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Theorem isblo 24354
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 24353 . . 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 5802 . . . 4  |-  ( t  =  T  ->  ( N `  t )  =  ( N `  T ) )
76breq1d 4413 . . 3  |-  ( t  =  T  ->  (
( N `  t
)  < +oo  <->  ( N `  T )  < +oo ) )
87elrab 3224 . 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 369    = wceq 1370    e. wcel 1758   {crab 2803   class class class wbr 4403   ` cfv 5529  (class class class)co 6203   +oocpnf 9529    < clt 9532   NrmCVeccnv 24134    LnOp clno 24312   normOpOLDcnmoo 24313    BLnOp cblo 24314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-iota 5492  df-fun 5531  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-blo 24318
This theorem is referenced by:  isblo2  24355  bloln  24356  nmblore  24358  isblo3i  24373  htthlem  24491
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