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	syl6bb	|- ( ( U e. NrmCVec /\ W e. NrmCVec ) -> ( T e. B <-> ( T e. L /\ ( N ` T ) < +oo ) ) )