Theorem bnsca 25291

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 25290	. 2 ⊢ (𝑊 ∈ Ban ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp))
3	2	simp3bi 1147	1 ⊢ (𝑊 ∈ Ban → 𝐹 ∈ CMetSp)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  ‘cfv 6531  Scalarcsca 17274  NrmVeccnvc 24520  CMetSpccms 25284  Bancbn 25285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-iota 6484  df-fv 6539  df-bn 25288

This theorem is referenced by:  lssbn  25304  hlprlem  25319  bncssbn  25326  cmslsschl  25329