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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | latmidm 18201 | Lattice meet is idempotent. Analogue of inidm 4153. (Contributed by NM, 8-Nov-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∧ = (meet‘𝐾) ⇒ ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ 𝑋) = 𝑋) | ||
Theorem | latabs1 18202 | Lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (chabs1 29887 analog.) (Contributed by NM, 8-Nov-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) ⇒ ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ (𝑋 ∧ 𝑌)) = 𝑋) | ||
Theorem | latabs2 18203 | Lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (chabs2 29888 analog.) (Contributed by NM, 8-Nov-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) ⇒ ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ (𝑋 ∨ 𝑌)) = 𝑋) | ||
Theorem | latledi 18204 | An ortholattice is distributive in one ordering direction. (ledi 29911 analog.) (Contributed by NM, 7-Nov-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) ⇒ ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ 𝑍)) ≤ (𝑋 ∧ (𝑌 ∨ 𝑍))) | ||
Theorem | latmlej11 18205 | Ordering of a meet and join with a common variable. (Contributed by NM, 4-Oct-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) ⇒ ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ∧ 𝑌) ≤ (𝑋 ∨ 𝑍)) | ||
Theorem | latmlej12 18206 | Ordering of a meet and join with a common variable. (Contributed by NM, 4-Oct-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) ⇒ ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ∧ 𝑌) ≤ (𝑍 ∨ 𝑋)) | ||
Theorem | latmlej21 18207 | Ordering of a meet and join with a common variable. (Contributed by NM, 4-Oct-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) ⇒ ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑌 ∧ 𝑋) ≤ (𝑋 ∨ 𝑍)) | ||
Theorem | latmlej22 18208 | Ordering of a meet and join with a common variable. (Contributed by NM, 4-Oct-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) ⇒ ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑌 ∧ 𝑋) ≤ (𝑍 ∨ 𝑋)) | ||
Theorem | lubsn 18209 | The least upper bound of a singleton. (chsupsn 29784 analog.) (Contributed by NM, 20-Oct-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝑈 = (lub‘𝐾) ⇒ ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → (𝑈‘{𝑋}) = 𝑋) | ||
Theorem | latjass 18210 | Lattice join is associative. Lemma 2.2 in [MegPav2002] p. 362. (chjass 29904 analog.) (Contributed by NM, 17-Sep-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∨ = (join‘𝐾) ⇒ ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ∨ 𝑌) ∨ 𝑍) = (𝑋 ∨ (𝑌 ∨ 𝑍))) | ||
Theorem | latj12 18211 | Swap 1st and 2nd members of lattice join. (chj12 29905 analog.) (Contributed by NM, 4-Jun-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∨ = (join‘𝐾) ⇒ ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ∨ (𝑌 ∨ 𝑍)) = (𝑌 ∨ (𝑋 ∨ 𝑍))) | ||
Theorem | latj32 18212 | Swap 2nd and 3rd members of lattice join. Lemma 2.2 in [MegPav2002] p. 362. (Contributed by NM, 2-Dec-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∨ = (join‘𝐾) ⇒ ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ∨ 𝑌) ∨ 𝑍) = ((𝑋 ∨ 𝑍) ∨ 𝑌)) | ||
Theorem | latj13 18213 | Swap 1st and 3rd members of lattice join. (Contributed by NM, 4-Jun-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∨ = (join‘𝐾) ⇒ ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ∨ (𝑌 ∨ 𝑍)) = (𝑍 ∨ (𝑌 ∨ 𝑋))) | ||
