P. 406 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 40501-40600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	omlop 40501	An orthomodular lattice is an orthoposet. (Contributed by NM, 6-Nov-2011.)
⊢ (𝐾 ∈ OML → 𝐾 ∈ OP)
Theorem	omllat 40502	An orthomodular lattice is a lattice. (Contributed by NM, 6-Nov-2011.)
⊢ (𝐾 ∈ OML → 𝐾 ∈ Lat)
Theorem	omllaw 40503	The orthomodular law. (Contributed by NM, 18-Sep-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → 𝑌 = (𝑋 ∨ (𝑌 ∧ ( ⊥ ‘𝑋)))))
Theorem	omllaw2N 40504	Variation of orthomodular law. Definition of OML law in [Kalmbach] p. 22. (pjoml2i 32220 analog.) (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → (𝑋 ∨ (( ⊥ ‘𝑋) ∧ 𝑌)) = 𝑌))
Theorem	omllaw3 40505	Orthomodular law equivalent. Theorem 2(ii) of [Kalmbach] p. 22. (pjoml 32071 analog.) (Contributed by NM, 19-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ (𝑌 ∧ ( ⊥ ‘𝑋)) = 0 ) → 𝑋 = 𝑌))
Theorem	omllaw4 40506	Orthomodular law equivalent. Remark in [Holland95] p. 223. (Contributed by NM, 19-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → (( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∧ 𝑌) = 𝑋))
Theorem	omllaw5N 40507	The orthomodular law. Remark in [Kalmbach] p. 22. (pjoml5 32248 analog.) (Contributed by NM, 14-Nov-2011.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ (( ⊥ ‘𝑋) ∧ (𝑋 ∨ 𝑌))) = (𝑋 ∨ 𝑌))
Theorem	cmtcomlemN 40508	Lemma for cmtcomN 40509. (cmcmlem 32226 analog.) (Contributed by NM, 7-Nov-2011.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐶 = (cm‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 → 𝑌𝐶𝑋))
Theorem	cmtcomN 40509	Commutation is symmetric. Theorem 2(v) in [Kalmbach] p. 22. (cmcmi 32227 analog.) (Contributed by NM, 7-Nov-2011.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐶 = (cm‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ 𝑌𝐶𝑋))
Theorem	cmt2N 40510	Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (cmcm2i 32228 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    &   ⊢ 𝐶 = (cm‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ 𝑋𝐶( ⊥ ‘𝑌)))
Theorem	cmt3N 40511	Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (cmcm4i 32230 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    &   ⊢ 𝐶 = (cm‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ ( ⊥ ‘𝑋)𝐶𝑌))
Theorem	cmt4N 40512	Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (cmcm4i 32230 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    &   ⊢ 𝐶 = (cm‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ ( ⊥ ‘𝑋)𝐶( ⊥ ‘𝑌)))
Theorem	cmtbr2N 40513	Alternate definition of the commutes relation. Remark in [Kalmbach] p. 23. (cmbr2i 32231 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    &   ⊢ 𝐶 = (cm‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ 𝑋 = ((𝑋 ∨ 𝑌) ∧ (𝑋 ∨ ( ⊥ ‘𝑌)))))
Theorem	cmtbr3N 40514	Alternate definition for the commutes relation. Lemma 3 of [Kalmbach] p. 23. (cmbr3 32243 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    &   ⊢ 𝐶 = (cm‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌)))
Theorem	cmtbr4N 40515	Alternate definition for the commutes relation. (cmbr4i 32236 analog.) (Contributed by NM, 10-Nov-2011.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    &   ⊢ 𝐶 = (cm‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑌))
Theorem	lecmtN 40516	Ordered elements commute. (lecmi 32237 analog.) (Contributed by NM, 10-Nov-2011.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐶 = (cm‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → 𝑋𝐶𝑌))
Theorem	cmtidN 40517	Any element commutes with itself. (cmidi 32245 analog.) (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐶 = (cm‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵) → 𝑋𝐶𝑋)
Theorem	omlfh1N 40518	Foulis-Holland Theorem, part 1. If any 2 pairs in a triple of orthomodular lattice elements commute, the triple is distributive. Part of Theorem 5 in [Kalmbach] p. 25. (fh1 32253 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐶 = (cm‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OML ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋𝐶𝑌 ∧ 𝑋𝐶𝑍)) → (𝑋 ∧ (𝑌 ∨ 𝑍)) = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ 𝑍)))
Theorem	omlfh3N 40519	Foulis-Holland Theorem, part 3. Dual of omlfh1N 40518. (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐶 = (cm‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OML ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋𝐶𝑌 ∧ 𝑋𝐶𝑍)) → (𝑋 ∨ (𝑌 ∧ 𝑍)) = ((𝑋 ∨ 𝑌) ∧ (𝑋 ∨ 𝑍)))
Theorem	omlmod1i2N 40520	Analogue of modular law atmod1i2 41119 that holds in any OML. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐶 = (cm‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OML ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ 𝑌𝐶𝑍)) → (𝑋 ∨ (𝑌 ∧ 𝑍)) = ((𝑋 ∨ 𝑌) ∧ 𝑍))
Theorem	omlspjN 40521	Contraction of a Sasaki projection. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OML ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → ((𝑋 ∨ ( ⊥ ‘𝑌)) ∧ 𝑌) = 𝑋)
21.30.10  Atomic lattices with covering property
Syntax	ccvr 40522	Extend class notation with covers relation.
