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Theorem isblo 25895
Description: The predicate "is a bounded linear operator." (Contributed by NM, 6-Nov-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
bloval.3  |-  N  =  ( U normOpOLD W
)
bloval.4  |-  L  =  ( U  LnOp  W
)
bloval.5  |-  B  =  ( U  BLnOp  W )
Assertion
Ref Expression
isblo  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( T  e.  B  <->  ( T  e.  L  /\  ( N `  T )  < +oo ) ) )

Proof of Theorem isblo
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 bloval.3 . . . 4  |-  N  =  ( U normOpOLD W
)
2 bloval.4 . . . 4  |-  L  =  ( U  LnOp  W
)
3 bloval.5 . . . 4  |-  B  =  ( U  BLnOp  W )
41, 2, 3bloval 25894 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  B  =  { t  e.  L  |  ( N `  t )  < +oo } )
54eleq2d 2524 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( T  e.  B  <->  T  e.  { t  e.  L  | 
( N `  t
)  < +oo } ) )
6 fveq2 5848 . . . 4  |-  ( t  =  T  ->  ( N `  t )  =  ( N `  T ) )
76breq1d 4449 . . 3  |-  ( t  =  T  ->  (
( N `  t
)  < +oo  <->  ( N `  T )  < +oo ) )
87elrab 3254 . 2  |-  ( T  e.  { t  e.  L  |  ( N `
 t )  < +oo }  <->  ( T  e.  L  /\  ( N `
 T )  < +oo ) )
95, 8syl6bb 261 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( T  e.  B  <->  ( T  e.  L  /\  ( N `  T )  < +oo ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   {crab 2808   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   +oocpnf 9614    < clt 9617   NrmCVeccnv 25675    LnOp clno 25853   normOpOLDcnmoo 25854    BLnOp cblo 25855
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-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  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-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-blo 25859
This theorem is referenced by:  isblo2  25896  bloln  25897  nmblore  25899  isblo3i  25914  htthlem  26032
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