Theorem hlph 31150

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 31148	. 2 ⊢ (𝑈 ∈ CHilOLD ↔ (𝑈 ∈ CBan ∧ 𝑈 ∈ CPreHilOLD))
2	1	simprbi 502	1 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ CPreHilOLD)

Syntax hints:   → wi 4   ∈ wcel 2145  CPreHil_OLDccphlo 31073  CBanccbn 31123  CHil_OLDchlo 31146

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-in 3914  df-hlo 31147

This theorem is referenced by:  hlpar2  31157  hlpar  31158  hlipdir  31173  hlipass  31174  htthlem  31178