Theorem ishil

Description: The predicate "is a Hilbert space" (over a *-division ring). A Hilbert space is a pre-Hilbert space such that all closed subspaces have a projection decomposition. (Contributed by NM, 7-Oct-2011) (Revised by Mario Carneiro, 22-Jun-2014)

	Ref	Expression
Hypotheses	ishil.k	⊢ 𝐾 = ( proj ‘ 𝐻 )
	ishil.c	⊢ 𝐶 = ( ClSubSp ‘ 𝐻 )
Assertion	ishil	⊢ ( 𝐻 ∈ Hil ↔ ( 𝐻 ∈ PreHil ∧ dom 𝐾 = 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		ishil.k	⊢ 𝐾 = ( proj ‘ 𝐻 )
2		ishil.c	⊢ 𝐶 = ( ClSubSp ‘ 𝐻 )
3		fveq2	⊢ ( ℎ = 𝐻 → ( proj ‘ ℎ ) = ( proj ‘ 𝐻 ) )
4	3 1	eqtr4di	⊢ ( ℎ = 𝐻 → ( proj ‘ ℎ ) = 𝐾 )
5	4	dmeqd	⊢ ( ℎ = 𝐻 → dom ( proj ‘ ℎ ) = dom 𝐾 )
6		fveq2	⊢ ( ℎ = 𝐻 → ( ClSubSp ‘ ℎ ) = ( ClSubSp ‘ 𝐻 ) )
7	6 2	eqtr4di	⊢ ( ℎ = 𝐻 → ( ClSubSp ‘ ℎ ) = 𝐶 )
8	5 7	eqeq12d	⊢ ( ℎ = 𝐻 → ( dom ( proj ‘ ℎ ) = ( ClSubSp ‘ ℎ ) ↔ dom 𝐾 = 𝐶 ) )
9		df-hil	⊢ Hil = { ℎ ∈ PreHil ∣ dom ( proj ‘ ℎ ) = ( ClSubSp ‘ ℎ ) }
10	8 9	elrab2	⊢ ( 𝐻 ∈ Hil ↔ ( 𝐻 ∈ PreHil ∧ dom 𝐾 = 𝐶 ) )