Theorem hlcph 25332

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 25330	. 2 ⊢ (𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil))
2	1	simprbi 497	1 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ ℂPreHil)

Syntax hints:   → wi 4   ∈ wcel 2114  ℂPreHilccph 25134  Bancbn 25301  ℂHilchl 25302

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-in 3910  df-hl 25305

This theorem is referenced by:  hlphl  25333  hlprlem  25335  cmslsschl  25345  chlcsschl  25346  pjthlem1  25405  pjthlem2  25406  cldcss  25409