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Theorem elbdop 23355
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 5720 . . 3  |-  ( t  =  T  ->  ( normop `  t )  =  (
normop `  T ) )
21breq1d 4214 . 2  |-  ( t  =  T  ->  (
( normop `  t )  <  +oo  <->  ( normop `  T
)  <  +oo ) )
3 df-bdop 23337 . 2  |-  BndLinOp  =  {
t  e.  LinOp  |  (
normop `  t )  <  +oo }
42, 3elrab2 3086 1  |-  ( T  e.  BndLinOp 
<->  ( T  e.  LinOp  /\  ( normop `  T )  <  +oo ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   class class class wbr 4204   ` cfv 5446    +oocpnf 9109    < clt 9112   normopcnop 22440   LinOpclo 22442   BndLinOpcbo 22443
This theorem is referenced by:  bdopln  23356  nmopre  23365  elbdop2  23366  0bdop  23488
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
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-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rex 2703  df-rab 2706  df-v 2950  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-iota 5410  df-fv 5454  df-bdop 23337
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