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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | latlem12 16901 | An element is less than or equal to a meet iff the element is less than or equal to each argument of the meet. (Contributed by NM, 21-Oct-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∧ = (meet‘𝐾) ⇒ ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ≤ 𝑌 ∧ 𝑋 ≤ 𝑍) ↔ 𝑋 ≤ (𝑌 ∧ 𝑍))) | ||
Theorem | latleeqm1 16902 | Less-than-or-equal-to in terms of meet. (Contributed by NM, 7-Nov-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∧ = (meet‘𝐾) ⇒ ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ↔ (𝑋 ∧ 𝑌) = 𝑋)) | ||
Theorem | latleeqm2 16903 | Less-than-or-equal-to in terms of meet. (Contributed by NM, 7-Nov-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∧ = (meet‘𝐾) ⇒ ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ↔ (𝑌 ∧ 𝑋) = 𝑋)) | ||
Theorem | latmlem1 16904 | Add meet to both sides of a lattice ordering. (Contributed by NM, 10-Nov-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∧ = (meet‘𝐾) ⇒ ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ≤ 𝑌 → (𝑋 ∧ 𝑍) ≤ (𝑌 ∧ 𝑍))) | ||
Theorem | latmlem2 16905 | Add meet to both sides of a lattice ordering. (sslin 3801 analog.) (Contributed by NM, 10-Nov-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∧ = (meet‘𝐾) ⇒ ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ≤ 𝑌 → (𝑍 ∧ 𝑋) ≤ (𝑍 ∧ 𝑌))) | ||
Theorem | latmlem12 16906 | Add join to both sides of a lattice ordering. (ss2in 3802 analog.) (Contributed by NM, 10-Nov-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∧ = (meet‘𝐾) ⇒ ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → ((𝑋 ≤ 𝑌 ∧ 𝑍 ≤ 𝑊) → (𝑋 ∧ 𝑍) ≤ (𝑌 ∧ 𝑊))) | ||
Theorem | latnlemlt 16907 | Negation of less-than-or-equal-to expressed in terms of meet and less-than. (nssinpss 3818 analog.) (Contributed by NM, 5-Feb-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ < = (lt‘𝐾) & ⊢ ∧ = (meet‘𝐾) ⇒ ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (¬ 𝑋 ≤ 𝑌 ↔ (𝑋 ∧ 𝑌) < 𝑋)) | ||
Theorem | latnle 16908 | Equivalent expressions for "not less than" in a lattice. (chnle 27757 analog.) (Contributed by NM, 16-Nov-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ < = (lt‘𝐾) & ⊢ ∨ = (join‘𝐾) ⇒ ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (¬ 𝑌 ≤ 𝑋 ↔ 𝑋 < (𝑋 ∨ 𝑌))) | ||
Theorem | latmidm 16909 | Lattice join is idempotent. (inidm 3784 analog.) (Contributed by NM, 8-Nov-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∧ = (meet‘𝐾) ⇒ ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ 𝑋) = 𝑋) | ||
Theorem | latabs1 16910 | Lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (chabs1 27759 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 16911 | Lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (chabs2 27760 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 16912 | An ortholattice is distributive in one ordering direction. (ledi 27783 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 16913 | 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 16914 | 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 16915 | 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 16916 | 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 16917 | The least upper bound of a singleton. (chsupsn 27656 analog.) (Contributed by NM, 20-Oct-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝑈 = (lub‘𝐾) ⇒ ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → (𝑈‘{𝑋}) = 𝑋) | ||
Theorem | latjass 16918 | Lattice join is associative. Lemma 2.2 in [MegPav2002] p. 362. (chjass 27776 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 16919 | Swap 1st and 2nd members of lattice join. (chj12 27777 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 16920 | 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 16921 | 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 16922 | 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 16923 | 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 16924 | Rearrangement of lattice join of 4 classes. (chj4 27778 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 16925 | 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 16926 | 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 16927 | 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 16928 | The weak direction of the modular law (e.g., pmod1i 34152, atmod1i1 34161) 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 16929 | The weak direction of the modular law (e.g., pmod2iN 34153) 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)))) | ||
Syntax | ccla 16930 | Extend class notation with complete lattices. |
class CLat | ||
Definition | df-clat 16931 | 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 16932 | 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 16933 | A complete lattice is a poset. (Contributed by NM, 8-Sep-2018.) |
⊢ (𝐾 ∈ CLat → 𝐾 ∈ Poset) | ||
Theorem | clatlem 16934 | 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 16935 | 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 16936 | 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 16937 | 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 16938 | 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 | clatl 16939 | A complete lattice is a lattice. (Contributed by NM, 18-Sep-2011.) TODO: use eqrelrdv2 5142 to shorten proof and eliminate joindmss 16830 and meetdmss 16844? |
⊢ (𝐾 ∈ CLat → 𝐾 ∈ Lat) | ||
Theorem | isglbd 16940* | 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 16941* | 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 16942 | 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 16943* | 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 16944 | Subset law for least upper bounds. (chsupss 27585 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 16945 | 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 16946 | 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 16947* | 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 16948 | 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 16949* | 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 16950 | 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 | codu 16951 | Class function defining dual orders. |
class ODual | ||
Definition | df-odu 16952 |
Define the dual of an ordered structure, which replaces the order
component of the structure with its reverse. See odubas 16956, oduleval 16954,
and oduleg 16955 for its principal properties.
