Theorem bdopln 31840

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 31839	. 2 ⊢ (𝑇 ∈ BndLinOp ↔ (𝑇 ∈ LinOp ∧ (normop‘𝑇) < +∞))
2	1	simplbi 497	1 ⊢ (𝑇 ∈ BndLinOp → 𝑇 ∈ LinOp)

Syntax hints:   → wi 4   ∈ wcel 2109   class class class wbr 5102  ‘cfv 6499  +∞cpnf 11181   < clt 11184  norm_opcnop 30924  LinOpclo 30926  BndLinOpcbo 30927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-iota 6452  df-fv 6507  df-bdop 31821

This theorem is referenced by:  bdopf  31841  nmbdoplbi  32003  bdophmi  32011  lncnopbd  32016  nmopcoi  32074  bdophsi  32075  bdopcoi  32077  nmopcoadj0i  32082  unierri  32083