P. 394 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 39301-39400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cvlsupr4 39301	Consequence of superposition condition (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅). (Contributed by NM, 9-Nov-2012.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    ⇒   ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))) → 𝑅 ≤ (𝑃 ∨ 𝑄))
Theorem	cvlsupr5 39302	Consequence of superposition condition (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅). (Contributed by NM, 9-Nov-2012.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    ⇒   ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))) → 𝑅 ≠ 𝑃)
Theorem	cvlsupr6 39303	Consequence of superposition condition (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅). (Contributed by NM, 9-Nov-2012.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    ⇒   ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))) → 𝑅 ≠ 𝑄)
Theorem	cvlsupr7 39304	Consequence of superposition condition (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅). (Contributed by NM, 24-Nov-2012.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    ⇒   ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))) → (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑄))
Theorem	cvlsupr8 39305	Consequence of superposition condition (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅). (Contributed by NM, 24-Nov-2012.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    ⇒   ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))) → (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑅))
21.28.11  Hilbert lattices
Syntax	chlt 39306	Extend class notation with Hilbert lattices.
class HL
Definition	df-hlat 39307*	Define the class of Hilbert lattices, which are complete, atomic lattices satisfying the superposition principle and minimum height. (Contributed by NM, 5-Nov-2012.)
⊢ HL = {𝑙 ∈ ((OML ∩ CLat) ∩ CvLat) ∣ (∀𝑎 ∈ (Atoms‘𝑙)∀𝑏 ∈ (Atoms‘𝑙)(𝑎 ≠ 𝑏 → ∃𝑐 ∈ (Atoms‘𝑙)(𝑐 ≠ 𝑎 ∧ 𝑐 ≠ 𝑏 ∧ 𝑐(le‘𝑙)(𝑎(join‘𝑙)𝑏))) ∧ ∃𝑎 ∈ (Base‘𝑙)∃𝑏 ∈ (Base‘𝑙)∃𝑐 ∈ (Base‘𝑙)(((0.‘𝑙)(lt‘𝑙)𝑎 ∧ 𝑎(lt‘𝑙)𝑏) ∧ (𝑏(lt‘𝑙)𝑐 ∧ 𝑐(lt‘𝑙)(1.‘𝑙))))}
Theorem	ishlat1 39308*	The predicate "is a Hilbert lattice", which is: is orthomodular (𝐾 ∈ OML), complete (𝐾 ∈ CLat), atomic and satisfies the exchange (or covering) property (𝐾 ∈ CvLat), satisfies the superposition principle, and has a minimum height of 4 (witnessed here by 0, x, y, z, 1). (Contributed by NM, 5-Nov-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 1 = (1.‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (𝐾 ∈ HL ↔ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑥 ∨ 𝑦))) ∧ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (( 0 < 𝑥 ∧ 𝑥 < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 )))))
Theorem	ishlat2 39309*	The predicate "is a Hilbert lattice". Here we replace 𝐾 ∈ CvLat with the weaker 𝐾 ∈ AtLat and show the exchange property explicitly. (Contributed by NM, 5-Nov-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 1 = (1.‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (𝐾 ∈ HL ↔ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥 ≠ 𝑦 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑥 ∨ 𝑦))) ∧ ∀𝑧 ∈ 𝐵 ((¬ 𝑥 ≤ 𝑧 ∧ 𝑥 ≤ (𝑧 ∨ 𝑦)) → 𝑦 ≤ (𝑧 ∨ 𝑥))) ∧ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (( 0 < 𝑥 ∧ 𝑥 < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 )))))
Theorem	ishlat3N 39310*	The predicate "is a Hilbert lattice". Note that the superposition principle is expressed in the compact form ∃𝑧 ∈ 𝐴(𝑥 ∨ 𝑧) = (𝑦 ∨ 𝑧). The exchange property and atomicity are provided by 𝐾 ∈ CvLat, and "minimum height 4" is shown explicitly. (Contributed by NM, 8-Nov-2012.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 1 = (1.‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (𝐾 ∈ HL ↔ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐴 (𝑥 ∨ 𝑧) = (𝑦 ∨ 𝑧) ∧ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (( 0 < 𝑥 ∧ 𝑥 < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 )))))
Theorem	ishlatiN 39311*	Properties that determine a Hilbert lattice. (Contributed by NM, 13-Nov-2011.) (New usage is discouraged.)
