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Theorem bdopln 26906
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 26905 . 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 1819   class class class wbr 4456   ` cfv 5594   +oocpnf 9642    < clt 9645   normopcnop 25988   LinOpclo 25990   BndLinOpcbo 25991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-iota 5557  df-fv 5602  df-bdop 26887
This theorem is referenced by:  bdopf  26907  nmbdoplbi  27069  bdophmi  27077  lncnopbd  27082  nmopcoi  27140  bdophsi  27141  bdopcoi  27143  nmopcoadj0i  27148  unierri  27149
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