Theorem | latj31 18214 | Swap 2nd and 3rd members of lattice join. Lemma 2.2 in [MegPav2002] p. 362. (Contributed by NM, 23-Jun-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∨ = (join‘𝐾) ⇒ ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ∨ 𝑌) ∨ 𝑍) = ((𝑍 ∨ 𝑌) ∨ 𝑋)) | ||
Theorem | latjrot 18215 | Rotate lattice join of 3 classes. (Contributed by NM, 23-Jul-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∨ = (join‘𝐾) ⇒ ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ∨ 𝑌) ∨ 𝑍) = ((𝑍 ∨ 𝑋) ∨ 𝑌)) | ||
Theorem | latj4 18216 | Rearrangement of lattice join of 4 classes. (chj4 29906 analog.) (Contributed by NM, 14-Jun-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∨ = (join‘𝐾) ⇒ ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → ((𝑋 ∨ 𝑌) ∨ (𝑍 ∨ 𝑊)) = ((𝑋 ∨ 𝑍) ∨ (𝑌 ∨ 𝑊))) | ||
Theorem | latj4rot 18217 | Rotate lattice join of 4 classes. (Contributed by NM, 11-Jul-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∨ = (join‘𝐾) ⇒ ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → ((𝑋 ∨ 𝑌) ∨ (𝑍 ∨ 𝑊)) = ((𝑊 ∨ 𝑋) ∨ (𝑌 ∨ 𝑍))) | ||
Theorem | latjjdi 18218 | Lattice join distributes over itself. (Contributed by NM, 30-Jul-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∨ = (join‘𝐾) ⇒ ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ∨ (𝑌 ∨ 𝑍)) = ((𝑋 ∨ 𝑌) ∨ (𝑋 ∨ 𝑍))) | ||
Theorem | latjjdir 18219 | Lattice join distributes over itself. (Contributed by NM, 2-Aug-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∨ = (join‘𝐾) ⇒ ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ∨ 𝑌) ∨ 𝑍) = ((𝑋 ∨ 𝑍) ∨ (𝑌 ∨ 𝑍))) | ||
Theorem | mod1ile 18220 | The weak direction of the modular law (e.g., pmod1i 37869, atmod1i1 37878) that holds in any lattice. (Contributed by NM, 11-May-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) ⇒ ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ≤ 𝑍 → (𝑋 ∨ (𝑌 ∧ 𝑍)) ≤ ((𝑋 ∨ 𝑌) ∧ 𝑍))) | ||
Theorem | mod2ile 18221 | The weak direction of the modular law (e.g., pmod2iN 37870) that holds in any lattice. (Contributed by NM, 11-May-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) ⇒ ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑍 ≤ 𝑋 → ((𝑋 ∧ 𝑌) ∨ 𝑍) ≤ (𝑋 ∧ (𝑌 ∨ 𝑍)))) | ||
Theorem | latmass 18222 | Lattice meet is associative. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∧ = (meet‘𝐾) ⇒ ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ∧ 𝑌) ∧ 𝑍) = (𝑋 ∧ (𝑌 ∧ 𝑍))) | ||
Theorem | latdisdlem 18223* | Lemma for latdisd 18224. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) ⇒ ⊢ (𝐾 ∈ Lat → (∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑢 ∨ (𝑣 ∧ 𝑤)) = ((𝑢 ∨ 𝑣) ∧ (𝑢 ∨ 𝑤)) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ∧ (𝑦 ∨ 𝑧)) = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧)))) | ||
Theorem | latdisd 18224* | In a lattice, joins distribute over meets if and only if meets distribute over joins; the distributive property is self-dual. (Contributed by Stefan O'Rear, 29-Jan-2015.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) ⇒ ⊢ (𝐾 ∈ Lat → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ∨ (𝑦 ∧ 𝑧)) = ((𝑥 ∨ 𝑦) ∧ (𝑥 ∨ 𝑧)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ∧ (𝑦 ∨ 𝑧)) = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧)))) | ||
Syntax | ccla 18225 | Extend class notation with complete lattices. |
class CLat | ||
Definition | df-clat 18226 | Define the class of all complete lattices, where every subset of the base set has an LUB and a GLB. (Contributed by NM, 18-Oct-2012.) (Revised by NM, 12-Sep-2018.) |
⊢ CLat = {𝑝 ∈ Poset ∣ (dom (lub‘𝑝) = 𝒫 (Base‘𝑝) ∧ dom (glb‘𝑝) = 𝒫 (Base‘𝑝))} | ||