⊢ 𝐾 ∈ OML    &   ⊢ 𝐾 ∈ CLat    &   ⊢ 𝐾 ∈ AtLat    &   ⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 1 = (1.‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥 ≠ 𝑦 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑥 ∨ 𝑦))) ∧ ∀𝑧 ∈ 𝐵 ((¬ 𝑥 ≤ 𝑧 ∧ 𝑥 ≤ (𝑧 ∨ 𝑦)) → 𝑦 ≤ (𝑧 ∨ 𝑥)))    &   ⊢ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (( 0 < 𝑥 ∧ 𝑥 < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 ))    ⇒   ⊢ 𝐾 ∈ HL
Theorem	hlomcmcv 39312	A Hilbert lattice is orthomodular, complete, and has the covering (exchange) property. (Contributed by NM, 5-Nov-2012.)
⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat))
Theorem	hloml 39313	A Hilbert lattice is orthomodular. (Contributed by NM, 20-Oct-2011.)
⊢ (𝐾 ∈ HL → 𝐾 ∈ OML)
Theorem	hlclat 39314	A Hilbert lattice is complete. (Contributed by NM, 20-Oct-2011.)
⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat)
Theorem	hlcvl 39315	A Hilbert lattice is an atomic lattice with the covering property. (Contributed by NM, 5-Nov-2012.)
⊢ (𝐾 ∈ HL → 𝐾 ∈ CvLat)
Theorem	hlatl 39316	A Hilbert lattice is atomic. (Contributed by NM, 20-Oct-2011.)
⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
Theorem	hlol 39317	A Hilbert lattice is an ortholattice. (Contributed by NM, 20-Oct-2011.)
⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
Theorem	hlop 39318	A Hilbert lattice is an orthoposet. (Contributed by NM, 20-Oct-2011.)
⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
Theorem	hllat 39319	A Hilbert lattice is a lattice. (Contributed by NM, 20-Oct-2011.)
⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
Theorem	hllatd 39320	Deduction form of hllat 39319. A Hilbert lattice is a lattice. (Contributed by BJ, 14-Aug-2022.)
⊢ (𝜑 → 𝐾 ∈ HL)    ⇒   ⊢ (𝜑 → 𝐾 ∈ Lat)
Theorem	hlomcmat 39321	A Hilbert lattice is orthomodular, complete, and atomic. (Contributed by NM, 5-Nov-2012.)
⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat))
Theorem	hlpos 39322	A Hilbert lattice is a poset. (Contributed by NM, 20-Oct-2011.)
⊢ (𝐾 ∈ HL → 𝐾 ∈ Poset)
Theorem	hlatjcl 39323	Closure of join operation. Frequently-used special case of latjcl 18509 for atoms. (Contributed by NM, 15-Jun-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 ∨ 𝑌) ∈ 𝐵)
Theorem	hlatjcom 39324	Commutatitivity of join operation. Frequently-used special case of latjcom 18517 for atoms. (Contributed by NM, 15-Jun-2012.)
⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 ∨ 𝑌) = (𝑌 ∨ 𝑋))
Theorem	hlatjidm 39325	Idempotence of join operation. Frequently-used special case of latjcom 18517 for atoms. (Contributed by NM, 15-Jul-2012.)
⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐴) → (𝑋 ∨ 𝑋) = 𝑋)
Theorem	hlatjass 39326	Lattice join is associative. Frequently-used special case of latjass 18553 for atoms. (Contributed by NM, 27-Jul-2012.)
⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) = (𝑃 ∨ (𝑄 ∨ 𝑅)))
Theorem	hlatj12 39327	Swap 1st and 2nd members of lattice join. Frequently-used special case of latj32 18555 for atoms. (Contributed by NM, 4-Jun-2012.)
⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (𝑃 ∨ (𝑄 ∨ 𝑅)) = (𝑄 ∨ (𝑃 ∨ 𝑅)))
Theorem	hlatj32 39328	Swap 2nd and 3rd members of lattice join. Frequently-used special case of latj32 18555 for atoms. (Contributed by NM, 21-Jul-2012.)
⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑃 ∨ 𝑅) ∨ 𝑄))
Theorem	hlatjrot 39329	Rotate lattice join of 3 classes. Frequently-used special case of latjrot 18558 for atoms. (Contributed by NM, 2-Aug-2012.)
⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑅 ∨ 𝑃) ∨ 𝑄))
Theorem	hlatj4 39330	Rearrangement of lattice join of 4 classes. Frequently-used special case of latj4 18559 for atoms. (Contributed by NM, 9-Aug-2012.)
⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∨ (𝑅 ∨ 𝑆)) = ((𝑃 ∨ 𝑅) ∨ (𝑄 ∨ 𝑆)))
Theorem	hlatlej1 39331	A join's first argument is less than or equal to the join. Special case of latlej1 18518 to show an atom is on a line. (Contributed by NM, 15-May-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑃 ≤ (𝑃 ∨ 𝑄))
Theorem	hlatlej2 39332	A join's second argument is less than or equal to the join. Special case of latlej2 18519 to show an atom is on a line. (Contributed by NM, 15-May-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑄 ≤ (𝑃 ∨ 𝑄))
Theorem	glbconN 39333*	De Morgan's law for GLB and LUB. This holds in any complete ortholattice, although we assume HL for convenience. (Contributed by NM, 17-Jan-2012.) New df-riota 7404. (Revised by SN, 3-Jan-2025.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝑈 = (lub‘𝐾)    &   ⊢ 𝐺 = (glb‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐵) → (𝐺‘𝑆) = ( ⊥ ‘(𝑈‘{𝑥 ∈ 𝐵 ∣ ( ⊥ ‘𝑥) ∈ 𝑆})))
Theorem	glbconNOLD 39334*	Obsolete version of glbconN 39333 as of 3-Jan-2025. (Contributed by NM, 17-Jan-2012.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝑈 = (lub‘𝐾)    &   ⊢ 𝐺 = (glb‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐵) → (𝐺‘𝑆) = ( ⊥ ‘(𝑈‘{𝑥 ∈ 𝐵 ∣ ( ⊥ ‘𝑥) ∈ 𝑆})))
Theorem	glbconxN 39335*	De Morgan's law for GLB and LUB. Index-set version of glbconN 39333, where we read 𝑆 as 𝑆(𝑖). (Contributed by NM, 17-Jan-2012.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝑈 = (lub‘𝐾)    &   ⊢ 𝐺 = (glb‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) → (𝐺‘{𝑥 ∣ ∃𝑖 ∈ 𝐼 𝑥 = 𝑆}) = ( ⊥ ‘(𝑈‘{𝑥 ∣ ∃𝑖 ∈ 𝐼 𝑥 = ( ⊥ ‘𝑆)})))
Theorem	atnlej1 39336	If an atom is not less than or equal to the join of two others, it is not equal to either. (This also holds for non-atoms, but in this form it is convenient.) (Contributed by NM, 8-Jan-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅)) → 𝑃 ≠ 𝑄)
Theorem	atnlej2 39337	If an atom is not less than or equal to the join of two others, it is not equal to either. (This also holds for non-atoms, but in this form it is convenient.) (Contributed by NM, 8-Jan-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅)) → 𝑃 ≠ 𝑅)
Theorem	hlsuprexch 39338*	A Hilbert lattice has the superposition and exchange properties. (Contributed by NM, 13-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ((𝑃 ≠ 𝑄 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑃 ∧ 𝑧 ≠ 𝑄 ∧ 𝑧 ≤ (𝑃 ∨ 𝑄))) ∧ ∀𝑧 ∈ 𝐵 ((¬ 𝑃 ≤ 𝑧 ∧ 𝑃 ≤ (𝑧 ∨ 𝑄)) → 𝑄 ≤ (𝑧 ∨ 𝑃))))
Theorem	hlexch1 39339	A Hilbert lattice has the exchange property. (Contributed by NM, 13-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) → 𝑄 ≤ (𝑋 ∨ 𝑃)))
Theorem	hlexch2 39340	A Hilbert lattice has the exchange property. (Contributed by NM, 6-May-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑄 ∨ 𝑋) → 𝑄 ≤ (𝑃 ∨ 𝑋)))
Theorem	hlexchb1 39341	A Hilbert lattice has the exchange property. (Contributed by NM, 16-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) ↔ (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄)))
Theorem	hlexchb2 39342	A Hilbert lattice has the exchange property. (Contributed by NM, 22-Jun-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑄 ∨ 𝑋) ↔ (𝑃 ∨ 𝑋) = (𝑄 ∨ 𝑋)))
Theorem	hlsupr 39343*	A Hilbert lattice has the superposition property. Theorem 13.2 in [Crawley] p. 107. (Contributed by NM, 30-Jan-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → ∃𝑟 ∈ 𝐴 (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄)))
Theorem	hlsupr2 39344*	A Hilbert lattice has the superposition property. (Contributed by NM, 25-Nov-2012.)
⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ∃𝑟 ∈ 𝐴 (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟))
Theorem	hlhgt4 39345*	A Hilbert lattice has a height of at least 4. (Contributed by NM, 4-Dec-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 1 = (1.‘𝐾)    ⇒   ⊢ (𝐾 ∈ HL → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (( 0 < 𝑥 ∧ 𝑥 < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 )))
Theorem	hlhgt2 39346*	A Hilbert lattice has a height of at least 2. (Contributed by NM, 4-Dec-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 1 = (1.‘𝐾)    ⇒   ⊢ (𝐾 ∈ HL → ∃𝑥 ∈ 𝐵 ( 0 < 𝑥 ∧ 𝑥 < 1 ))
Theorem	hl0lt1N 39347	Lattice 0 is less than lattice 1 in a Hilbert lattice. (Contributed by NM, 4-Dec-2011.) (New usage is discouraged.)
⊢ < = (lt‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 1 = (1.‘𝐾)    ⇒   ⊢ (𝐾 ∈ HL → 0 < 1 )
Theorem	hlexch3 39348	A Hilbert lattice has the exchange property. (atexch 32413 analog.) (Contributed by NM, 15-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∧ 𝑋) = 0 ) → (𝑃 ≤ (𝑋 ∨ 𝑄) → 𝑄 ≤ (𝑋 ∨ 𝑃)))
Theorem	hlexch4N 39349	A Hilbert lattice has the exchange property. Part of Definition 7.8 of [MaedaMaeda] p. 32. (Contributed by NM, 15-Nov-2011.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∧ 𝑋) = 0 ) → (𝑃 ≤ (𝑋 ∨ 𝑄) ↔ (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄)))
Theorem	hlatexchb1 39350	A version of hlexchb1 39341 for atoms. (Contributed by NM, 15-Nov-2011.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑅) → (𝑃 ≤ (𝑅 ∨ 𝑄) ↔ (𝑅 ∨ 𝑃) = (𝑅 ∨ 𝑄)))
Theorem	hlatexchb2 39351	A version of hlexchb2 39342 for atoms. (Contributed by NM, 7-Feb-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑅) → (𝑃 ≤ (𝑄 ∨ 𝑅) ↔ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)))
Theorem	hlatexch1 39352	Atom exchange property. (Contributed by NM, 7-Jan-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑅) → (𝑃 ≤ (𝑅 ∨ 𝑄) → 𝑄 ≤ (𝑅 ∨ 𝑃)))
Theorem	hlatexch2 39353	Atom exchange property. (Contributed by NM, 8-Jan-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑅) → (𝑃 ≤ (𝑄 ∨ 𝑅) → 𝑄 ≤ (𝑃 ∨ 𝑅)))
Theorem	hlatmstcOLDN 39354*	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 32394 analog.) (Contributed by NM, 21-Oct-2011.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝑈 = (lub‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑈‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}) = 𝑋)
Theorem	hlatle 39355*	The ordering of two Hilbert lattice elements is determined by the atoms under them. (chrelat3 32403 analog.) (Contributed by NM, 4-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ↔ ∀𝑝 ∈ 𝐴 (𝑝 ≤ 𝑋 → 𝑝 ≤ 𝑌)))
Theorem	hlateq 39356*	The equality of two Hilbert lattice elements is determined by the atoms under them. (chrelat4i 32405 analog.) (Contributed by NM, 24-May-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑝 ∈ 𝐴 (𝑝 ≤ 𝑋 ↔ 𝑝 ≤ 𝑌) ↔ 𝑋 = 𝑌))
