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Theorem elbdop 22456
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 5541 . . 3  |-  ( t  =  T  ->  ( normop `  t )  =  (
normop `  T ) )
21breq1d 4049 . 2  |-  ( t  =  T  ->  (
( normop `  t )  <  +oo  <->  ( normop `  T
)  <  +oo ) )
3 df-bdop 22438 . 2  |-  BndLinOp  =  {
t  e.  LinOp  |  (
normop `  t )  <  +oo }
42, 3elrab2 2938 1  |-  ( T  e.  BndLinOp 
<->  ( T  e.  LinOp  /\  ( normop `  T )  <  +oo ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271    +oocpnf 8880    < clt 8883   normopcnop 21541   LinOpclo 21543   BndLinOpcbo 21544
This theorem is referenced by:  bdopln  22457  nmopre  22466  elbdop2  22467  0bdop  22589
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-bdop 22438
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