P. 395 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 39401-39500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	hlateq 39401*	The equality of two Hilbert lattice elements is determined by the atoms under them. (chrelat4i 32392 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 39402*	An atomistic lattice with 0 is relatively atomic. Part of Lemma 7.2 of [MaedaMaeda] p. 30. (chpssati 32382, 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 39403*	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 39404*	A Hilbert lattice is relatively atomic. Remark 2 of [Kalmbach] p. 149. (chrelati 32383 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 39405*	A consequence of relative atomicity. (chrelat2i 32384 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 39406	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 39407*	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 39408	At least one atom exists. (Contributed by NM, 15-Jul-2012.)
⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (𝐾 ∈ HL → 𝐴 ≠ ∅)
Theorem	intnatN 39409	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 39410	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 39411	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 39412	A Hilbert lattice has the covering property. Proposition 1(ii) in [Kalmbach] p. 140 (and its converse). (chcv1 32374 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 39413	Less-than and covers equivalence in a Hilbert lattice. (chcv2 32375 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 39414*	The Hilbert lattice is relatively atomic. Stronger version of hlrelat 39404. (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 39415*	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 39416*	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 39417*	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 39418	A Hilbert lattice satisfies the covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 32394 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 39419	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 39420	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 39421	Lemma for cvrexch 39422. (cvexchlem 32387 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 39422	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 32388 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 39423	Lemma for cvrat 39424. (atcvatlem 32404 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 39424	A nonzero Hilbert lattice element less than the join of two atoms is an atom. (atcvati 32405 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 39425	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 39426	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 39427	The covers relation is not transitive. (cvntr 32311 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 39428	The covers relation is not transitive. (atcv0eq 32398 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 39429	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 39430	Two atoms covering the zero subspace are equal. (atcv1 32399 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 39431	A Hilbert lattice element covered by the join of two distinct atoms is an atom. (atcvat2i 32406 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 39432	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 39433	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 39434	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 39435	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 39436	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 39437	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 39438*	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 39439	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 39440	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 39441*	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 39442	Atom exchange property. Version of hlatexch2 39398 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 39443	Atom exchange property. Version of hlatexch2 39398 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 39444	A condition implying that a certain lattice element is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 32415 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 39445*	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 32416 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))))
Theorem	cvrat42 39446*	Commuted version of cvrat4 39445. (Contributed by NM, 28-Jan-2012.)
