Theorem bnnvc 25667

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 2760	. . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
2	1	isbn 25665	. 2 ⊢ (𝑊 ∈ Ban ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ (Scalar‘𝑊) ∈ CMetSp))
3	2	simp1bi 1163	1 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ NrmVec)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144  ‘cfv 6539  Scalarcsca 17437  NrmVeccnvc 24906  CMetSpccms 25659  Bancbn 25660

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-iota 6494  df-fv 6547  df-bn 25663

This theorem is used by:  bnnlm  25668  lssbn  25679  bncssbn  25701  cmslsschl  25704