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	2polpmapN 40501	Double polarity of a projective map. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝑀 = (pmap‘𝐾)    &   ⊢ ⊥ = (⊥𝑃‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘(𝑀‘𝑋))) = (𝑀‘𝑋))
Theorem	2polvalN 40502	Value of double polarity. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
⊢ 𝑈 = (lub‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑀 = (pmap‘𝐾)    &   ⊢ ⊥ = (⊥𝑃‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑋)) = (𝑀‘(𝑈‘𝑋)))
Theorem	2polssN 40503	A set of atoms is a subset of its double polarity. (Contributed by NM, 29-Jan-2012.) (New usage is discouraged.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ ⊥ = (⊥𝑃‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → 𝑋 ⊆ ( ⊥ ‘( ⊥ ‘𝑋)))
Theorem	3polN 40504	Triple polarity cancels to a single polarity. (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ ⊥ = (⊥𝑃‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑆))) = ( ⊥ ‘𝑆))
Theorem	polcon3N 40505	Contraposition law for polarity. Remark in [Holland95] p. 223. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ ⊥ = (⊥𝑃‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴 ∧ 𝑋 ⊆ 𝑌) → ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋))
Theorem	2polcon4bN 40506	Contraposition law for polarity. (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ ⊥ = (⊥𝑃‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (( ⊥ ‘( ⊥ ‘𝑋)) ⊆ ( ⊥ ‘( ⊥ ‘𝑌)) ↔ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)))
Theorem	polcon2N 40507	Contraposition law for polarity. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ ⊥ = (⊥𝑃‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) → 𝑌 ⊆ ( ⊥ ‘𝑋))
Theorem	polcon2bN 40508	Contraposition law for polarity. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ ⊥ = (⊥𝑃‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 ⊆ ( ⊥ ‘𝑌) ↔ 𝑌 ⊆ ( ⊥ ‘𝑋)))
Theorem	pclss2polN 40509	The projective subspace closure is a subset of closed subspace closure. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ ⊥ = (⊥𝑃‘𝐾)    &   ⊢ 𝑈 = (PCl‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑈‘𝑋) ⊆ ( ⊥ ‘( ⊥ ‘𝑋)))
Theorem	pcl0N 40510	The projective subspace closure of the empty subspace. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)
⊢ 𝑈 = (PCl‘𝐾)    ⇒   ⊢ (𝐾 ∈ HL → (𝑈‘∅) = ∅)
Theorem	pcl0bN 40511	The projective subspace closure of the empty subspace. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑈 = (PCl‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑃 ⊆ 𝐴) → ((𝑈‘𝑃) = ∅ ↔ 𝑃 = ∅))
Theorem	pmaplubN 40512	The LUB of a projective map is the projective map's argument. (Contributed by NM, 13-Mar-2012.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝑈 = (lub‘𝐾)    &   ⊢ 𝑀 = (pmap‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑈‘(𝑀‘𝑋)) = 𝑋)
Theorem	sspmaplubN 40513	A set of atoms is a subset of the projective map of its LUB. (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)
⊢ 𝑈 = (lub‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑀 = (pmap‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → 𝑆 ⊆ (𝑀‘(𝑈‘𝑆)))
Theorem	2pmaplubN 40514	Double projective map of an LUB. (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)
⊢ 𝑈 = (lub‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑀 = (pmap‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → (𝑀‘(𝑈‘(𝑀‘(𝑈‘𝑆)))) = (𝑀‘(𝑈‘𝑆)))
