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Type | Label | Description |
---|---|---|
Statement | ||
Syntax | catm 36401 | Extend class notation with atoms. |
class Atoms | ||
Syntax | cal 36402 | Extend class notation with atomic lattices. |
class AtLat | ||
Syntax | clc 36403 | Extend class notation with lattices with the covering property. |
class CvLat | ||
Definition | df-covers 36404* | Define the covers relation ("is covered by") for posets. "𝑎 is covered by 𝑏 " means that 𝑎 is strictly less than 𝑏 and there is nothing in between. See cvrval 36407 for the relation form. (Contributed by NM, 18-Sep-2011.) |
⊢ ⋖ = (𝑝 ∈ V ↦ {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (Base‘𝑝) ∧ 𝑏 ∈ (Base‘𝑝)) ∧ 𝑎(lt‘𝑝)𝑏 ∧ ¬ ∃𝑧 ∈ (Base‘𝑝)(𝑎(lt‘𝑝)𝑧 ∧ 𝑧(lt‘𝑝)𝑏))}) | ||
Definition | df-ats 36405* | Define the class of poset atoms. (Contributed by NM, 18-Sep-2011.) |
⊢ Atoms = (𝑝 ∈ V ↦ {𝑎 ∈ (Base‘𝑝) ∣ (0.‘𝑝)( ⋖ ‘𝑝)𝑎}) | ||
Theorem | cvrfval 36406* | Value of covers relation "is covered by". (Contributed by NM, 18-Sep-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ < = (lt‘𝐾) & ⊢ 𝐶 = ( ⋖ ‘𝐾) ⇒ ⊢ (𝐾 ∈ 𝐴 → 𝐶 = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧 ∈ 𝐵 (𝑥 < 𝑧 ∧ 𝑧 < 𝑦))}) | ||
Theorem | cvrval 36407* | Binary relation expressing 𝐵 covers 𝐴, which means that 𝐵 is larger than 𝐴 and there is nothing in between. Definition 3.2.18 of [PtakPulmannova] p. 68. (cvbr 30061 analog.) (Contributed by NM, 18-Sep-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ < = (lt‘𝐾) & ⊢ 𝐶 = ( ⋖ ‘𝐾) ⇒ ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 < 𝑌 ∧ ¬ ∃𝑧 ∈ 𝐵 (𝑋 < 𝑧 ∧ 𝑧 < 𝑌)))) | ||
Theorem | cvrlt 36408 | The covers relation implies the less-than relation. (cvpss 30064 analog.) (Contributed by NM, 8-Oct-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ < = (lt‘𝐾) & ⊢ 𝐶 = ( ⋖ ‘𝐾) ⇒ ⊢ (((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌) | ||
Theorem | cvrnbtwn 36409 | There is no element between the two arguments of the covers relation. (cvnbtwn 30065 analog.) (Contributed by NM, 18-Oct-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ < = (lt‘𝐾) & ⊢ 𝐶 = ( ⋖ ‘𝐾) ⇒ ⊢ ((𝐾 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ¬ (𝑋 < 𝑍 ∧ 𝑍 < 𝑌)) | ||
Theorem | ncvr1 36410 | No element covers the lattice unit. (Contributed by NM, 8-Jul-2013.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ 1 = (1.‘𝐾) & ⊢ 𝐶 = ( ⋖ ‘𝐾) ⇒ ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ¬ 1 𝐶𝑋) | ||
Theorem | cvrletrN 36411 | Property of an element above a covering. (Contributed by NM, 7-Dec-2012.) (New usage is discouraged.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ < = (lt‘𝐾) & ⊢ 𝐶 = ( ⋖ ‘𝐾) ⇒ ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋𝐶𝑌 ∧ 𝑌 ≤ 𝑍) → 𝑋 < 𝑍)) | ||
Theorem | cvrval2 36412* | Binary relation expressing 𝑌 covers 𝑋. Definition of covers in [Kalmbach] p. 15. (cvbr2 30062 analog.) (Contributed by NM, 16-Nov-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ < = (lt‘𝐾) & ⊢ 𝐶 = ( ⋖ ‘𝐾) ⇒ ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 < 𝑌 ∧ ∀𝑧 ∈ 𝐵 ((𝑋 < 𝑧 ∧ 𝑧 ≤ 𝑌) → 𝑧 = 𝑌)))) | ||
Theorem | cvrnbtwn2 36413 | The covers relation implies no in-betweenness. (cvnbtwn2 30066 analog.) (Contributed by NM, 17-Nov-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ < = (lt‘𝐾) & ⊢ 𝐶 = ( ⋖ ‘𝐾) ⇒ ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌) ↔ 𝑍 = 𝑌)) | ||
