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

Theorem leopnmid 23490
Description: A bounded Hermitian operator is less than or equal to its norm times the identity operator. (Contributed by NM, 11-Aug-2006.) (New usage is discouraged.)
Assertion
Ref Expression
leopnmid  |-  ( ( T  e.  HrmOp  /\  ( normop `  T )  e.  RR )  ->  T  <_op  (
( normop `  T )  .op  Iop  ) )

Proof of Theorem leopnmid
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 hmopre 23275 . . . . 5  |-  ( ( T  e.  HrmOp  /\  x  e.  ~H )  ->  (
( T `  x
)  .ih  x )  e.  RR )
21adantlr 696 . . . 4  |-  ( ( ( T  e.  HrmOp  /\  ( normop `  T )  e.  RR )  /\  x  e.  ~H )  ->  (
( T `  x
)  .ih  x )  e.  RR )
31recnd 9048 . . . . . 6  |-  ( ( T  e.  HrmOp  /\  x  e.  ~H )  ->  (
( T `  x
)  .ih  x )  e.  CC )
43abscld 12166 . . . . 5  |-  ( ( T  e.  HrmOp  /\  x  e.  ~H )  ->  ( abs `  ( ( T `
 x )  .ih  x ) )  e.  RR )
54adantlr 696 . . . 4  |-  ( ( ( T  e.  HrmOp  /\  ( normop `  T )  e.  RR )  /\  x  e.  ~H )  ->  ( abs `  ( ( T `
 x )  .ih  x ) )  e.  RR )
6 idhmop 23334 . . . . . . 7  |-  Iop  e.  HrmOp
7 hmopm 23373 . . . . . . 7  |-  ( ( ( normop `  T )  e.  RR  /\  Iop  e.  HrmOp
)  ->  ( ( normop `  T )  .op  Iop  )  e.  HrmOp )
86, 7mpan2 653 . . . . . 6  |-  ( (
normop `  T )  e.  RR  ->  ( ( normop `  T )  .op  Iop  )  e.  HrmOp )
9 hmopre 23275 . . . . . 6  |-  ( ( ( ( normop `  T
)  .op  Iop  )  e. 
HrmOp  /\  x  e.  ~H )  ->  ( ( ( ( normop `  T )  .op  Iop  ) `  x
)  .ih  x )  e.  RR )
108, 9sylan 458 . . . . 5  |-  ( ( ( normop `  T )  e.  RR  /\  x  e. 
~H )  ->  (
( ( ( normop `  T )  .op  Iop  ) `  x )  .ih  x )  e.  RR )
1110adantll 695 . . . 4  |-  ( ( ( T  e.  HrmOp  /\  ( normop `  T )  e.  RR )  /\  x  e.  ~H )  ->  (
( ( ( normop `  T )  .op  Iop  ) `  x )  .ih  x )  e.  RR )
121leabsd 12145 . . . . 5  |-  ( ( T  e.  HrmOp  /\  x  e.  ~H )  ->  (
( T `  x
)  .ih  x )  <_  ( abs `  (
( T `  x
)  .ih  x )
) )
1312adantlr 696 . . . 4  |-  ( ( ( T  e.  HrmOp  /\  ( normop `  T )  e.  RR )  /\  x  e.  ~H )  ->  (
( T `  x
)  .ih  x )  <_  ( abs `  (
( T `  x
)  .ih  x )
) )
14 hmopf 23226 . . . . . . . 8  |-  ( T  e.  HrmOp  ->  T : ~H
--> ~H )
15 ffvelrn 5808 . . . . . . . . 9  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ~H )
