Theorem bnnvc 25267

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 2731	. . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
2	1	isbn 25265	. 2 ⊢ (𝑊 ∈ Ban ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ (Scalar‘𝑊) ∈ CMetSp))
3	2	simp1bi 1145	1 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ NrmVec)

Syntax hints:   → wi 4   ∈ wcel 2111  ‘cfv 6481  Scalarcsca 17164  NrmVeccnvc 24496  CMetSpccms 25259  Bancbn 25260

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-iota 6437  df-fv 6489  df-bn 25263

This theorem is referenced by:  bnnlm  25268  lssbn  25279  bncssbn  25301  cmslsschl  25304