Theorem hlph 30976

Description: Every complex Hilbert space is an inner product space (also called a pre-Hilbert space). (Contributed by NM, 28-Apr-2007.) (New usage is discouraged.)

Assertion

Ref	Expression
hlph	⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ CPreHilOLD)

Proof of Theorem hlph

Step	Hyp	Ref	Expression
1		ishlo 30974	. 2 ⊢ (𝑈 ∈ CHilOLD ↔ (𝑈 ∈ CBan ∧ 𝑈 ∈ CPreHilOLD))
2	1	simprbi 497	1 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ CPreHilOLD)

Syntax hints:   → wi 4   ∈ wcel 2114  CPreHil_OLDccphlo 30899  CBanccbn 30949  CHil_OLDchlo 30972

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-in 3910  df-hlo 30973

This theorem is referenced by:  hlpar2  30983  hlpar  30984  hlipdir  30999  hlipass  31000  htthlem  31004