class ⋖
Syntax	catm 40523	Extend class notation with atoms.
class Atoms
Syntax	cal 40524	Extend class notation with atomic lattices.
class AtLat
Syntax	clc 40525	Extend class notation with lattices with the covering property.
class CvLat
Definition	df-covers 40526*	Define the covers relation ("is covered by") for posets. "𝑎 is covered by 𝑏 " means that 𝑎 is strictly less than 𝑏 and there is nothing in between. See cvrval 40529 for the relation form. (Contributed by NM, 18-Sep-2011.)
⊢ ⋖ = (𝑝 ∈ V ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (Base‘𝑝) ∧ 𝑏 ∈ (Base‘𝑝)) ∧ 𝑎(lt‘𝑝)𝑏 ∧ ¬ ∃𝑧 ∈ (Base‘𝑝)(𝑎(lt‘𝑝)𝑧 ∧ 𝑧(lt‘𝑝)𝑏))})
Definition	df-ats 40527*	Define the class of poset atoms. (Contributed by NM, 18-Sep-2011.)
⊢ Atoms = (𝑝 ∈ V ↦ {𝑎 ∈ (Base‘𝑝) ∣ (0.‘𝑝)( ⋖ ‘𝑝)𝑎})
Theorem	cvrfval 40528*	Value of covers relation "is covered by". (Contributed by NM, 18-Sep-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    ⇒   ⊢ (𝐾 ∈ 𝐴 → 𝐶 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧 ∈ 𝐵 (𝑥 < 𝑧 ∧ 𝑧 < 𝑦))})
Theorem	cvrval 40529*	Binary relation expressing 𝐵 covers 𝐴, which means that 𝐵 is larger than 𝐴 and there is nothing in between. Definition 3.2.18 of [PtakPulmannova] p. 68. (cvbr 32917 analog.) (Contributed by NM, 18-Sep-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    ⇒   ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 < 𝑌 ∧ ¬ ∃𝑧 ∈ 𝐵 (𝑋 < 𝑧 ∧ 𝑧 < 𝑌))))
Theorem	cvrlt 40530	The covers relation implies the less-than relation. (cvpss 32920 analog.) (Contributed by NM, 8-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    ⇒   ⊢ (((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
Theorem	cvrnbtwn 40531	There is no element between the two arguments of the covers relation. (cvnbtwn 32921 analog.) (Contributed by NM, 18-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    ⇒   ⊢ ((𝐾 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ¬ (𝑋 < 𝑍 ∧ 𝑍 < 𝑌))
Theorem	ncvr1 40532	No element covers the lattice unity. (Contributed by NM, 8-Jul-2013.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 1 = (1.‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ¬ 1 𝐶𝑋)
Theorem	cvrletrN 40533	Property of an element above a covering. (Contributed by NM, 7-Dec-2012.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋𝐶𝑌 ∧ 𝑌 ≤ 𝑍) → 𝑋 < 𝑍))
Theorem	cvrval2 40534*	Binary relation expressing 𝑌 covers 𝑋. Definition of covers in [Kalmbach] p. 15. (cvbr2 32918 analog.) (Contributed by NM, 16-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    ⇒   ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 < 𝑌 ∧ ∀𝑧 ∈ 𝐵 ((𝑋 < 𝑧 ∧ 𝑧 ≤ 𝑌) → 𝑧 = 𝑌))))
Theorem	cvrnbtwn2 40535	The covers relation implies no in-betweenness. (cvnbtwn2 32922 analog.) (Contributed by NM, 17-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌) ↔ 𝑍 = 𝑌))
Theorem	cvrnbtwn3 40536	The covers relation implies no in-betweenness. (cvnbtwn3 32923 analog.) (Contributed by NM, 4-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) ↔ 𝑋 = 𝑍))