Theorem	hlrelat1 39357*	An atomistic lattice with 0 is relatively atomic. Part of Lemma 7.2 of [MaedaMaeda] p. 30. (chpssati 32395, with ∧ swapped, analog.) (Contributed by NM, 4-Dec-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 → ∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌)))
Theorem	hlrelat5N 39358*	An atomistic lattice with 0 is relatively atomic, using the definition in Remark 2 of [Kalmbach] p. 149. (Contributed by NM, 21-Oct-2011.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) → ∃𝑝 ∈ 𝐴 (𝑋 < (𝑋 ∨ 𝑝) ∧ 𝑝 ≤ 𝑌))
Theorem	hlrelat 39359*	A Hilbert lattice is relatively atomic. Remark 2 of [Kalmbach] p. 149. (chrelati 32396 analog.) (Contributed by NM, 4-Feb-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) → ∃𝑝 ∈ 𝐴 (𝑋 < (𝑋 ∨ 𝑝) ∧ (𝑋 ∨ 𝑝) ≤ 𝑌))
Theorem	hlrelat2 39360*	A consequence of relative atomicity. (chrelat2i 32397 analog.) (Contributed by NM, 5-Feb-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (¬ 𝑋 ≤ 𝑌 ↔ ∃𝑝 ∈ 𝐴 (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑌)))
Theorem	exatleN 39361	A condition for an atom to be less than or equal to a lattice element. Part of proof of Lemma A in [Crawley] p. 112. (Contributed by NM, 28-Apr-2012.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝑅 ≤ 𝑋 ↔ 𝑅 = 𝑃))
Theorem	hl2at 39362*	A Hilbert lattice has at least 2 atoms. (Contributed by NM, 5-Dec-2011.)
⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (𝐾 ∈ HL → ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 𝑝 ≠ 𝑞)
Theorem	atex 39363	At least one atom exists. (Contributed by NM, 15-Jul-2012.)
⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (𝐾 ∈ HL → 𝐴 ≠ ∅)
Theorem	intnatN 39364	If the intersection with a non-majorizing element is an atom, the intersecting element is not an atom. (Contributed by NM, 26-Jun-2012.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (¬ 𝑌 ≤ 𝑋 ∧ (𝑋 ∧ 𝑌) ∈ 𝐴)) → ¬ 𝑌 ∈ 𝐴)
Theorem	2llnne2N 39365	Condition implying that two intersecting lines are different. (Contributed by NM, 13-Jun-2012.) (New usage is discouraged.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ¬ 𝑃 ≤ (𝑅 ∨ 𝑄)) → (𝑅 ∨ 𝑃) ≠ (𝑅 ∨ 𝑄))
Theorem	2llnneN 39366	Condition implying that two intersecting lines are different. (Contributed by NM, 29-May-2012.) (New usage is discouraged.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝑅 ∨ 𝑃) ≠ (𝑅 ∨ 𝑄))
Theorem	cvr1 39367	A Hilbert lattice has the covering property. Proposition 1(ii) in [Kalmbach] p. 140 (and its converse). (chcv1 32387 analog.) (Contributed by NM, 17-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (¬ 𝑃 ≤ 𝑋 ↔ 𝑋𝐶(𝑋 ∨ 𝑃)))
Theorem	cvr2N 39368	Less-than and covers equivalence in a Hilbert lattice. (chcv2 32388 analog.) (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (𝑋 < (𝑋 ∨ 𝑃) ↔ 𝑋𝐶(𝑋 ∨ 𝑃)))
Theorem	hlrelat3 39369*	The Hilbert lattice is relatively atomic. Stronger version of hlrelat 39359. (Contributed by NM, 2-May-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) → ∃𝑝 ∈ 𝐴 (𝑋𝐶(𝑋 ∨ 𝑝) ∧ (𝑋 ∨ 𝑝) ≤ 𝑌))
