Theorem elbdop 29639

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 ∧ (normop‘𝑇) < +∞))

Proof of Theorem elbdop

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6672	. . 3 ⊢ (𝑡 = 𝑇 → (normop‘𝑡) = (normop‘𝑇))
2	1	breq1d 5078	. 2 ⊢ (𝑡 = 𝑇 → ((normop‘𝑡) < +∞ ↔ (normop‘𝑇) < +∞))
3		df-bdop 29621	. 2 ⊢ BndLinOp = {𝑡 ∈ LinOp ∣ (normop‘𝑡) < +∞}
4	2, 3	elrab2 3685	1 ⊢ (𝑇 ∈ BndLinOp ↔ (𝑇 ∈ LinOp ∧ (normop‘𝑇) < +∞))

Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114   class class class wbr 5068  ‘cfv 6357  +∞cpnf 10674   < clt 10677  norm_opcnop 28724  LinOpclo 28726  BndLinOpcbo 28727

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-iota 6316  df-fv 6365  df-bdop 29621

This theorem is referenced by:  bdopln  29640  nmopre  29649  elbdop2  29650  0bdop  29772