Theorem chsupval2

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 ⊆ Cℋ → ( ∨ℋ ‘ 𝐴 ) = ∩ { 𝑥 ∈ Cℋ ∣ ∪ 𝐴 ⊆ 𝑥 } )

Proof

Step	Hyp	Ref	Expression
1		chsspwh	⊢ Cℋ ⊆ 𝒫 ℋ
2		sstr2	⊢ ( 𝐴 ⊆ Cℋ → ( Cℋ ⊆ 𝒫 ℋ → 𝐴 ⊆ 𝒫 ℋ ) )
3	1 2	mpi	⊢ ( 𝐴 ⊆ Cℋ → 𝐴 ⊆ 𝒫 ℋ )
4		hsupval2	⊢ ( 𝐴 ⊆ 𝒫 ℋ → ( ∨ℋ ‘ 𝐴 ) = ∩ { 𝑥 ∈ Cℋ ∣ ∪ 𝐴 ⊆ 𝑥 } )
5	3 4	syl	⊢ ( 𝐴 ⊆ Cℋ → ( ∨ℋ ‘ 𝐴 ) = ∩ { 𝑥 ∈ Cℋ ∣ ∪ 𝐴 ⊆ 𝑥 } )