Theorem | isclat 18227 | The predicate "is a complete lattice". (Contributed by NM, 18-Oct-2012.) (Revised by NM, 12-Sep-2018.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝑈 = (lub‘𝐾) & ⊢ 𝐺 = (glb‘𝐾) ⇒ ⊢ (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom 𝑈 = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵))) | ||
Theorem | clatpos 18228 | A complete lattice is a poset. (Contributed by NM, 8-Sep-2018.) |
⊢ (𝐾 ∈ CLat → 𝐾 ∈ Poset) | ||
Theorem | clatlem 18229 | Lemma for properties of a complete lattice. (Contributed by NM, 14-Sep-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝑈 = (lub‘𝐾) & ⊢ 𝐺 = (glb‘𝐾) ⇒ ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → ((𝑈‘𝑆) ∈ 𝐵 ∧ (𝐺‘𝑆) ∈ 𝐵)) | ||
Theorem | clatlubcl 18230 | Any subset of the base set has an LUB in a complete lattice. (Contributed by NM, 14-Sep-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝑈 = (lub‘𝐾) ⇒ ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → (𝑈‘𝑆) ∈ 𝐵) | ||
Theorem | clatlubcl2 18231 | Any subset of the base set has an LUB in a complete lattice. (Contributed by NM, 13-Sep-2018.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝑈 = (lub‘𝐾) ⇒ ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝑆 ∈ dom 𝑈) | ||
Theorem | clatglbcl 18232 | Any subset of the base set has a GLB in a complete lattice. (Contributed by NM, 14-Sep-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝐺 = (glb‘𝐾) ⇒ ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → (𝐺‘𝑆) ∈ 𝐵) | ||
Theorem | clatglbcl2 18233 | Any subset of the base set has a GLB in a complete lattice. (Contributed by NM, 13-Sep-2018.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝐺 = (glb‘𝐾) ⇒ ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝑆 ∈ dom 𝐺) | ||
Theorem | oduclatb 18234 | Being a complete lattice is self-dual. (Contributed by Stefan O'Rear, 29-Jan-2015.) |
⊢ 𝐷 = (ODual‘𝑂) ⇒ ⊢ (𝑂 ∈ CLat ↔ 𝐷 ∈ CLat) | ||
Theorem | clatl 18235 | A complete lattice is a lattice. (Contributed by NM, 18-Sep-2011.) TODO: use eqrelrdv2 5707 to shorten proof and eliminate joindmss 18106 and meetdmss 18120? |
⊢ (𝐾 ∈ CLat → 𝐾 ∈ Lat) | ||
Theorem | isglbd 18236* | Properties that determine the greatest lower bound of a complete lattice. (Contributed by Mario Carneiro, 19-Mar-2014.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐺 = (glb‘𝐾) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝐻 ≤ 𝑦) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) → 𝑥 ≤ 𝐻) & ⊢ (𝜑 → 𝐾 ∈ CLat) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) & ⊢ (𝜑 → 𝐻 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐺‘𝑆) = 𝐻) | ||
Theorem | lublem 18237* | Lemma for the least upper bound properties in a complete lattice. (Contributed by NM, 19-Oct-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝑈 = (lub‘𝐾) ⇒ ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → (∀𝑦 ∈ 𝑆 𝑦 ≤ (𝑈‘𝑆) ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → (𝑈‘𝑆) ≤ 𝑧))) | ||
Theorem | lubub 18238 | The LUB of a complete lattice subset is an upper bound. (Contributed by NM, 19-Oct-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝑈 = (lub‘𝐾) ⇒ ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ∧ 𝑋 ∈ 𝑆) → 𝑋 ≤ (𝑈‘𝑆)) | ||
Theorem | lubl 18239* | The LUB of a complete lattice subset is the least bound. (Contributed by NM, 19-Oct-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝑈 = (lub‘𝐾) ⇒ ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ∧ 𝑋 ∈ 𝐵) → (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑋 → (𝑈‘𝑆) ≤ 𝑋)) | ||
