Theorem hlphl 24262

Description: Every subcomplex Hilbert space is an inner product space (also called a pre-Hilbert space). (Contributed by NM, 28-Apr-2007.) (Revised by Mario Carneiro, 15-Oct-2015.)

Assertion

Ref	Expression
hlphl	⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ PreHil)

Proof of Theorem hlphl

Step	Hyp	Ref	Expression
1		hlcph 24261	. 2 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ ℂPreHil)
2		cphphl 24068	. 2 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil)
3	1, 2	syl 17	1 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ PreHil)

Syntax hints:   → wi 4   ∈ wcel 2110  PreHilcphl 20586  ℂPreHilccph 24063  ℂHilchl 24231

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-nul 5199

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-xp 5557  df-cnv 5559  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fv 6388  df-ov 7216  df-cph 24065  df-hl 24234

This theorem is referenced by:  chlcsschl  24275  pjthlem1  24334  pjth  24336  pjth2  24337  cldcss  24338  hlhil  24340