Theorem	cvrcon3b 40537	Contraposition law for the covers relation. (cvcon3 32919 analog.) (Contributed by NM, 4-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ ( ⊥ ‘𝑌)𝐶( ⊥ ‘𝑋)))
Theorem	cvrle 40538	The covers relation implies the "less than or equal to" relation. (Contributed by NM, 12-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    ⇒   ⊢ (((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 ≤ 𝑌)
Theorem	cvrnbtwn4 40539	The covers relation implies no in-betweenness. Part of proof of Lemma 7.5.1 of [MaedaMaeda] p. 31. (cvnbtwn4 32924 analog.) (Contributed by NM, 18-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 ≤ 𝑍 ∧ 𝑍 ≤ 𝑌) ↔ (𝑋 = 𝑍 ∨ 𝑍 = 𝑌)))
Theorem	cvrnle 40540	The covers relation implies the negation of the converse "less than or equal to" relation. (Contributed by NM, 18-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    ⇒   ⊢ (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ¬ 𝑌 ≤ 𝑋)
Theorem	cvrne 40541	The covers relation implies inequality. (Contributed by NM, 13-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    ⇒   ⊢ (((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 ≠ 𝑌)
Theorem	cvrnrefN 40542	The covers relation is not reflexive. (cvnref 32926 analog.) (Contributed by NM, 1-Nov-2012.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    ⇒   ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → ¬ 𝑋𝐶𝑋)
Theorem	cvrcmp 40543	If two lattice elements that cover a third are comparable, then they are equal. (Contributed by NM, 6-Feb-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) → (𝑋 ≤ 𝑌 ↔ 𝑋 = 𝑌))
Theorem	cvrcmp2 40544	If two lattice elements covered by a third are comparable, then they are equal. (Contributed by NM, 20-Jun-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OP ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋𝐶𝑍 ∧ 𝑌𝐶𝑍)) → (𝑋 ≤ 𝑌 ↔ 𝑋 = 𝑌))
Theorem	pats 40545*	The set of atoms in a poset. (Contributed by NM, 18-Sep-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (𝐾 ∈ 𝐷 → 𝐴 = {𝑥 ∈ 𝐵 ∣ 0 𝐶𝑥})
Theorem	isat 40546	The predicate "is an atom". (ela 32974 analog.) (Contributed by NM, 18-Sep-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (𝐾 ∈ 𝐷 → (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ 𝐵 ∧ 0 𝐶𝑃)))
Theorem	isat2 40547	The predicate "is an atom". (elatcv0 32976 analog.) (Contributed by NM, 18-Jun-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑃 ∈ 𝐵) → (𝑃 ∈ 𝐴 ↔ 0 𝐶𝑃))
Theorem	atcvr0 40548	An atom covers zero. (atcv0 32977 analog.) (Contributed by NM, 4-Nov-2011.)
⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑃 ∈ 𝐴) → 0 𝐶𝑃)
Theorem	atbase 40549	An atom is a member of the lattice base set (i.e. a lattice element). (atelch 32979 analog.) (Contributed by NM, 10-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵)
Theorem	atssbase 40550	The set of atoms is a subset of the base set. (atssch 32978 analog.) (Contributed by NM, 21-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ 𝐴 ⊆ 𝐵
Theorem	0ltat 40551	An atom is greater than zero. (Contributed by NM, 4-Jul-2012.)
