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Theorem elbdop 29053
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 ∧ (normop𝑇) < +∞))

Proof of Theorem elbdop
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6332 . . 3 (𝑡 = 𝑇 → (normop𝑡) = (normop𝑇))
21breq1d 4794 . 2 (𝑡 = 𝑇 → ((normop𝑡) < +∞ ↔ (normop𝑇) < +∞))
3 df-bdop 29035 . 2 BndLinOp = {𝑡 ∈ LinOp ∣ (normop𝑡) < +∞}
42, 3elrab2 3516 1 (𝑇 ∈ BndLinOp ↔ (𝑇 ∈ LinOp ∧ (normop𝑇) < +∞))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382   = wceq 1630  wcel 2144   class class class wbr 4784  cfv 6031  +∞cpnf 10272   < clt 10275  normopcnop 28136  LinOpclo 28138  BndLinOpcbo 28139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-iota 5994  df-fv 6039  df-bdop 29035
This theorem is referenced by:  bdopln  29054  nmopre  29063  elbdop2  29064  0bdop  29186
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