Theorem	paddunN 40515	The closure of the projective sum of two sets of atoms is the same as the closure of their union. (Closure is actually double polarity, which can be trivially inferred from this theorem using fveq2d 6867.) (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ + = (+𝑃‘𝐾)    &   ⊢ ⊥ = (⊥𝑃‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘(𝑆 + 𝑇)) = ( ⊥ ‘(𝑆 ∪ 𝑇)))
Theorem	poldmj1N 40516	De Morgan's law for polarity of projective sum. (oldmj1 39809 analog.) (Contributed by NM, 7-Mar-2012.) (New usage is discouraged.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ + = (+𝑃‘𝐾)    &   ⊢ ⊥ = (⊥𝑃‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘(𝑆 + 𝑇)) = (( ⊥ ‘𝑆) ∩ ( ⊥ ‘𝑇)))
Theorem	pmapj2N 40517	The projective map of the join of two lattice elements. (Contributed by NM, 14-Mar-2012.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝑀 = (pmap‘𝐾)    &   ⊢ + = (+𝑃‘𝐾)    &   ⊢ ⊥ = (⊥𝑃‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑀‘(𝑋 ∨ 𝑌)) = ( ⊥ ‘( ⊥ ‘((𝑀‘𝑋) + (𝑀‘𝑌)))))
Theorem	pmapocjN 40518	The projective map of the orthocomplement of the join of two lattice elements. (Contributed by NM, 14-Mar-2012.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    &   ⊢ 𝐹 = (pmap‘𝐾)    &   ⊢ + = (+𝑃‘𝐾)    &   ⊢ 𝑁 = (⊥𝑃‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘( ⊥ ‘(𝑋 ∨ 𝑌))) = (𝑁‘((𝐹‘𝑋) + (𝐹‘𝑌))))
Theorem	polatN 40519	The polarity of the singleton of an atom (i.e. a point). (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)
⊢ ⊥ = (oc‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑀 = (pmap‘𝐾)    &   ⊢ 𝑃 = (⊥𝑃‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → (𝑃‘{𝑄}) = (𝑀‘( ⊥ ‘𝑄)))
Theorem	2polatN 40520	Double polarity of the singleton of an atom (i.e. a point). (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑃 = (⊥𝑃‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → (𝑃‘(𝑃‘{𝑄})) = {𝑄})
Theorem	pnonsingN 40521	The intersection of a set of atoms and its polarity is empty. Definition of nonsingular in [Holland95] p. 214. (Contributed by NM, 29-Jan-2012.) (New usage is discouraged.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑃 = (⊥𝑃‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑋 ∩ (𝑃‘𝑋)) = ∅)
Syntax	cpscN 40522	Extend class notation with set of all closed projective subspaces for a Hilbert lattice.
class PSubCl
Definition	df-psubclN 40523*	Define set of all closed projective subspaces, which are those sets of atoms that equal their double polarity. Based on definition in [Holland95] p. 223. (Contributed by NM, 23-Jan-2012.)
⊢ PSubCl = (𝑘 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ (Atoms‘𝑘) ∧ ((⊥𝑃‘𝑘)‘((⊥𝑃‘𝑘)‘𝑠)) = 𝑠)})
Theorem	psubclsetN 40524*	The set of closed projective subspaces in a Hilbert lattice. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ ⊥ = (⊥𝑃‘𝐾)    &   ⊢ 𝐶 = (PSubCl‘𝐾)    ⇒   ⊢ (𝐾 ∈ 𝐵 → 𝐶 = {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠)})
Theorem	ispsubclN 40525	The predicate "is a closed projective subspace". (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ ⊥ = (⊥𝑃‘𝐾)    &   ⊢ 𝐶 = (PSubCl‘𝐾)    ⇒   ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝐶 ↔ (𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)))
Theorem	psubcliN 40526	Property of a closed projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ ⊥ = (⊥𝑃‘𝐾)    &   ⊢ 𝐶 = (PSubCl‘𝐾)    ⇒   ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐶) → (𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋))
Theorem	psubcli2N 40527	Property of a closed projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
