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Theorem isblo 22275
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 normOp OLD 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 normOp OLD W
)
2 bloval.4 . . . 4  |-  L  =  ( U  LnOp  W
)
3 bloval.5 . . . 4  |-  B  =  ( U  BLnOp  W )
41, 2, 3bloval 22274 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  B  =  { t  e.  L  |  ( N `  t )  <  +oo } )
54eleq2d 2502 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( T  e.  B  <->  T  e.  { t  e.  L  | 
( N `  t
)  <  +oo } ) )
6 fveq2 5720 . . . 4  |-  ( t  =  T  ->  ( N `  t )  =  ( N `  T ) )
76breq1d 4214 . . 3  |-  ( t  =  T  ->  (
( N `  t
)  <  +oo  <->  ( N `  T )  <  +oo ) )
87elrab 3084 . 2  |-  ( T  e.  { t  e.  L  |  ( N `
 t )  <  +oo }  <->  ( T  e.  L  /\  ( N `
 T )  <  +oo ) )
95, 8syl6bb 253 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( T  e.  B  <->  ( T  e.  L  /\  ( N `  T )  <  +oo ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   {crab 2701   class class class wbr 4204   ` cfv 5446  (class class class)co 6073    +oocpnf 9109    < clt 9112   NrmCVeccnv 22055    LnOp clno 22233   normOp OLDcnmoo 22234    BLnOp cblo 22235
This theorem is referenced by:  isblo2  22276  bloln  22277  nmblore  22279  isblo3i  22294  htthlem  22412
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  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-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-blo 22239
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