Theorem bncms 25386

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 2761	. . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
2	1	isbn 25380	. 2 ⊢ (𝑊 ∈ Ban ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ (Scalar‘𝑊) ∈ CMetSp))
3	2	simp2bi 1158	1 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ CMetSp)

Syntax hints:   → wi 4   ∈ wcel 2141  ‘cfv 6517  Scalarcsca 17272  NrmVeccnvc 24621  CMetSpccms 25374  Bancbn 25375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-iota 6473  df-fv 6525  df-bn 25378

This theorem is referenced by:  bncmet  25389  lssbn  25394  hlcms  25408  bncssbn  25416  sitgclbn  34601