⊢ ⊥ = (⊥𝑃‘𝐾)    &   ⊢ 𝐶 = (PSubCl‘𝐾)    ⇒   ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐶) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
Theorem	psubclsubN 40528	A closed projective subspace is a projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
⊢ 𝑆 = (PSubSp‘𝐾)    &   ⊢ 𝐶 = (PSubCl‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ 𝑆)
Theorem	psubclssatN 40529	A closed projective subspace is a set of atoms. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐶 = (PSubCl‘𝐾)    ⇒   ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐶) → 𝑋 ⊆ 𝐴)
Theorem	pmapidclN 40530	Projective map of the LUB of a closed subspace. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
⊢ 𝑈 = (lub‘𝐾)    &   ⊢ 𝑀 = (pmap‘𝐾)    &   ⊢ 𝐶 = (PSubCl‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → (𝑀‘(𝑈‘𝑋)) = 𝑋)
Theorem	0psubclN 40531	The empty set is a closed projective subspace. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
⊢ 𝐶 = (PSubCl‘𝐾)    ⇒   ⊢ (𝐾 ∈ HL → ∅ ∈ 𝐶)
Theorem	1psubclN 40532	The set of all atoms is a closed projective subspace. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐶 = (PSubCl‘𝐾)    ⇒   ⊢ (𝐾 ∈ HL → 𝐴 ∈ 𝐶)
Theorem	atpsubclN 40533	A point (singleton of an atom) is a closed projective subspace. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐶 = (PSubCl‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → {𝑄} ∈ 𝐶)
Theorem	pmapsubclN 40534	A projective map value is a closed projective subspace. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝑀 = (pmap‘𝐾)    &   ⊢ 𝐶 = (PSubCl‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) ∈ 𝐶)
Theorem	ispsubcl2N 40535*	Alternate predicate for "is a closed projective subspace". Remark in [Holland95] p. 223. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝑀 = (pmap‘𝐾)    &   ⊢ 𝐶 = (PSubCl‘𝐾)    ⇒   ⊢ (𝐾 ∈ HL → (𝑋 ∈ 𝐶 ↔ ∃𝑦 ∈ 𝐵 𝑋 = (𝑀‘𝑦)))
Theorem	psubclinN 40536	The intersection of two closed subspaces is closed. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
⊢ 𝐶 = (PSubCl‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) → (𝑋 ∩ 𝑌) ∈ 𝐶)
Theorem	paddatclN 40537	The projective sum of a closed subspace and an atom is a closed projective subspace. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ + = (+𝑃‘𝐾)    &   ⊢ 𝐶 = (PSubCl‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑄 ∈ 𝐴) → (𝑋 + {𝑄}) ∈ 𝐶)
Theorem	pclfinclN 40538	The projective subspace closure of a finite set of atoms is a closed subspace. Compare the (non-closed) subspace version pclfinN 40488 and also pclcmpatN 40489. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑈 = (PCl‘𝐾)    &   ⊢ 𝑆 = (PSubCl‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑋 ∈ Fin) → (𝑈‘𝑋) ∈ 𝑆)
Theorem	linepsubclN 40539	A line is a closed projective subspace. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
⊢ 𝑁 = (Lines‘𝐾)    &   ⊢ 𝐶 = (PSubCl‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) → 𝑋 ∈ 𝐶)
Theorem	polsubclN 40540	A polarity is a closed projective subspace. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ ⊥ = (⊥𝑃‘𝐾)    &   ⊢ 𝐶 = (PSubCl‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) ∈ 𝐶)
Theorem	poml4N 40541	Orthomodular law for projective lattices. Lemma 3.3(1) in [Holland95] p. 215. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ ⊥ = (⊥𝑃‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → ((𝑋 ⊆ 𝑌 ∧ ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌) → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌) = ( ⊥ ‘( ⊥ ‘𝑋))))
