Theorem ishlo 27743

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 e. CHilOLD <-> ( U e. CBan /\ U e. CPreHil OLD ) )

Proof of Theorem ishlo

Step	Hyp	Ref	Expression
1		df-hlo 27742	. 2 |- CHilOLD = ( CBan i^i CPreHil OLD )
2	1	elin2 3801	1 |- ( U e. CHilOLD <-> ( U e. CBan /\ U e. CPreHil OLD ) )

Syntax hints:   <-> wb 196   /\ wa 384   e. wcel 1990   CPreHil OLDccphlo 27667   CBanccbn 27718   CHilOLDchlo 27741

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581  df-hlo 27742

This theorem is referenced by:  hlobn  27744  hlph  27745  cnchl  27772  ssphl  27773  hhhl  28061  hhsshl  28138