P. 400 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 39901-40000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	polsubN 39901	The polarity of a set of atoms is a projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑆 = (PSubSp‘𝐾)    &   ⊢ ⊥ = (⊥𝑃‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) ∈ 𝑆)
Theorem	polssatN 39902	The polarity of a set of atoms is a set of atoms. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ ⊥ = (⊥𝑃‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) ⊆ 𝐴)
Theorem	pol0N 39903	The polarity of the empty projective subspace is the whole space. (Contributed by NM, 29-Oct-2011.) (New usage is discouraged.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ ⊥ = (⊥𝑃‘𝐾)    ⇒   ⊢ (𝐾 ∈ 𝐵 → ( ⊥ ‘∅) = 𝐴)
Theorem	pol1N 39904	The polarity of the whole projective subspace is the empty space. Remark in [Holland95] p. 223. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ ⊥ = (⊥𝑃‘𝐾)    ⇒   ⊢ (𝐾 ∈ HL → ( ⊥ ‘𝐴) = ∅)
Theorem	2pol0N 39905	The closed subspace closure of the empty set. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)
⊢ ⊥ = (⊥𝑃‘𝐾)    ⇒   ⊢ (𝐾 ∈ HL → ( ⊥ ‘( ⊥ ‘∅)) = ∅)
Theorem	polpmapN 39906	The polarity of a projective map. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    &   ⊢ 𝑀 = (pmap‘𝐾)    &   ⊢ 𝑃 = (⊥𝑃‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑃‘(𝑀‘𝑋)) = (𝑀‘( ⊥ ‘𝑋)))
Theorem	2polpmapN 39907	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 39908	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 39909	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 39910	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 39911	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 39912	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 39913	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 39914	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 39915	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 39916	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 39917	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 39918	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 39919	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 39920	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 39921	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 6862.) (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 39922	De Morgan's law for polarity of projective sum. (oldmj1 39214 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 39923	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 39924	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 39925	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 39926	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 39927	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 39928	Extend class notation with set of all closed projective subspaces for a Hilbert lattice.
class PSubCl
Definition	df-psubclN 39929*	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 39930*	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 39931	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 39932	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 39933	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 39934	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 39935	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 39936	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 39937	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 39938	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 39939	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 39940	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 39941*	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 39942	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 39943	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 39944	The projective subspace closure of a finite set of atoms is a closed subspace. Compare the (non-closed) subspace version pclfinN 39894 and also pclcmpatN 39895. (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 39945	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 39946	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 39947	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 39948	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 39949	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 39950	Lemma for osumclN 39961. (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 39951	Lemma for osumclN 39961. (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 39952	Lemma for osumclN 39961. (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 39953	Lemma for osumclN 39961. (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 39954	Lemma for osumclN 39961. (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 39955	Lemma for osumclN 39961. Use atom exchange hlatexch1 39389 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 39956*	Lemma for osumclN 39961. (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 39957	Lemma for osumclN 39961. (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 39958	Lemma for osumclN 39961. (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 39959	Lemma for osumclN 39961. Contradict osumcllem9N 39958. (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 39960	Lemma for osumclN 39961. (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 39961	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 39962	For orthogonal elements, projective map of join equals projective sum. Compare pmapjoin 39846 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 39963	Excluded middle law for closed projective subspaces, which can be shown to be equivalent to (and derivable from) the orthomodular law poml4N 39947. Lemma 3.3(2) in [Holland95] p. 215, which we prove as a special case of osumclN 39961. (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 39964	Lemma for pexmidN 39963. 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 39965	Lemma for pexmidN 39963. (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 39966	Lemma for pexmidN 39963. Use atom exchange hlatexch1 39389 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 39967*	Lemma for pexmidN 39963. (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 39968	Lemma for pexmidN 39963. (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 39969	Lemma for pexmidN 39963. (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 39970	Lemma for pexmidN 39963. Contradict pexmidlem6N 39969. (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 39971	Lemma for pexmidN 39963. The contradiction of pexmidlem6N 39969 and pexmidlem7N 39970 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 39972	Excluded middle law for closed projective subspaces, which is equivalent to (and derived from) the orthomodular law poml4N 39947. 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 39973	Lemma for pl42N 39977. (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 39974	Lemma for pl42N 39977. (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 39975	Lemma for pl42N 39977. (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 39976	Lemma for pl42N 39977. (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 39977	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 39978	Extend class notation with set of all co-atoms (lattice hyperplanes).
class LHyp
Syntax	claut 39979	Extend class notation with set of all lattice automorphisms.
class LAut
Syntax	cwpointsN 39980	Extend class notation with W points.
class WAtoms
Syntax	cpautN 39981	Extend class notation with set of all projective automorphisms.
class PAut
Definition	df-lhyp 39982*	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 39983*	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 39984*	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 39985*	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 39986*	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 39987	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 39988	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 39989*	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 39990	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 39991	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 39992	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 39993	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 39994	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 39995	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 39996*	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 39997	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 39998	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 39999*	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 40000*	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)