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Metamath Proof Explorer

Theorem ela

Description: Atoms in a Hilbert lattice are the elements that cover the zero subspace. Definition of atom in Kalmbach p. 15. (Contributed by NM, 9-Jun-2004) (New usage is discouraged.)

		Ref	Expression
	Assertion	ela	⊢ ( 𝐴 ∈ HAtoms ↔ ( 𝐴 ∈ C_ℋ ∧ 0_ℋ ⋖_ℋ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		breq2	⊢ ( 𝑥 = 𝐴 → ( 0_ℋ ⋖_ℋ 𝑥 ↔ 0_ℋ ⋖_ℋ 𝐴 ) )
2		df-at	⊢ HAtoms = { 𝑥 ∈ C_ℋ ∣ 0_ℋ ⋖_ℋ 𝑥 }
3	1 2	elrab2	⊢ ( 𝐴 ∈ HAtoms ↔ ( 𝐴 ∈ C_ℋ ∧ 0_ℋ ⋖_ℋ 𝐴 ) )