EDITORIAL: likely usable to simplify many lattice proofs, as it allows for duality arguments to be formalized; for instance latmass 17011. (Contributed by Stefan O'Rear, 29-Jan-2015.) |
⊢ ODual = (𝑤 ∈ V ↦ (𝑤 sSet 〈(le‘ndx), ◡(le‘𝑤)〉)) | ||
Theorem | oduval 16953 | Value of an order dual structure. (Contributed by Stefan O'Rear, 29-Jan-2015.) |
⊢ 𝐷 = (ODual‘𝑂) & ⊢ ≤ = (le‘𝑂) ⇒ ⊢ 𝐷 = (𝑂 sSet 〈(le‘ndx), ◡ ≤ 〉) | ||
Theorem | oduleval 16954 | Value of the less-equal relation in an order dual structure. (Contributed by Stefan O'Rear, 29-Jan-2015.) |
⊢ 𝐷 = (ODual‘𝑂) & ⊢ ≤ = (le‘𝑂) ⇒ ⊢ ◡ ≤ = (le‘𝐷) | ||
Theorem | oduleg 16955 | Truth of the less-equal relation in an order dual structure. (Contributed by Stefan O'Rear, 29-Jan-2015.) |
⊢ 𝐷 = (ODual‘𝑂) & ⊢ ≤ = (le‘𝑂) & ⊢ 𝐺 = (le‘𝐷) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴𝐺𝐵 ↔ 𝐵 ≤ 𝐴)) | ||
Theorem | odubas 16956 | Base set of an order dual structure. (Contributed by Stefan O'Rear, 29-Jan-2015.) |
⊢ 𝐷 = (ODual‘𝑂) & ⊢ 𝐵 = (Base‘𝑂) ⇒ ⊢ 𝐵 = (Base‘𝐷) | ||
Theorem | pospropd 16957* | Posethood is determined only by structure components and only by the value of the relation within the base set. (Contributed by Stefan O'Rear, 29-Jan-2015.) |
⊢ (𝜑 → 𝐾 ∈ 𝑉) & ⊢ (𝜑 → 𝐿 ∈ 𝑊) & ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(le‘𝐾)𝑦 ↔ 𝑥(le‘𝐿)𝑦)) ⇒ ⊢ (𝜑 → (𝐾 ∈ Poset ↔ 𝐿 ∈ Poset)) | ||
Theorem | odupos 16958 | Being a poset is a self-dual property. (Contributed by Stefan O'Rear, 29-Jan-2015.) |
⊢ 𝐷 = (ODual‘𝑂) ⇒ ⊢ (𝑂 ∈ Poset → 𝐷 ∈ Poset) | ||
Theorem | oduposb 16959 | Being a poset is a self-dual property. (Contributed by Stefan O'Rear, 29-Jan-2015.) |
⊢ 𝐷 = (ODual‘𝑂) ⇒ ⊢ (𝑂 ∈ 𝑉 → (𝑂 ∈ Poset ↔ 𝐷 ∈ Poset)) | ||
Theorem | meet0 16960 | Lemma for odujoin 16965. (Contributed by Stefan O'Rear, 29-Jan-2015.) TODO (df-riota 6511 update): This proof increased from 152 bytes to 547 bytes after the df-riota 6511 change. Any way to shorten it? join0 16961 also. |
⊢ (meet‘∅) = ∅ | ||
Theorem | join0 16961 | Lemma for odumeet 16963. (Contributed by Stefan O'Rear, 29-Jan-2015.) |
⊢ (join‘∅) = ∅ | ||
Theorem | oduglb 16962 | Greatest lower bounds in a dual order are least upper bounds in the original order. (Contributed by Stefan O'Rear, 29-Jan-2015.) |
⊢ 𝐷 = (ODual‘𝑂) & ⊢ 𝑈 = (lub‘𝑂) ⇒ ⊢ (𝑂 ∈ 𝑉 → 𝑈 = (glb‘𝐷)) | ||
Theorem | odumeet 16963 | Meets in a dual order are joins in the original. (Contributed by Stefan O'Rear, 29-Jan-2015.) |
⊢ 𝐷 = (ODual‘𝑂) & ⊢ ∨ = (join‘𝑂) ⇒ ⊢ ∨ = (meet‘𝐷) | ||
Theorem | odulub 16964 | Least upper bounds in a dual order are greatest lower bounds in the original order. (Contributed by Stefan O'Rear, 29-Jan-2015.) |
⊢ 𝐷 = (ODual‘𝑂) & ⊢ 𝐿 = (glb‘𝑂) ⇒ ⊢ (𝑂 ∈ 𝑉 → 𝐿 = (lub‘𝐷)) | ||
Theorem | odujoin 16965 | Joins in a dual order are meets in the original. (Contributed by Stefan O'Rear, 29-Jan-2015.) |
⊢ 𝐷 = (ODual‘𝑂) & ⊢ ∧ = (meet‘𝑂) ⇒ ⊢ ∧ = (join‘𝐷) | ||
Theorem | odulatb 16966 | Being a lattice is self-dual. (Contributed by Stefan O'Rear, 29-Jan-2015.) |
⊢ 𝐷 = (ODual‘𝑂) ⇒ ⊢ (𝑂 ∈ 𝑉 → (𝑂 ∈ Lat ↔ 𝐷 ∈ Lat)) | ||
Theorem | oduclatb 16967 | Being a complete lattice is self-dual. (Contributed by Stefan O'Rear, 29-Jan-2015.) |
⊢ 𝐷 = (ODual‘𝑂) ⇒ ⊢ (𝑂 ∈ CLat ↔ 𝐷 ∈ CLat) | ||
Theorem | odulat 16968 | Being a lattice is self-dual. (Contributed by Stefan O'Rear, 29-Jan-2015.) |
⊢ 𝐷 = (ODual‘𝑂) ⇒ ⊢ (𝑂 ∈ Lat → 𝐷 ∈ Lat) | ||
Theorem | poslubmo 16969* | Least upper bounds in a poset are unique if they exist. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by NM, 16-Jun-2017.) |
⊢ ≤ = (le‘𝐾) & ⊢ 𝐵 = (Base‘𝐾) ⇒ ⊢ ((𝐾 ∈ Poset ∧ 𝑆 ⊆ 𝐵) → ∃*𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))) | ||
Theorem | posglbmo 16970* | Greatest lower bounds in a poset are unique if they exist. (Contributed by NM, 20-Sep-2018.) |
⊢ ≤ = (le‘𝐾) & ⊢ 𝐵 = (Base‘𝐾) ⇒ ⊢ ((𝐾 ∈ Poset ∧ 𝑆 ⊆ 𝐵) → ∃*𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))) | ||
Theorem | poslubd 16971* | Properties which determine the least upper bound in a poset. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
