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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 Scalar
Assertion
Ref Expression
isbn Ban NrmVec CMetSp CMetSp

Proof of Theorem isbn
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elin 3530 . . 3 NrmVec CMetSp NrmVec CMetSp
21anbi1i 677 . 2 NrmVec CMetSp CMetSp NrmVec CMetSp CMetSp
3 fveq2 5728 . . . . 5 Scalar Scalar
4 isbn.1 . . . . 5 Scalar
53, 4syl6eqr 2486 . . . 4 Scalar
65eleq1d 2502 . . 3 Scalar CMetSp CMetSp
7 df-bn 19289 . . 3 Ban NrmVec CMetSp Scalar CMetSp
86, 7elrab2 3094 . 2 Ban NrmVec CMetSp CMetSp
9 df-3an 938 . 2 NrmVec CMetSp CMetSp NrmVec CMetSp CMetSp
102, 8, 93bitr4i 269 1 Ban NrmVec CMetSp CMetSp
 Colors of variables: wff set class Syntax hints:   wb 177   wa 359   w3a 936   wceq 1652   wcel 1725   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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