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Theorem isbn 25386
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 3979 . . 3 (𝑊 ∈ (NrmVec ∩ CMetSp) ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp))
21anbi1i 624 . 2 ((𝑊 ∈ (NrmVec ∩ CMetSp) ∧ 𝐹 ∈ CMetSp) ↔ ((𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp) ∧ 𝐹 ∈ CMetSp))
3 fveq2 6907 . . . . 5 (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
4 isbn.1 . . . . 5 𝐹 = (Scalar‘𝑊)
53, 4eqtr4di 2793 . . . 4 (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
65eleq1d 2824 . . 3 (𝑤 = 𝑊 → ((Scalar‘𝑤) ∈ CMetSp ↔ 𝐹 ∈ CMetSp))
7 df-bn 25384 . . 3 Ban = {𝑤 ∈ (NrmVec ∩ CMetSp) ∣ (Scalar‘𝑤) ∈ CMetSp}
86, 7elrab2 3698 . 2 (𝑊 ∈ Ban ↔ (𝑊 ∈ (NrmVec ∩ CMetSp) ∧ 𝐹 ∈ CMetSp))
9 df-3an 1088 . 2 ((𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp) ↔ ((𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp) ∧ 𝐹 ∈ CMetSp))
102, 8, 93bitr4i 303 1 (𝑊 ∈ Ban ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  cin 3962  cfv 6563  Scalarcsca 17301  NrmVeccnvc 24610  CMetSpccms 25380  Bancbn 25381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-bn 25384
This theorem is referenced by:  bnsca  25387  bnnvc  25388  bncms  25392  lssbn  25400  srabn  25408  ishl2  25418  cmslssbn  25420
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