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Theorem isblo 21360
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 21359 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  B  =  { t  e.  L  |  ( N `  t )  <  +oo } )
54eleq2d 2350 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( T  e.  B  <->  T  e.  { t  e.  L  | 
( N `  t
)  <  +oo } ) )
6 fveq2 5525 . . . 4  |-  ( t  =  T  ->  ( N `  t )  =  ( N `  T ) )
76breq1d 4033 . . 3  |-  ( t  =  T  ->  (
( N `  t
)  <  +oo  <->  ( N `  T )  <  +oo ) )
87elrab 2923 . 2  |-  ( T  e.  { t  e.  L  |  ( N `
 t )  <  +oo }  <->  ( T  e.  L  /\  ( N `
 T )  <  +oo ) )
95, 8syl6bb 252 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547   class class class wbr 4023   ` cfv 5255  (class class class)co 5858    +oocpnf 8864    < clt 8867   NrmCVeccnv 21140    LnOp clno 21318   normOp OLDcnmoo 21319    BLnOp cblo 21320
This theorem is referenced by:  isblo2  21361  bloln  21362  nmblore  21364  isblo3i  21379  htthlem  21497
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-blo 21324
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