Theorem bnnvc 25297

Description: A Banach space is a normed vector space. (Contributed by Mario Carneiro, 15-Oct-2015.)

Assertion

Ref	Expression
bnnvc	⊢ (𝑊 ∈ Ban → 𝑊 ∈ NrmVec)

Proof of Theorem bnnvc

Step	Hyp	Ref	Expression
1		eqid 2736	. . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
2	1	isbn 25295	. 2 ⊢ (𝑊 ∈ Ban ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ (Scalar‘𝑊) ∈ CMetSp))
3	2	simp1bi 1145	1 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ NrmVec)

Syntax hints:   → wi 4   ∈ wcel 2109  ‘cfv 6536  Scalarcsca 17279  NrmVeccnvc 24525  CMetSpccms 25289  Bancbn 25290

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-iota 6489  df-fv 6544  df-bn 25293

This theorem is referenced by:  bnnlm  25298  lssbn  25309  bncssbn  25331  cmslsschl  25334