⊢ 0 = (0.‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OP ∧ 𝑃 ∈ 𝐴) → 0 < 𝑃)
Theorem	leatb 40552	A poset element less than or equal to an atom equals either zero or the atom. (atss 32981 analog.) (Contributed by NM, 17-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (𝑋 ≤ 𝑃 ↔ (𝑋 = 𝑃 ∨ 𝑋 = 0 )))
Theorem	leat 40553	A poset element less than or equal to an atom equals either zero or the atom. (Contributed by NM, 15-Oct-2013.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑋 ≤ 𝑃) → (𝑋 = 𝑃 ∨ 𝑋 = 0 ))
Theorem	leat2 40554	A nonzero poset element less than or equal to an atom equals the atom. (Contributed by NM, 6-Mar-2013.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ (𝑋 ≠ 0 ∧ 𝑋 ≤ 𝑃)) → 𝑋 = 𝑃)
Theorem	leat3 40555	A poset element less than or equal to an atom is either an atom or zero. (Contributed by NM, 2-Dec-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑋 ≤ 𝑃) → (𝑋 ∈ 𝐴 ∨ 𝑋 = 0 ))
Theorem	meetat 40556	The meet of any element with an atom is either the atom or zero. (Contributed by NM, 28-Aug-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → ((𝑋 ∧ 𝑃) = 𝑃 ∨ (𝑋 ∧ 𝑃) = 0 ))
Theorem	meetat2 40557	The meet of any element with an atom is either the atom or zero. (Contributed by NM, 30-Aug-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → ((𝑋 ∧ 𝑃) ∈ 𝐴 ∨ (𝑋 ∧ 𝑃) = 0 ))
Definition	df-atl 40558*	Define the class of atomic lattices, in which every nonzero element is greater than or equal to an atom. We also ensure the existence of a lattice zero, since a lattice by itself may not have a zero. (Contributed by NM, 18-Sep-2011.) (Revised by NM, 14-Sep-2018.)
⊢ AtLat = {𝑘 ∈ Lat ∣ ((Base‘𝑘) ∈ dom (glb‘𝑘) ∧ ∀𝑥 ∈ (Base‘𝑘)(𝑥 ≠ (0.‘𝑘) → ∃𝑝 ∈ (Atoms‘𝑘)𝑝(le‘𝑘)𝑥))}
Theorem	isatl 40559*	The predicate "is an atomic lattice." Every nonzero element is less than or equal to an atom. (Contributed by NM, 18-Sep-2011.) (Revised by NM, 14-Sep-2018.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐺 = (glb‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ 𝐵 ∈ dom 𝐺 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)))
Theorem	atllat 40560	An atomic lattice is a lattice. (Contributed by NM, 21-Oct-2011.)
⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Lat)
Theorem	atlpos 40561	An atomic lattice is a poset. (Contributed by NM, 5-Nov-2012.)
⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Poset)
Theorem	atl0dm 40562	Condition necessary for zero element to exist. (Contributed by NM, 14-Sep-2018.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝑈 = (lub‘𝐾)    &   ⊢ 𝐺 = (glb‘𝐾)    ⇒   ⊢ (𝐾 ∈ AtLat → 𝐵 ∈ dom 𝐺)
Theorem	atl0cl 40563	An atomic lattice has a zero element. We can use this in place of op0cl 40444 for lattices without orthocomplements. (Contributed by NM, 5-Nov-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    ⇒   ⊢ (𝐾 ∈ AtLat → 0 ∈ 𝐵)
Theorem	atl0le 40564	Orthoposet zero is less than or equal to any element. (ch0le 32076 analog.) (Contributed by NM, 12-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    ⇒   ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵) → 0 ≤ 𝑋)
Theorem	atlle0 40565	An element less than or equal to zero equals zero. (chle0 32078 analog.) (Contributed by NM, 21-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    ⇒   ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵) → (𝑋 ≤ 0 ↔ 𝑋 = 0 ))
Theorem	atlltn0 40566	A lattice element greater than zero is nonzero. (Contributed by NM, 1-Jun-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    ⇒   ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵) → ( 0 < 𝑋 ↔ 𝑋 ≠ 0 ))
Theorem	isat3 40567*	The predicate "is an atom". (elat2 32975 analog.) (Contributed by NM, 27-Apr-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (𝐾 ∈ AtLat → (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ 𝐵 ∧ 𝑃 ≠ 0 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑃 → (𝑥 = 𝑃 ∨ 𝑥 = 0 )))))
Theorem	atn0 40568	An atom is not zero. (atne0 32980 analog.) (Contributed by NM, 5-Nov-2012.)
⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → 𝑃 ≠ 0 )
Theorem	atnle0 40569	An atom is not less than or equal to zero. (Contributed by NM, 17-Oct-2011.)
