Theorem bnsca 25266

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 25265	. 2 ⊢ (𝑊 ∈ Ban ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp))
3	2	simp3bi 1145	1 ⊢ (𝑊 ∈ Ban → 𝐹 ∈ CMetSp)

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099  ‘cfv 6548  Scalarcsca 17235  NrmVeccnvc 24489  CMetSpccms 25259  Bancbn 25260

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-iota 6500  df-fv 6556  df-bn 25263

This theorem is referenced by:  lssbn  25279  hlprlem  25294  bncssbn  25301  cmslsschl  25304