16 normcl 22476 . . . . . . . . 9  |-  ( ( T `  x )  e.  ~H  ->  ( normh `  ( T `  x ) )  e.  RR )
1715, 16syl 16 . . . . . . . 8  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( normh `  ( T `  x ) )  e.  RR )
1814, 17sylan 458 . . . . . . 7  |-  ( ( T  e.  HrmOp  /\  x  e.  ~H )  ->  ( normh `  ( T `  x ) )  e.  RR )
1918adantlr 696 . . . . . 6  |-  ( ( ( T  e.  HrmOp  /\  ( normop `  T )  e.  RR )  /\  x  e.  ~H )  ->  ( normh `  ( T `  x ) )  e.  RR )
20 normcl 22476 . . . . . . 7  |-  ( x  e.  ~H  ->  ( normh `  x )  e.  RR )
2120adantl 453 . . . . . 6  |-  ( ( ( T  e.  HrmOp  /\  ( normop `  T )  e.  RR )  /\  x  e.  ~H )  ->  ( normh `  x )  e.  RR )
2219, 21remulcld 9050 . . . . 5  |-  ( ( ( T  e.  HrmOp  /\  ( normop `  T )  e.  RR )  /\  x  e.  ~H )  ->  (
( normh `  ( T `  x ) )  x.  ( normh `  x )
)  e.  RR )
2314, 15sylan 458 . . . . . . 7  |-  ( ( T  e.  HrmOp  /\  x  e.  ~H )  ->  ( T `  x )  e.  ~H )
24 bcs 22532 . . . . . . 7  |-  ( ( ( T `  x
)  e.  ~H  /\  x  e.  ~H )  ->  ( abs `  (
( T `  x
)  .ih  x )
)  <_  ( ( normh `  ( T `  x ) )  x.  ( normh `  x )
) )
2523, 24sylancom 649 . . . . . 6  |-  ( ( T  e.  HrmOp  /\  x  e.  ~H )  ->  ( abs `  ( ( T `
 x )  .ih  x ) )  <_ 
( ( normh `  ( T `  x )
)  x.  ( normh `  x ) ) )
2625adantlr 696 . . . . 5  |-  ( ( ( T  e.  HrmOp  /\  ( normop `  T )  e.  RR )  /\  x  e.  ~H )  ->  ( abs `  ( ( T `
 x )  .ih  x ) )  <_ 
( ( normh `  ( T `  x )
)  x.  ( normh `  x ) ) )
27 remulcl 9009 . . . . . . . . 9  |-  ( ( ( normop `  T )  e.  RR  /\  ( normh `  x )  e.  RR )  ->  ( ( normop `  T )  x.  ( normh `  x ) )  e.  RR )
2820, 27sylan2 461 . . . . . . . 8  |-  ( ( ( normop `  T )  e.  RR  /\  x  e. 
~H )  ->  (
( normop `  T )  x.  ( normh `  x )
)  e.  RR )
2928adantll 695 . . . . . . 7  |-  ( ( ( T  e.  HrmOp  /\  ( normop `  T )  e.  RR )  /\  x  e.  ~H )  ->  (
( normop `  T )  x.  ( normh `  x )
)  e.  RR )
30 normge0 22477 . . . . . . . . 9  |-  ( x  e.  ~H  ->  0  <_  ( normh `  x )
)
3120, 30jca 519 . . . . . . . 8  |-  ( x  e.  ~H  ->  (
( normh `  x )  e.  RR  /\  0  <_ 
( normh `  x )
) )
3231adantl 453 . . . . . . 7  |-  ( ( ( T  e.  HrmOp  /\  ( normop `  T )  e.  RR )  /\  x  e.  ~H )  ->  (
