Theorem bncms 24508

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 24502	. 2 ⊢ (𝑊 ∈ Ban ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ (Scalar‘𝑊) ∈ CMetSp))
3	2	simp2bi 1145	1 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ CMetSp)

Syntax hints:   → wi 4   ∈ wcel 2106  ‘cfv 6433  Scalarcsca 16965  NrmVeccnvc 23737  CMetSpccms 24496  Bancbn 24497

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-bn 24500

This theorem is referenced by:  bncmet  24511  lssbn  24516  hlcms  24530  bncssbn  24538  sitgclbn  32310