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Theorem isbn 20871
Description: A Banach space is a normed vector space with a complete induced metric. (Contributed by NM, 5-Dec-2006.) (Revised by Mario Carneiro, 15-Oct-2015.)
Hypothesis
Ref Expression
isbn.1  |-  F  =  (Scalar `  W )
Assertion
Ref Expression
isbn  |-  ( W  e. Ban 
<->  ( W  e. NrmVec  /\  W  e. CMetSp  /\  F  e. CMetSp )
)

Proof of Theorem isbn
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 elin 3560 . . 3  |-  ( W  e.  (NrmVec  i^i CMetSp )  <->  ( W  e. NrmVec  /\  W  e. CMetSp )
)
21anbi1i 695 . 2  |-  ( ( W  e.  (NrmVec  i^i CMetSp )  /\  F  e. CMetSp )  <->  ( ( W  e. NrmVec  /\  W  e. CMetSp )  /\  F  e. CMetSp
) )
3 fveq2 5712 . . . . 5  |-  ( w  =  W  ->  (Scalar `  w )  =  (Scalar `  W ) )
4 isbn.1 . . . . 5  |-  F  =  (Scalar `  W )
53, 4syl6eqr 2493 . . . 4  |-  ( w  =  W  ->  (Scalar `  w )  =  F )
65eleq1d 2509 . . 3  |-  ( w  =  W  ->  (
(Scalar `  w )  e. CMetSp  <-> 
F  e. CMetSp ) )
7 df-bn 20869 . . 3  |- Ban  =  {
w  e.  (NrmVec  i^i CMetSp )  |  (Scalar `  w
)  e. CMetSp }
86, 7elrab2 3140 . 2  |-  ( W  e. Ban 
<->  ( W  e.  (NrmVec 
i^i CMetSp )  /\  F  e. CMetSp
) )
9 df-3an 967 . 2  |-  ( ( W  e. NrmVec  /\  W  e. CMetSp  /\  F  e. CMetSp )  <->  ( ( W  e. NrmVec  /\  W  e. CMetSp
)  /\  F  e. CMetSp ) )
102, 8, 93bitr4i 277 1  |-  ( W  e. Ban 
<->  ( W  e. NrmVec  /\  W  e. CMetSp  /\  F  e. CMetSp )
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    i^i cin 3348   ` cfv 5439  Scalarcsca 14262  NrmVeccnvc 20196  CMetSpccms 20865  Bancbn 20866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-rex 2742  df-rab 2745  df-v 2995  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-iota 5402  df-fv 5447  df-bn 20869
This theorem is referenced by:  bnsca  20872  bnnvc  20873  bncms  20877  lssbn  20884  srabn  20894  ishl2  20904
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