Theorem hlph 31378

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 31376	. 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 2145  CPreHil_OLDccphlo 31301  CBanccbn 31351  CHil_OLDchlo 31374

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3455  df-in 3909  df-hlo 31375

This theorem is used by:  hlpar2  31385  hlpar  31386  hlipdir  31401  hlipass  31402  htthlem  31406