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Theorem leopnmid 23633
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 23418 . . . . 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 9106 . . . . . 6  |-  ( ( T  e.  HrmOp  /\  x  e.  ~H )  ->  (
( T `  x
)  .ih  x )  e.  CC )
43abscld 12230 . . . . 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 23477 . . . . . . 7  |-  Iop  e.  HrmOp
7 hmopm 23516 . . . . . . 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 23418 . . . . . 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 12209 . . . . 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 23369 . . . . . . . 8  |-  ( T  e.  HrmOp  ->  T : ~H
--> ~H )
15 ffvelrn 5860 . . . . . . . . 9  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ~H )
16 normcl 22619 . . . . . . . . 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 22619 . . . . . . 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 9108 . . . . 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 22675 . . . . . . 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 9067 . . . . . . . . 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 22620 . . . . . . . . 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 23437 . . . . . . . . 9  |-  ( T  e.  HrmOp  ->  T  e.  LinOp
)
34 elbdop2 23366 . . . . . . . . . 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 23520 . . . . . . . 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 9856 . . . . . . 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 9072 . . . . . . . . . 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 9106 . . . . . . . . 9  |-  ( ( ( T  e.  HrmOp  /\  ( normop `  T )  e.  RR )  /\  x  e.  ~H )  ->  ( normh `  x )  e.  CC )
4442, 43, 43mulassd 9103 . . . . . . . 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 22578 . . . . . . . . . 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 9106 . . . . . . . . . . . . 13  |-  ( x  e.  ~H  ->  ( normh `  x )  e.  CC )
4948sqvald 11512 . . . . . . . . . . . 12  |-  ( x  e.  ~H  ->  (
( normh `  x ) ^ 2 )  =  ( ( normh `  x
)  x.  ( normh `  x ) ) )
50 normsq 22628 . . . . . . . . . . . 12  |-  ( x  e.  ~H  ->  (
( normh `  x ) ^ 2 )  =  ( x  .ih  x
) )
5149, 50eqtr3d 2469 . . . . . . . . . . 11  |-  ( x  e.  ~H  ->  (
( normh `  x )  x.  ( normh `  x )
)  =  ( x 
.ih  x ) )
5251oveq2d 6089 . . . . . . . . . 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 2470 . . . . . . . 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 2470 . . . . . . 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 23249 . . . . . . . . . . 11  |-  Iop  : ~H
-1-1-onto-> ~H
57 f1of 5666 . . . . . . . . . . 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 23236 . . . . . . . . . 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 23250 . . . . . . . . . . 11  |-  ( x  e.  ~H  ->  (  Iop  `  x )  =  x )
6261oveq2d 6089 . . . . . . . . . 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 2467 . . . . . . . 8  |-  ( ( ( T  e.  HrmOp  /\  ( normop `  T )  e.  RR )  /\  x  e.  ~H )  ->  (
( ( normop `  T
)  .op  Iop  ) `  x )  =  ( ( normop `  T )  .h  x ) )
6564oveq1d 6088 . . . . . . 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 2470 . . . . . 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 4228 . . . . 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 9219 . . . 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 9219 . . 3  |-  ( ( ( T  e.  HrmOp  /\  ( normop `  T )  e.  RR )  /\  x  e.  ~H )  ->  (
( T `  x
)  .ih  x )  <_  ( ( ( (
normop `  T )  .op  Iop  ) `  x ) 
.ih  x ) )
7069ralrimiva 2781 . 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 23619 . . 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 1652    e. wcel 1725   A.wral 2697   class class class wbr 4204   -->wf 5442   -1-1-onto->wf1o 5445   ` cfv 5446  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982    x. cmul 8987    <_ cle 9113   2c2 10041   ^cexp 11374   abscabs 12031   ~Hchil 22414    .h csm 22416    .ih csp 22417   normhcno 22418    .op chot 22434    Iop chio 22439   normopcnop 22440   LinOpclo 22442   BndLinOpcbo 22443   HrmOpcho 22445    <_op cleo 22453
This theorem is referenced by:  nmopleid  23634
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cc 8307  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061  ax-mulf 9062  ax-hilex 22494  ax-hfvadd 22495  ax-hvcom 22496  ax-hvass 22497  ax-hv0cl 22498  ax-hvaddid 22499  ax-hfvmul 22500  ax-hvmulid 22501  ax-hvmulass 22502  ax-hvdistr1 22503  ax-hvdistr2 22504  ax-hvmul0 22505  ax-hfi 22573  ax-his1 22576  ax-his2 22577  ax-his3 22578  ax-his4 22579  ax-hcompl 22696
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-omul 6721  df-er 6897  df-map 7012  df-pm 7013  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-sup 7438  df-oi 7471  df-card 7818  df-acn 7821  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-q 10567  df-rp 10605  df-xneg 10702  df-xadd 10703  df-xmul 10704  df-ioo 10912  df-ico 10914  df-icc 10915  df-fz 11036  df-fzo 11128  df-fl 11194  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-rlim 12275  df-sum 12472  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-starv 13536  df-sca 13537  df-vsca 13538  df-tset 13540  df-ple 13541  df-ds 13543  df-unif 13544  df-hom 13545  df-cco 13546  df-rest 13642  df-topn 13643  df-topgen 13659  df-pt 13660  df-prds 13663  df-xrs 13718  df-0g 13719  df-gsum 13720  df-qtop 13725  df-imas 13726  df-xps 13728  df-mre 13803  df-mrc 13804  df-acs 13806  df-mnd 14682  df-submnd 14731  df-mulg 14807  df-cntz 15108  df-cmn 15406  df-psmet 16686  df-xmet 16687  df-met 16688  df-bl 16689  df-mopn 16690  df-fbas 16691  df-fg 16692  df-cnfld 16696  df-top 16955  df-bases 16957  df-topon 16958  df-topsp 16959  df-cld 17075  df-ntr 17076  df-cls 17077  df-nei 17154  df-cn 17283  df-cnp 17284  df-lm 17285  df-t1 17370  df-haus 17371  df-tx 17586  df-hmeo 17779  df-fil 17870  df-fm 17962  df-flim 17963  df-flf 17964  df-xms 18342  df-ms 18343  df-tms 18344  df-cfil 19200  df-cau 19201  df-cmet 19202  df-grpo 21771  df-gid 21772  df-ginv 21773  df-gdiv 21774  df-ablo 21862  df-subgo 21882  df-vc 22017  df-nv 22063  df-va 22066  df-ba 22067  df-sm 22068  df-0v 22069  df-vs 22070  df-nmcv 22071  df-ims 22072  df-dip 22189  df-ssp 22213  df-ph 22306  df-cbn 22357  df-hnorm 22463  df-hba 22464  df-hvsub 22466  df-hlim 22467  df-hcau 22468  df-sh 22701  df-ch 22716  df-oc 22746  df-ch0 22747  df-shs 22802  df-pjh 22889  df-hosum 23225  df-homul 23226  df-hodif 23227  df-h0op 23243  df-iop 23244  df-nmop 23334  df-lnop 23336  df-bdop 23337  df-hmop 23339  df-leop 23347
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