Theorem sshjval

Description: Value of join for subsets of Hilbert space. (Contributed by NM, 1-Nov-2000) (Revised by Mario Carneiro, 23-Dec-2013) (New usage is discouraged.)

	Ref	Expression
Assertion	sshjval	⊢ ( ( 𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ ) → ( 𝐴 ∨ℋ 𝐵 ) = ( ⊥ ‘ ( ⊥ ‘ ( 𝐴 ∪ 𝐵 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		ax-hilex	⊢ ℋ ∈ V
2	1	elpw2	⊢ ( 𝐴 ∈ 𝒫 ℋ ↔ 𝐴 ⊆ ℋ )
3	1	elpw2	⊢ ( 𝐵 ∈ 𝒫 ℋ ↔ 𝐵 ⊆ ℋ )
4		uneq12	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝑥 ∪ 𝑦 ) = ( 𝐴 ∪ 𝐵 ) )
5	4	fveq2d	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( ⊥ ‘ ( 𝑥 ∪ 𝑦 ) ) = ( ⊥ ‘ ( 𝐴 ∪ 𝐵 ) ) )
6	5	fveq2d	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( ⊥ ‘ ( ⊥ ‘ ( 𝑥 ∪ 𝑦 ) ) ) = ( ⊥ ‘ ( ⊥ ‘ ( 𝐴 ∪ 𝐵 ) ) ) )
7		df-chj	⊢ ∨ℋ = ( 𝑥 ∈ 𝒫 ℋ , 𝑦 ∈ 𝒫 ℋ ↦ ( ⊥ ‘ ( ⊥ ‘ ( 𝑥 ∪ 𝑦 ) ) ) )
8		fvex	⊢ ( ⊥ ‘ ( ⊥ ‘ ( 𝐴 ∪ 𝐵 ) ) ) ∈ V
9	6 7 8	ovmpoa	⊢ ( ( 𝐴 ∈ 𝒫 ℋ ∧ 𝐵 ∈ 𝒫 ℋ ) → ( 𝐴 ∨ℋ 𝐵 ) = ( ⊥ ‘ ( ⊥ ‘ ( 𝐴 ∪ 𝐵 ) ) ) )
10	2 3 9	syl2anbr	⊢ ( ( 𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ ) → ( 𝐴 ∨ℋ 𝐵 ) = ( ⊥ ‘ ( ⊥ ‘ ( 𝐴 ∪ 𝐵 ) ) ) )