( normh `  x )  e.  RR  /\  0  <_ 
( normh `  x )
) )
33 hmoplin 23294 . . . . . . . . 9  |-  ( T  e.  HrmOp  ->  T  e.  LinOp
)
34 elbdop2 23223 . . . . . . . . . 10  |-  ( T  e.  BndLinOp 
<->  ( T  e.  LinOp  /\  ( normop `  T )  e.  RR ) )
3534biimpri 198 . . . . . . . . 9  |-  ( ( T  e.  LinOp  /\  ( normop `  T )  e.  RR )  ->  T  e.  BndLinOp )
3633, 35sylan 458 . . . . . . . 8  |-  ( ( T  e.  HrmOp  /\  ( normop `  T )  e.  RR )  ->  T  e.  BndLinOp )
37 nmbdoplb 23377 . . . . . . . 8  |-  ( ( T  e.  BndLinOp  /\  x  e.  ~H )  ->  ( normh `  ( T `  x ) )  <_ 
( ( normop `  T
)  x.  ( normh `  x ) ) )
3836, 37sylan 458 . . . . . . 7  |-  ( ( ( T  e.  HrmOp  /\  ( normop `  T )  e.  RR )  /\  x  e.  ~H )  ->  ( normh `  ( T `  x ) )  <_ 
( ( normop `  T
)  x.  ( normh `  x ) ) )
39 lemul1a 9797 . . . . . . 7  |-  ( ( ( ( normh `  ( T `  x )
)  e.  RR  /\  ( ( normop `  T
)  x.  ( normh `  x ) )  e.  RR  /\  ( (
normh `  x )  e.  RR  /\  0  <_ 
( normh `  x )
) )  /\  ( normh `  ( T `  x ) )  <_ 
( ( normop `  T
)  x.  ( normh `  x ) ) )  ->  ( ( normh `  ( T `  x
) )  x.  ( normh `  x ) )  <_  ( ( (
normop `  T )  x.  ( normh `  x )
)  x.  ( normh `  x ) ) )
4019, 29, 32, 38, 39syl31anc 1187 . . . . . 6  |-  ( ( ( T  e.  HrmOp  /\  ( normop `  T )  e.  RR )  /\  x  e.  ~H )  ->  (
( normh `  ( T `  x ) )  x.  ( normh `  x )
)  <_  ( (
( normop `  T )  x.  ( normh `  x )
)  x.  ( normh `  x ) ) )
41 recn 9014 . . . . . . . . . 10  |-  ( (
normop `  T )  e.  RR  ->  ( normop `  T
)  e.  CC )
4241ad2antlr 708 . . . . . . . . 9  |-  ( ( ( T  e.  HrmOp  /\  ( normop `  T )  e.  RR )  /\  x  e.  ~H )  ->  ( normop `  T )  e.  CC )
4321recnd 9048 . . . . . . . . 9  |-  ( ( ( T  e.  HrmOp  /\  ( normop `  T )  e.  RR )  /\  x  e.  ~H )  ->  ( normh `  x )  e.  CC )
4442, 43, 43mulassd 9045 . . . . . . . 8  |-  ( ( ( T  e.  HrmOp  /\  ( normop `  T )  e.  RR )  /\  x  e.  ~H )  ->  (
( ( normop `  T
)  x.  ( normh `  x ) )  x.  ( normh `  x )
)  =  ( (
normop `  T )  x.  ( ( normh `  x
)  x.  ( normh `  x ) ) ) )
45 simpr 448 . . . . . . . . . 10  |-  ( ( ( T  e.  HrmOp  /\  ( normop `  T )  e.  RR )  /\  x  e.  ~H )  ->  x  e.  ~H )
46 ax-his3 22435 . . . . . . . . . 10  |-  ( ( ( normop `  T )  e.  CC  /\  x  e. 