Theorem | lubss 18240 | Subset law for least upper bounds. (chsupss 29713 analog.) (Contributed by NM, 20-Oct-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝑈 = (lub‘𝐾) ⇒ ⊢ ((𝐾 ∈ CLat ∧ 𝑇 ⊆ 𝐵 ∧ 𝑆 ⊆ 𝑇) → (𝑈‘𝑆) ≤ (𝑈‘𝑇)) | ||
Theorem | lubel 18241 | An element of a set is less than or equal to the least upper bound of the set. (Contributed by NM, 21-Oct-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝑈 = (lub‘𝐾) ⇒ ⊢ ((𝐾 ∈ CLat ∧ 𝑋 ∈ 𝑆 ∧ 𝑆 ⊆ 𝐵) → 𝑋 ≤ (𝑈‘𝑆)) | ||
Theorem | lubun 18242 | The LUB of a union. (Contributed by NM, 5-Mar-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝑈 = (lub‘𝐾) ⇒ ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝐵) → (𝑈‘(𝑆 ∪ 𝑇)) = ((𝑈‘𝑆) ∨ (𝑈‘𝑇))) | ||
Theorem | clatglb 18243* | Properties of greatest lower bound of a complete lattice. (Contributed by NM, 5-Dec-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐺 = (glb‘𝐾) ⇒ ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → (∀𝑦 ∈ 𝑆 (𝐺‘𝑆) ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ (𝐺‘𝑆)))) | ||
Theorem | clatglble 18244 | The greatest lower bound is the least element. (Contributed by NM, 5-Dec-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐺 = (glb‘𝐾) ⇒ ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ∧ 𝑋 ∈ 𝑆) → (𝐺‘𝑆) ≤ 𝑋) | ||
Theorem | clatleglb 18245* | Two ways of expressing "less than or equal to the greatest lower bound." (Contributed by NM, 5-Dec-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐺 = (glb‘𝐾) ⇒ ⊢ ((𝐾 ∈ CLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑆 ⊆ 𝐵) → (𝑋 ≤ (𝐺‘𝑆) ↔ ∀𝑦 ∈ 𝑆 𝑋 ≤ 𝑦)) | ||
Theorem | clatglbss 18246 | Subset law for greatest lower bound. (Contributed by Mario Carneiro, 16-Apr-2014.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐺 = (glb‘𝐾) ⇒ ⊢ ((𝐾 ∈ CLat ∧ 𝑇 ⊆ 𝐵 ∧ 𝑆 ⊆ 𝑇) → (𝐺‘𝑇) ≤ (𝐺‘𝑆)) | ||
Syntax | cdlat 18247 | The class of distributive lattices. |
class DLat | ||
Definition | df-dlat 18248* | A distributive lattice is a lattice in which meets distribute over joins, or equivalently (latdisd 18224) joins distribute over meets. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
⊢ DLat = {𝑘 ∈ Lat ∣ [(Base‘𝑘) / 𝑏][(join‘𝑘) / 𝑗][(meet‘𝑘) / 𝑚]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧))} | ||
Theorem | isdlat 18249* | Property of being a distributive lattice. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) ⇒ ⊢ (𝐾 ∈ DLat ↔ (𝐾 ∈ Lat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ∧ (𝑦 ∨ 𝑧)) = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧)))) | ||
Theorem | dlatmjdi 18250 | In a distributive lattice, meets distribute over joins. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) ⇒ ⊢ ((𝐾 ∈ DLat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ∧ (𝑌 ∨ 𝑍)) = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ 𝑍))) | ||
Theorem | dlatl 18251 | A distributive lattice is a lattice. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
⊢ (𝐾 ∈ DLat → 𝐾 ∈ Lat) | ||
Theorem | odudlatb 18252 | The dual of a distributive lattice is a distributive lattice and conversely. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
⊢ 𝐷 = (ODual‘𝐾) ⇒ ⊢ (𝐾 ∈ 𝑉 → (𝐾 ∈ DLat ↔ 𝐷 ∈ DLat)) | ||
Theorem | dlatjmdi 18253 | In a distributive lattice, joins distribute over meets. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) ⇒ ⊢ ((𝐾 ∈ DLat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ∨ (𝑌 ∧ 𝑍)) = ((𝑋 ∨ 𝑌) ∧ (𝑋 ∨ 𝑍))) | ||
Syntax | cipo 18254 | Class function defining inclusion posets. |
class toInc | ||
