Theorem bnsca 25331

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 25330	. 2 ⊢ (𝑊 ∈ Ban ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp))
3	2	simp3bi 1153	1 ⊢ (𝑊 ∈ Ban → 𝐹 ∈ CMetSp)

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  ‘cfv 6492  Scalarcsca 17221  NrmVeccnvc 24571  CMetSpccms 25324  Bancbn 25325

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-iota 6448  df-fv 6500  df-bn 25328

This theorem is referenced by:  lssbn  25344  hlprlem  25359  bncssbn  25366  cmslsschl  25369