Theorem bdopln 32350

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 32349	. 2 ⊢ (𝑇 ∈ BndLinOp ↔ (𝑇 ∈ LinOp ∧ (normop‘𝑇) < +∞))
2	1	simplbi 502	1 ⊢ (𝑇 ∈ BndLinOp → 𝑇 ∈ LinOp)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145   class class class wbr 5107  ‘cfv 6537  +∞cpnf 11268   < clt 11271  norm_opcnop 31434  LinOpclo 31436  BndLinOpcbo 31437

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-iota 6493  df-fv 6545  df-bdop 32331

This theorem is used by:  bdopf  32351  nmbdoplbi  32513  bdophmi  32521  lncnopbd  32526  nmopcoi  32584  bdophsi  32585  bdopcoi  32587  nmopcoadj0i  32592  unierri  32593