Definition | df-ipo 18255* |
For any family of sets, define the poset of that family ordered by
inclusion. See ipobas 18258, ipolerval 18259, and ipole 18261 for its contract.
EDITORIAL: I'm not thrilled with the name. Any suggestions? (Contributed by Stefan O'Rear, 30-Jan-2015.) (New usage is discouraged.) |
⊢ toInc = (𝑓 ∈ V ↦ ⦋{〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦)} / 𝑜⦌({〈(Base‘ndx), 𝑓〉, 〈(TopSet‘ndx), (ordTop‘𝑜)〉} ∪ {〈(le‘ndx), 𝑜〉, 〈(oc‘ndx), (𝑥 ∈ 𝑓 ↦ ∪ {𝑦 ∈ 𝑓 ∣ (𝑦 ∩ 𝑥) = ∅})〉})) | ||
Theorem | ipostr 18256 | The structure of df-ipo 18255 is a structure defining indices up to 11. (Contributed by Mario Carneiro, 25-Oct-2015.) |
⊢ ({〈(Base‘ndx), 𝐵〉, 〈(TopSet‘ndx), 𝐽〉} ∪ {〈(le‘ndx), ≤ 〉, 〈(oc‘ndx), ⊥ 〉}) Struct 〈1, ;11〉 | ||
Theorem | ipoval 18257* | Value of the inclusion poset. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
⊢ 𝐼 = (toInc‘𝐹) & ⊢ ≤ = {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)} ⇒ ⊢ (𝐹 ∈ 𝑉 → 𝐼 = ({〈(Base‘ndx), 𝐹〉, 〈(TopSet‘ndx), (ordTop‘ ≤ )〉} ∪ {〈(le‘ndx), ≤ 〉, 〈(oc‘ndx), (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})〉})) | ||
Theorem | ipobas 18258 | Base set of the inclusion poset. (Contributed by Stefan O'Rear, 30-Jan-2015.) (Revised by Mario Carneiro, 25-Oct-2015.) |
⊢ 𝐼 = (toInc‘𝐹) ⇒ ⊢ (𝐹 ∈ 𝑉 → 𝐹 = (Base‘𝐼)) | ||
Theorem | ipolerval 18259* | Relation of the inclusion poset. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
⊢ 𝐼 = (toInc‘𝐹) ⇒ ⊢ (𝐹 ∈ 𝑉 → {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)} = (le‘𝐼)) | ||
Theorem | ipotset 18260 | Topology of the inclusion poset. (Contributed by Mario Carneiro, 24-Oct-2015.) |
⊢ 𝐼 = (toInc‘𝐹) & ⊢ ≤ = (le‘𝐼) ⇒ ⊢ (𝐹 ∈ 𝑉 → (ordTop‘ ≤ ) = (TopSet‘𝐼)) | ||
Theorem | ipole 18261 | Weak order condition of the inclusion poset. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
⊢ 𝐼 = (toInc‘𝐹) & ⊢ ≤ = (le‘𝐼) ⇒ ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝐹 ∧ 𝑌 ∈ 𝐹) → (𝑋 ≤ 𝑌 ↔ 𝑋 ⊆ 𝑌)) | ||
Theorem | ipolt 18262 | Strict order condition of the inclusion poset. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
⊢ 𝐼 = (toInc‘𝐹) & ⊢ < = (lt‘𝐼) ⇒ ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝐹 ∧ 𝑌 ∈ 𝐹) → (𝑋 < 𝑌 ↔ 𝑋 ⊊ 𝑌)) | ||
Theorem | ipopos 18263 | The inclusion poset on a family of sets is actually a poset. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
⊢ 𝐼 = (toInc‘𝐹) ⇒ ⊢ 𝐼 ∈ Poset | ||
Theorem | isipodrs 18264* | Condition for a family of sets to be directed by inclusion. (Contributed by Stefan O'Rear, 2-Apr-2015.) |
⊢ ((toInc‘𝐴) ∈ Dirset ↔ (𝐴 ∈ V ∧ 𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐴 (𝑥 ∪ 𝑦) ⊆ 𝑧)) | ||
Theorem | ipodrscl 18265 | Direction by inclusion as used here implies sethood. (Contributed by Stefan O'Rear, 2-Apr-2015.) |
⊢ ((toInc‘𝐴) ∈ Dirset → 𝐴 ∈ V) | ||
Theorem | ipodrsfi 18266* | Finite upper bound property for directed collections of sets. (Contributed by Stefan O'Rear, 2-Apr-2015.) |
⊢ (((toInc‘𝐴) ∈ Dirset ∧ 𝑋 ⊆ 𝐴 ∧ 𝑋 ∈ Fin) → ∃𝑧 ∈ 𝐴 ∪ 𝑋 ⊆ 𝑧) | ||
Theorem | fpwipodrs 18267 | The finite subsets of any set are directed by inclusion. (Contributed by Stefan O'Rear, 2-Apr-2015.) |
⊢ (𝐴 ∈ 𝑉 → (toInc‘(𝒫 𝐴 ∩ Fin)) ∈ Dirset) | ||
Theorem | ipodrsima 18268* | The monotone image of a directed set. (Contributed by Stefan O'Rear, 2-Apr-2015.) |
⊢ (𝜑 → 𝐹 Fn 𝒫 𝐴) & ⊢ ((𝜑 ∧ (𝑢 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝐴)) → (𝐹‘𝑢) ⊆ (𝐹‘𝑣)) & ⊢ (𝜑 → (toInc‘𝐵) ∈ Dirset) & ⊢ (𝜑 → 𝐵 ⊆ 𝒫 𝐴) & ⊢ (𝜑 → (𝐹 “ 𝐵) ∈ 𝑉) ⇒ ⊢ (𝜑 → (toInc‘(𝐹 “ 𝐵)) ∈ Dirset) | ||
