Theorem hlph 31184

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 31182	. 2 ⊢ (𝑈 ∈ CHilOLD ↔ (𝑈 ∈ CBan ∧ 𝑈 ∈ CPreHilOLD))
2	1	simprbi 502	1 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ CPreHilOLD)

Syntax hints:   → wi 4   ∈ wcel 2149  CPreHil_OLDccphlo 31107  CBanccbn 31157  CHil_OLDchlo 31180

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-in 3920  df-hlo 31181

This theorem is referenced by:  hlpar2  31191  hlpar  31192  hlipdir  31207  hlipass  31208  htthlem  31212