Theorem | cvrnbtwn3 36414 | The covers relation implies no in-betweenness. (cvnbtwn3 30067 analog.) (Contributed by NM, 4-Nov-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ < = (lt‘𝐾) & ⊢ 𝐶 = ( ⋖ ‘𝐾) ⇒ ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) ↔ 𝑋 = 𝑍)) | ||
Theorem | cvrcon3b 36415 | Contraposition law for the covers relation. (cvcon3 30063 analog.) (Contributed by NM, 4-Nov-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ⊥ = (oc‘𝐾) & ⊢ 𝐶 = ( ⋖ ‘𝐾) ⇒ ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ ( ⊥ ‘𝑌)𝐶( ⊥ ‘𝑋))) | ||
Theorem | cvrle 36416 | The covers relation implies the "less than or equal to" relation. (Contributed by NM, 12-Oct-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐶 = ( ⋖ ‘𝐾) ⇒ ⊢ (((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 ≤ 𝑌) | ||
Theorem | cvrnbtwn4 36417 | The covers relation implies no in-betweenness. Part of proof of Lemma 7.5.1 of [MaedaMaeda] p. 31. (cvnbtwn4 30068 analog.) (Contributed by NM, 18-Oct-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐶 = ( ⋖ ‘𝐾) ⇒ ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 ≤ 𝑍 ∧ 𝑍 ≤ 𝑌) ↔ (𝑋 = 𝑍 ∨ 𝑍 = 𝑌))) | ||
Theorem | cvrnle 36418 | The covers relation implies the negation of the converse "less than or equal to" relation. (Contributed by NM, 18-Oct-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐶 = ( ⋖ ‘𝐾) ⇒ ⊢ (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ¬ 𝑌 ≤ 𝑋) | ||
Theorem | cvrne 36419 | The covers relation implies inequality. (Contributed by NM, 13-Oct-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝐶 = ( ⋖ ‘𝐾) ⇒ ⊢ (((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 ≠ 𝑌) | ||
Theorem | cvrnrefN 36420 | The covers relation is not reflexive. (cvnref 30070 analog.) (Contributed by NM, 1-Nov-2012.) (New usage is discouraged.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝐶 = ( ⋖ ‘𝐾) ⇒ ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → ¬ 𝑋𝐶𝑋) | ||
Theorem | cvrcmp 36421 | If two lattice elements that cover a third are comparable, then they are equal. (Contributed by NM, 6-Feb-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐶 = ( ⋖ ‘𝐾) ⇒ ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) → (𝑋 ≤ 𝑌 ↔ 𝑋 = 𝑌)) | ||
Theorem | cvrcmp2 36422 | If two lattice elements covered by a third are comparable, then they are equal. (Contributed by NM, 20-Jun-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐶 = ( ⋖ ‘𝐾) ⇒ ⊢ ((𝐾 ∈ OP ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋𝐶𝑍 ∧ 𝑌𝐶𝑍)) → (𝑋 ≤ 𝑌 ↔ 𝑋 = 𝑌)) | ||
Theorem | pats 36423* | The set of atoms in a poset. (Contributed by NM, 18-Sep-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ 0 = (0.‘𝐾) & ⊢ 𝐶 = ( ⋖ ‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ (𝐾 ∈ 𝐷 → 𝐴 = {𝑥 ∈ 𝐵 ∣ 0 𝐶𝑥}) | ||
Theorem | isat 36424 | The predicate "is an atom". (ela 30118 analog.) (Contributed by NM, 18-Sep-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ 0 = (0.‘𝐾) & ⊢ 𝐶 = ( ⋖ ‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ (𝐾 ∈ 𝐷 → (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ 𝐵 ∧ 0 𝐶𝑃))) | ||
Theorem | isat2 36425 | The predicate "is an atom". (elatcv0 30120 analog.) (Contributed by NM, 18-Jun-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ 0 = (0.‘𝐾) & ⊢ 𝐶 = ( ⋖ ‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑃 ∈ 𝐵) → (𝑃 ∈ 𝐴 ↔ 0 𝐶𝑃)) | ||
