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Theorem elbdop 25199
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 5688 . . 3  |-  ( t  =  T  ->  ( normop `  t )  =  (
normop `  T ) )
21breq1d 4299 . 2  |-  ( t  =  T  ->  (
( normop `  t )  < +oo  <->  ( normop `  T
)  < +oo )
)
3 df-bdop 25181 . 2  |-  BndLinOp  =  {
t  e.  LinOp  |  (
normop `  t )  < +oo }
42, 3elrab2 3116 1  |-  ( T  e.  BndLinOp 
<->  ( T  e.  LinOp  /\  ( normop `  T )  < +oo ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761   class class class wbr 4289   ` cfv 5415   +oocpnf 9411    < clt 9414   normopcnop 24282   LinOpclo 24284   BndLinOpcbo 24285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-iota 5378  df-fv 5423  df-bdop 25181
This theorem is referenced by:  bdopln  25200  nmopre  25209  elbdop2  25210  0bdop  25332
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