Theorem hlcph 25690

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 25688	. 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 2144  ℂPreHilccph 25492  Bancbn 25659  ℂHilchl 25660

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-in 4091  df-hl 25663

This theorem is used by:  hlphl  25691  hlprlem  25693  cmslsschl  25703  chlcsschl  25704  pjthlem1  25763  pjthlem2  25764  cldcss  25767