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Theorem isbn 23942
 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 ∈ Ban ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp))

Proof of Theorem isbn
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 elin 3897 . . 3 (𝑊 ∈ (NrmVec ∩ CMetSp) ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp))
21anbi1i 626 . 2 ((𝑊 ∈ (NrmVec ∩ CMetSp) ∧ 𝐹 ∈ CMetSp) ↔ ((𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp) ∧ 𝐹 ∈ CMetSp))
3 fveq2 6645 . . . . 5 (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
4 isbn.1 . . . . 5 𝐹 = (Scalar‘𝑊)
53, 4eqtr4di 2851 . . . 4 (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
65eleq1d 2874 . . 3 (𝑤 = 𝑊 → ((Scalar‘𝑤) ∈ CMetSp ↔ 𝐹 ∈ CMetSp))
7 df-bn 23940 . . 3 Ban = {𝑤 ∈ (NrmVec ∩ CMetSp) ∣ (Scalar‘𝑤) ∈ CMetSp}
86, 7elrab2 3631 . 2 (𝑊 ∈ Ban ↔ (𝑊 ∈ (NrmVec ∩ CMetSp) ∧ 𝐹 ∈ CMetSp))
9 df-3an 1086 . 2 ((𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp) ↔ ((𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp) ∧ 𝐹 ∈ CMetSp))
102, 8, 93bitr4i 306 1 (𝑊 ∈ Ban ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ∩ cin 3880  ‘cfv 6324  Scalarcsca 16560  NrmVeccnvc 23188  CMetSpccms 23936  Bancbn 23937 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-rab 3115  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-iota 6283  df-fv 6332  df-bn 23940 This theorem is referenced by:  bnsca  23943  bnnvc  23944  bncms  23948  lssbn  23956  srabn  23964  ishl2  23974  cmslssbn  23976
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