Theorem hlcms 25238

Description: Every subcomplex Hilbert space is a complete metric space. (Contributed by Mario Carneiro, 17-Oct-2015.)

Assertion

Ref	Expression
hlcms	⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ CMetSp)

Proof of Theorem hlcms

Step	Hyp	Ref	Expression
1		hlbn 25235	. 2 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ Ban)
2		bncms 25216	. 2 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ CMetSp)
3	1, 2	syl 17	1 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ CMetSp)

Syntax hints:   → wi 4   ∈ wcel 2098  CMetSpccms 25204  Bancbn 25205  ℂHilchl 25206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-iota 6486  df-fv 6542  df-bn 25208  df-hl 25209

This theorem is referenced by:  pjthlem2  25310