Theorem hlphl 24529

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 24528	. 2 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ ℂPreHil)
2		cphphl 24335	. 2 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil)
3	1, 2	syl 17	1 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ PreHil)

Syntax hints:   → wi 4   ∈ wcel 2106  PreHilcphl 20829  ℂPreHilccph 24330  ℂHilchl 24498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-nul 5230

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-xp 5595  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fv 6441  df-ov 7278  df-cph 24332  df-hl 24501

This theorem is referenced by:  chlcsschl  24542  pjthlem1  24601  pjth  24603  pjth2  24604  cldcss  24605  hlhil  24607