HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  bdopln Structured version   Unicode version

Theorem bdopln 23356
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 23355 . 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 1725   class class class wbr 4204   ` cfv 5446    +oocpnf 9109    < clt 9112   normopcnop 22440   LinOpclo 22442   BndLinOpcbo 22443
This theorem is referenced by:  bdopf  23357  nmbdoplbi  23519  bdophmi  23527  lncnopbd  23532  nmopcoi  23590  bdophsi  23591  bdopcoi  23593  nmopcoadj0i  23598  unierri  23599
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-bdop 23337
  Copyright terms: Public domain W3C validator