . . . 9  |-  ( normh  o. 
-h  )  =  (
normh  o.  -h  )
1917, 18hhims 21751 . . . . . . . 8  |-  ( normh  o. 
-h  )  =  (
IndMet `  <. <.  +h  ,  .h  >. ,  normh >. )
20 eqid 2283 . . . . . . . 8  |-  ( MetOpen `  ( normh  o.  -h  )
)  =  ( MetOpen `  ( normh  o.  -h  )
)
2117, 19, 20hhlm 21778 . . . . . . 7  |-  ~~>v  =  ( ( ~~> t `  ( MetOpen
`  ( normh  o.  -h  ) ) )  |`  ( ~H  ^m  NN ) )
22 resss 4979 . . . . . . 7  |-  ( ( ~~> t `  ( MetOpen `  ( normh  o.  -h  )
) )  |`  ( ~H  ^m  NN ) ) 
C_  ( ~~> t `  ( MetOpen `  ( normh  o. 
-h  ) ) )
2321, 22eqsstri 3208 . . . . . 6  |-  ~~>v  C_  ( ~~> t `  ( MetOpen `  ( normh  o.  -h  ) ) )
2423ssbri 4065 . . . . 5  |-  ( F 
~~>v  (  ~~>v  `  F )  ->  F ( ~~> t `  ( MetOpen `  ( normh  o. 
-h  ) ) ) (  ~~>v  `  F )
)
2516, 24syl 15 . . . 4  |-  ( ph  ->  F ( ~~> t `  ( MetOpen `  ( normh  o. 
-h  ) ) ) (  ~~>v  `  F )
)
2618hilxmet 21774 . . . . . 6  |-  ( normh  o. 
-h  )  e.  ( * Met `  ~H )
2720mopntopon 17985 . . . . . 6  |-  ( (
normh  o.  -h  )  e.  ( * Met `  ~H )  ->  ( MetOpen `  ( normh  o.  -h  ) )  e.  (TopOn `  ~H ) )
2826, 27mp1i 11 . . . . 5  |-  ( ph  ->  ( MetOpen `  ( normh  o. 
-h  ) )  e.  (TopOn `  ~H )
)
2928cnmptid 17355 . . . . 5  |-  ( ph  ->  ( x  e.  ~H  |->  x )  e.  ( ( MetOpen `  ( normh  o. 
-h  ) )  Cn  ( MetOpen `  ( normh  o. 
-h  ) ) ) )
30 occl.1 . . . . . . 7  |-  ( ph  ->  A  C_  ~H )
31 occl.4 . . . . . . 7  |-  ( ph  ->  B  e.  A )
3230, 31sseldd 3181 . . . . . 6  |-  ( ph  ->  B  e.  ~H )
3328, 28, 32cnmptc 17356 . . . . 5  |-  ( ph  ->  ( x  e.  ~H  |->  B )  e.  ( ( MetOpen `  ( normh  o. 
-h  ) )  Cn  ( MetOpen `  ( normh  o. 
-h  ) ) ) )
3417hhnv 21744 . . . . . 6  |-  <. <.  +h  ,  .h  >. ,  normh >.  e.  NrmCVec
3517hhip 21756 . . . . . . 7  |-  .ih  =  ( .i OLD `  <. <.  +h  ,  .h  >. ,  normh >.
)
3635, 19, 20, 1dipcn 21296 . . . . . 6  |-  ( <. <.  +h  ,  .h  >. , 
normh >.  e.  NrmCVec  ->  .ih  e.  ( ( ( MetOpen `  ( normh  o.  -h  )
)  tX  ( MetOpen `  ( normh  o.  -h  )
) )  Cn  ( TopOpen
` fld
) ) )
3734, 36mp1i 11 . . . . 5  |-  ( ph  ->  .ih  e.  ( ( ( MetOpen `  ( normh  o. 
-h  ) )  tX  ( MetOpen `  ( normh  o. 
-h  ) ) )  Cn  ( TopOpen ` fld ) ) )
3828, 29, 33, 37cnmpt12f 17360 . . . 4  |-  ( ph  ->  ( x  e.  ~H  |->  ( x  .ih  B ) )  e.  ( (
MetOpen `  ( normh  o.  -h  ) )  Cn  ( TopOpen
` fld
) ) )
3925, 38lmcn 17033 . . 3  |-  ( ph  ->  ( ( x  e. 