~H  /\  x  e.  ~H )  ->  ( ( ( normop `  T )  .h  x )  .ih  x
)  =  ( (
normop `  T )  x.  ( x  .ih  x
) ) )
4742, 45, 45, 46syl3anc 1184 . . . . . . . . 9  |-  ( ( ( T  e.  HrmOp  /\  ( normop `  T )  e.  RR )  /\  x  e.  ~H )  ->  (
( ( normop `  T
)  .h  x ) 
.ih  x )  =  ( ( normop `  T
)  x.  ( x 
.ih  x ) ) )
4820recnd 9048 . . . . . . . . . . . . 13  |-  ( x  e.  ~H  ->  ( normh `  x )  e.  CC )
4948sqvald 11448 . . . . . . . . . . . 12  |-  ( x  e.  ~H  ->  (
( normh `  x ) ^ 2 )  =  ( ( normh `  x
)  x.  ( normh `  x ) ) )
50 normsq 22485 . . . . . . . . . . . 12  |-  ( x  e.  ~H  ->  (
( normh `  x ) ^ 2 )  =  ( x  .ih  x
) )
5149, 50eqtr3d 2422 . . . . . . . . . . 11  |-  ( x  e.  ~H  ->  (
( normh `  x )  x.  ( normh `  x )
)  =  ( x 
.ih  x ) )
5251oveq2d 6037 . . . . . . . . . 10  |-  ( x  e.  ~H  ->  (
( normop `  T )  x.  ( ( normh `  x
)  x.  ( normh `  x ) ) )  =  ( ( normop `  T )  x.  (
x  .ih  x )
) )
5352adantl 453 . . . . . . . . 9  |-  ( ( ( T  e.  HrmOp  /\  ( normop `  T )  e.  RR )  /\  x  e.  ~H )  ->  (
( normop `  T )  x.  ( ( normh `  x
)  x.  ( normh `  x ) ) )  =  ( ( normop `  T )  x.  (
x  .ih  x )
) )
5447, 53eqtr4d 2423 . . . . . . . 8  |-  ( ( ( T  e.  HrmOp  /\  ( normop `  T )  e.  RR )  /\  x  e.  ~H )  ->  (
( ( normop `  T
)  .h  x ) 
.ih  x )  =  ( ( normop `  T
)  x.  ( (
normh `  x )  x.  ( normh `  x )
) ) )
5544, 54eqtr4d 2423 . . . . . . 7  |-  ( ( ( T  e.  HrmOp  /\  ( normop `  T )  e.  RR )  /\  x  e.  ~H )  ->  (
( ( normop `  T
)  x.  ( normh `  x ) )  x.  ( normh `  x )
)  =  ( ( ( normop `  T )  .h  x )  .ih  x
) )
56 hoif 23106 . . . . . . . . . . 11  |-  Iop  : ~H
-1-1-onto-> ~H
57 f1of 5615 . . . . . . . . . . 11  |-  (  Iop 
: ~H -1-1-onto-> ~H  ->  Iop  : ~H --> ~H )
5856, 57mp1i 12 . . . . . . . . . 10  |-  ( ( ( T  e.  HrmOp  /\  ( normop `  T )  e.  RR )  /\  x  e.  ~H )  ->  Iop  : ~H --> ~H )
59 homval 23093 . . . . . . . . . 10  |-  ( ( ( normop `  T )  e.  CC  /\  Iop  : ~H
--> ~H  /\  x  e. 
~H )  ->  (
( ( normop `  T
)  .op  Iop  ) `  x )  =  ( ( normop `  T )  .h  (  Iop  `  x
) ) )
6042, 58, 45, 59syl3anc 1184 . . . . . . . . 9  |-  ( ( ( T  e.  HrmOp  /\  ( normop `  T )  e.  RR )  /\  x  e.  ~H )  ->  (
( ( normop `  T
)  .op  Iop  ) `  x )  =  ( ( normop `  T )  .h  (  Iop  `  x
) ) )
61 hoival 23107 . . . . . . . . . . 11  |-  ( x  e.  ~H  ->  (  Iop  `  x )  =  x )
6261oveq2d 6037 . . . . . . . . . 10  |-  ( x  e.  ~H  ->  (
( normop `  T )  .h  (  Iop  `  x
) )  =  ( ( normop `  T )  .h  x ) )
6362adantl 453 . . . . . . . . 9  |-  ( ( ( T  e.  HrmOp  /\  ( normop `  T )  e.  RR )  /\  x  e.  ~H )  ->  (
( normop `  T )  .h  (  Iop  `  x
) )  =  ( ( normop `  T )  .h  x ) )
