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Metamath Proof Explorer

Theorem isblo

Description: The predicate "is a bounded linear operator." (Contributed by NM, 6-Nov-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 isblo 
|- ( ( U e. NrmCVec /\ W e. NrmCVec ) -> ( T e. B <-> ( T e. L /\ ( N ` T ) < +oo ) ) )

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 bloval 
 |-  ( ( U e. NrmCVec /\ W e. NrmCVec ) -> B = { t e. L | ( N ` t ) < +oo } )
5 4 eleq2d 
 |-  ( ( U e. NrmCVec /\ W e. NrmCVec ) -> ( T e. B <-> T e. { t e. L | ( N ` t ) < +oo } ) )
6 fveq2 
 |-  ( t = T -> ( N ` t ) = ( N ` T ) )
7 6 breq1d 
 |-  ( t = T -> ( ( N ` t ) < +oo <-> ( N ` T ) < +oo ) )
8 7 elrab 
 |-  ( T e. { t e. L | ( N ` t ) < +oo } <-> ( T e. L /\ ( N ` T ) < +oo ) )
9 5 8 bitrdi 
 |-  ( ( U e. NrmCVec /\ W e. NrmCVec ) -> ( T e. B <-> ( T e. L /\ ( N ` T ) < +oo ) ) )