Theorem bdopln 32493

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 32492	. 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 2147   class class class wbr 5106  ‘cfv 6558  +∞cpnf 11367   < clt 11370  norm_opcnop 31577  LinOpclo 31579  BndLinOpcbo 31580

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-iota 6512  df-fv 6566  df-bdop 32474

This theorem is used by:  bdopf  32494  nmbdoplbi  32656  bdophmi  32664  lncnopbd  32669  nmopcoi  32727  bdophsi  32728  bdopcoi  32730  nmopcoadj0i  32735  unierri  32736