Definition df-hil 22015

Description: Define class of all Hilbert spaces. Based on Proposition 4.5, p. 176, Gudrun Kalmbach, Quantum Measures and Spaces, Kluwer, Dordrecht, 1998. (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 16-Oct-2015.)

Assertion

Ref	Expression
df-hil	⊢ Hil = {ℎ ∈ PreHil ∣ dom (proj‘ℎ) = (ClSubSp‘ℎ)}

Detailed syntax breakdown of Definition df-hil

Step	Hyp	Ref	Expression
1		chil 22012	. 2 class Hil
2		vh	. . . . . . 7 setvar ℎ
3	2	cv 1568	. . . . . 6 class ℎ
4		cpj 22011	. . . . . 6 class proj
5	3, 4	cfv 6538	. . . . 5 class (proj‘ℎ)
6	5	cdm 5650	. . . 4 class dom (proj‘ℎ)
7		ccss 21972	. . . . 5 class ClSubSp
8	3, 7	cfv 6538	. . . 4 class (ClSubSp‘ℎ)
9	6, 8	wceq 1569	. . 3 wff dom (proj‘ℎ) = (ClSubSp‘ℎ)
10		cphl 21935	. . 3 class PreHil
11	9, 2, 10	crab 3412	. 2 class {ℎ ∈ PreHil ∣ dom (proj‘ℎ) = (ClSubSp‘ℎ)}
12	1, 11	wceq 1569	1 wff Hil = {ℎ ∈ PreHil ∣ dom (proj‘ℎ) = (ClSubSp‘ℎ)}

This definition is used by:  ishil  22029