Theorem bncms 25260

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 2729	. . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
2	1	isbn 25254	. 2 ⊢ (𝑊 ∈ Ban ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ (Scalar‘𝑊) ∈ CMetSp))
3	2	simp2bi 1146	1 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ CMetSp)

Syntax hints:   → wi 4   ∈ wcel 2109  ‘cfv 6486  Scalarcsca 17182  NrmVeccnvc 24485  CMetSpccms 25248  Bancbn 25249

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701

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 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-iota 6442  df-fv 6494  df-bn 25252

This theorem is referenced by:  bncmet  25263  lssbn  25268  hlcms  25282  bncssbn  25290  sitgclbn  34310