Theorem	poml5N 40542	Orthomodular law for projective lattices. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ ⊥ = (⊥𝑃‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌))) ∩ ( ⊥ ‘𝑌)) = ( ⊥ ‘( ⊥ ‘𝑋)))
Theorem	poml6N 40543	Orthomodular law for projective lattices. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
⊢ 𝐶 = (PSubCl‘𝐾)    &   ⊢ ⊥ = (⊥𝑃‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ 𝑋 ⊆ 𝑌) → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌) = 𝑋)
Theorem	osumcllem1N 40544	Lemma for osumclN 40555. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ + = (+𝑃‘𝐾)    &   ⊢ ⊥ = (⊥𝑃‘𝐾)    &   ⊢ 𝐶 = (PSubCl‘𝐾)    &   ⊢ 𝑀 = (𝑋 + {𝑝})    &   ⊢ 𝑈 = ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌)))    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑈) → (𝑈 ∩ 𝑀) = 𝑀)
Theorem	osumcllem2N 40545	Lemma for osumclN 40555. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ + = (+𝑃‘𝐾)    &   ⊢ ⊥ = (⊥𝑃‘𝐾)    &   ⊢ 𝐶 = (PSubCl‘𝐾)    &   ⊢ 𝑀 = (𝑋 + {𝑝})    &   ⊢ 𝑈 = ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌)))    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑈) → 𝑋 ⊆ (𝑈 ∩ 𝑀))
Theorem	osumcllem3N 40546	Lemma for osumclN 40555. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ + = (+𝑃‘𝐾)    &   ⊢ ⊥ = (⊥𝑃‘𝐾)    &   ⊢ 𝐶 = (PSubCl‘𝐾)    &   ⊢ 𝑀 = (𝑋 + {𝑝})    &   ⊢ 𝑈 = ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌)))    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝐶 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) → (( ⊥ ‘𝑋) ∩ 𝑈) = 𝑌)
Theorem	osumcllem4N 40547	Lemma for osumclN 40555. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ + = (+𝑃‘𝐾)    &   ⊢ ⊥ = (⊥𝑃‘𝐾)    &   ⊢ 𝐶 = (PSubCl‘𝐾)    &   ⊢ 𝑀 = (𝑋 + {𝑝})    &   ⊢ 𝑈 = ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌)))    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) ∧ (𝑟 ∈ 𝑋 ∧ 𝑞 ∈ 𝑌)) → 𝑞 ≠ 𝑟)
Theorem	osumcllem5N 40548	Lemma for osumclN 40555. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ + = (+𝑃‘𝐾)    &   ⊢ ⊥ = (⊥𝑃‘𝐾)    &   ⊢ 𝐶 = (PSubCl‘𝐾)    &   ⊢ 𝑀 = (𝑋 + {𝑝})    &   ⊢ 𝑈 = ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌)))    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝐴 ∧ (𝑟 ∈ 𝑋 ∧ 𝑞 ∈ 𝑌 ∧ 𝑝 ≤ (𝑟 ∨ 𝑞))) → 𝑝 ∈ (𝑋 + 𝑌))
Theorem	osumcllem6N 40549	Lemma for osumclN 40555. Use atom exchange hlatexch1 39983 to swap 𝑝 and 𝑞. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ + = (+𝑃‘𝐾)    &   ⊢ ⊥ = (⊥𝑃‘𝐾)    &   ⊢ 𝐶 = (PSubCl‘𝐾)    &   ⊢ 𝑀 = (𝑋 + {𝑝})    &   ⊢ 𝑈 = ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌)))    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑝 ∈ 𝐴) ∧ (𝑟 ∈ 𝑋 ∧ 𝑞 ∈ 𝑌 ∧ 𝑞 ≤ (𝑟 ∨ 𝑝))) → 𝑝 ∈ (𝑋 + 𝑌))
Theorem	osumcllem7N 40550*	Lemma for osumclN 40555. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ + = (+𝑃‘𝐾)    &   ⊢ ⊥ = (⊥𝑃‘𝐾)    &   ⊢ 𝐶 = (PSubCl‘𝐾)    &   ⊢ 𝑀 = (𝑋 + {𝑝})    &   ⊢ 𝑈 = ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌)))    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝐴) ∧ 𝑞 ∈ (𝑌 ∩ 𝑀)) → 𝑝 ∈ (𝑋 + 𝑌))
Theorem	osumcllem8N 40551	Lemma for osumclN 40555. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ + = (+𝑃‘𝐾)    &   ⊢ ⊥ = (⊥𝑃‘𝐾)    &   ⊢ 𝐶 = (PSubCl‘𝐾)    &   ⊢ 𝑀 = (𝑋 + {𝑝})    &   ⊢ 𝑈 = ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌)))    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝐴) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (𝑌 ∩ 𝑀) = ∅)