Theorem	cvrval3 39370*	Binary relation expressing 𝑌 covers 𝑋. (Contributed by NM, 16-Jun-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ ∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑋 ∧ (𝑋 ∨ 𝑝) = 𝑌)))
Theorem	cvrval4N 39371*	Binary relation expressing 𝑌 covers 𝑋. (Contributed by NM, 16-Jun-2012.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 < 𝑌 ∧ ∃𝑝 ∈ 𝐴 (𝑋 ∨ 𝑝) = 𝑌)))
Theorem	cvrval5 39372*	Binary relation expressing 𝑋 covers 𝑋 ∧ 𝑌. (Contributed by NM, 7-Dec-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ 𝑌)𝐶𝑋 ↔ ∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑌 ∧ (𝑝 ∨ (𝑋 ∧ 𝑌)) = 𝑋)))
Theorem	cvrp 39373	A Hilbert lattice satisfies the covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 32407 analog.) (Contributed by NM, 18-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → ((𝑋 ∧ 𝑃) = 0 ↔ 𝑋𝐶(𝑋 ∨ 𝑃)))
Theorem	atcvr1 39374	An atom is covered by its join with a different atom. (Contributed by NM, 7-Feb-2012.)
⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ≠ 𝑄 ↔ 𝑃𝐶(𝑃 ∨ 𝑄)))
Theorem	atcvr2 39375	An atom is covered by its join with a different atom. (Contributed by NM, 7-Feb-2012.)
⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ≠ 𝑄 ↔ 𝑃𝐶(𝑄 ∨ 𝑃)))
Theorem	cvrexchlem 39376	Lemma for cvrexch 39377. (cvexchlem 32400 analog.) (Contributed by NM, 18-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ 𝑌)𝐶𝑌 → 𝑋𝐶(𝑋 ∨ 𝑌)))
Theorem	cvrexch 39377	A Hilbert lattice satisfies the exchange axiom. Proposition 1(iii) of [Kalmbach] p. 140 and its converse. Originally proved by Garrett Birkhoff in 1933. (cvexchi 32401 analog.) (Contributed by NM, 18-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ 𝑌)𝐶𝑌 ↔ 𝑋𝐶(𝑋 ∨ 𝑌)))
Theorem	cvratlem 39378	Lemma for cvrat 39379. (atcvatlem 32417 analog.) (Contributed by NM, 22-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ (𝑋 ≠ 0 ∧ 𝑋 < (𝑃 ∨ 𝑄))) → (¬ 𝑃(le‘𝐾)𝑋 → 𝑋 ∈ 𝐴))
Theorem	cvrat 39379	A nonzero Hilbert lattice element less than the join of two atoms is an atom. (atcvati 32418 analog.) (Contributed by NM, 22-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑋 ≠ 0 ∧ 𝑋 < (𝑃 ∨ 𝑄)) → 𝑋 ∈ 𝐴))
Theorem	ltltncvr 39380	A chained strong ordering is not a covers relation. (Contributed by NM, 18-Jun-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    ⇒   ⊢ ((𝐾 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 < 𝑌 ∧ 𝑌 < 𝑍) → ¬ 𝑋𝐶𝑍))
Theorem	ltcvrntr 39381	Non-transitive condition for the covers relation. (Contributed by NM, 18-Jun-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    ⇒   ⊢ ((𝐾 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 < 𝑌 ∧ 𝑌𝐶𝑍) → ¬ 𝑋𝐶𝑍))
Theorem	cvrntr 39382	The covers relation is not transitive. (cvntr 32324 analog.) (Contributed by NM, 18-Jun-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    ⇒   ⊢ ((𝐾 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋𝐶𝑌 ∧ 𝑌𝐶𝑍) → ¬ 𝑋𝐶𝑍))
Theorem	atcvr0eq 39383	The covers relation is not transitive. (atcv0eq 32411 analog.) (Contributed by NM, 29-Nov-2011.)
