Theorem ishl 23966

Description: The predicate "is a subcomplex 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.) (Revised by Mario Carneiro, 15-Oct-2015.)

Assertion

Ref	Expression
ishl	⊢ (𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil))

Proof of Theorem ishl

Step	Hyp	Ref	Expression
1		df-hl 23941	. 2 ⊢ ℂHil = (Ban ∩ ℂPreHil)
2	1	elin2 4124	1 ⊢ (𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil))

Syntax hints:   ↔ wb 209   ∧ wa 399   ∈ wcel 2111  ℂPreHilccph 23771  Bancbn 23937  ℂHilchl 23938

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-in 3888  df-hl 23941

This theorem is referenced by:  hlbn  23967  hlcph  23968  ishl2  23974  cphssphl  23975  cmslsschl  23981  chlcsschl  23982