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Theorem isblo 22124
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 22123 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  B  =  { t  e.  L  |  ( N `  t )  <  +oo } )
54eleq2d 2447 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( T  e.  B  <->  T  e.  { t  e.  L  | 
( N `  t
)  <  +oo } ) )
6 fveq2 5661 . . . 4  |-  ( t  =  T  ->  ( N `  t )  =  ( N `  T ) )
76breq1d 4156 . . 3  |-  ( t  =  T  ->  (
( N `  t
)  <  +oo  <->  ( N `  T )  <  +oo ) )
87elrab 3028 . 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 1649    e. wcel 1717   {crab 2646   class class class wbr 4146   ` cfv 5387  (class class class)co 6013    +oocpnf 9043    < clt 9046   NrmCVeccnv 21904    LnOp clno 22082   normOp OLDcnmoo 22083    BLnOp cblo 22084
This theorem is referenced by:  isblo2  22125  bloln  22126  nmblore  22128  isblo3i  22143  htthlem  22261
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-iota 5351  df-fun 5389  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-blo 22088
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