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Metamath Proof Explorer

Theorem chocvali

Description: Value of the orthogonal complement of a Hilbert lattice element. The orthogonal complement of A is the set of vectors that are orthogonal to all vectors in A . (Contributed by NM, 8-Aug-2004) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	chocval.1	⊢ 𝐴 ∈ C_ℋ
	Assertion	chocvali	⊢ ( ⊥ ‘ 𝐴 ) = { 𝑥 ∈ ℋ ∣ ∀ 𝑦 ∈ 𝐴 ( 𝑥 ·_ih 𝑦 ) = 0 }

Proof

Step	Hyp	Ref	Expression
1		chocval.1	⊢ 𝐴 ∈ C_ℋ
2	1	chssii	⊢ 𝐴 ⊆ ℋ
3		ocval	⊢ ( 𝐴 ⊆ ℋ → ( ⊥ ‘ 𝐴 ) = { 𝑥 ∈ ℋ ∣ ∀ 𝑦 ∈ 𝐴 ( 𝑥 ·_ih 𝑦 ) = 0 } )
4	2 3	ax-mp	⊢ ( ⊥ ‘ 𝐴 ) = { 𝑥 ∈ ℋ ∣ ∀ 𝑦 ∈ 𝐴 ( 𝑥 ·_ih 𝑦 ) = 0 }