Theorem	osumcllem9N 40552	Lemma for osumclN 40555. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ + = (+𝑃‘𝐾)    &   ⊢ ⊥ = (⊥𝑃‘𝐾)    &   ⊢ 𝐶 = (PSubCl‘𝐾)    &   ⊢ 𝑀 = (𝑋 + {𝑝})    &   ⊢ 𝑈 = ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌)))    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑀 = 𝑋)
Theorem	osumcllem10N 40553	Lemma for osumclN 40555. Contradict osumcllem9N 40552. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ + = (+𝑃‘𝐾)    &   ⊢ ⊥ = (⊥𝑃‘𝐾)    &   ⊢ 𝐶 = (PSubCl‘𝐾)    &   ⊢ 𝑀 = (𝑋 + {𝑝})    &   ⊢ 𝑈 = ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌)))    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑀 ≠ 𝑋)
Theorem	osumcllem11N 40554	Lemma for osumclN 40555. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
⊢ + = (+𝑃‘𝐾)    &   ⊢ ⊥ = (⊥𝑃‘𝐾)    &   ⊢ 𝐶 = (PSubCl‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅)) → (𝑋 + 𝑌) = ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))))
Theorem	osumclN 40555	Closure of orthogonal sum. If 𝑋 and 𝑌 are orthogonal closed projective subspaces, then their sum is closed. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
⊢ + = (+𝑃‘𝐾)    &   ⊢ ⊥ = (⊥𝑃‘𝐾)    &   ⊢ 𝐶 = (PSubCl‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) → (𝑋 + 𝑌) ∈ 𝐶)
Theorem	pmapojoinN 40556	For orthogonal elements, projective map of join equals projective sum. Compare pmapjoin 40440 where only one direction holds. (Contributed by NM, 11-Apr-2012.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝑀 = (pmap‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    &   ⊢ + = (+𝑃‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ ( ⊥ ‘𝑌)) → (𝑀‘(𝑋 ∨ 𝑌)) = ((𝑀‘𝑋) + (𝑀‘𝑌)))
Theorem	pexmidN 40557	Excluded middle law for closed projective subspaces, which can be shown to be equivalent to (and derivable from) the orthomodular law poml4N 40541. Lemma 3.3(2) in [Holland95] p. 215, which we prove as a special case of osumclN 40555. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ + = (+𝑃‘𝐾)    &   ⊢ ⊥ = (⊥𝑃‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → (𝑋 + ( ⊥ ‘𝑋)) = 𝐴)
Theorem	pexmidlem1N 40558	Lemma for pexmidN 40557. Holland's proof implicitly requires 𝑞 ≠ 𝑟, which we prove here. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ + = (+𝑃‘𝐾)    &   ⊢ ⊥ = (⊥𝑃‘𝐾)    &   ⊢ 𝑀 = (𝑋 + {𝑝})    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ (𝑟 ∈ 𝑋 ∧ 𝑞 ∈ ( ⊥ ‘𝑋))) → 𝑞 ≠ 𝑟)
Theorem	pexmidlem2N 40559	Lemma for pexmidN 40557. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ + = (+𝑃‘𝐾)    &   ⊢ ⊥ = (⊥𝑃‘𝐾)    &   ⊢ 𝑀 = (𝑋 + {𝑝})    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (𝑟 ∈ 𝑋 ∧ 𝑞 ∈ ( ⊥ ‘𝑋) ∧ 𝑝 ≤ (𝑟 ∨ 𝑞))) → 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))
Theorem	pexmidlem3N 40560	Lemma for pexmidN 40557. Use atom exchange hlatexch1 39983 to swap 𝑝 and 𝑞. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ + = (+𝑃‘𝐾)    &   ⊢ ⊥ = (⊥𝑃‘𝐾)    &   ⊢ 𝑀 = (𝑋 + {𝑝})    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (𝑟 ∈ 𝑋 ∧ 𝑞 ∈ ( ⊥ ‘𝑋)) ∧ 𝑞 ≤ (𝑟 ∨ 𝑝)) → 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))
Theorem	pexmidlem4N 40561*	Lemma for pexmidN 40557. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ + = (+𝑃‘𝐾)    &   ⊢ ⊥ = (⊥𝑃‘𝐾)    &   ⊢ 𝑀 = (𝑋 + {𝑝})    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (𝑋 ≠ ∅ ∧ 𝑞 ∈ (( ⊥ ‘𝑋) ∩ 𝑀))) → 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))
Theorem	pexmidlem5N 40562	Lemma for pexmidN 40557. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ + = (+𝑃‘𝐾)    &   ⊢ ⊥ = (⊥𝑃‘𝐾)    &   ⊢ 𝑀 = (𝑋 + {𝑝})    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → (( ⊥ ‘𝑋) ∩ 𝑀) = ∅)