Theorem | isacs3lem 18269* | An algebraic closure system satisfies isacs3 18277. (Contributed by Stefan O'Rear, 2-Apr-2015.) |
⊢ (𝐶 ∈ (ACS‘𝑋) → (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝐶))) | ||
Theorem | acsdrsel 18270 | An algebraic closure system contains all directed unions of closed sets. (Contributed by Stefan O'Rear, 2-Apr-2015.) |
⊢ ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑌 ⊆ 𝐶 ∧ (toInc‘𝑌) ∈ Dirset) → ∪ 𝑌 ∈ 𝐶) | ||
Theorem | isacs4lem 18271* | In a closure system in which directed unions of closed sets are closed, closure commutes with directed unions. (Contributed by Stefan O'Rear, 2-Apr-2015.) |
⊢ 𝐹 = (mrCls‘𝐶) ⇒ ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝐶)) → (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑡 ∈ 𝒫 𝒫 𝑋((toInc‘𝑡) ∈ Dirset → (𝐹‘∪ 𝑡) = ∪ (𝐹 “ 𝑡)))) | ||
Theorem | isacs5lem 18272* | If closure commutes with directed unions, then the closure of a set is the closure of its finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.) |
⊢ 𝐹 = (mrCls‘𝐶) ⇒ ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑡 ∈ 𝒫 𝒫 𝑋((toInc‘𝑡) ∈ Dirset → (𝐹‘∪ 𝑡) = ∪ (𝐹 “ 𝑡))) → (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝑋(𝐹‘𝑠) = ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)))) | ||
Theorem | acsdrscl 18273 | In an algebraic closure system, closure commutes with directed unions. (Contributed by Stefan O'Rear, 2-Apr-2015.) |
⊢ 𝐹 = (mrCls‘𝐶) ⇒ ⊢ ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑌 ⊆ 𝒫 𝑋 ∧ (toInc‘𝑌) ∈ Dirset) → (𝐹‘∪ 𝑌) = ∪ (𝐹 “ 𝑌)) | ||
Theorem | acsficl 18274 | A closure in an algebraic closure system is the union of the closures of finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.) |
⊢ 𝐹 = (mrCls‘𝐶) ⇒ ⊢ ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (𝐹‘𝑆) = ∪ (𝐹 “ (𝒫 𝑆 ∩ Fin))) | ||
Theorem | isacs5 18275* | A closure system is algebraic iff the closure of a generating set is the union of the closures of its finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.) |
⊢ 𝐹 = (mrCls‘𝐶) ⇒ ⊢ (𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝑋(𝐹‘𝑠) = ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)))) | ||
Theorem | isacs4 18276* | A closure system is algebraic iff closure commutes with directed unions. (Contributed by Stefan O'Rear, 2-Apr-2015.) |
⊢ 𝐹 = (mrCls‘𝐶) ⇒ ⊢ (𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝒫 𝑋((toInc‘𝑠) ∈ Dirset → (𝐹‘∪ 𝑠) = ∪ (𝐹 “ 𝑠)))) | ||
Theorem | isacs3 18277* | A closure system is algebraic iff directed unions of closed sets are closed. (Contributed by Stefan O'Rear, 2-Apr-2015.) |
⊢ (𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝐶))) | ||
Theorem | acsficld 18278 | In an algebraic closure system, the closure of a set is the union of the closures of its finite subsets. Deduction form of acsficl 18274. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (ACS‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) & ⊢ (𝜑 → 𝑆 ⊆ 𝑋) ⇒ ⊢ (𝜑 → (𝑁‘𝑆) = ∪ (𝑁 “ (𝒫 𝑆 ∩ Fin))) | ||
Theorem | acsficl2d 18279* | In an algebraic closure system, an element is in the closure of a set if and only if it is in the closure of a finite subset. Alternate form of acsficl 18274. Deduction form. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (ACS‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) & ⊢ (𝜑 → 𝑆 ⊆ 𝑋) ⇒ ⊢ (𝜑 → (𝑌 ∈ (𝑁‘𝑆) ↔ ∃𝑥 ∈ (𝒫 𝑆 ∩ Fin)𝑌 ∈ (𝑁‘𝑥))) | ||
