Theorem ishl 8522

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 8521	. . 3 |- CHil = (CBan i^i CPreHil)
2	1	eleq2i 1530	. 2 |- (U e. CHil <-> U e. (CBan i^i CPreHil))
3		elin 2197	. 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 955   i^i cin 2036  CPreHilcphl 8402  CBancbn 8453  CHilchl 8520

This theorem is referenced by:  hlbn 8523  hlph 8524  cnhl 8548  ssphl 8549  hhhl 8994

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-in 2041  df-hl 8521

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