Theorem bdopln 31889

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 31888	. 2 ⊢ (𝑇 ∈ BndLinOp ↔ (𝑇 ∈ LinOp ∧ (normop‘𝑇) < +∞))
2	1	simplbi 497	1 ⊢ (𝑇 ∈ BndLinOp → 𝑇 ∈ LinOp)

Syntax hints:   → wi 4   ∈ wcel 2105   class class class wbr 5147  ‘cfv 6562  +∞cpnf 11289   < clt 11292  norm_opcnop 30973  LinOpclo 30975  BndLinOpcbo 30976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-iota 6515  df-fv 6570  df-bdop 31870

This theorem is referenced by:  bdopf  31890  nmbdoplbi  32052  bdophmi  32060  lncnopbd  32065  nmopcoi  32123  bdophsi  32124  bdopcoi  32126  nmopcoadj0i  32131  unierri  32132