Theorem	pexmidlem6N 40563	Lemma for pexmidN 40557. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ + = (+𝑃‘𝐾)    &   ⊢ ⊥ = (⊥𝑃‘𝐾)    &   ⊢ 𝑀 = (𝑋 + {𝑝})    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → 𝑀 = 𝑋)
Theorem	pexmidlem7N 40564	Lemma for pexmidN 40557. Contradict pexmidlem6N 40563. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ + = (+𝑃‘𝐾)    &   ⊢ ⊥ = (⊥𝑃‘𝐾)    &   ⊢ 𝑀 = (𝑋 + {𝑝})    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → 𝑀 ≠ 𝑋)
Theorem	pexmidlem8N 40565	Lemma for pexmidN 40557. The contradiction of pexmidlem6N 40563 and pexmidlem7N 40564 shows that there can be no atom 𝑝 that is not in 𝑋 + ( ⊥ ‘𝑋), which is therefore the whole atom space. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ + = (+𝑃‘𝐾)    &   ⊢ ⊥ = (⊥𝑃‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅)) → (𝑋 + ( ⊥ ‘𝑋)) = 𝐴)
Theorem	pexmidALTN 40566	Excluded middle law for closed projective subspaces, which is equivalent to (and derived from) the orthomodular law poml4N 40541. Lemma 3.3(2) in [Holland95] p. 215. In our proof, we use the variables 𝑋, 𝑀, 𝑝, 𝑞, 𝑟 in place of Hollands' l, m, P, Q, L respectively. TODO: should we make this obsolete? (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ + = (+𝑃‘𝐾)    &   ⊢ ⊥ = (⊥𝑃‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → (𝑋 + ( ⊥ ‘𝑋)) = 𝐴)
Theorem	pl42lem1N 40567	Lemma for pl42N 40571. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    &   ⊢ 𝐹 = (pmap‘𝐾)    &   ⊢ + = (+𝑃‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)) → ((𝑋 ≤ ( ⊥ ‘𝑌) ∧ 𝑍 ≤ ( ⊥ ‘𝑊)) → (𝐹‘((((𝑋 ∨ 𝑌) ∧ 𝑍) ∨ 𝑊) ∧ 𝑉)) = (((((𝐹‘𝑋) + (𝐹‘𝑌)) ∩ (𝐹‘𝑍)) + (𝐹‘𝑊)) ∩ (𝐹‘𝑉))))
Theorem	pl42lem2N 40568	Lemma for pl42N 40571. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    &   ⊢ 𝐹 = (pmap‘𝐾)    &   ⊢ + = (+𝑃‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)) → (((𝐹‘𝑋) + (𝐹‘𝑌)) + (((𝐹‘𝑋) + (𝐹‘𝑊)) ∩ ((𝐹‘𝑌) + (𝐹‘𝑉)))) ⊆ (𝐹‘((𝑋 ∨ 𝑌) ∨ ((𝑋 ∨ 𝑊) ∧ (𝑌 ∨ 𝑉)))))
Theorem	pl42lem3N 40569	Lemma for pl42N 40571. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    &   ⊢ 𝐹 = (pmap‘𝐾)    &   ⊢ + = (+𝑃‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)) → (((((𝐹‘𝑋) + (𝐹‘𝑌)) ∩ (𝐹‘𝑍)) + (𝐹‘𝑊)) ∩ (𝐹‘𝑉)) ⊆ ((((𝐹‘𝑋) + (𝐹‘𝑌)) + (𝐹‘𝑊)) ∩ (((𝐹‘𝑋) + (𝐹‘𝑌)) + (𝐹‘𝑉))))
Theorem	pl42lem4N 40570	Lemma for pl42N 40571. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    &   ⊢ 𝐹 = (pmap‘𝐾)    &   ⊢ + = (+𝑃‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)) → ((𝑋 ≤ ( ⊥ ‘𝑌) ∧ 𝑍 ≤ ( ⊥ ‘𝑊)) → (𝐹‘((((𝑋 ∨ 𝑌) ∧ 𝑍) ∨ 𝑊) ∧ 𝑉)) ⊆ (𝐹‘((𝑋 ∨ 𝑌) ∨ ((𝑋 ∨ 𝑊) ∧ (𝑌 ∨ 𝑉))))))
Theorem	pl42N 40571	Law holding in a Hilbert lattice that fails in orthomodular lattice L42 (Figure 7 in [MegPav2000] p. 2366). (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)) → ((𝑋 ≤ ( ⊥ ‘𝑌) ∧ 𝑍 ≤ ( ⊥ ‘𝑊)) → ((((𝑋 ∨ 𝑌) ∧ 𝑍) ∨ 𝑊) ∧ 𝑉) ≤ ((𝑋 ∨ 𝑌) ∨ ((𝑋 ∨ 𝑊) ∧ (𝑌 ∨ 𝑉)))))
Syntax	clh 40572	Extend class notation with set of all co-atoms (lattice hyperplanes).
class LHyp
Syntax	claut 40573	Extend class notation with set of all lattice automorphisms.
class LAut
Syntax	cwpointsN 40574	Extend class notation with W points.
class WAtoms
Syntax	cpautN 40575	Extend class notation with set of all projective automorphisms.
class PAut
Definition	df-lhyp 40576*	Define the set of lattice hyperplanes, which are all lattice elements covered by 1 (i.e., all co-atoms). We call them "hyperplanes" instead of "co-atoms" in analogy with projective geometry hyperplanes. (Contributed by NM, 11-May-2012.)
