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Theorem bdopln 31880
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 31879 . 2 (𝑇 ∈ BndLinOp ↔ (𝑇 ∈ LinOp ∧ (normop𝑇) < +∞))
21simplbi 497 1 (𝑇 ∈ BndLinOp → 𝑇 ∈ LinOp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108   class class class wbr 5143  cfv 6561  +∞cpnf 11292   < clt 11295  normopcnop 30964  LinOpclo 30966  BndLinOpcbo 30967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-bdop 31861
This theorem is referenced by:  bdopf  31881  nmbdoplbi  32043  bdophmi  32051  lncnopbd  32056  nmopcoi  32114  bdophsi  32115  bdopcoi  32117  nmopcoadj0i  32122  unierri  32123
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