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Theorem bdopln 22441
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 22440 . 2  |-  ( T  e.  BndLinOp 
<->  ( T  e.  LinOp  /\  ( normop `  T )  <  +oo ) )
21simplbi 446 1  |-  ( T  e.  BndLinOp  ->  T  e.  LinOp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    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:  bdopf  22442  nmbdoplbi  22604  bdophmi  22612  lncnopbd  22617  nmopcoi  22675  bdophsi  22676  bdopcoi  22678  nmopcoadj0i  22683  unierri  22684
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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