Theorem hlcph 25504

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 25502	. 2 ⊢ (𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil))
2	1	simprbi 502	1 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ ℂPreHil)

Syntax hints:   → wi 4   ∈ wcel 2143  ℂPreHilccph 25306  Bancbn 25473  ℂHilchl 25474

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-in 3913  df-hl 25477

This theorem is referenced by:  hlphl  25505  hlprlem  25507  cmslsschl  25517  chlcsschl  25518  pjthlem1  25577  pjthlem2  25578  cldcss  25581