Theorem hlcph 25718

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 25716	. 2 ⊢ (𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil))
2	1	simprbi 503	1 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ ℂPreHil)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147  ℂPreHilccph 25520  Bancbn 25687  ℂHilchl 25688

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-in 4094  df-hl 25691

This theorem is used by:  hlphl  25719  hlprlem  25721  cmslsschl  25731  chlcsschl  25732  pjthlem1  25791  pjthlem2  25792  cldcss  25795