Theorem hlph 31497

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 31495	. 2 ⊢ (𝑈 ∈ CHilOLD ↔ (𝑈 ∈ CBan ∧ 𝑈 ∈ CPreHilOLD))
2	1	simprbi 503	1 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ CPreHilOLD)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144  CPreHil_OLDccphlo 31420  CBanccbn 31470  CHil_OLDchlo 31493

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-in 4091  df-hlo 31494

This theorem is used by:  hlpar2  31504  hlpar  31505  hlipdir  31520  hlipass  31521  htthlem  31525