⊢ 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 ≤ (𝑟 ∨ 𝑄))))
Theorem	2atjm 39447	The meet of a line (expressed with 2 atoms) and a lattice element. (Contributed by NM, 30-Jul-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → ((𝑃 ∨ 𝑄) ∧ 𝑋) = 𝑃)
Theorem	atbtwn 39448	Property of a 3rd atom 𝑅 on a line 𝑃 ∨ 𝑄 intersecting element 𝑋 at 𝑃. (Contributed by NM, 30-Jul-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝑅 ≠ 𝑃 ↔ ¬ 𝑅 ≤ 𝑋))
Theorem	atbtwnexOLDN 39449*	There exists a 3rd atom 𝑟 on a line 𝑃 ∨ 𝑄 intersecting element 𝑋 at 𝑃, such that 𝑟 is different from 𝑄 and not in 𝑋. (Contributed by NM, 30-Jul-2012.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → ∃𝑟 ∈ 𝐴 (𝑟 ≠ 𝑄 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄)))
Theorem	atbtwnex 39450*	Given atoms 𝑃 in 𝑋 and 𝑄 not in 𝑋, there exists an atom 𝑟 not in 𝑋 such that the line 𝑄 ∨ 𝑟 intersects 𝑋 at 𝑃. (Contributed by NM, 1-Aug-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → ∃𝑟 ∈ 𝐴 (𝑟 ≠ 𝑄 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟)))
Theorem	3noncolr2 39451	Two ways to express 3 non-colinear atoms (rotated right 2 places). (Contributed by NM, 12-Jul-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝑄 ≠ 𝑅 ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅)))
Theorem	3noncolr1N 39452	Two ways to express 3 non-colinear atoms (rotated right 1 place). (Contributed by NM, 12-Jul-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 ∧ ¬ 𝑄 ≤ (𝑅 ∨ 𝑃)))
Theorem	hlatcon3 39453	Atom exchange combined with contraposition. (Contributed by NM, 13-Jun-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ¬ 𝑃 ≤ (𝑄 ∨ 𝑅))
Theorem	hlatcon2 39454	Atom exchange combined with contraposition. (Contributed by NM, 13-Jun-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ¬ 𝑃 ≤ (𝑅 ∨ 𝑄))
Theorem	4noncolr3 39455	A way to express 4 non-colinear atoms (rotated right 3 places). (Contributed by NM, 11-Jul-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → (𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆)))
Theorem	4noncolr2 39456	A way to express 4 non-colinear atoms (rotated right 2 places). (Contributed by NM, 11-Jul-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → (𝑅 ≠ 𝑆 ∧ ¬ 𝑃 ≤ (𝑅 ∨ 𝑆) ∧ ¬ 𝑄 ≤ ((𝑅 ∨ 𝑆) ∨ 𝑃)))
Theorem	4noncolr1 39457	A way to express 4 non-colinear atoms (rotated right 1 places). (Contributed by NM, 11-Jul-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → (𝑆 ≠ 𝑃 ∧ ¬ 𝑄 ≤ (𝑆 ∨ 𝑃) ∧ ¬ 𝑅 ≤ ((𝑆 ∨ 𝑃) ∨ 𝑄)))
Theorem	athgt 39458*	A Hilbert lattice, whose height is at least 4, has a chain of 4 successively covering atom joins. (Contributed by NM, 3-May-2012.)
⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (𝐾 ∈ HL → ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 (𝑝𝐶(𝑝 ∨ 𝑞) ∧ ∃𝑟 ∈ 𝐴 ((𝑝 ∨ 𝑞)𝐶((𝑝 ∨ 𝑞) ∨ 𝑟) ∧ ∃𝑠 ∈ 𝐴 ((𝑝 ∨ 𝑞) ∨ 𝑟)𝐶(((𝑝 ∨ 𝑞) ∨ 𝑟) ∨ 𝑠))))
Theorem	3dim0 39459*	There exists a 3-dimensional (height-4) element i.e. a volume. (Contributed by NM, 25-Jul-2012.)
⊢ ∨ = (join‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (𝐾 ∈ HL → ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝 ∨ 𝑞) ∧ ¬ 𝑠 ≤ ((𝑝 ∨ 𝑞) ∨ 𝑟)))
Theorem	3dimlem1 39460	Lemma for 3dim1 39469. (Contributed by NM, 25-Jul-2012.)
⊢ ∨ = (join‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆)) ∧ 𝑃 = 𝑄) → (𝑃 ≠ 𝑅 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑅) ∧ ¬ 𝑇 ≤ ((𝑃 ∨ 𝑅) ∨ 𝑆)))
Theorem	3dimlem2 39461	Lemma for 3dim1 39469. (Contributed by NM, 25-Jul-2012.)
⊢ ∨ = (join‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑆 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) → (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑆)))
Theorem	3dimlem3a 39462	Lemma for 3dim3 39471. (Contributed by NM, 27-Jul-2012.)
⊢ ∨ = (join‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → ¬ 𝑇 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
Theorem	3dimlem3 39463	Lemma for 3dim1 39469. (Contributed by NM, 25-Jul-2012.)
⊢ ∨ = (join‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ ¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)))
Theorem	3dimlem3OLDN 39464	Lemma for 3dim1 39469. (Contributed by NM, 25-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∨ = (join‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ ¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)))
Theorem	3dimlem4a 39465	Lemma for 3dim3 39471. (Contributed by NM, 27-Jul-2012.)
⊢ ∨ = (join‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (¬ 𝑆 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
Theorem	3dimlem4 39466	Lemma for 3dim1 39469. (Contributed by NM, 25-Jul-2012.)
