Theorem bdopln 32463

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 32462	. 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 2144   class class class wbr 5102  ‘cfv 6538  +∞cpnf 11340   < clt 11343  norm_opcnop 31547  LinOpclo 31549  BndLinOpcbo 31550

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-iota 6494  df-fv 6546  df-bdop 32444

This theorem is used by:  bdopf  32464  nmbdoplbi  32626  bdophmi  32634  lncnopbd  32639  nmopcoi  32697  bdophsi  32698  bdopcoi  32700  nmopcoadj0i  32705  unierri  32706