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Theorem bdopln 31114
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 31113 . 2 (𝑇 ∈ BndLinOp ↔ (𝑇 ∈ LinOp ∧ (normop𝑇) < +∞))
21simplbi 499 1 (𝑇 ∈ BndLinOp → 𝑇 ∈ LinOp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107   class class class wbr 5149  cfv 6544  +∞cpnf 11245   < clt 11248  normopcnop 30198  LinOpclo 30200  BndLinOpcbo 30201
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-bdop 31095
This theorem is referenced by:  bdopf  31115  nmbdoplbi  31277  bdophmi  31285  lncnopbd  31290  nmopcoi  31348  bdophsi  31349  bdopcoi  31351  nmopcoadj0i  31356  unierri  31357
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