Theorem ishlo 31036

Description: The predicate "is a complex Hilbert space." A Hilbert space is a Banach space which is also an inner product space, i.e. whose norm satisfies the parallelogram law. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)

Assertion

Ref	Expression
ishlo	⊢ (𝑈 ∈ CHilOLD ↔ (𝑈 ∈ CBan ∧ 𝑈 ∈ CPreHilOLD))

Proof of Theorem ishlo

Step	Hyp	Ref	Expression
1		df-hlo 31035	. 2 ⊢ CHilOLD = (CBan ∩ CPreHilOLD)
2	1	elin2 4155	1 ⊢ (𝑈 ∈ CHilOLD ↔ (𝑈 ∈ CBan ∧ 𝑈 ∈ CPreHilOLD))

Syntax hints:   ↔ wb 208   ∧ wa 399   ∈ wcel 2141  CPreHil_OLDccphlo 30961  CBanccbn 31011  CHil_OLDchlo 31034

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-in 3911  df-hlo 31035

This theorem is referenced by:  hlobn  31037  hlph  31038  cnchl  31065  hhhl  31353