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Theorem bdopln 31369
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 31368 . 2 (𝑇 ∈ BndLinOp ↔ (𝑇 ∈ LinOp ∧ (normop𝑇) < +∞))
21simplbi 498 1 (𝑇 ∈ BndLinOp → 𝑇 ∈ LinOp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106   class class class wbr 5148  cfv 6543  +∞cpnf 11249   < clt 11252  normopcnop 30453  LinOpclo 30455  BndLinOpcbo 30456
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-bdop 31350
This theorem is referenced by:  bdopf  31370  nmbdoplbi  31532  bdophmi  31540  lncnopbd  31545  nmopcoi  31603  bdophsi  31604  bdopcoi  31606  nmopcoadj0i  31611  unierri  31612
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