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Theorem bdopln 23212
Description: A bounded linear Hilbert space operator is a linear operator. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
Assertion
Ref Expression
bdopln  |-  ( T  e.  BndLinOp  ->  T  e.  LinOp )

Proof of Theorem bdopln
StepHypRef Expression
1 elbdop 23211 . 2  |-  ( T  e.  BndLinOp 
<->  ( T  e.  LinOp  /\  ( normop `  T )  <  +oo ) )
21simplbi 447 1  |-  ( T  e.  BndLinOp  ->  T  e.  LinOp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1717   class class class wbr 4153   ` cfv 5394    +oocpnf 9050    < clt 9053   normopcnop 22296   LinOpclo 22298   BndLinOpcbo 22299
This theorem is referenced by:  bdopf  23213  nmbdoplbi  23375  bdophmi  23383  lncnopbd  23388  nmopcoi  23446  bdophsi  23447  bdopcoi  23449  nmopcoadj0i  23454  unierri  23455
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 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
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 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-iota 5358  df-fv 5402  df-bdop 23193
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