Metamath Proof Explorer


Definition df-sh

Description: Define the set of subspaces of a Hilbert space. See issh for its membership relation. Basically, a subspace is a subset of a Hilbert space that acts like a vector space. From Definition of Beran p. 95. (Contributed by Mario Carneiro, 23-Dec-2013) (New usage is discouraged.)

Ref Expression
Assertion df-sh S = h 𝒫 | 0 h + h × h h × h h

Detailed syntax breakdown

Step Hyp Ref Expression
0 csh class S
1 vh setvar h
2 chba class
3 2 cpw class 𝒫
4 c0v class 0
5 1 cv setvar h
6 4 5 wcel wff 0 h
7 cva class +
8 5 5 cxp class h × h
9 7 8 cima class + h × h
10 9 5 wss wff + h × h h
11 csm class
12 cc class
13 12 5 cxp class × h
14 11 13 cima class × h
15 14 5 wss wff × h h
16 6 10 15 w3a wff 0 h + h × h h × h h
17 16 1 3 crab class h 𝒫 | 0 h + h × h h × h h
18 0 17 wceq wff S = h 𝒫 | 0 h + h × h h × h h