~H  |->  ( x  .ih  B ) )  o.  F
) ( ~~> t `  ( TopOpen ` fld ) ) ( ( x  e.  ~H  |->  ( x  .ih  B ) ) `  (  ~~>v  `  F ) ) )
40 occl.3 . . . . . . . . . . 11  |-  ( ph  ->  F : NN --> ( _|_ `  A ) )
41 ffvelrn 5663 . . . . . . . . . . 11  |-  ( ( F : NN --> ( _|_ `  A )  /\  k  e.  NN )  ->  ( F `  k )  e.  ( _|_ `  A
) )
4240, 41sylan 457 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 k )  e.  ( _|_ `  A
) )
43 ocel 21860 . . . . . . . . . . . 12  |-  ( A 
C_  ~H  ->  ( ( F `  k )  e.  ( _|_ `  A
)  <->  ( ( F `
 k )  e. 
~H  /\  A. x  e.  A  ( ( F `  k )  .ih  x )  =  0 ) ) )
4430, 43syl 15 . . . . . . . . . . 11  |-  ( ph  ->  ( ( F `  k )  e.  ( _|_ `  A )  <-> 
( ( F `  k )  e.  ~H  /\ 
A. x  e.  A  ( ( F `  k )  .ih  x
)  =  0 ) ) )
4544adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( F `  k )  e.  ( _|_ `  A
)  <->  ( ( F `
 k )  e. 
~H  /\  A. x  e.  A  ( ( F `  k )  .ih  x )  =  0 ) ) )
4642, 45mpbid 201 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( F `  k )  e.  ~H  /\  A. x  e.  A  (
( F `  k
)  .ih  x )  =  0 ) )
4746simpld 445 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 k )  e. 
~H )
48 oveq1 5865 . . . . . . . . 9  |-  ( x  =  ( F `  k )  ->  (
x  .ih  B )  =  ( ( F `
 k )  .ih  B ) )
49 eqid 2283 . . . . . . . . 9  |-  ( x  e.  ~H  |->  ( x 
.ih  B ) )  =  ( x  e. 
~H  |->  ( x  .ih  B ) )
50 ovex 5883 . . . . . . . . 9  |-  ( ( F `  k ) 
.ih  B )  e. 
_V
5148, 49, 50fvmpt 5602 . . . . . . . 8  |-  ( ( F `  k )  e.  ~H  ->  (
( x  e.  ~H  |->  ( x  .ih  B ) ) `  ( F `
 k ) )  =  ( ( F `
 k )  .ih  B ) )
5247, 51syl 15 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( x  e.  ~H  |->  ( x  .ih  B ) ) `  ( F `
 k ) )  =  ( ( F `
 k )  .ih  B ) )
5331adantr 451 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  B  e.  A )
5446simprd 449 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  A. x  e.  A  ( ( F `  k )  .ih  x )  =  0 )
55 oveq2 5866 . . . . . . . . . 10  |-  ( x  =  B  ->  (
( F `  k
)  .ih  x )  =  ( ( F `
 k )  .ih  B ) )
5655eqeq1d 2291 . . . . . . . . 9  |-  ( x  =  B  ->  (
( ( F `  k )  .ih  x
)  =  0  <->  (
( F `  k
)  .ih  B )  =  0 ) )
5756rspcv 2880 . . . . . . . 8  |-  ( B  e.  A  ->  ( A. x  e.  A  ( ( F `  k )  .ih  x
)  =  0  -> 
( ( F `  k )  .ih  B
)  =  0 ) )
5853, 54, 57sylc 56 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( F `  k ) 
.ih  B )  =  0 )
5952, 58eqtrd 2315 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( x  e.  ~H  |->  ( x  .ih  B ) ) `  ( F `
 k ) )  =  0 )
60 ocss 21864 . . . . . . . . 9  |-  ( A 
C_  ~H  ->  ( _|_ `  A )  C_  ~H )
6130, 60syl 15 . . . . . . . 8  |-  ( ph  ->  ( _|_ `  A
)  C_  ~H )
62 fss 5397 . . . . . . . 8  |-  ( ( F : NN --> ( _|_ `  A )  /\  ( _|_ `  A )  C_  ~H )  ->  F : NN
--> ~H )
6340, 61, 62syl2anc 642 . . . . . . 7  |-  ( ph  ->  F : NN --> ~H )
64 fvco3 5596 . . . . . . 7  |-  ( ( F : NN --> ~H  /\  k  e.  NN )  ->  ( ( ( x  e.  ~H  |->  ( x 
.ih  B ) )  o.  F ) `  k )  =  ( ( x  e.  ~H  |->  ( x  .ih  B ) ) `  ( F `
 k ) ) )
6563, 64sylan 457 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( ( x  e.  ~H  |->  ( x  .ih  B ) )  o.  F ) `
 k )  =  ( ( x  e. 
