Metamath Proof Explorer

Theorem isblo2

Description: The predicate "is a bounded linear operator." (Contributed by NM, 8-Dec-2007) (New usage is discouraged.)

Ref Expression
Hypotheses bloval.3 
|- N = ( U normOpOLD W )
bloval.4 
|- L = ( U LnOp W )
bloval.5 
|- B = ( U BLnOp W )
Assertion isblo2 
|- ( ( U e. NrmCVec /\ W e. NrmCVec ) -> ( T e. B <-> ( T e. L /\ ( N ` T ) e. RR ) ) )

Proof

Step Hyp Ref Expression
1 bloval.3 
 |-  N = ( U normOpOLD W )
2 bloval.4 
 |-  L = ( U LnOp W )
3 bloval.5 
 |-  B = ( U BLnOp W )
4 1 2 3 isblo 
 |-  ( ( U e. NrmCVec /\ W e. NrmCVec ) -> ( T e. B <-> ( T e. L /\ ( N ` T ) < +oo ) ) )
5 eqid 
 |-  ( BaseSet ` U ) = ( BaseSet ` U )
6 eqid 
 |-  ( BaseSet ` W ) = ( BaseSet ` W )
7 5 6 2 lnof 
 |-  ( ( U e. NrmCVec /\ W e. NrmCVec /\ T e. L ) -> T : ( BaseSet ` U ) --> ( BaseSet ` W ) )
8 5 6 1 nmoreltpnf 
 |-  ( ( U e. NrmCVec /\ W e. NrmCVec /\ T : ( BaseSet ` U ) --> ( BaseSet ` W ) ) -> ( ( N ` T ) e. RR <-> ( N ` T ) < +oo ) )
9 7 8 syld3an3 
 |-  ( ( U e. NrmCVec /\ W e. NrmCVec /\ T e. L ) -> ( ( N ` T ) e. RR <-> ( N ` T ) < +oo ) )
10 9 3expa 
 |-  ( ( ( U e. NrmCVec /\ W e. NrmCVec ) /\ T e. L ) -> ( ( N ` T ) e. RR <-> ( N ` T ) < +oo ) )
11 10 pm5.32da 
 |-  ( ( U e. NrmCVec /\ W e. NrmCVec ) -> ( ( T e. L /\ ( N ` T ) e. RR ) <-> ( T e. L /\ ( N ` T ) < +oo ) ) )
12 4 11 bitr4d 
 |-  ( ( U e. NrmCVec /\ W e. NrmCVec ) -> ( T e. B <-> ( T e. L /\ ( N ` T ) e. RR ) ) )