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Theorem isbn 23354
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 3947 . . 3 (𝑊 ∈ (NrmVec ∩ CMetSp) ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp))
21anbi1i 610 . 2 ((𝑊 ∈ (NrmVec ∩ CMetSp) ∧ 𝐹 ∈ CMetSp) ↔ ((𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp) ∧ 𝐹 ∈ CMetSp))
3 fveq2 6333 . . . . 5 (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
4 isbn.1 . . . . 5 𝐹 = (Scalar‘𝑊)
53, 4syl6eqr 2823 . . . 4 (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
65eleq1d 2835 . . 3 (𝑤 = 𝑊 → ((Scalar‘𝑤) ∈ CMetSp ↔ 𝐹 ∈ CMetSp))
7 df-bn 23352 . . 3 Ban = {𝑤 ∈ (NrmVec ∩ CMetSp) ∣ (Scalar‘𝑤) ∈ CMetSp}
86, 7elrab2 3518 . 2 (𝑊 ∈ Ban ↔ (𝑊 ∈ (NrmVec ∩ CMetSp) ∧ 𝐹 ∈ CMetSp))
9 df-3an 1073 . 2 ((𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp) ↔ ((𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp) ∧ 𝐹 ∈ CMetSp))
102, 8, 93bitr4i 292 1 (𝑊 ∈ Ban ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  cin 3722  cfv 6030  Scalarcsca 16152  NrmVeccnvc 22606  CMetSpccms 23348  Bancbn 23349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-iota 5993  df-fv 6038  df-bn 23352
This theorem is referenced by:  bnsca  23355  bnnvc  23356  bncms  23360  lssbn  23367  srabn  23375  ishl2  23385
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