~H  |->  ( x  .ih  B ) ) `  ( F `  k )
) )
66 c0ex 8832 . . . . . . . 8  |-  0  e.  _V
6766fvconst2 5729 . . . . . . 7  |-  ( k  e.  NN  ->  (
( NN  X.  {
0 } ) `  k )  =  0 )
6867adantl 452 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( NN  X.  { 0 } ) `  k
)  =  0 )
6959, 65, 683eqtr4d 2325 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( ( x  e.  ~H  |->  ( x  .ih  B ) )  o.  F ) `
 k )  =  ( ( NN  X.  { 0 } ) `
 k ) )
7069ralrimiva 2626 . . . 4  |-  ( ph  ->  A. k  e.  NN  ( ( ( x  e.  ~H  |->  ( x 
.ih  B ) )  o.  F ) `  k )  =  ( ( NN  X.  {
0 } ) `  k ) )
71 ovex 5883 . . . . . . . 8  |-  ( x 
.ih  B )  e. 
_V
7271, 49fnmpti 5372 . . . . . . 7  |-  ( x  e.  ~H  |->  ( x 
.ih  B ) )  Fn  ~H
7372a1i 10 . . . . . 6  |-  ( ph  ->  ( x  e.  ~H  |->  ( x  .ih  B ) )  Fn  ~H )
74 fnfco 5407 . . . . . 6  |-  ( ( ( x  e.  ~H  |->  ( x  .ih  B ) )  Fn  ~H  /\  F : NN --> ~H )  ->  ( ( x  e. 
~H  |->  ( x  .ih  B ) )  o.  F
)  Fn  NN )
7573, 63, 74syl2anc 642 . . . . 5  |-  ( ph  ->  ( ( x  e. 
~H  |->  ( x  .ih  B ) )  o.  F
)  Fn  NN )
7666fconst 5427 . . . . . 6  |-  ( NN 
X.  { 0 } ) : NN --> { 0 }
77 ffn 5389 . . . . . 6  |-  ( ( NN  X.  { 0 } ) : NN --> { 0 }  ->  ( NN  X.  { 0 } )  Fn  NN )
7876, 77ax-mp 8 . . . . 5  |-  ( NN 
X.  { 0 } )  Fn  NN
79 eqfnfv 5622 . . . . 5  |-  ( ( ( ( x  e. 
~H  |->  ( x  .ih  B ) )  o.  F
)  Fn  NN  /\  ( NN  X.  { 0 } )  Fn  NN )  ->  ( ( ( x  e.  ~H  |->  ( x  .ih  B ) )  o.  F )  =  ( NN  X.  { 0 } )  <->  A. k  e.  NN  ( ( ( x  e.  ~H  |->  ( x 
.ih  B ) )  o.  F ) `  k )  =  ( ( NN  X.  {
0 } ) `  k ) ) )
8075, 78, 79sylancl 643 . . . 4  |-  ( ph  ->  ( ( ( x  e.  ~H  |->  ( x 
.ih  B ) )  o.  F )  =  ( NN  X.  {
0 } )  <->  A. k  e.  NN  ( ( ( x  e.  ~H  |->  ( x  .ih  B ) )  o.  F ) `
 k )  =  ( ( NN  X.  { 0 } ) `
 k ) ) )
8170, 80mpbird 223 . . 3  |-  ( ph  ->  ( ( x  e. 
