Theorem ishl 8535

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 8534	. . 3 |- CHil = (CBan i^i CPreHil)
2	1	eleq2i 1535	. 2 |- (U e. CHil <-> U e. (CBan i^i CPreHil))
3		elin 2203	. 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 956   i^i cin 2042  CPreHilcphl 8415  CBancbn 8466  CHilchl 8533

This theorem is referenced by:  hlbn 8536  hlph 8537  cnhl 8561  ssphl 8562  hhhl 9012  hhsshl 9091

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-in 2047  df-hl 8534

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