Theorem hlcms 25486

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 25483	. 2 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ Ban)
2		bncms 25464	. 2 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ CMetSp)
3	1, 2	syl 18	1 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ CMetSp)

Syntax hints:   → wi 4   ∈ wcel 2145  CMetSpccms 25452  Bancbn 25453  ℂHilchl 25454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-iota 6481  df-fv 6533  df-bn 25456  df-hl 25457

This theorem is referenced by:  pjthlem2  25558