⊢ ∨ = (join‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ( 0 𝐶(𝑃 ∨ 𝑄) ↔ 𝑃 = 𝑄))
Theorem	lnnat 39384	A line (the join of two distinct atoms) is not an atom. (Contributed by NM, 14-Jun-2012.)
⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ≠ 𝑄 ↔ ¬ (𝑃 ∨ 𝑄) ∈ 𝐴))
Theorem	atcvrj0 39385	Two atoms covering the zero subspace are equal. (atcv1 32412 analog.) (Contributed by NM, 29-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋𝐶(𝑃 ∨ 𝑄)) → (𝑋 = 0 ↔ 𝑃 = 𝑄))
Theorem	cvrat2 39386	A Hilbert lattice element covered by the join of two distinct atoms is an atom. (atcvat2i 32419 analog.) (Contributed by NM, 30-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ 𝑋𝐶(𝑃 ∨ 𝑄))) → 𝑋 ∈ 𝐴)
Theorem	atcvrneN 39387	Inequality derived from atom condition. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)
⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑄 ≠ 𝑅)
Theorem	atcvrj1 39388	Condition for an atom to be covered by the join of two others. (Contributed by NM, 7-Feb-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) → 𝑃𝐶(𝑄 ∨ 𝑅))
Theorem	atcvrj2b 39389	Condition for an atom to be covered by the join of two others. (Contributed by NM, 7-Feb-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ((𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅)) ↔ 𝑃𝐶(𝑄 ∨ 𝑅)))
Theorem	atcvrj2 39390	Condition for an atom to be covered by the join of two others. (Contributed by NM, 7-Feb-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) → 𝑃𝐶(𝑄 ∨ 𝑅))
Theorem	atleneN 39391	Inequality derived from atom condition. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) → 𝑄 ≠ 𝑅)
Theorem	atltcvr 39392	An equivalence of less-than ordering and covers relation. (Contributed by NM, 7-Feb-2012.)
⊢ < = (lt‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (𝑃 < (𝑄 ∨ 𝑅) ↔ 𝑃𝐶(𝑄 ∨ 𝑅)))
Theorem	atle 39393*	Any nonzero element has an atom under it. (Contributed by NM, 28-Jun-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ∃𝑝 ∈ 𝐴 𝑝 ≤ 𝑋)
Theorem	atlt 39394	Two atoms are unequal iff their join is greater than one of them. (Contributed by NM, 6-May-2012.)
⊢ < = (lt‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 < (𝑃 ∨ 𝑄) ↔ 𝑃 ≠ 𝑄))
Theorem	atlelt 39395	Transfer less-than relation from one atom to another. (Contributed by NM, 7-May-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋)) → 𝑃 < 𝑋)
Theorem	2atlt 39396*	Given an atom less than an element, there is another atom less than the element. (Contributed by NM, 6-May-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑃 < 𝑋) → ∃𝑞 ∈ 𝐴 (𝑞 ≠ 𝑃 ∧ 𝑞 < 𝑋))
Theorem	atexchcvrN 39397	Atom exchange property. Version of hlatexch2 39353 with covers relation. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)
⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑅) → (𝑃𝐶(𝑄 ∨ 𝑅) → 𝑄𝐶(𝑃 ∨ 𝑅)))
Theorem	atexchltN 39398	Atom exchange property. Version of hlatexch2 39353 with less-than ordering. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)
⊢ < = (lt‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑅) → (𝑃 < (𝑄 ∨ 𝑅) → 𝑄 < (𝑃 ∨ 𝑅)))
Theorem	cvrat3 39399	A condition implying that a certain lattice element is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 32428 analog.) (Contributed by NM, 30-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴))
Theorem	cvrat4 39400*	A condition implying existence of an atom with the properties shown. Lemma 3.2.20 in [PtakPulmannova] p. 68. Also Lemma 9.2(delta) in [MaedaMaeda] p. 41. (atcvat4i 32429 analog.) (Contributed by NM, 30-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑋 ≠ 0 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → ∃𝑟 ∈ 𝐴 (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟))))