Theorem hlcph 24078

Description: Every subcomplex Hilbert space is a subcomplex pre-Hilbert space. (Contributed by Mario Carneiro, 15-Oct-2015.)

Assertion

Ref	Expression
hlcph	⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ ℂPreHil)

Proof of Theorem hlcph

Step	Hyp	Ref	Expression
1		ishl 24076	. 2 ⊢ (𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil))
2	1	simprbi 500	1 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ ℂPreHil)

Syntax hints:   → wi 4   ∈ wcel 2111  ℂPreHilccph 23881  Bancbn 24047  ℂHilchl 24048

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-in 3867  df-hl 24051

This theorem is referenced by:  hlphl  24079  hlprlem  24081  cmslsschl  24091  chlcsschl  24092  pjthlem1  24151  pjthlem2  24152  cldcss  24155