Theorem | atcvr0 36426 | An atom covers zero. (atcv0 30121 analog.) (Contributed by NM, 4-Nov-2011.) |
⊢ 0 = (0.‘𝐾) & ⊢ 𝐶 = ( ⋖ ‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑃 ∈ 𝐴) → 0 𝐶𝑃) | ||
Theorem | atbase 36427 | An atom is a member of the lattice base set (i.e. a lattice element). (atelch 30123 analog.) (Contributed by NM, 10-Oct-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵) | ||
Theorem | atssbase 36428 | The set of atoms is a subset of the base set. (atssch 30122 analog.) (Contributed by NM, 21-Oct-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ 𝐴 ⊆ 𝐵 | ||
Theorem | 0ltat 36429 | An atom is greater than zero. (Contributed by NM, 4-Jul-2012.) |
⊢ 0 = (0.‘𝐾) & ⊢ < = (lt‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ OP ∧ 𝑃 ∈ 𝐴) → 0 < 𝑃) | ||
Theorem | leatb 36430 | A poset element less than or equal to an atom equals either zero or the atom. (atss 30125 analog.) (Contributed by NM, 17-Nov-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 0 = (0.‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (𝑋 ≤ 𝑃 ↔ (𝑋 = 𝑃 ∨ 𝑋 = 0 ))) | ||
Theorem | leat 36431 | A poset element less than or equal to an atom equals either zero or the atom. (Contributed by NM, 15-Oct-2013.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 0 = (0.‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ (((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑋 ≤ 𝑃) → (𝑋 = 𝑃 ∨ 𝑋 = 0 )) | ||
Theorem | leat2 36432 | A nonzero poset element less than or equal to an atom equals the atom. (Contributed by NM, 6-Mar-2013.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 0 = (0.‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ (((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ (𝑋 ≠ 0 ∧ 𝑋 ≤ 𝑃)) → 𝑋 = 𝑃) | ||
Theorem | leat3 36433 | A poset element less than or equal to an atom is either an atom or zero. (Contributed by NM, 2-Dec-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 0 = (0.‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ (((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑋 ≤ 𝑃) → (𝑋 ∈ 𝐴 ∨ 𝑋 = 0 )) | ||
Theorem | meetat 36434 | The meet of any element with an atom is either the atom or zero. (Contributed by NM, 28-Aug-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 0 = (0.‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → ((𝑋 ∧ 𝑃) = 𝑃 ∨ (𝑋 ∧ 𝑃) = 0 )) | ||
Theorem | meetat2 36435 | The meet of any element with an atom is either the atom or zero. (Contributed by NM, 30-Aug-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 0 = (0.‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → ((𝑋 ∧ 𝑃) ∈ 𝐴 ∨ (𝑋 ∧ 𝑃) = 0 )) | ||
Definition | df-atl 36436* | Define the class of atomic lattices, in which every nonzero element is greater than or equal to an atom. We also ensure the existence of a lattice zero, since a lattice by itself may not have a zero. (Contributed by NM, 18-Sep-2011.) (Revised by NM, 14-Sep-2018.) |
⊢ AtLat = {𝑘 ∈ Lat ∣ ((Base‘𝑘) ∈ dom (glb‘𝑘) ∧ ∀𝑥 ∈ (Base‘𝑘)(𝑥 ≠ (0.‘𝑘) → ∃𝑝 ∈ (Atoms‘𝑘)𝑝(le‘𝑘)𝑥))} | ||
Theorem | isatl 36437* | The predicate "is an atomic lattice." Every nonzero element is less than or equal to an atom. (Contributed by NM, 18-Sep-2011.) (Revised by NM, 14-Sep-2018.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝐺 = (glb‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 0 = (0.‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ 𝐵 ∈ dom 𝐺 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))) | ||
