Theorem hlcph 25484

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 25482	. 2 ⊢ (𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil))
2	1	simprbi 502	1 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ ℂPreHil)

Syntax hints:   → wi 4   ∈ wcel 2145  ℂPreHilccph 25286  Bancbn 25453  ℂHilchl 25454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-in 3914  df-hl 25457

This theorem is referenced by:  hlphl  25485  hlprlem  25487  cmslsschl  25497  chlcsschl  25498  pjthlem1  25557  pjthlem2  25558  cldcss  25561