Description: The value of the supremum of a set of closed subspaces of Hilbert space. Definition of supremum in Proposition 1 of Kalmbach p. 65. (Contributed by NM, 13-Aug-2002) (New usage is discouraged.)
Ref | Expression | ||
---|---|---|---|
Assertion | chsupval2 | |- ( A C_ CH -> ( \/H ` A ) = |^| { x e. CH | U. A C_ x } ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | chsspwh | |- CH C_ ~P ~H |
|
2 | sstr2 | |- ( A C_ CH -> ( CH C_ ~P ~H -> A C_ ~P ~H ) ) |
|
3 | 1 2 | mpi | |- ( A C_ CH -> A C_ ~P ~H ) |
4 | hsupval2 | |- ( A C_ ~P ~H -> ( \/H ` A ) = |^| { x e. CH | U. A C_ x } ) |
|
5 | 3 4 | syl | |- ( A C_ CH -> ( \/H ` A ) = |^| { x e. CH | U. A C_ x } ) |