Theorem | atllat 36438 | An atomic lattice is a lattice. (Contributed by NM, 21-Oct-2011.) |
⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Lat) | ||
Theorem | atlpos 36439 | An atomic lattice is a poset. (Contributed by NM, 5-Nov-2012.) |
⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Poset) | ||
Theorem | atl0dm 36440 | Condition necessary for zero element to exist. (Contributed by NM, 14-Sep-2018.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝑈 = (lub‘𝐾) & ⊢ 𝐺 = (glb‘𝐾) ⇒ ⊢ (𝐾 ∈ AtLat → 𝐵 ∈ dom 𝐺) | ||
Theorem | atl0cl 36441 | An atomic lattice has a zero element. We can use this in place of op0cl 36322 for lattices without orthocomplements. (Contributed by NM, 5-Nov-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ 0 = (0.‘𝐾) ⇒ ⊢ (𝐾 ∈ AtLat → 0 ∈ 𝐵) | ||
Theorem | atl0le 36442 | Orthoposet zero is less than or equal to any element. (ch0le 29220 analog.) (Contributed by NM, 12-Oct-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 0 = (0.‘𝐾) ⇒ ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵) → 0 ≤ 𝑋) | ||
Theorem | atlle0 36443 | An element less than or equal to zero equals zero. (chle0 29222 analog.) (Contributed by NM, 21-Oct-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 0 = (0.‘𝐾) ⇒ ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵) → (𝑋 ≤ 0 ↔ 𝑋 = 0 )) | ||
Theorem | atlltn0 36444 | A lattice element greater than zero is nonzero. (Contributed by NM, 1-Jun-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ < = (lt‘𝐾) & ⊢ 0 = (0.‘𝐾) ⇒ ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵) → ( 0 < 𝑋 ↔ 𝑋 ≠ 0 )) | ||
Theorem | isat3 36445* | The predicate "is an atom". (elat2 30119 analog.) (Contributed by NM, 27-Apr-2014.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 0 = (0.‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ (𝐾 ∈ AtLat → (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ 𝐵 ∧ 𝑃 ≠ 0 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑃 → (𝑥 = 𝑃 ∨ 𝑥 = 0 ))))) | ||
Theorem | atn0 36446 | An atom is not zero. (atne0 30124 analog.) (Contributed by NM, 5-Nov-2012.) |
⊢ 0 = (0.‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → 𝑃 ≠ 0 ) | ||
Theorem | atnle0 36447 | An atom is not less than or equal to zero. (Contributed by NM, 17-Oct-2011.) |
⊢ ≤ = (le‘𝐾) & ⊢ 0 = (0.‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → ¬ 𝑃 ≤ 0 ) | ||
Theorem | atlen0 36448 | A lattice element is nonzero if an atom is under it. (Contributed by NM, 26-May-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 0 = (0.‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 𝑋 ≠ 0 ) | ||
Theorem | atcmp 36449 | If two atoms are comparable, they are equal. (atsseq 30126 analog.) (Contributed by NM, 13-Oct-2011.) |
⊢ ≤ = (le‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ≤ 𝑄 ↔ 𝑃 = 𝑄)) | ||
Theorem | atncmp 36450 | Frequently-used variation of atcmp 36449. (Contributed by NM, 29-Jun-2012.) |
⊢ ≤ = (le‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (¬ 𝑃 ≤ 𝑄 ↔ 𝑃 ≠ 𝑄)) | ||
Theorem | atnlt 36451 | Two atoms cannot satisfy the less than relation. (Contributed by NM, 7-Feb-2012.) |
