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Theorem elbdop 28607
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 6158 . . 3 (𝑡 = 𝑇 → (normop𝑡) = (normop𝑇))
21breq1d 4633 . 2 (𝑡 = 𝑇 → ((normop𝑡) < +∞ ↔ (normop𝑇) < +∞))
3 df-bdop 28589 . 2 BndLinOp = {𝑡 ∈ LinOp ∣ (normop𝑡) < +∞}
42, 3elrab2 3353 1 (𝑇 ∈ BndLinOp ↔ (𝑇 ∈ LinOp ∧ (normop𝑇) < +∞))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wcel 1987   class class class wbr 4623  cfv 5857  +∞cpnf 10031   < clt 10034  normopcnop 27690  LinOpclo 27692  BndLinOpcbo 27693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-iota 5820  df-fv 5865  df-bdop 28589
This theorem is referenced by:  bdopln  28608  nmopre  28617  elbdop2  28618  0bdop  28740
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