Theorem bdopln 32066

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 32065	. 2 ⊢ (𝑇 ∈ BndLinOp ↔ (𝑇 ∈ LinOp ∧ (normop‘𝑇) < +∞))
2	1	simplbi 500	1 ⊢ (𝑇 ∈ BndLinOp → 𝑇 ∈ LinOp)

Syntax hints:   → wi 4   ∈ wcel 2144   class class class wbr 5102  ‘cfv 6523  +∞cpnf 11215   < clt 11218  norm_opcnop 31150  LinOpclo 31152  BndLinOpcbo 31153

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-iota 6479  df-fv 6531  df-bdop 32047

This theorem is referenced by:  bdopf  32067  nmbdoplbi  32229  bdophmi  32237  lncnopbd  32242  nmopcoi  32300  bdophsi  32301  bdopcoi  32303  nmopcoadj0i  32308  unierri  32309