⊢ < = (lt‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ¬ 𝑃 < 𝑄) | ||
Theorem | atcvreq0 36452 | An element covered by an atom must be zero. (atcveq0 30127 analog.) (Contributed by NM, 4-Nov-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 0 = (0.‘𝐾) & ⊢ 𝐶 = ( ⋖ ‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (𝑋𝐶𝑃 ↔ 𝑋 = 0 )) | ||
Theorem | atncvrN 36453 | Two atoms cannot satisfy the covering relation. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.) |
⊢ 𝐶 = ( ⋖ ‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ¬ 𝑃𝐶𝑄) | ||
Theorem | atlex 36454* | Every nonzero element of an atomic lattice is greater than or equal to an atom. (hatomic 30139 analog.) (Contributed by NM, 21-Oct-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 0 = (0.‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑋) | ||
Theorem | atnle 36455 | Two ways of expressing "an atom is not less than or equal to a lattice element." (atnssm0 30155 analog.) (Contributed by NM, 5-Nov-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 0 = (0.‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (¬ 𝑃 ≤ 𝑋 ↔ (𝑃 ∧ 𝑋) = 0 )) | ||
Theorem | atnem0 36456 | The meet of distinct atoms is zero. (atnemeq0 30156 analog.) (Contributed by NM, 5-Nov-2012.) |
⊢ ∧ = (meet‘𝐾) & ⊢ 0 = (0.‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ≠ 𝑄 ↔ (𝑃 ∧ 𝑄) = 0 )) | ||
Theorem | atlatmstc 36457* | 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 30141 analog.) (Contributed by NM, 5-Nov-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 1 = (lub‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → ( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}) = 𝑋) | ||
Theorem | atlatle 36458* | The ordering of two Hilbert lattice elements is determined by the atoms under them. (chrelat3 30150 analog.) (Contributed by NM, 5-Nov-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ↔ ∀𝑝 ∈ 𝐴 (𝑝 ≤ 𝑋 → 𝑝 ≤ 𝑌))) | ||
Theorem | atlrelat1 36459* | An atomistic lattice with 0 is relatively atomic. Part of Lemma 7.2 of [MaedaMaeda] p. 30. (chpssati 30142, with ∧ swapped, analog.) (Contributed by NM, 4-Dec-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ < = (lt‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 → ∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌))) | ||
Definition | df-cvlat 36460* | Define the class of atomic lattices with the covering property. (This is actually the exchange property, but they are equivalent. The literature usually uses the covering property terminology.) (Contributed by NM, 5-Nov-2012.) |
⊢ CvLat = {𝑘 ∈ AtLat ∣ ∀𝑎 ∈ (Atoms‘𝑘)∀𝑏 ∈ (Atoms‘𝑘)∀𝑐 ∈ (Base‘𝑘)((¬ 𝑎(le‘𝑘)𝑐 ∧ 𝑎(le‘𝑘)(𝑐(join‘𝑘)𝑏)) → 𝑏(le‘𝑘)(𝑐(join‘𝑘)𝑎))} | ||
Theorem | iscvlat 36461* | The predicate "is an atomic lattice with the covering (or exchange) property". (Contributed by NM, 5-Nov-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ (𝐾 ∈ CvLat ↔ (𝐾 ∈ AtLat ∧ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ∀𝑥 ∈ 𝐵 ((¬ 𝑝 ≤ 𝑥 ∧ 𝑝 ≤ (𝑥 ∨ 𝑞)) → 𝑞 ≤ (𝑥 ∨ 𝑝)))) | ||
Theorem | iscvlat2N 36462* | The predicate "is an atomic lattice with the covering (or exchange) property". (Contributed by NM, 5-Nov-2012.) (New usage is discouraged.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 0 = (0.‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ (𝐾 ∈ CvLat ↔ (𝐾 ∈ AtLat ∧ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ∀𝑥 ∈ 𝐵 (((𝑝 ∧ 𝑥) = 0 ∧ 𝑝 ≤ (𝑥 ∨ 𝑞)) → 𝑞 ≤ (𝑥 ∨ 𝑝)))) | ||