Theorem | acsfiindd 18280 | In an algebraic closure system, a set is independent if and only if all its finite subsets are independent. Part of Proposition 4.1.3 in [FaureFrolicher] p. 83. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (ACS‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) & ⊢ 𝐼 = (mrInd‘𝐴) & ⊢ (𝜑 → 𝑆 ⊆ 𝑋) ⇒ ⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ (𝒫 𝑆 ∩ Fin) ⊆ 𝐼)) | ||
Theorem | acsmapd 18281* | In an algebraic closure system, if 𝑇 is contained in the closure of 𝑆, there is a map 𝑓 from 𝑇 into the set of finite subsets of 𝑆 such that the closure of ∪ ran 𝑓 contains 𝑇. This is proven by applying acsficl2d 18279 to each element of 𝑇. See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (ACS‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) & ⊢ (𝜑 → 𝑆 ⊆ 𝑋) & ⊢ (𝜑 → 𝑇 ⊆ (𝑁‘𝑆)) ⇒ ⊢ (𝜑 → ∃𝑓(𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) | ||
Theorem | acsmap2d 18282* | In an algebraic closure system, if 𝑆 and 𝑇 have the same closure and 𝑆 is independent, then there is a map 𝑓 from 𝑇 into the set of finite subsets of 𝑆 such that 𝑆 equals the union of ran 𝑓. This is proven by taking the map 𝑓 from acsmapd 18281 and observing that, since 𝑆 and 𝑇 have the same closure, the closure of ∪ ran 𝑓 must contain 𝑆. Since 𝑆 is independent, by mrissmrcd 17358, ∪ ran 𝑓 must equal 𝑆. See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (ACS‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) & ⊢ 𝐼 = (mrInd‘𝐴) & ⊢ (𝜑 → 𝑆 ∈ 𝐼) & ⊢ (𝜑 → 𝑇 ⊆ 𝑋) & ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇)) ⇒ ⊢ (𝜑 → ∃𝑓(𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑆 = ∪ ran 𝑓)) | ||
Theorem | acsinfd 18283 | In an algebraic closure system, if 𝑆 and 𝑇 have the same closure and 𝑆 is infinite independent, then 𝑇 is infinite. This follows from applying unirnffid 9120 to the map given in acsmap2d 18282. See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (ACS‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) & ⊢ 𝐼 = (mrInd‘𝐴) & ⊢ (𝜑 → 𝑆 ∈ 𝐼) & ⊢ (𝜑 → 𝑇 ⊆ 𝑋) & ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇)) & ⊢ (𝜑 → ¬ 𝑆 ∈ Fin) ⇒ ⊢ (𝜑 → ¬ 𝑇 ∈ Fin) | ||
Theorem | acsdomd 18284 | In an algebraic closure system, if 𝑆 and 𝑇 have the same closure and 𝑆 is infinite independent, then 𝑇 dominates 𝑆. This follows from applying acsinfd 18283 and then applying unirnfdomd 10332 to the map given in acsmap2d 18282. See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (ACS‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) & ⊢ 𝐼 = (mrInd‘𝐴) & ⊢ (𝜑 → 𝑆 ∈ 𝐼) & ⊢ (𝜑 → 𝑇 ⊆ 𝑋) & ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇)) & ⊢ (𝜑 → ¬ 𝑆 ∈ Fin) ⇒ ⊢ (𝜑 → 𝑆 ≼ 𝑇) | ||
Theorem | acsinfdimd 18285 | In an algebraic closure system, if two independent sets have equal closure and one is infinite, then they are equinumerous. This is proven by using acsdomd 18284 twice with acsinfd 18283. See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (ACS‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) & ⊢ 𝐼 = (mrInd‘𝐴) & ⊢ (𝜑 → 𝑆 ∈ 𝐼) & ⊢ (𝜑 → 𝑇 ∈ 𝐼) & ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇)) & ⊢ (𝜑 → ¬ 𝑆 ∈ Fin) ⇒ ⊢ (𝜑 → 𝑆 ≈ 𝑇) | ||
Theorem | acsexdimd 18286* | In an algebraic closure system whose closure operator has the exchange property, if two independent sets have equal closure, they are equinumerous. See mreexfidimd 17368 for the finite case and acsinfdimd 18285 for the infinite case. This is a special case of Theorem 4.2.2 in [FaureFrolicher] p. 87. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (ACS‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) & ⊢ 𝐼 = (mrInd‘𝐴) & ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧}))) & ⊢ (𝜑 → 𝑆 ∈ 𝐼) & ⊢ (𝜑 → 𝑇 ∈ 𝐼) & ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇)) ⇒ ⊢ (𝜑 → 𝑆 ≈ 𝑇) | ||
