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Theorem elbdop 28103
 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 6103 . . 3 (𝑡 = 𝑇 → (normop𝑡) = (normop𝑇))
21breq1d 4593 . 2 (𝑡 = 𝑇 → ((normop𝑡) < +∞ ↔ (normop𝑇) < +∞))
3 df-bdop 28085 . 2 BndLinOp = {𝑡 ∈ LinOp ∣ (normop𝑡) < +∞}
42, 3elrab2 3333 1 (𝑇 ∈ BndLinOp ↔ (𝑇 ∈ LinOp ∧ (normop𝑇) < +∞))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   class class class wbr 4583  ‘cfv 5804  +∞cpnf 9950   < clt 9953  normopcnop 27186  LinOpclo 27188  BndLinOpcbo 27189 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-bdop 28085 This theorem is referenced by:  bdopln  28104  nmopre  28113  elbdop2  28114  0bdop  28236
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