Theorem hlcph 25665

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 25663	. 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 2145  ℂPreHilccph 25467  Bancbn 25634  ℂHilchl 25635

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3453  df-in 3906  df-hl 25638

This theorem is used by:  hlphl  25666  hlprlem  25668  cmslsschl  25678  chlcsschl  25679  pjthlem1  25738  pjthlem2  25739  cldcss  25742