⊢ ≤ = (le‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → ¬ 𝑃 ≤ 0 )
Theorem	atlen0 40570	A lattice element is nonzero if an atom is under it. (Contributed by NM, 26-May-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 𝑋 ≠ 0 )
Theorem	atcmp 40571	If two atoms are comparable, they are equal. (atsseq 32982 analog.) (Contributed by NM, 13-Oct-2011.)
⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ≤ 𝑄 ↔ 𝑃 = 𝑄))
Theorem	atncmp 40572	Frequently-used variation of atcmp 40571. (Contributed by NM, 29-Jun-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (¬ 𝑃 ≤ 𝑄 ↔ 𝑃 ≠ 𝑄))
Theorem	atnlt 40573	Two atoms cannot satisfy the less than relation. (Contributed by NM, 7-Feb-2012.)
⊢ < = (lt‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ¬ 𝑃 < 𝑄)
Theorem	atcvreq0 40574	An element covered by an atom must be zero. (atcveq0 32983 analog.) (Contributed by NM, 4-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (𝑋𝐶𝑃 ↔ 𝑋 = 0 ))
Theorem	atncvrN 40575	Two atoms cannot satisfy the covering relation. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)
⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ¬ 𝑃𝐶𝑄)
Theorem	atlex 40576*	Every nonzero element of an atomic lattice is greater than or equal to an atom. (hatomic 32995 analog.) (Contributed by NM, 21-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑋)
Theorem	atnle 40577	Two ways of expressing "an atom is not less than or equal to a lattice element." (atnssm0 33011 analog.) (Contributed by NM, 5-Nov-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (¬ 𝑃 ≤ 𝑋 ↔ (𝑃 ∧ 𝑋) = 0 ))
Theorem	atnem0 40578	The meet of distinct atoms is zero. (atnemeq0 33012 analog.) (Contributed by NM, 5-Nov-2012.)
⊢ ∧ = (meet‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ≠ 𝑄 ↔ (𝑃 ∧ 𝑄) = 0 ))
Theorem	atlatmstc 40579*	An atomic, complete, orthomodular lattice is atomistic i.e. every element is the join of the atoms under it. See remark before Proposition 1 in [Kalmbach] p. 140; also remark in [BeltramettiCassinelli] p. 98. (hatomistici 32997 analog.) (Contributed by NM, 5-Nov-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 1 = (lub‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → ( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}) = 𝑋)
Theorem	atlatle 40580*	The ordering of two Hilbert lattice elements is determined by the atoms under them. (chrelat3 33006 analog.) (Contributed by NM, 5-Nov-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ↔ ∀𝑝 ∈ 𝐴 (𝑝 ≤ 𝑋 → 𝑝 ≤ 𝑌)))
Theorem	atlrelat1 40581*	An atomistic lattice with 0 is relatively atomic. Part of Lemma 7.2 of [MaedaMaeda] p. 30. (chpssati 32998, with ∧ swapped, analog.) (Contributed by NM, 4-Dec-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 → ∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌)))
Definition	df-cvlat 40582*	Define the class of atomic lattices with the covering property. (This is actually the exchange property, but they are equivalent. The literature usually uses the covering property terminology.) (Contributed by NM, 5-Nov-2012.)
⊢ CvLat = {𝑘 ∈ AtLat ∣ ∀𝑎 ∈ (Atoms‘𝑘)∀𝑏 ∈ (Atoms‘𝑘)∀𝑐 ∈ (Base‘𝑘)((¬ 𝑎(le‘𝑘)𝑐 ∧ 𝑎(le‘𝑘)(𝑐(join‘𝑘)𝑏)) → 𝑏(le‘𝑘)(𝑐(join‘𝑘)𝑎))}
Theorem	iscvlat 40583*	The predicate "is an atomic lattice with the covering (or exchange) property". (Contributed by NM, 5-Nov-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (𝐾 ∈ CvLat ↔ (𝐾 ∈ AtLat ∧ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ∀𝑥 ∈ 𝐵 ((¬ 𝑝 ≤ 𝑥 ∧ 𝑝 ≤ (𝑥 ∨ 𝑞)) → 𝑞 ≤ (𝑥 ∨ 𝑝))))
Theorem	iscvlat2N 40584*	The predicate "is an atomic lattice with the covering (or exchange) property". (Contributed by NM, 5-Nov-2012.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (𝐾 ∈ CvLat ↔ (𝐾 ∈ AtLat ∧ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ∀𝑥 ∈ 𝐵 (((𝑝 ∧ 𝑥) = 0 ∧ 𝑝 ≤ (𝑥 ∨ 𝑞)) → 𝑞 ≤ (𝑥 ∨ 𝑝))))
Theorem	cvlatl 40585	An atomic lattice with the covering property is an atomic lattice. (Contributed by NM, 5-Nov-2012.)
