Theorem ishlo 30980

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 30979	. 2 ⊢ CHilOLD = (CBan ∩ CPreHilOLD)
2	1	elin2 4135	1 ⊢ (𝑈 ∈ CHilOLD ↔ (𝑈 ∈ CBan ∧ 𝑈 ∈ CPreHilOLD))

Syntax hints:   ↔ wb 208   ∧ wa 397   ∈ wcel 2121  CPreHil_OLDccphlo 30905  CBanccbn 30955  CHil_OLDchlo 30978

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-tru 1551  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-v 3435  df-in 3892  df-hlo 30979

This theorem is referenced by:  hlobn  30981  hlph  30982  cnchl  31009  hhhl  31297