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Theorem elbdop 22440
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 5525 . . 3  |-  ( t  =  T  ->  ( normop `  t )  =  (
normop `  T ) )
21breq1d 4033 . 2  |-  ( t  =  T  ->  (
( normop `  t )  <  +oo  <->  ( normop `  T
)  <  +oo ) )
3 df-bdop 22422 . 2  |-  BndLinOp  =  {
t  e.  LinOp  |  (
normop `  t )  <  +oo }
42, 3elrab2 2925 1  |-  ( T  e.  BndLinOp 
<->  ( T  e.  LinOp  /\  ( normop `  T )  <  +oo ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255    +oocpnf 8864    < clt 8867   normopcnop 21525   LinOpclo 21527   BndLinOpcbo 21528
This theorem is referenced by:  bdopln  22441  nmopre  22450  elbdop2  22451  0bdop  22573
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
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-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  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-iota 5219  df-fv 5263  df-bdop 22422
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