~H  |->  ( x  .ih  B ) )  o.  F
)  =  ( NN 
X.  { 0 } ) )
82 fvex 5539 . . . . 5  |-  (  ~~>v  `  F )  e.  _V
8382hlimveci 21769 . . . 4  |-  ( F 
~~>v  (  ~~>v  `  F )  ->  (  ~~>v  `  F )  e.  ~H )
84 oveq1 5865 . . . . 5  |-  ( x  =  (  ~~>v  `  F
)  ->  ( x  .ih  B )  =  ( (  ~~>v  `  F )  .ih  B ) )
85 ovex 5883 . . . . 5  |-  ( ( 
~~>v  `  F )  .ih  B )  e.  _V
8684, 49, 85fvmpt 5602 . . . 4  |-  ( ( 
~~>v  `  F )  e. 
~H  ->  ( ( x  e.  ~H  |->  ( x 
.ih  B ) ) `
 (  ~~>v  `  F
) )  =  ( (  ~~>v  `  F )  .ih  B ) )
8716, 83, 863syl 18 . . 3  |-  ( ph  ->  ( ( x  e. 
~H  |->  ( x  .ih  B ) ) `  (  ~~>v 
`  F ) )  =  ( (  ~~>v  `  F )  .ih  B
) )
8839, 81, 873brtr3d 4052 . 2  |-  ( ph  ->  ( NN  X.  {
0 } ) ( ~~> t `  ( TopOpen ` fld )
) ( (  ~~>v  `  F )  .ih  B
) )
891cnfldtopon 18292 . . . 4  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
9089a1i 10 . . 3  |-  ( ph  ->  ( TopOpen ` fld )  e.  (TopOn `  CC ) )
91 0cn 8831 . . . 4  |-  0  e.  CC
9291a1i 10 . . 3  |-  ( ph  ->  0  e.  CC )
93 1z 10053 . . . 4  |-  1  e.  ZZ
9493a1i 10 . . 3  |-  ( ph  ->  1  e.  ZZ )
95 nnuz 10263 . . . 4  |-  NN  =  ( ZZ>= `  1 )
9695lmconst 16991 . . 3  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  0  e.  CC  /\  1  e.  ZZ )  ->  ( NN  X.  { 0 } ) ( ~~> t `  ( TopOpen ` fld ) ) 0 )
9790, 92, 94, 96syl3anc 1182 . 2  |-  ( ph  ->  ( NN  X.  {
0 } ) ( ~~> t `  ( TopOpen ` fld )
) 0 )
983, 88, 97lmmo 17108 1  |-  ( ph  ->  ( (  ~~>v  `  F
)  .ih  B )  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    C_ wss 3152   {csn 3640   <.cop 3643   class class class wbr 4023    e. cmpt 4077    X. cxp 4687   dom cdm 4689    |` cres 4691    o. ccom 4693   Fun wfun 5249    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   CCcc 8735   0cc0 8737   1c1 8738   NNcn 9746   ZZcz 10024   TopOpenctopn 13326   * Metcxmt 16369   MetOpencmopn 16372  ℂfldccnfld 16377  TopOnctopon 16632    Cn ccn 16954   ~~> tclm 16956   Hauscha 17036    tX ctx 17255   NrmCVeccnv 21140   ~Hchil 21499    +h cva 21500    .h csm 21501    .ih csp 21502   normhcno 21503    -h cmv 21505   Cauchyccau 21506    ~~>v chli 21507   _|_cort 21510
This theorem is referenced by:  occl  21883
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817  ax-hilex 21579  ax-hfvadd 21580  ax-hvcom 21581  ax-hvass 21582  ax-hv0cl 21583  ax-hvaddid 21584  ax-hfvmul 21585  ax-hvmulid 21586  ax-hvmulass 21587  ax-hvdistr1 21588  ax-hvdistr2 21589  ax-hvmul0 21590  ax-hfi 21658  ax-his1 21661  ax-his2 21662  ax-his3 21663  ax-his4 21664  ax-hcompl 21781
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-icc 10663  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cn 16957  df-cnp 16958  df-lm 16959  df-haus 17043  df-tx 17257  df-hmeo 17446  df-xms 17885  df-ms 17886  df-tms 17887  df-grpo 20858  df-gid 20859  df-ginv 20860  df-gdiv 20861  df-ablo 20949  df-vc 21102  df-nv 21148  df-va 21151  df-ba 21152  df-sm 21153  df-0v 21154  df-vs 21155  df-nmcv 21156  df-ims 21157  df-dip 21274  df-hnorm 21548  df-hvsub 21551  df-hlim 21552  df-sh 21786  df-oc 21831
  Copyright terms: Public domain W3C validator