Theorem | cvlatl 36463 | An atomic lattice with the covering property is an atomic lattice. (Contributed by NM, 5-Nov-2012.) |
⊢ (𝐾 ∈ CvLat → 𝐾 ∈ AtLat) | ||
Theorem | cvllat 36464 | An atomic lattice with the covering property is a lattice. (Contributed by NM, 5-Nov-2012.) |
⊢ (𝐾 ∈ CvLat → 𝐾 ∈ Lat) | ||
Theorem | cvlposN 36465 | An atomic lattice with the covering property is a poset. (Contributed by NM, 5-Nov-2012.) (New usage is discouraged.) |
⊢ (𝐾 ∈ CvLat → 𝐾 ∈ Poset) | ||
Theorem | cvlexch1 36466 | An atomic covering lattice has the exchange property. (Contributed by NM, 6-Nov-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) → 𝑄 ≤ (𝑋 ∨ 𝑃))) | ||
Theorem | cvlexch2 36467 | An atomic covering lattice has the exchange property. (Contributed by NM, 6-May-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑄 ∨ 𝑋) → 𝑄 ≤ (𝑃 ∨ 𝑋))) | ||
Theorem | cvlexchb1 36468 | An atomic covering lattice has the exchange property. (Contributed by NM, 16-Nov-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) ↔ (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄))) | ||
Theorem | cvlexchb2 36469 | An atomic covering lattice has the exchange property. (Contributed by NM, 22-Jun-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑄 ∨ 𝑋) ↔ (𝑃 ∨ 𝑋) = (𝑄 ∨ 𝑋))) | ||
Theorem | cvlexch3 36470 | An atomic covering lattice has the exchange property. (atexch 30160 analog.) (Contributed by NM, 5-Nov-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 0 = (0.‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∧ 𝑋) = 0 ) → (𝑃 ≤ (𝑋 ∨ 𝑄) → 𝑄 ≤ (𝑋 ∨ 𝑃))) | ||
Theorem | cvlexch4N 36471 | An atomic covering lattice has the exchange property. Part of Definition 7.8 of [MaedaMaeda] p. 32. (Contributed by NM, 5-Nov-2012.) (New usage is discouraged.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 0 = (0.‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∧ 𝑋) = 0 ) → (𝑃 ≤ (𝑋 ∨ 𝑄) ↔ (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄))) | ||
Theorem | cvlatexchb1 36472 | A version of cvlexchb1 36468 for atoms. (Contributed by NM, 5-Nov-2012.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑅) → (𝑃 ≤ (𝑅 ∨ 𝑄) ↔ (𝑅 ∨ 𝑃) = (𝑅 ∨ 𝑄))) | ||
Theorem | cvlatexchb2 36473 | A version of cvlexchb2 36469 for atoms. (Contributed by NM, 5-Nov-2012.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑅) → (𝑃 ≤ (𝑄 ∨ 𝑅) ↔ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))) | ||
Theorem | cvlatexch1 36474 | Atom exchange property. (Contributed by NM, 5-Nov-2012.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑅) → (𝑃 ≤ (𝑅 ∨ 𝑄) → 𝑄 ≤ (𝑅 ∨ 𝑃))) | ||
Theorem | cvlatexch2 36475 | Atom exchange property. (Contributed by NM, 5-Nov-2012.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑅) → (𝑃 ≤ (𝑄 ∨ 𝑅) → 𝑄 ≤ (𝑃 ∨ 𝑅))) | ||
Theorem | cvlatexch3 36476 | Atom exchange property. (Contributed by NM, 29-Nov-2012.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) → (𝑃 ≤ (𝑄 ∨ 𝑅) → (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑅))) | ||
Theorem | cvlcvr1 36477 | The covering property. Proposition 1(ii) in [Kalmbach] p. 140 (and its converse). (chcv1 30134 analog.) (Contributed by NM, 5-Nov-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐶 = ( ⋖ ‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (¬ 𝑃 ≤ 𝑋 ↔ 𝑋𝐶(𝑋 ∨ 𝑃))) | ||
