Theorem hlcms 25425

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 25422	. 2 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ Ban)
2		bncms 25403	. 2 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ CMetSp)
3	1, 2	syl 17	1 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ CMetSp)

Syntax hints:   → wi 4   ∈ wcel 2108  CMetSpccms 25391  Bancbn 25392  ℂHilchl 25393

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3483  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-nul 4343  df-if 4535  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-br 5152  df-iota 6522  df-fv 6577  df-bn 25395  df-hl 25396

This theorem is referenced by:  pjthlem2  25497