6460, 63eqtrd 2420 . . . . . . . 8  |-  ( ( ( T  e.  HrmOp  /\  ( normop `  T )  e.  RR )  /\  x  e.  ~H )  ->  (
( ( normop `  T
)  .op  Iop  ) `  x )  =  ( ( normop `  T )  .h  x ) )
6564oveq1d 6036 . . . . . . 7  |-  ( ( ( T  e.  HrmOp  /\  ( normop `  T )  e.  RR )  /\  x  e.  ~H )  ->  (
( ( ( normop `  T )  .op  Iop  ) `  x )  .ih  x )  =  ( ( ( normop `  T
)  .h  x ) 
.ih  x ) )
6655, 65eqtr4d 2423 . . . . . 6  |-  ( ( ( T  e.  HrmOp  /\  ( normop `  T )  e.  RR )  /\  x  e.  ~H )  ->  (
( ( normop `  T
)  x.  ( normh `  x ) )  x.  ( normh `  x )
)  =  ( ( ( ( normop `  T
)  .op  Iop  ) `  x )  .ih  x
) )
6740, 66breqtrd 4178 . . . . 5  |-  ( ( ( T  e.  HrmOp  /\  ( normop `  T )  e.  RR )  /\  x  e.  ~H )  ->  (
( normh `  ( T `  x ) )  x.  ( normh `  x )
)  <_  ( (
( ( normop `  T
)  .op  Iop  ) `  x )  .ih  x
) )
685, 22, 11, 26, 67letrd 9160 . . . 4  |-  ( ( ( T  e.  HrmOp  /\  ( normop `  T )  e.  RR )  /\  x  e.  ~H )  ->  ( abs `  ( ( T `
 x )  .ih  x ) )  <_ 
( ( ( (
normop `  T )  .op  Iop  ) `  x ) 
.ih  x ) )
692, 5, 11, 13, 68letrd 9160 . . 3  |-  ( ( ( T  e.  HrmOp  /\  ( normop `  T )  e.  RR )  /\  x  e.  ~H )  ->  (
( T `  x
)  .ih  x )  <_  ( ( ( (
normop `  T )  .op  Iop  ) `  x ) 
.ih  x ) )
7069ralrimiva 2733 . 2  |-  ( ( T  e.  HrmOp  /\  ( normop `  T )  e.  RR )  ->  A. x  e.  ~H  ( ( T `  x )  .ih  x
)  <_  ( (
( ( normop `  T
)  .op  Iop  ) `  x )  .ih  x
) )
71 leop2 23476 . . 3  |-  ( ( T  e.  HrmOp  /\  (
( normop `  T )  .op  Iop  )  e.  HrmOp )  ->  ( T  <_op  ( ( normop `  T )  .op  Iop  )  <->  A. x  e.  ~H  ( ( T `
 x )  .ih  x )  <_  (
( ( ( normop `  T )  .op  Iop  ) `  x )  .ih  x ) ) )
728, 71sylan2 461 . 2  |-  ( ( T  e.  HrmOp  /\  ( normop `  T )  e.  RR )  ->  ( T  <_op  ( ( normop `  T )  .op  Iop  )  <->  A. x  e.  ~H  ( ( T `
 x )  .ih  x )  <_  (
( ( ( normop `  T )  .op  Iop  ) `  x )  .ih  x ) ) )
7370, 72mpbird 224 1  |-  ( ( T  e.  HrmOp  /\  ( normop `  T )  e.  RR )  ->  T  <_op  (
( normop `  T )  .op  Iop  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2650   class class class wbr 4154   -->wf 5391   -1-1-onto->wf1o 5394   ` cfv 5395  (class class class)co 6021   CCcc 8922   RRcr 8923   0cc0 8924    x. cmul 8929    <_ cle 9055   2c2 9982   ^cexp 11310   abscabs 11967   ~Hchil 22271    .h csm 22273    .ih csp 22274   normhcno 22275    .op chot 22291    Iop chio 22296   normopcnop 22297   LinOpclo 22299   BndLinOpcbo 22300   HrmOpcho 22302    <_op cleo 22310
This theorem is referenced by:  nmopleid  23491
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530  ax-cc 8249  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002  ax-addf 9003  ax-mulf 9004  ax-hilex 22351  ax-hfvadd 22352  ax-hvcom 22353  ax-hvass 22354  ax-hv0cl 22355  ax-hvaddid 22356  ax-hfvmul 22357  ax-hvmulid 22358  ax-hvmulass 22359  ax-hvdistr1 22360  ax-hvdistr2 22361  ax-hvmul0 22362  ax-hfi 22430  ax-his1 22433  ax-his2 22434  ax-his3 22435  ax-his4 22436  ax-hcompl 22553
