Theorem bncms 24413

Description: A Banach space is a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015.)

Assertion

Ref	Expression
bncms	⊢ (𝑊 ∈ Ban → 𝑊 ∈ CMetSp)

Proof of Theorem bncms

Step	Hyp	Ref	Expression
1		eqid 2738	. . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
2	1	isbn 24407	. 2 ⊢ (𝑊 ∈ Ban ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ (Scalar‘𝑊) ∈ CMetSp))
3	2	simp2bi 1144	1 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ CMetSp)

Syntax hints:   → wi 4   ∈ wcel 2108  ‘cfv 6418  Scalarcsca 16891  NrmVeccnvc 23643  CMetSpccms 24401  Bancbn 24402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-iota 6376  df-fv 6426  df-bn 24405

This theorem is referenced by:  bncmet  24416  lssbn  24421  hlcms  24435  bncssbn  24443  sitgclbn  32210