⊢ (𝐾 ∈ CvLat → 𝐾 ∈ AtLat)
Theorem	cvllat 40586	An atomic lattice with the covering property is a lattice. (Contributed by NM, 5-Nov-2012.)
⊢ (𝐾 ∈ CvLat → 𝐾 ∈ Lat)
Theorem	cvlposN 40587	An atomic lattice with the covering property is a poset. (Contributed by NM, 5-Nov-2012.) (New usage is discouraged.)
⊢ (𝐾 ∈ CvLat → 𝐾 ∈ Poset)
Theorem	cvlexch1 40588	An atomic covering lattice has the exchange property. (Contributed by NM, 6-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) → 𝑄 ≤ (𝑋 ∨ 𝑃)))
Theorem	cvlexch2 40589	An atomic covering lattice has the exchange property. (Contributed by NM, 6-May-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑄 ∨ 𝑋) → 𝑄 ≤ (𝑃 ∨ 𝑋)))
Theorem	cvlexchb1 40590	An atomic covering lattice has the exchange property. (Contributed by NM, 16-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) ↔ (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄)))
Theorem	cvlexchb2 40591	An atomic covering lattice has the exchange property. (Contributed by NM, 22-Jun-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑄 ∨ 𝑋) ↔ (𝑃 ∨ 𝑋) = (𝑄 ∨ 𝑋)))
Theorem	cvlexch3 40592	An atomic covering lattice has the exchange property. (atexch 33016 analog.) (Contributed by NM, 5-Nov-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∧ 𝑋) = 0 ) → (𝑃 ≤ (𝑋 ∨ 𝑄) → 𝑄 ≤ (𝑋 ∨ 𝑃)))
Theorem	cvlexch4N 40593	An atomic covering lattice has the exchange property. Part of Definition 7.8 of [MaedaMaeda] p. 32. (Contributed by NM, 5-Nov-2012.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∧ 𝑋) = 0 ) → (𝑃 ≤ (𝑋 ∨ 𝑄) ↔ (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄)))
Theorem	cvlatexchb1 40594	A version of cvlexchb1 40590 for atoms. (Contributed by NM, 5-Nov-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑅) → (𝑃 ≤ (𝑅 ∨ 𝑄) ↔ (𝑅 ∨ 𝑃) = (𝑅 ∨ 𝑄)))
Theorem	cvlatexchb2 40595	A version of cvlexchb2 40591 for atoms. (Contributed by NM, 5-Nov-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑅) → (𝑃 ≤ (𝑄 ∨ 𝑅) ↔ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)))
Theorem	cvlatexch1 40596	Atom exchange property. (Contributed by NM, 5-Nov-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑅) → (𝑃 ≤ (𝑅 ∨ 𝑄) → 𝑄 ≤ (𝑅 ∨ 𝑃)))
Theorem	cvlatexch2 40597	Atom exchange property. (Contributed by NM, 5-Nov-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑅) → (𝑃 ≤ (𝑄 ∨ 𝑅) → 𝑄 ≤ (𝑃 ∨ 𝑅)))
Theorem	cvlatexch3 40598	Atom exchange property. (Contributed by NM, 29-Nov-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) → (𝑃 ≤ (𝑄 ∨ 𝑅) → (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑅)))
Theorem	cvlcvr1 40599	The covering property. Proposition 1(ii) in [Kalmbach] p. 140 (and its converse). (chcv1 32990 analog.) (Contributed by NM, 5-Nov-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (¬ 𝑃 ≤ 𝑋 ↔ 𝑋𝐶(𝑋 ∨ 𝑃)))
Theorem	cvlcvrp 40600	A Hilbert lattice satisfies the covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 33010 analog.) (Contributed by NM, 5-Nov-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → ((𝑋 ∧ 𝑃) = 0 ↔ 𝑋𝐶(𝑋 ∨ 𝑃)))