⊢ LHyp = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ 𝑥( ⋖ ‘𝑘)(1.‘𝑘)})
Definition	df-laut 40577*	Define set of lattice autoisomorphisms. (Contributed by NM, 11-May-2012.)
⊢ LAut = (𝑘 ∈ V ↦ {𝑓 ∣ (𝑓:(Base‘𝑘)–1-1-onto→(Base‘𝑘) ∧ ∀𝑥 ∈ (Base‘𝑘)∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦 ↔ (𝑓‘𝑥)(le‘𝑘)(𝑓‘𝑦)))})
Definition	df-watsN 40578*	Define W-atoms corresponding to an arbitrary "fiducial (i.e. reference) atom" 𝑑. These are all atoms not in the polarity of {𝑑}), which is the hyperplane determined by 𝑑. Definition of set W in [Crawley] p. 111. (Contributed by NM, 26-Jan-2012.)
⊢ WAtoms = (𝑘 ∈ V ↦ (𝑑 ∈ (Atoms‘𝑘) ↦ ((Atoms‘𝑘) ∖ ((⊥𝑃‘𝑘)‘{𝑑}))))
Definition	df-pautN 40579*	Define set of all projective automorphisms. This is the intended definition of automorphism in [Crawley] p. 112. (Contributed by NM, 26-Jan-2012.)
⊢ PAut = (𝑘 ∈ V ↦ {𝑓 ∣ (𝑓:(PSubSp‘𝑘)–1-1-onto→(PSubSp‘𝑘) ∧ ∀𝑥 ∈ (PSubSp‘𝑘)∀𝑦 ∈ (PSubSp‘𝑘)(𝑥 ⊆ 𝑦 ↔ (𝑓‘𝑥) ⊆ (𝑓‘𝑦)))})
Theorem	watfvalN 40580*	The W atoms function. (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑃 = (⊥𝑃‘𝐾)    &   ⊢ 𝑊 = (WAtoms‘𝐾)    ⇒   ⊢ (𝐾 ∈ 𝐵 → 𝑊 = (𝑑 ∈ 𝐴 ↦ (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝑑}))))
Theorem	watvalN 40581	Value of the W atoms function. (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑃 = (⊥𝑃‘𝐾)    &   ⊢ 𝑊 = (WAtoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴) → (𝑊‘𝐷) = (𝐴 ∖ ((⊥𝑃‘𝐾)‘{𝐷})))
Theorem	iswatN 40582	The predicate "is a W atom" (corresponding to fiducial atom 𝐷). (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑃 = (⊥𝑃‘𝐾)    &   ⊢ 𝑊 = (WAtoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴) → (𝑃 ∈ (𝑊‘𝐷) ↔ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ∈ ((⊥𝑃‘𝐾)‘{𝐷}))))
Theorem	lhpset 40583*	The set of co-atoms (lattice hyperplanes). (Contributed by NM, 11-May-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 1 = (1.‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    ⇒   ⊢ (𝐾 ∈ 𝐴 → 𝐻 = {𝑤 ∈ 𝐵 ∣ 𝑤𝐶 1 })
Theorem	islhp 40584	The predicate "is a co-atom (lattice hyperplane)". (Contributed by NM, 11-May-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 1 = (1.‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    ⇒   ⊢ (𝐾 ∈ 𝐴 → (𝑊 ∈ 𝐻 ↔ (𝑊 ∈ 𝐵 ∧ 𝑊𝐶 1 )))
Theorem	islhp2 40585	The predicate "is a co-atom (lattice hyperplane)". (Contributed by NM, 18-May-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 1 = (1.‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    ⇒   ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐵) → (𝑊 ∈ 𝐻 ↔ 𝑊𝐶 1 ))
Theorem	lhpbase 40586	A co-atom is a member of the lattice base set (i.e., a lattice element). (Contributed by NM, 18-May-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    ⇒   ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵)
Theorem	lhp1cvr 40587	The lattice unity covers a co-atom (lattice hyperplane). (Contributed by NM, 18-May-2012.)
