Theorem hlcms 24530

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 24527	. 2 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ Ban)
2		bncms 24508	. 2 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ CMetSp)
3	1, 2	syl 17	1 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ CMetSp)

Syntax hints:   → wi 4   ∈ wcel 2106  CMetSpccms 24496  Bancbn 24497  ℂHilchl 24498

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  df-hl 24501

This theorem is referenced by:  pjthlem2  24602