Theorem sshjval2

Description: Value of join in the set of closed subspaces of Hilbert space CH . (Contributed by NM, 1-Nov-2000) (Revised by Mario Carneiro, 23-Dec-2013) (New usage is discouraged.)

	Ref	Expression
Assertion	sshjval2	⊢ ( ( 𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ ) → ( 𝐴 ∨ℋ 𝐵 ) = ∩ { 𝑥 ∈ Cℋ ∣ ( 𝐴 ∪ 𝐵 ) ⊆ 𝑥 } )

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		ococin	⊢ ( ( 𝐴 ∪ 𝐵 ) ⊆ ℋ → ( ⊥ ‘ ( ⊥ ‘ ( 𝐴 ∪ 𝐵 ) ) ) = ∩ { 𝑥 ∈ Cℋ ∣ ( 𝐴 ∪ 𝐵 ) ⊆ 𝑥 } )
4	2 3	sylbi	⊢ ( ( 𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ ) → ( ⊥ ‘ ( ⊥ ‘ ( 𝐴 ∪ 𝐵 ) ) ) = ∩ { 𝑥 ∈ Cℋ ∣ ( 𝐴 ∪ 𝐵 ) ⊆ 𝑥 } )
5	1 4	eqtrd	⊢ ( ( 𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ ) → ( 𝐴 ∨ℋ 𝐵 ) = ∩ { 𝑥 ∈ Cℋ ∣ ( 𝐴 ∪ 𝐵 ) ⊆ 𝑥 } )