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Theorem elbdop 31879
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 6906 . . 3 (𝑡 = 𝑇 → (normop𝑡) = (normop𝑇))
21breq1d 5153 . 2 (𝑡 = 𝑇 → ((normop𝑡) < +∞ ↔ (normop𝑇) < +∞))
3 df-bdop 31861 . 2 BndLinOp = {𝑡 ∈ LinOp ∣ (normop𝑡) < +∞}
42, 3elrab2 3695 1 (𝑇 ∈ BndLinOp ↔ (𝑇 ∈ LinOp ∧ (normop𝑇) < +∞))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108   class class class wbr 5143  cfv 6561  +∞cpnf 11292   < clt 11295  normopcnop 30964  LinOpclo 30966  BndLinOpcbo 30967
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 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-bdop 31861
This theorem is referenced by:  bdopln  31880  nmopre  31889  elbdop2  31890  0bdop  32012
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