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	$$⊢A\in \mathrm{HAtoms}\leftrightarrow \left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge 0_{\mathscr{H}}{⋖}_{\mathscr{H}}A\right)$$

Proof

Step	Hyp	Ref	Expression
1		breq2	$$⊢x=A\to \left(0_{\mathscr{H}}{⋖}_{\mathscr{H}}x\leftrightarrow 0_{\mathscr{H}}{⋖}_{\mathscr{H}}A\right)$$
2		df-at	$$⊢\mathrm{HAtoms}=\left\{x\in {\mathbf{C}}_{\mathscr{H}}|0_{\mathscr{H}}{⋖}_{\mathscr{H}}x\right\}$$
3	1 2	elrab2	$$⊢A\in \mathrm{HAtoms}\leftrightarrow \left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge 0_{\mathscr{H}}{⋖}_{\mathscr{H}}A\right)$$