Theorem ishl 9936

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 9935	. . 3 |- CHil = (CBan i^i CPreHil)
2	1	eleq2i 1961	. 2 |- (U e. CHil <-> U e. (CBan i^i CPreHil))
3		elin 2786	. 2 |- (U e. (CBan i^i CPreHil) <-> (U e. CBan /\ U e. CPreHil))
4	2, 3	bitri 190	1 |- (U e. CHil <-> (U e. CBan /\ U e. CPreHil))

Syntax hints:   <-> wb 163   /\ wa 240   e. wcel 1300   i^i cin 2592  CPreHilcphl 9812  CBancbn 9864  CHilchl 9934

This theorem is referenced by:  hlbn 9937  hlph 9938  cnhl 9965  ssphl 9966  hhhl 10706  hhsshl 10785

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-in 2603  df-hl 9935

Copyright terms: Public domain