Theorem elbdop 31804

Description: Property defining a bounded linear Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
elbdop	⊢ (𝑇 ∈ BndLinOp ↔ (𝑇 ∈ LinOp ∧ (normop‘𝑇) < +∞))

Proof of Theorem elbdop

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6822	. . 3 ⊢ (𝑡 = 𝑇 → (normop‘𝑡) = (normop‘𝑇))
2	1	breq1d 5102	. 2 ⊢ (𝑡 = 𝑇 → ((normop‘𝑡) < +∞ ↔ (normop‘𝑇) < +∞))
3		df-bdop 31786	. 2 ⊢ BndLinOp = {𝑡 ∈ LinOp ∣ (normop‘𝑡) < +∞}
4	2, 3	elrab2 3651	1 ⊢ (𝑇 ∈ BndLinOp ↔ (𝑇 ∈ LinOp ∧ (normop‘𝑇) < +∞))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   class class class wbr 5092  ‘cfv 6482  +∞cpnf 11146   < clt 11149  norm_opcnop 30889  LinOpclo 30891  BndLinOpcbo 30892

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 3395  df-v 3438  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-iota 6438  df-fv 6490  df-bdop 31786

This theorem is referenced by:  bdopln  31805  nmopre  31814  elbdop2  31815  0bdop  31937