⊢ ∨ = (join‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ≤ (𝑄 ∨ 𝑅))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅)) ∧ ¬ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆)) → (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)))
Theorem	3dimlem4OLDN 39467	Lemma for 3dim1 39469. (Contributed by NM, 25-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∨ = (join‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ≤ (𝑄 ∨ 𝑅))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅)) ∧ ¬ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆)) → (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)))
Theorem	3dim1lem5 39468*	Lemma for 3dim1 39469. (Contributed by NM, 26-Jul-2012.)
⊢ ∨ = (join‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑃 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑃 ∨ 𝑢) ∨ 𝑣))) → ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (𝑃 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑞) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑞) ∨ 𝑟)))
Theorem	3dim1 39469*	Construct a 3-dimensional volume (height-4 element) on top of a given atom 𝑃. (Contributed by NM, 25-Jul-2012.)
⊢ ∨ = (join‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) → ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (𝑃 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑞) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑞) ∨ 𝑟)))
Theorem	3dim2 39470*	Construct 2 new layers on top of 2 given atoms. (Contributed by NM, 27-Jul-2012.)
⊢ ∨ = (join‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)))
Theorem	3dim3 39471*	Construct a new layer on top of 3 given atoms. (Contributed by NM, 27-Jul-2012.)
⊢ ∨ = (join‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ∃𝑠 ∈ 𝐴 ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
Theorem	2dim 39472*	Generate a height-3 element (2-dimensional plane) from an atom. (Contributed by NM, 3-May-2012.)
⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) → ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑃𝐶(𝑃 ∨ 𝑞) ∧ (𝑃 ∨ 𝑞)𝐶((𝑃 ∨ 𝑞) ∨ 𝑟)))
Theorem	1dimN 39473*	An atom is covered by a height-2 element (1-dimensional line). (Contributed by NM, 3-May-2012.) (New usage is discouraged.)
⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) → ∃𝑞 ∈ 𝐴 𝑃𝐶(𝑃 ∨ 𝑞))
Theorem	1cvrco 39474	The orthocomplement of an element covered by 1 is an atom. (Contributed by NM, 7-May-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 1 = (1.‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑋𝐶 1 ↔ ( ⊥ ‘𝑋) ∈ 𝐴))
Theorem	1cvratex 39475*	There exists an atom less than an element covered by 1. (Contributed by NM, 7-May-2012.) (Revised by Mario Carneiro, 13-Jun-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ 1 = (1.‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑋𝐶 1 ) → ∃𝑝 ∈ 𝐴 𝑝 < 𝑋)
Theorem	1cvratlt 39476	An atom less than or equal to an element covered by 1 is less than the element. (Contributed by NM, 7-May-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ 1 = (1.‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑋𝐶 1 ∧ 𝑃 ≤ 𝑋)) → 𝑃 < 𝑋)
Theorem	1cvrjat 39477	An element covered by the lattice unity, when joined with an atom not under it, equals the lattice unity. (Contributed by NM, 30-Apr-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 1 = (1.‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋)) → (𝑋 ∨ 𝑃) = 1 )
Theorem	1cvrat 39478	Create an atom under an element covered by the lattice unity. Part of proof of Lemma B in [Crawley] p. 112. (Contributed by NM, 30-Apr-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 1 = (1.‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≠ 𝑄 ∧ 𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋)) → ((𝑃 ∨ 𝑄) ∧ 𝑋) ∈ 𝐴)
Theorem	ps-1 39479	The join of two atoms 𝑅 ∨ 𝑆 (specifying a projective geometry line) is determined uniquely by any two atoms (specifying two points) less than or equal to that join. Part of Lemma 16.4 of [MaedaMaeda] p. 69, showing projective space postulate PS1 in [MaedaMaeda] p. 67. (Contributed by NM, 15-Nov-2011.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ≤ (𝑅 ∨ 𝑆) ↔ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆)))
Theorem	ps-2 39480*	Lattice analogue for the projective geometry axiom, "if a line intersects two sides of a triangle at different points then it also intersects the third side." Projective space condition PS2 in [MaedaMaeda] p. 68 and part of Theorem 16.4 in [MaedaMaeda] p. 69. (Contributed by NM, 1-Dec-2011.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇) ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) → ∃𝑢 ∈ 𝐴 (𝑢 ≤ (𝑃 ∨ 𝑅) ∧ 𝑢 ≤ (𝑆 ∨ 𝑇)))
Theorem	2atjlej 39481	Two atoms are different if their join majorizes the join of two different atoms. (Contributed by NM, 4-Jun-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ (𝑃 ∨ 𝑄) ≤ (𝑅 ∨ 𝑆))) → 𝑅 ≠ 𝑆)
Theorem	hlatexch3N 39482	Rearrange join of atoms in an equality. (Contributed by NM, 29-Jul-2013.) (New usage is discouraged.)
⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑅))) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑅))
Theorem	hlatexch4 39483	Exchange 2 atoms. (Contributed by NM, 13-May-2013.)
⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑅 ∧ 𝑄 ≠ 𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) → (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑆))
Theorem	ps-2b 39484	Variation of projective geometry axiom ps-2 39480. (Contributed by NM, 3-Jul-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇 ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) → ((𝑃 ∨ 𝑅) ∧ (𝑆 ∨ 𝑇)) ≠ 0 )
Theorem	3atlem1 39485	Lemma for 3at 39492. (Contributed by NM, 22-Jun-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑆 ∨ 𝑇) ∨ 𝑈))
Theorem	3atlem2 39486	Lemma for 3at 39492. (Contributed by NM, 22-Jun-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑆 ∨ 𝑇) ∨ 𝑈))
Theorem	3atlem3 39487	Lemma for 3at 39492. (Contributed by NM, 23-Jun-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≠ 𝑈 ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑆 ∨ 𝑇) ∨ 𝑈))
Theorem	3atlem4 39488	Lemma for 3at 39492. (Contributed by NM, 23-Jun-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≠ 𝑄) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑅)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑆 ∨ 𝑇) ∨ 𝑅))
Theorem	3atlem5 39489	Lemma for 3at 39492. (Contributed by NM, 23-Jun-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑆 ∨ 𝑇) ∨ 𝑈))
Theorem	3atlem6 39490	Lemma for 3at 39492. (Contributed by NM, 23-Jun-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≠ 𝑄 ∧ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑆 ∨ 𝑇) ∨ 𝑈))
Theorem	3atlem7 39491	Lemma for 3at 39492. (Contributed by NM, 23-Jun-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≠ 𝑄) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑆 ∨ 𝑇) ∨ 𝑈))
Theorem	3at 39492	Any three non-colinear atoms in a (lattice) plane determine the plane uniquely. This is the 2-dimensional analogue of ps-1 39479 for lines and 4at 39615 for volumes. I could not find this proof in the literature on projective geometry (where it is either given as an axiom or stated as an unproved fact), but it is similar to Theorem 15 of Veblen, "The Foundations of Geometry" (1911), p. 18, which uses different axioms. This proof was written before I became aware of Veblen's, and it is possible that a shorter proof could be obtained by using Veblen's proof for hints. (Contributed by NM, 23-Jun-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≠ 𝑄)) → (((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈) ↔ ((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑆 ∨ 𝑇) ∨ 𝑈)))
21.28.12  Projective geometries based on Hilbert lattices
Syntax	clln 39493	Extend class notation with set of all "lattice lines" (lattice elements which cover an atom) in a Hilbert lattice.
class LLines
Syntax	clpl 39494	Extend class notation with set of all "lattice planes" (lattice elements which cover a line) in a Hilbert lattice.
class LPlanes
Syntax	clvol 39495	Extend class notation with set of all 3-dimensional "lattice volumes" (lattice elements which cover a plane) in a Hilbert lattice.
class LVols
Syntax	clines 39496	Extend class notation with set of all projective lines for a Hilbert lattice.
class Lines
Syntax	cpointsN 39497	Extend class notation with set of all projective points.
class Points
Syntax	cpsubsp 39498	Extend class notation with set of all projective subspaces.
class PSubSp
Syntax	cpmap 39499	Extend class notation with projective map.
class pmap
Definition	df-llines 39500*	Define the set of all "lattice lines" (lattice elements which cover an atom) in a Hilbert lattice 𝑘, in other words all elements of height 2 (or lattice dimension 2 or projective dimension 1). (Contributed by NM, 16-Jun-2012.)
⊢ LLines = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (Atoms‘𝑘)𝑝( ⋖ ‘𝑘)𝑥})