Theorem bnnvc 23137

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 2622	. . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
2	1	isbn 23135	. 2 ⊢ (𝑊 ∈ Ban ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ (Scalar‘𝑊) ∈ CMetSp))
3	2	simp1bi 1076	1 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ NrmVec)

Syntax hints:   → wi 4   ∈ wcel 1990  ‘cfv 5888  Scalarcsca 15944  NrmVeccnvc 22386  CMetSpccms 23129  Bancbn 23130

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-bn 23133

This theorem is referenced by:  bnnlm  23138  lssbn  23148