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Theorem isbn 21943
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 3673 . . 3  |-  ( W  e.  (NrmVec  i^i CMetSp )  <->  ( W  e. NrmVec  /\  W  e. CMetSp )
)
21anbi1i 693 . 2  |-  ( ( W  e.  (NrmVec  i^i CMetSp )  /\  F  e. CMetSp )  <->  ( ( W  e. NrmVec  /\  W  e. CMetSp )  /\  F  e. CMetSp
) )
3 fveq2 5848 . . . . 5  |-  ( w  =  W  ->  (Scalar `  w )  =  (Scalar `  W ) )
4 isbn.1 . . . . 5  |-  F  =  (Scalar `  W )
53, 4syl6eqr 2513 . . . 4  |-  ( w  =  W  ->  (Scalar `  w )  =  F )
65eleq1d 2523 . . 3  |-  ( w  =  W  ->  (
(Scalar `  w )  e. CMetSp  <-> 
F  e. CMetSp ) )
7 df-bn 21941 . . 3  |- Ban  =  {
w  e.  (NrmVec  i^i CMetSp )  |  (Scalar `  w
)  e. CMetSp }
86, 7elrab2 3256 . 2  |-  ( W  e. Ban 
<->  ( W  e.  (NrmVec 
i^i CMetSp )  /\  F  e. CMetSp
) )
9 df-3an 973 . 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823    i^i cin 3460   ` cfv 5570  Scalarcsca 14787  NrmVeccnvc 21268  CMetSpccms 21937  Bancbn 21938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-iota 5534  df-fv 5578  df-bn 21941
This theorem is referenced by:  bnsca  21944  bnnvc  21945  bncms  21949  lssbn  21956  srabn  21966  ishl2  21976
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