Theorem | mrelatglb 18287 | Greatest lower bounds in a Moore space are realized by intersections. (Contributed by Stefan O'Rear, 31-Jan-2015.) See mrelatglbALT 46293 for an alternate proof. |
⊢ 𝐼 = (toInc‘𝐶) & ⊢ 𝐺 = (glb‘𝐼) ⇒ ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶 ∧ 𝑈 ≠ ∅) → (𝐺‘𝑈) = ∩ 𝑈) | ||
Theorem | mrelatglb0 18288 | The empty intersection in a Moore space is realized by the base set. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
⊢ 𝐼 = (toInc‘𝐶) & ⊢ 𝐺 = (glb‘𝐼) ⇒ ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝐺‘∅) = 𝑋) | ||
Theorem | mrelatlub 18289 | Least upper bounds in a Moore space are realized by the closure of the union. (Contributed by Stefan O'Rear, 31-Jan-2015.) See mrelatlubALT 46292 for an alternate proof. |
⊢ 𝐼 = (toInc‘𝐶) & ⊢ 𝐹 = (mrCls‘𝐶) & ⊢ 𝐿 = (lub‘𝐼) ⇒ ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → (𝐿‘𝑈) = (𝐹‘∪ 𝑈)) | ||
Theorem | mreclatBAD 18290* | A Moore space is a complete lattice under inclusion. (Contributed by Stefan O'Rear, 31-Jan-2015.) TODO (df-riota 7241 update): Reprove using isclat 18227 instead of the isclatBAD. hypothesis. See commented-out mreclat above. See mreclat 46294 for a good version. |
⊢ 𝐼 = (toInc‘𝐶) & ⊢ (𝐼 ∈ CLat ↔ (𝐼 ∈ Poset ∧ ∀𝑥(𝑥 ⊆ (Base‘𝐼) → (((lub‘𝐼)‘𝑥) ∈ (Base‘𝐼) ∧ ((glb‘𝐼)‘𝑥) ∈ (Base‘𝐼))))) ⇒ ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐼 ∈ CLat) | ||
See commented-out notes for lattices as relations. | ||
Syntax | cps 18291 | Extend class notation with the class of all posets. |
class PosetRel | ||
Syntax | ctsr 18292 | Extend class notation with the class of all totally ordered sets. |
class TosetRel | ||
Definition | df-ps 18293 | Define the class of all posets (partially ordered sets) with weak ordering (e.g., "less than or equal to" instead of "less than"). A poset is a relation which is transitive, reflexive, and antisymmetric. (Contributed by NM, 11-May-2008.) |
⊢ PosetRel = {𝑟 ∣ (Rel 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟 ∧ (𝑟 ∩ ◡𝑟) = ( I ↾ ∪ ∪ 𝑟))} | ||
Definition | df-tsr 18294 | Define the class of all totally ordered sets. (Contributed by FL, 1-Nov-2009.) |
⊢ TosetRel = {𝑟 ∈ PosetRel ∣ (dom 𝑟 × dom 𝑟) ⊆ (𝑟 ∪ ◡𝑟)} | ||
Theorem | isps 18295 | The predicate "is a poset" i.e. a transitive, reflexive, antisymmetric relation. (Contributed by NM, 11-May-2008.) |
⊢ (𝑅 ∈ 𝐴 → (𝑅 ∈ PosetRel ↔ (Rel 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅)))) | ||
Theorem | psrel 18296 | A poset is a relation. (Contributed by NM, 12-May-2008.) |
⊢ (𝐴 ∈ PosetRel → Rel 𝐴) | ||
Theorem | psref2 18297 | A poset is antisymmetric and reflexive. (Contributed by FL, 3-Aug-2009.) |
⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅)) | ||
Theorem | pstr2 18298 | A poset is transitive. (Contributed by FL, 3-Aug-2009.) |
⊢ (𝑅 ∈ PosetRel → (𝑅 ∘ 𝑅) ⊆ 𝑅) | ||
Theorem | pslem 18299 | Lemma for psref 18301 and others. (Contributed by NM, 12-May-2008.) (Revised by Mario Carneiro, 30-Apr-2015.) |
⊢ (𝑅 ∈ PosetRel → (((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶) ∧ (𝐴 ∈ ∪ ∪ 𝑅 → 𝐴𝑅𝐴) ∧ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴) → 𝐴 = 𝐵))) | ||
Theorem | psdmrn 18300 | The domain and range of a poset equal its field. (Contributed by NM, 13-May-2008.) |
⊢ (𝑅 ∈ PosetRel → (dom 𝑅 = ∪ ∪ 𝑅 ∧ ran 𝑅 = ∪ ∪ 𝑅)) |
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