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Theorem elbdop 31100
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 6888 . . 3 (𝑡 = 𝑇 → (normop𝑡) = (normop𝑇))
21breq1d 5157 . 2 (𝑡 = 𝑇 → ((normop𝑡) < +∞ ↔ (normop𝑇) < +∞))
3 df-bdop 31082 . 2 BndLinOp = {𝑡 ∈ LinOp ∣ (normop𝑡) < +∞}
42, 3elrab2 3685 1 (𝑇 ∈ BndLinOp ↔ (𝑇 ∈ LinOp ∧ (normop𝑇) < +∞))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1541  wcel 2106   class class class wbr 5147  cfv 6540  +∞cpnf 11241   < clt 11244  normopcnop 30185  LinOpclo 30187  BndLinOpcbo 30188
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6492  df-fv 6548  df-bdop 31082
This theorem is referenced by:  bdopln  31101  nmopre  31110  elbdop2  31111  0bdop  31233
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