Theorem elbdop 28719

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 e. BndLinOp <-> ( T e. LinOp /\ ( normop ` T ) < +oo ) )

Proof of Theorem elbdop

Dummy variable t is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . 3 |- ( t = T -> ( normop ` t ) = ( normop ` T ) )
2	1	breq1d 4663	. 2 |- ( t = T -> ( ( normop ` t ) < +oo <-> ( normop ` T ) < +oo ) )
3		df-bdop 28701	. 2 |- BndLinOp = { t e. LinOp | ( normop ` t ) < +oo }
4	2, 3	elrab2 3366	1 |- ( T e. BndLinOp <-> ( T e. LinOp /\ ( normop ` T ) < +oo ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888   +oocpnf 10071   < clt 10074   normopcnop 27802   LinOpclo 27804   BndLinOpcbo 27805

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-bdop 28701

This theorem is referenced by:  bdopln  28720  nmopre  28729  elbdop2  28730  0bdop  28852