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Metamath Proof Explorer

Theorem blocn

Description: A linear operator is continuous iff it is bounded. Theorem 2.7-9(a) of Kreyszig p. 97. (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
blocn.4 
|- L = ( U LnOp W )
Assertion blocn 
|- ( T e. L -> ( T e. ( J Cn K ) <-> T e. B ) )

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 blocn.4 
 |-  L = ( U LnOp W )
9 eleq1 
 |-  ( T = if ( T e. L , T , ( U 0op W ) ) -> ( T e. ( J Cn K ) <-> if ( T e. L , T , ( U 0op W ) ) e. ( J Cn K ) ) )
10 eleq1 
 |-  ( T = if ( T e. L , T , ( U 0op W ) ) -> ( T e. B <-> if ( T e. L , T , ( U 0op W ) ) e. B ) )
11 9 10 bibi12d 
 |-  ( T = if ( T e. L , T , ( U 0op W ) ) -> ( ( T e. ( J Cn K ) <-> T e. B ) <-> ( if ( T e. L , T , ( U 0op W ) ) e. ( J Cn K ) <-> if ( T e. L , T , ( U 0op W ) ) e. B ) ) )
12 eqid 
 |-  ( U 0op W ) = ( U 0op W )
13 12 8 0lno 
 |-  ( ( U e. NrmCVec /\ W e. NrmCVec ) -> ( U 0op W ) e. L )
14 6 7 13 mp2an 
 |-  ( U 0op W ) e. L
15 14 elimel 
 |-  if ( T e. L , T , ( U 0op W ) ) e. L
16 1 2 3 4 8 5 6 7 15 blocni 
 |-  ( if ( T e. L , T , ( U 0op W ) ) e. ( J Cn K ) <-> if ( T e. L , T , ( U 0op W ) ) e. B )
17 11 16 dedth 
 |-  ( T e. L -> ( T e. ( J Cn K ) <-> T e. B ) )