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-iin 4039  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-se 4484  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-isom 5404  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-of 6245  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-2o 6662  df-oadd 6665  df-omul 6666  df-er 6842  df-map 6957  df-pm 6958  df-ixp 7001  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-fi 7352  df-sup 7382  df-oi 7413  df-card 7760  df-acn 7763  df-cda 7982  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-4 9993  df-5 9994  df-6 9995  df-7 9996  df-8 9997  df-9 9998  df-10 9999  df-n0 10155  df-z 10216  df-dec 10316  df-uz 10422  df-q 10508  df-rp 10546  df-xneg 10643  df-xadd 10644  df-xmul 10645  df-ioo 10853  df-ico 10855  df-icc 10856  df-fz 10977  df-fzo 11067  df-fl 11130  df-seq 11252  df-exp 11311  df-hash 11547  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-clim 12210  df-rlim 12211  df-sum 12408  df-struct 13399  df-ndx 13400  df-slot 13401  df-base 13402  df-sets 13403  df-ress 13404  df-plusg 13470  df-mulr 13471  df-starv 13472  df-sca 13473  df-vsca 13474  df-tset 13476  df-ple 13477  df-ds 13479  df-unif 13480  df-hom 13481  df-cco 13482  df-rest 13578  df-topn 13579  df-topgen 13595  df-pt 13596  df-prds 13599  df-xrs 13654  df-0g 13655  df-gsum 13656  df-qtop 13661  df-imas 13662  df-xps 13664  df-mre 13739  df-mrc 13740  df-acs 13742  df-mnd 14618  df-submnd 14667  df-mulg 14743  df-cntz 15044  df-cmn 15342  df-xmet 16620  df-met 16621  df-bl 16622  df-mopn 16623  df-fbas 16624  df-fg 16625  df-cnfld 16628  df-top 16887  df-bases 16889  df-topon 16890  df-topsp 16891  df-cld 17007  df-ntr 17008  df-cls 17009  df-nei 17086  df-cn 17214  df-cnp 17215  df-lm 17216  df-t1 17301  df-haus 17302  df-tx 17516  df-hmeo 17709  df-fil 17800  df-fm 17892  df-flim 17893  df-flf 17894  df-xms 18260  df-ms 18261  df-tms 18262  df-cfil 19080  df-cau 19081  df-cmet 19082  df-grpo 21628  df-gid 21629  df-ginv 21630  df-gdiv 21631  df-ablo 21719  df-subgo 21739  df-vc 21874  df-nv 21920  df-va 21923  df-ba 21924  df-sm 21925  df-0v 21926  df-vs 21927  df-nmcv 21928  df-ims 21929  df-dip 22046  df-ssp 22070  df-ph 22163  df-cbn 22214  df-hnorm 22320  df-hba 22321  df-hvsub 22323  df-hlim 22324  df-hcau 22325  df-sh 22558  df-ch 22573  df-oc 22603  df-ch0 22604  df-shs 22659  df-pjh 22746  df-hosum 23082  df-homul 23083  df-hodif 23084  df-h0op 23100  df-iop 23101  df-nmop 23191  df-lnop 23193  df-bdop 23194  df-hmop 23196  df-leop 23204
  Copyright terms: Public domain W3C validator