Theorem bnnvc 25300

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 2737	. . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
2	1	isbn 25298	. 2 ⊢ (𝑊 ∈ Ban ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ (Scalar‘𝑊) ∈ CMetSp))
3	2	simp1bi 1146	1 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ NrmVec)

Syntax hints:   → wi 4   ∈ wcel 2114  ‘cfv 6493  Scalarcsca 17184  NrmVeccnvc 24529  CMetSpccms 25292  Bancbn 25293

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-iota 6449  df-fv 6501  df-bn 25296

This theorem is referenced by:  bnnlm  25301  lssbn  25312  bncssbn  25334  cmslsschl  25337