Theorem | cvlcvrp 36478 | A Hilbert lattice satisfies the covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 30154 analog.) (Contributed by NM, 5-Nov-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 0 = (0.‘𝐾) & ⊢ 𝐶 = ( ⋖ ‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → ((𝑋 ∧ 𝑃) = 0 ↔ 𝑋𝐶(𝑋 ∨ 𝑃))) | ||
Theorem | cvlatcvr1 36479 | An atom is covered by its join with a different atom. (Contributed by NM, 5-Nov-2012.) |
⊢ ∨ = (join‘𝐾) & ⊢ 𝐶 = ( ⋖ ‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ≠ 𝑄 ↔ 𝑃𝐶(𝑃 ∨ 𝑄))) | ||
Theorem | cvlatcvr2 36480 | An atom is covered by its join with a different atom. (Contributed by NM, 5-Nov-2012.) |
⊢ ∨ = (join‘𝐾) & ⊢ 𝐶 = ( ⋖ ‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ≠ 𝑄 ↔ 𝑃𝐶(𝑄 ∨ 𝑃))) | ||
Theorem | cvlsupr2 36481 | Two equivalent ways of expressing that 𝑅 is a superposition of 𝑃 and 𝑄. (Contributed by NM, 5-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 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)))) | ||
Theorem | cvlsupr3 36482 | Two equivalent ways of expressing that 𝑅 is a superposition of 𝑃 and 𝑄, which can replace the superposition part of ishlat1 36490, (𝑥 ≠ 𝑦 → ∃𝑧 ∈ 𝐴(𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑥 ∨ 𝑦)) ), with the simpler ∃𝑧 ∈ 𝐴(𝑥 ∨ 𝑧) = (𝑦 ∨ 𝑧) as shown in ishlat3N 36492. (Contributed by NM, 5-Nov-2012.) |
⊢ 𝐴 = (Atoms‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) ⇒ ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ↔ (𝑃 ≠ 𝑄 → (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))))) | ||
Theorem | cvlsupr4 36483 | 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 36484 | 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 36485 | 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 36486 | 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 36487 | 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)) | ||
Syntax | chlt 36488 | Extend class notation with Hilbert lattices. |
class HL | ||
Definition | df-hlat 36489* | 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 36490* | 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 36491* | 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 36492* | 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 36493* | 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 36494 | 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 36495 | A Hilbert lattice is orthomodular. (Contributed by NM, 20-Oct-2011.) |
⊢ (𝐾 ∈ HL → 𝐾 ∈ OML) | ||
Theorem | hlclat 36496 | A Hilbert lattice is complete. (Contributed by NM, 20-Oct-2011.) |
⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat) | ||
Theorem | hlcvl 36497 | A Hilbert lattice is an atomic lattice with the covering property. (Contributed by NM, 5-Nov-2012.) |
⊢ (𝐾 ∈ HL → 𝐾 ∈ CvLat) | ||
Theorem | hlatl 36498 | A Hilbert lattice is atomic. (Contributed by NM, 20-Oct-2011.) |
⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) | ||
Theorem | hlol 36499 | A Hilbert lattice is an ortholattice. (Contributed by NM, 20-Oct-2011.) |
⊢ (𝐾 ∈ HL → 𝐾 ∈ OL) | ||
Theorem | hlop 36500 | A Hilbert lattice is an orthoposet. (Contributed by NM, 20-Oct-2011.) |
⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) |
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