Theorem hlcph 24433

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 24431	. 2 ⊢ (𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil))
2	1	simprbi 496	1 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ ℂPreHil)

Syntax hints:   → wi 4   ∈ wcel 2108  ℂPreHilccph 24235  Bancbn 24402  ℂHilchl 24403

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-in 3890  df-hl 24406

This theorem is referenced by:  hlphl  24434  hlprlem  24436  cmslsschl  24446  chlcsschl  24447  pjthlem1  24506  pjthlem2  24507  cldcss  24510