Theorem bnsca 25387

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 25386	. 2 ⊢ (𝑊 ∈ Ban ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp))
3	2	simp3bi 1146	1 ⊢ (𝑊 ∈ Ban → 𝐹 ∈ CMetSp)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106  ‘cfv 6563  Scalarcsca 17301  NrmVeccnvc 24610  CMetSpccms 25380  Bancbn 25381

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-bn 25384

This theorem is referenced by:  lssbn  25400  hlprlem  25415  bncssbn  25422  cmslsschl  25425