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Theorem bdopln 25264
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 25263 . 2  |-  ( T  e.  BndLinOp 
<->  ( T  e.  LinOp  /\  ( normop `  T )  < +oo ) )
21simplbi 460 1  |-  ( T  e.  BndLinOp  ->  T  e.  LinOp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1756   class class class wbr 4291   ` cfv 5417   +oocpnf 9414    < clt 9417   normopcnop 24346   LinOpclo 24348   BndLinOpcbo 24349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-rex 2720  df-rab 2723  df-v 2973  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-iota 5380  df-fv 5425  df-bdop 25245
This theorem is referenced by:  bdopf  25265  nmbdoplbi  25427  bdophmi  25435  lncnopbd  25440  nmopcoi  25498  bdophsi  25499  bdopcoi  25501  nmopcoadj0i  25506  unierri  25507
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