Theorem bdopln 29942

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 29941	. 2 ⊢ (𝑇 ∈ BndLinOp ↔ (𝑇 ∈ LinOp ∧ (normop‘𝑇) < +∞))
2	1	simplbi 501	1 ⊢ (𝑇 ∈ BndLinOp → 𝑇 ∈ LinOp)

Syntax hints:   → wi 4   ∈ wcel 2110   class class class wbr 5053  ‘cfv 6380  +∞cpnf 10864   < clt 10867  norm_opcnop 29026  LinOpclo 29028  BndLinOpcbo 29029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-iota 6338  df-fv 6388  df-bdop 29923

This theorem is referenced by:  bdopf  29943  nmbdoplbi  30105  bdophmi  30113  lncnopbd  30118  nmopcoi  30176  bdophsi  30177  bdopcoi  30179  nmopcoadj0i  30184  unierri  30185