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Metamath Proof Explorer

Theorem blocn2

Description: A bounded linear operator is continuous. (Contributed by NM, 25-Dec-2007) (New usage is discouraged.)

Ref Expression
Hypotheses blocn.8 
|- C = ( IndMet ` U )
blocn.d 
|- D = ( IndMet ` W )
blocn.j 
|- J = ( MetOpen ` C )
blocn.k 
|- K = ( MetOpen ` D )
blocn.5 
|- B = ( U BLnOp W )
blocn.u 
|- U e. NrmCVec
blocn.w 
|- W e. NrmCVec
Assertion blocn2 
|- ( T e. B -> T e. ( J Cn K ) )

Proof

Step Hyp Ref Expression
1 blocn.8 
 |-  C = ( IndMet ` U )
2 blocn.d 
 |-  D = ( IndMet ` W )
3 blocn.j 
 |-  J = ( MetOpen ` C )
4 blocn.k 
 |-  K = ( MetOpen ` D )
5 blocn.5 
 |-  B = ( U BLnOp W )
6 blocn.u 
 |-  U e. NrmCVec
7 blocn.w 
 |-  W e. NrmCVec
8 eqid 
 |-  ( U LnOp W ) = ( U LnOp W )
9 8 5 bloln 
 |-  ( ( U e. NrmCVec /\ W e. NrmCVec /\ T e. B ) -> T e. ( U LnOp W ) )
10 6 7 9 mp3an12 
 |-  ( T e. B -> T e. ( U LnOp W ) )
11 1 2 3 4 5 6 7 8 blocn 
 |-  ( T e. ( U LnOp W ) -> ( T e. ( J Cn K ) <-> T e. B ) )
12 11 biimprd 
 |-  ( T e. ( U LnOp W ) -> ( T e. B -> T e. ( J Cn K ) ) )
13 10 12 mpcom 
 |-  ( T e. B -> T e. ( J Cn K ) )