Theorem bdopln 31940

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

Step	Hyp	Ref	Expression
1		elbdop 31939	. 2 ⊢ (𝑇 ∈ BndLinOp ↔ (𝑇 ∈ LinOp ∧ (normop‘𝑇) < +∞))
2	1	simplbi 497	1 ⊢ (𝑇 ∈ BndLinOp → 𝑇 ∈ LinOp)

Syntax hints:   → wi 4   ∈ wcel 2114   class class class wbr 5099  ‘cfv 6493  +∞cpnf 11167   < clt 11170  norm_opcnop 31024  LinOpclo 31026  BndLinOpcbo 31027

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-iota 6449  df-fv 6501  df-bdop 31921

This theorem is referenced by:  bdopf  31941  nmbdoplbi  32103  bdophmi  32111  lncnopbd  32116  nmopcoi  32174  bdophsi  32175  bdopcoi  32177  nmopcoadj0i  32182  unierri  32183