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Theorem elbdop 23316
Description: Property defining a bounded linear Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
elbdop  |-  ( T  e.  BndLinOp 
<->  ( T  e.  LinOp  /\  ( normop `  T )  <  +oo ) )

Proof of Theorem elbdop
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 fveq2 5687 . . 3  |-  ( t  =  T  ->  ( normop `  t )  =  (
normop `  T ) )
21breq1d 4182 . 2  |-  ( t  =  T  ->  (
( normop `  t )  <  +oo  <->  ( normop `  T
)  <  +oo ) )
3 df-bdop 23298 . 2  |-  BndLinOp  =  {
t  e.  LinOp  |  (
normop `  t )  <  +oo }
42, 3elrab2 3054 1  |-  ( T  e.  BndLinOp 
<->  ( T  e.  LinOp  /\  ( normop `  T )  <  +oo ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   class class class wbr 4172   ` cfv 5413    +oocpnf 9073    < clt 9076   normopcnop 22401   LinOpclo 22403   BndLinOpcbo 22404
This theorem is referenced by:  bdopln  23317  nmopre  23326  elbdop2  23327  0bdop  23449
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
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-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-iota 5377  df-fv 5421  df-bdop 23298
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