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Theorem bdopln 29688
 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 ∈ BndLinOp → 𝑇 ∈ LinOp)

Proof of Theorem bdopln
StepHypRef Expression
1 elbdop 29687 . 2 (𝑇 ∈ BndLinOp ↔ (𝑇 ∈ LinOp ∧ (normop𝑇) < +∞))
21simplbi 501 1 (𝑇 ∈ BndLinOp → 𝑇 ∈ LinOp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2111   class class class wbr 5034  ‘cfv 6332  +∞cpnf 10679   < clt 10682  normopcnop 28772  LinOpclo 28774  BndLinOpcbo 28775 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-rab 3115  df-v 3444  df-un 3888  df-in 3890  df-ss 3900  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-iota 6291  df-fv 6340  df-bdop 29669 This theorem is referenced by:  bdopf  29689  nmbdoplbi  29851  bdophmi  29859  lncnopbd  29864  nmopcoi  29922  bdophsi  29923  bdopcoi  29925  nmopcoadj0i  29930  unierri  29931
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