Theorem bnsca 23942

Description: The scalar field of a Banach space is complete. (Contributed by NM, 8-Sep-2007.) (Revised by Mario Carneiro, 15-Oct-2015.)

Hypothesis

Ref	Expression
isbn.1	⊢ 𝐹 = (Scalar‘𝑊)

Assertion

Ref	Expression
bnsca	⊢ (𝑊 ∈ Ban → 𝐹 ∈ CMetSp)

Proof of Theorem bnsca

Step	Hyp	Ref	Expression
1		isbn.1	. . 3 ⊢ 𝐹 = (Scalar‘𝑊)
2	1	isbn 23941	. 2 ⊢ (𝑊 ∈ Ban ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp))
3	2	simp3bi 1143	1 ⊢ (𝑊 ∈ Ban → 𝐹 ∈ CMetSp)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  ‘cfv 6355  Scalarcsca 16568  NrmVeccnvc 23191  CMetSpccms 23935  Bancbn 23936

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-iota 6314  df-fv 6363  df-bn 23939

This theorem is referenced by:  lssbn  23955  hlprlem  23970  bncssbn  23977  cmslsschl  23980