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Theorem isbn 18776
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 3371 . . 3  |-  ( W  e.  (NrmVec  i^i CMetSp )  <->  ( W  e. NrmVec  /\  W  e. CMetSp )
)
21anbi1i 676 . 2  |-  ( ( W  e.  (NrmVec  i^i CMetSp )  /\  F  e. CMetSp )  <->  ( ( W  e. NrmVec  /\  W  e. CMetSp )  /\  F  e. CMetSp
) )
3 fveq2 5541 . . . . 5  |-  ( w  =  W  ->  (Scalar `  w )  =  (Scalar `  W ) )
4 isbn.1 . . . . 5  |-  F  =  (Scalar `  W )
53, 4syl6eqr 2346 . . . 4  |-  ( w  =  W  ->  (Scalar `  w )  =  F )
65eleq1d 2362 . . 3  |-  ( w  =  W  ->  (
(Scalar `  w )  e. CMetSp  <-> 
F  e. CMetSp ) )
7 df-bn 18774 . . 3  |- Ban  =  {
w  e.  (NrmVec  i^i CMetSp )  |  (Scalar `  w
)  e. CMetSp }
86, 7elrab2 2938 . 2  |-  ( W  e. Ban 
<->  ( W  e.  (NrmVec 
i^i CMetSp )  /\  F  e. CMetSp
) )
9 df-3an 936 . 2  |-  ( ( W  e. NrmVec  /\  W  e. CMetSp  /\  F  e. CMetSp )  <->  ( ( W  e. NrmVec  /\  W  e. CMetSp
)  /\  F  e. CMetSp ) )
102, 8, 93bitr4i 268 1  |-  ( W  e. Ban 
<->  ( W  e. NrmVec  /\  W  e. CMetSp  /\  F  e. CMetSp )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    i^i cin 3164   ` cfv 5271  Scalarcsca 13227  NrmVeccnvc 18120  CMetSpccms 18770  Bancbn 18771
This theorem is referenced by:  bnsca  18777  bnnvc  18778  bncms  18782  lssbn  18789  srabn  18793  ishl2  18803
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-bn 18774
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