Theorem ishl 8591

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.)

Assertion

Ref	Expression
ishl	|- (U e. CHil <-> (U e. CBan /\ U e. CPreHil))

Proof of Theorem ishl

Step	Hyp	Ref	Expression
1		df-hl 8590	. . 3 |- CHil = (CBan i^i CPreHil)
2	1	eleq2i 1538	. 2 |- (U e. CHil <-> U e. (CBan i^i CPreHil))
3		elin 2207	. 2 |- (U e. (CBan i^i CPreHil) <-> (U e. CBan /\ U e. CPreHil))
4	2, 3	bitr 173	1 |- (U e. CHil <-> (U e. CBan /\ U e. CPreHil))

Syntax hints:   <-> wb 146   /\ wa 223   e. wcel 958   i^i cin 2046  CPreHilcphl 8471  CBancbn 8522  CHilchl 8589

This theorem is referenced by:  hlbn 8592  hlph 8593  cnhl 8618  ssphl 8619  hhhl 9073  hhsshl 9152

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-in 2051  df-hl 8590

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