⊢ ≤ = (le‘𝐾) & ⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝑈 = (lub‘𝐾) & ⊢ (𝜑 → 𝐾 ∈ Poset) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) & ⊢ (𝜑 → 𝑇 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ≤ 𝑇) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦) → 𝑇 ≤ 𝑦) ⇒ ⊢ (𝜑 → (𝑈‘𝑆) = 𝑇) | ||
Theorem | poslubdg 16972* | Properties which determine the least upper bound in a poset. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
⊢ ≤ = (le‘𝐾) & ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) & ⊢ (𝜑 → 𝑈 = (lub‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Poset) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) & ⊢ (𝜑 → 𝑇 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ≤ 𝑇) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦) → 𝑇 ≤ 𝑦) ⇒ ⊢ (𝜑 → (𝑈‘𝑆) = 𝑇) | ||
Theorem | posglbd 16973* | Properties which determine the greatest lower bound in a poset. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
⊢ ≤ = (le‘𝐾) & ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐺 = (glb‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Poset) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) & ⊢ (𝜑 → 𝑇 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑇 ≤ 𝑥) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝑆 𝑦 ≤ 𝑥) → 𝑦 ≤ 𝑇) ⇒ ⊢ (𝜑 → (𝐺‘𝑆) = 𝑇) | ||
Syntax | cipo 16974 | Class function defining inclusion posets. |
class toInc | ||
Definition | df-ipo 16975* |
For any family of sets, define the poset of that family ordered by
inclusion. See ipobas 16978, ipolerval 16979, and ipole 16981 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 16976 | The structure of df-ipo 16975 is a structure defining indexes 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 16977* | 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 16978 | 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 16979* | 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 16980 | Topology of the inclusion poset. (Contributed by Mario Carneiro, 24-Oct-2015.) |
⊢ 𝐼 = (toInc‘𝐹) & ⊢ ≤ = (le‘𝐼) ⇒ ⊢ (𝐹 ∈ 𝑉 → (ordTop‘ ≤ ) = (TopSet‘𝐼)) | ||
Theorem | ipole 16981 | 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 16982 | 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 16983 | 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 16984* | 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 16985 | Direction by inclusion as used here implies sethood. (Contributed by Stefan O'Rear, 2-Apr-2015.) |
⊢ ((toInc‘𝐴) ∈ Dirset → 𝐴 ∈ V) | ||
Theorem | ipodrsfi 16986* | 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 16987 | 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 16988* | 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 16989* | An algebraic closure system satisfies isacs3 16997. (Contributed by Stefan O'Rear, 2-Apr-2015.) |
⊢ (𝐶 ∈ (ACS‘𝑋) → (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝐶))) | ||
Theorem | acsdrsel 16990 | 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 16991* | 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 16992* | 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 16993 | 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 16994 | 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 16995* | 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 16996* | 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 16997* | 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 16998 | In an algebraic closure system, the closure of a set is the union of the closures of its finite subsets. Deduction form of acsficl 16994. (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 16999* | 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 16994. 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 17000 | 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)) |
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