Theorem sshjcl

Description: Closure of join for subsets of Hilbert space. (Contributed by NM, 1-Nov-2000) (New usage is discouraged.)

	Ref	Expression
Assertion	sshjcl	⊢ ( ( 𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ ) → ( 𝐴 ∨ℋ 𝐵 ) ∈ Cℋ )

Proof

Step	Hyp	Ref	Expression
1		sshjval	⊢ ( ( 𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ ) → ( 𝐴 ∨ℋ 𝐵 ) = ( ⊥ ‘ ( ⊥ ‘ ( 𝐴 ∪ 𝐵 ) ) ) )
2		unss	⊢ ( ( 𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ ) ↔ ( 𝐴 ∪ 𝐵 ) ⊆ ℋ )
3		ocss	⊢ ( ( 𝐴 ∪ 𝐵 ) ⊆ ℋ → ( ⊥ ‘ ( 𝐴 ∪ 𝐵 ) ) ⊆ ℋ )
4		occl	⊢ ( ( ⊥ ‘ ( 𝐴 ∪ 𝐵 ) ) ⊆ ℋ → ( ⊥ ‘ ( ⊥ ‘ ( 𝐴 ∪ 𝐵 ) ) ) ∈ Cℋ )
5	3 4	syl	⊢ ( ( 𝐴 ∪ 𝐵 ) ⊆ ℋ → ( ⊥ ‘ ( ⊥ ‘ ( 𝐴 ∪ 𝐵 ) ) ) ∈ Cℋ )
6	2 5	sylbi	⊢ ( ( 𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ ) → ( ⊥ ‘ ( ⊥ ‘ ( 𝐴 ∪ 𝐵 ) ) ) ∈ Cℋ )
7	1 6	eqeltrd	⊢ ( ( 𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ ) → ( 𝐴 ∨ℋ 𝐵 ) ∈ Cℋ )