Theorem hlcms 25600

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 25597	. 2 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ Ban)
2		bncms 25578	. 2 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ CMetSp)
3	1, 2	syl 18	1 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ CMetSp)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  CMetSpccms 25566  Bancbn 25567  ℂHilchl 25568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-iota 6493  df-fv 6545  df-bn 25570  df-hl 25571

This theorem is used by:  pjthlem2  25672