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	polcon2N 39901	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 39902	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 39903	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 39904	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 39905	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 39906	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 39907	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 39908	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 39909	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 6910.) (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 39910	De Morgan's law for polarity of projective sum. (oldmj1 39202 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 39911	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 39912	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 39913	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 39914	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 39915	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 39916	Extend class notation with set of all closed projective subspaces for a Hilbert lattice.
class PSubCl
Definition	df-psubclN 39917*	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 39918*	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 39919	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 39920	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 39921	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 39922	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 39923	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 39924	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 39925	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 39926	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 39927	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 39928	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 39929*	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 39930	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 39931	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 39932	The projective subspace closure of a finite set of atoms is a closed subspace. Compare the (non-closed) subspace version pclfinN 39882 and also pclcmpatN 39883. (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 39933	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 39934	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 39935	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 39936	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 39937	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 39938	Lemma for osumclN 39949. (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 39939	Lemma for osumclN 39949. (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 39940	Lemma for osumclN 39949. (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 39941	Lemma for osumclN 39949. (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 39942	Lemma for osumclN 39949. (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 39943	Lemma for osumclN 39949. Use atom exchange hlatexch1 39377 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 39944*	Lemma for osumclN 39949. (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 39945	Lemma for osumclN 39949. (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 39946	Lemma for osumclN 39949. (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 39947	Lemma for osumclN 39949. Contradict osumcllem9N 39946. (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 39948	Lemma for osumclN 39949. (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 39949	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 39950	For orthogonal elements, projective map of join equals projective sum. Compare pmapjoin 39834 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 39951	Excluded middle law for closed projective subspaces, which can be shown to be equivalent to (and derivable from) the orthomodular law poml4N 39935. Lemma 3.3(2) in [Holland95] p. 215, which we prove as a special case of osumclN 39949. (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 39952	Lemma for pexmidN 39951. 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 39953	Lemma for pexmidN 39951. (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 39954	Lemma for pexmidN 39951. Use atom exchange hlatexch1 39377 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 39955*	Lemma for pexmidN 39951. (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 39956	Lemma for pexmidN 39951. (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 39957	Lemma for pexmidN 39951. (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 39958	Lemma for pexmidN 39951. Contradict pexmidlem6N 39957. (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 39959	Lemma for pexmidN 39951. The contradiction of pexmidlem6N 39957 and pexmidlem7N 39958 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 39960	Excluded middle law for closed projective subspaces, which is equivalent to (and derived from) the orthomodular law poml4N 39935. 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 39961	Lemma for pl42N 39965. (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 39962	Lemma for pl42N 39965. (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 39963	Lemma for pl42N 39965. (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 39964	Lemma for pl42N 39965. (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 39965	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 39966	Extend class notation with set of all co-atoms (lattice hyperplanes).
class LHyp
Syntax	claut 39967	Extend class notation with set of all lattice automorphisms.
class LAut
Syntax	cwpointsN 39968	Extend class notation with W points.
class WAtoms
Syntax	cpautN 39969	Extend class notation with set of all projective automorphisms.
class PAut
Definition	df-lhyp 39970*	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 39971*	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 39972*	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 39973*	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 39974*	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 39975	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 39976	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 39977*	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 39978	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 39979	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 39980	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 39981	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 39982	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 39983	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 39984*	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 39985	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 39986	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 39987*	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 39988*	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 39989*	Lemma for lhpexle1 39990 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 39990*	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 39991*	Lemma for lhpexle2 39992. (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 39992*	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 39993*	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 39994*	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)))
Theorem	lhpex2leN 39995*	There exist at least two different atoms under a co-atom. This allows to create a line under the co-atom. TODO: is this needed? (Contributed by NM, 1-Jun-2012.) (New usage is discouraged.)
⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 (𝑝 ≤ 𝑊 ∧ 𝑞 ≤ 𝑊 ∧ 𝑝 ≠ 𝑞))
Theorem	lhpoc 39996	The orthocomplement of a co-atom (lattice hyperplane) is an atom. (Contributed by NM, 18-May-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐵) → (𝑊 ∈ 𝐻 ↔ ( ⊥ ‘𝑊) ∈ 𝐴))
Theorem	lhpoc2N 39997	The orthocomplement of an atom is a co-atom (lattice hyperplane). (Contributed by NM, 20-Jun-2012.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐵) → (𝑊 ∈ 𝐴 ↔ ( ⊥ ‘𝑊) ∈ 𝐻))
Theorem	lhpocnle 39998	The orthocomplement of a co-atom is not under it. (Contributed by NM, 22-May-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ¬ ( ⊥ ‘𝑊) ≤ 𝑊)
Theorem	lhpocat 39999	The orthocomplement of a co-atom is an atom. (Contributed by NM, 9-Feb-2013.)
⊢ ⊥ = (oc‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘𝑊) ∈ 𝐴)
Theorem	lhpocnel 40000	The orthocomplement of a co-atom is an atom not under it. Provides a convenient construction when we need the existence of any object with this property. (Contributed by NM, 25-May-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (( ⊥ ‘𝑊) ∈ 𝐴 ∧ ¬ ( ⊥ ‘𝑊) ≤ 𝑊))