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Theorem isbn 19291
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 3530 . . 3  |-  ( W  e.  (NrmVec  i^i CMetSp )  <->  ( W  e. NrmVec  /\  W  e. CMetSp )
)
21anbi1i 677 . 2  |-  ( ( W  e.  (NrmVec  i^i CMetSp )  /\  F  e. CMetSp )  <->  ( ( W  e. NrmVec  /\  W  e. CMetSp )  /\  F  e. CMetSp
) )
3 fveq2 5728 . . . . 5  |-  ( w  =  W  ->  (Scalar `  w )  =  (Scalar `  W ) )
4 isbn.1 . . . . 5  |-  F  =  (Scalar `  W )
53, 4syl6eqr 2486 . . . 4  |-  ( w  =  W  ->  (Scalar `  w )  =  F )
65eleq1d 2502 . . 3  |-  ( w  =  W  ->  (
(Scalar `  w )  e. CMetSp  <-> 
F  e. CMetSp ) )
7 df-bn 19289 . . 3  |- Ban  =  {
w  e.  (NrmVec  i^i CMetSp )  |  (Scalar `  w
)  e. CMetSp }
86, 7elrab2 3094 . 2  |-  ( W  e. Ban 
<->  ( W  e.  (NrmVec 
i^i CMetSp )  /\  F  e. CMetSp
) )
9 df-3an 938 . 2  |-  ( ( W  e. NrmVec  /\  W  e. CMetSp  /\  F  e. CMetSp )  <->  ( ( W  e. NrmVec  /\  W  e. CMetSp
)  /\  F  e. CMetSp ) )
102, 8, 93bitr4i 269 1  |-  ( W  e. Ban 
<->  ( W  e. NrmVec  /\  W  e. CMetSp  /\  F  e. CMetSp )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    i^i cin 3319   ` cfv 5454  Scalarcsca 13532  NrmVeccnvc 18629  CMetSpccms 19285  Bancbn 19286
This theorem is referenced by:  bnsca  19292  bnnvc  19293  bncms  19297  lssbn  19304  srabn  19314  ishl2  19324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-bn 19289
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