Theorem bdopln 32224

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 32223	. 2 ⊢ (𝑇 ∈ BndLinOp ↔ (𝑇 ∈ LinOp ∧ (normop‘𝑇) < +∞))
2	1	simplbi 501	1 ⊢ (𝑇 ∈ BndLinOp → 𝑇 ∈ LinOp)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2142   class class class wbr 5108  ‘cfv 6536  +∞cpnf 11246   < clt 11249  norm_opcnop 31308  LinOpclo 31310  BndLinOpcbo 31311

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-iota 6492  df-fv 6544  df-bdop 32205

This theorem is used by:  bdopf  32225  nmbdoplbi  32387  bdophmi  32395  lncnopbd  32400  nmopcoi  32458  bdophsi  32459  bdopcoi  32461  nmopcoadj0i  32466  unierri  32467