Theorem isblo 27637

Description: The predicate "is a bounded linear operator." (Contributed by NM, 6-Nov-2007.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bloval.3	|- N = ( U normOpOLD W )
bloval.4	|- L = ( U LnOp W )
bloval.5	|- B = ( U BLnOp W )

Assertion

Ref	Expression
isblo	|- ( ( U e. NrmCVec /\ W e. NrmCVec ) -> ( T e. B <-> ( T e. L /\ ( N ` T ) < +oo ) ) )

Proof of Theorem isblo

Dummy variable t is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bloval.3	. . . 4 |- N = ( U normOpOLD W )
2		bloval.4	. . . 4 |- L = ( U LnOp W )
3		bloval.5	. . . 4 |- B = ( U BLnOp W )
4	1, 2, 3	bloval 27636	. . 3 |- ( ( U e. NrmCVec /\ W e. NrmCVec ) -> B = { t e. L | ( N ` t ) < +oo } )
5	4	eleq2d 2687	. 2 |- ( ( U e. NrmCVec /\ W e. NrmCVec ) -> ( T e. B <-> T e. { t e. L | ( N ` t ) < +oo } ) )
6		fveq2 6191	. . . 4 |- ( t = T -> ( N ` t ) = ( N ` T ) )
7	6	breq1d 4663	. . 3 |- ( t = T -> ( ( N ` t ) < +oo <-> ( N ` T ) < +oo ) )
8	7	elrab 3363	. 2 |- ( T e. { t e. L | ( N ` t ) < +oo } <-> ( T e. L /\ ( N ` T ) < +oo ) )
9	5, 8	syl6bb 276	1 |- ( ( U e. NrmCVec /\ W e. NrmCVec ) -> ( T e. B <-> ( T e. L /\ ( N ` T ) < +oo ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   +oocpnf 10071   < clt 10074   NrmCVeccnv 27439   LnOp clno 27595   normOpOLDcnmoo 27596   BLnOp cblo 27597

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  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-blo 27601

This theorem is referenced by:  isblo2  27638  bloln  27639  nmblore  27641  isblo3i  27656  htthlem  27774