Theorem hlphl 25213

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 25212	. 2 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ ℂPreHil)
2		cphphl 25019	. 2 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil)
3	1, 2	syl 17	1 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ PreHil)

Syntax hints:   → wi 4   ∈ wcel 2105  PreHilcphl 21487  ℂPreHilccph 25014  ℂHilchl 25182

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-nul 5306

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-xp 5682  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fv 6551  df-ov 7415  df-cph 25016  df-hl 25185

This theorem is referenced by:  chlcsschl  25226  pjthlem1  25285  pjth  25287  pjth2  25288  cldcss  25289  hlhil  25291