⊢ 1 = (1.‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    ⇒   ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻) → 𝑊𝐶 1 )
Theorem	lhplt 40588	An atom under a co-atom is strictly less than it. TODO: is this needed? (Contributed by NM, 1-Jun-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊)) → 𝑃 < 𝑊)
Theorem	lhp2lt 40589	The join of two atoms under a co-atom is strictly less than it. (Contributed by NM, 8-Jul-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → (𝑃 ∨ 𝑄) < 𝑊)
Theorem	lhpexlt 40590*	There exists an atom less than a co-atom. TODO: is this needed? (Contributed by NM, 1-Jun-2012.)
⊢ < = (lt‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ∃𝑝 ∈ 𝐴 𝑝 < 𝑊)
Theorem	lhp0lt 40591	A co-atom is greater than zero. TODO: is this needed? (Contributed by NM, 1-Jun-2012.)
⊢ < = (lt‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 < 𝑊)
Theorem	lhpn0 40592	A co-atom is nonzero. TODO: is this needed? (Contributed by NM, 26-Apr-2013.)
⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑊 ≠ 0 )
Theorem	lhpexle 40593*	There exists an atom under a co-atom. (Contributed by NM, 26-Apr-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ∃𝑝 ∈ 𝐴 𝑝 ≤ 𝑊)
Theorem	lhpexnle 40594*	There exists an atom not under a co-atom. (Contributed by NM, 12-Apr-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ∃𝑝 ∈ 𝐴 ¬ 𝑝 ≤ 𝑊)
Theorem	lhpexle1lem 40595*	Lemma for lhpexle1 40596 and others that eliminates restrictions on 𝑋. (Contributed by NM, 24-Jul-2013.)
⊢ (𝜑 → ∃𝑝 ∈ 𝐴 (𝑝 ≤ 𝑊 ∧ 𝜓))    &   ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐴 ∧ 𝑋 ≤ 𝑊)) → ∃𝑝 ∈ 𝐴 (𝑝 ≤ 𝑊 ∧ 𝜓 ∧ 𝑝 ≠ 𝑋))    ⇒   ⊢ (𝜑 → ∃𝑝 ∈ 𝐴 (𝑝 ≤ 𝑊 ∧ 𝜓 ∧ 𝑝 ≠ 𝑋))
Theorem	lhpexle1 40596*	There exists an atom under a co-atom different from any given element. (Contributed by NM, 24-Jul-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ∃𝑝 ∈ 𝐴 (𝑝 ≤ 𝑊 ∧ 𝑝 ≠ 𝑋))
Theorem	lhpexle2lem 40597*	Lemma for lhpexle2 40598. (Contributed by NM, 19-Jun-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐴 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐴 ∧ 𝑌 ≤ 𝑊)) → ∃𝑝 ∈ 𝐴 (𝑝 ≤ 𝑊 ∧ 𝑝 ≠ 𝑋 ∧ 𝑝 ≠ 𝑌))
Theorem	lhpexle2 40598*	There exists atom under a co-atom different from any two other elements. (Contributed by NM, 24-Jul-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ∃𝑝 ∈ 𝐴 (𝑝 ≤ 𝑊 ∧ 𝑝 ≠ 𝑋 ∧ 𝑝 ≠ 𝑌))
Theorem	lhpexle3lem 40599*	There exists atom under a co-atom different from any three other atoms. TODO: study if adant*, simp* usage can be improved. (Contributed by NM, 9-Jul-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑍 ≤ 𝑊)) → ∃𝑝 ∈ 𝐴 (𝑝 ≤ 𝑊 ∧ (𝑝 ≠ 𝑋 ∧ 𝑝 ≠ 𝑌 ∧ 𝑝 ≠ 𝑍)))
Theorem	lhpexle3 40600*	There exists atom under a co-atom different from any three other elements. (Contributed by NM, 24-Jul-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ∃𝑝 ∈ 𝐴 (𝑝 ≤ 𝑊 ∧ (𝑝 ≠ 𝑋 ∧ 𝑝 ≠ 𝑌 ∧ 𝑝 ≠ 𝑍)))