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Theorem List for Metamath Proof Explorer - 18401-18500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlattr 18401 A lattice ordering is transitive. (sstr 3989 analog.) (Contributed by NM, 17-Nov-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ ((𝑋 ≀ π‘Œ ∧ π‘Œ ≀ 𝑍) β†’ 𝑋 ≀ 𝑍))
 
Theoremlatasymd 18402 Deduce equality from lattice ordering. (eqssd 3998 analog.) (Contributed by NM, 18-Nov-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   (πœ‘ β†’ 𝐾 ∈ Lat)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ 𝑋 ≀ π‘Œ)    &   (πœ‘ β†’ π‘Œ ≀ 𝑋)    β‡’   (πœ‘ β†’ 𝑋 = π‘Œ)
 
Theoremlattrd 18403 A lattice ordering is transitive. Deduction version of lattr 18401. (Contributed by NM, 3-Sep-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   (πœ‘ β†’ 𝐾 ∈ Lat)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ 𝑍 ∈ 𝐡)    &   (πœ‘ β†’ 𝑋 ≀ π‘Œ)    &   (πœ‘ β†’ π‘Œ ≀ 𝑍)    β‡’   (πœ‘ β†’ 𝑋 ≀ 𝑍)
 
Theoremlatjcom 18404 The join of a lattice commutes. (chjcom 31026 analog.) (Contributed by NM, 16-Sep-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∨ π‘Œ) = (π‘Œ ∨ 𝑋))
 
Theoremlatlej1 18405 A join's first argument is less than or equal to the join. (chub1 31027 analog.) (Contributed by NM, 17-Sep-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝑋 ≀ (𝑋 ∨ π‘Œ))
 
Theoremlatlej2 18406 A join's second argument is less than or equal to the join. (chub2 31028 analog.) (Contributed by NM, 17-Sep-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ π‘Œ ≀ (𝑋 ∨ π‘Œ))
 
Theoremlatjle12 18407 A join is less than or equal to a third value iff each argument is less than or equal to the third value. (chlub 31029 analog.) (Contributed by NM, 17-Sep-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ ((𝑋 ≀ 𝑍 ∧ π‘Œ ≀ 𝑍) ↔ (𝑋 ∨ π‘Œ) ≀ 𝑍))
 
Theoremlatleeqj1 18408 "Less than or equal to" in terms of join. (chlejb1 31032 analog.) (Contributed by NM, 21-Oct-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ≀ π‘Œ ↔ (𝑋 ∨ π‘Œ) = π‘Œ))
 
Theoremlatleeqj2 18409 "Less than or equal to" in terms of join. (chlejb2 31033 analog.) (Contributed by NM, 14-Nov-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ≀ π‘Œ ↔ (π‘Œ ∨ 𝑋) = π‘Œ))
 
Theoremlatjlej1 18410 Add join to both sides of a lattice ordering. (chlej1i 30993 analog.) (Contributed by NM, 8-Nov-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ (𝑋 ≀ π‘Œ β†’ (𝑋 ∨ 𝑍) ≀ (π‘Œ ∨ 𝑍)))
 
Theoremlatjlej2 18411 Add join to both sides of a lattice ordering. (chlej2i 30994 analog.) (Contributed by NM, 8-Nov-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ (𝑋 ≀ π‘Œ β†’ (𝑍 ∨ 𝑋) ≀ (𝑍 ∨ π‘Œ)))
 
Theoremlatjlej12 18412 Add join to both sides of a lattice ordering. (chlej12i 30995 analog.) (Contributed by NM, 8-Nov-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ (𝑍 ∈ 𝐡 ∧ π‘Š ∈ 𝐡)) β†’ ((𝑋 ≀ π‘Œ ∧ 𝑍 ≀ π‘Š) β†’ (𝑋 ∨ 𝑍) ≀ (π‘Œ ∨ π‘Š)))
 
Theoremlatnlej 18413 An idiom to express that a lattice element differs from two others. (Contributed by NM, 28-May-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡) ∧ Β¬ 𝑋 ≀ (π‘Œ ∨ 𝑍)) β†’ (𝑋 β‰  π‘Œ ∧ 𝑋 β‰  𝑍))
 
Theoremlatnlej1l 18414 An idiom to express that a lattice element differs from two others. (Contributed by NM, 19-Jul-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡) ∧ Β¬ 𝑋 ≀ (π‘Œ ∨ 𝑍)) β†’ 𝑋 β‰  π‘Œ)
 
Theoremlatnlej1r 18415 An idiom to express that a lattice element differs from two others. (Contributed by NM, 19-Jul-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡) ∧ Β¬ 𝑋 ≀ (π‘Œ ∨ 𝑍)) β†’ 𝑋 β‰  𝑍)
 
Theoremlatnlej2 18416 An idiom to express that a lattice element differs from two others. (Contributed by NM, 10-Jul-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡) ∧ Β¬ 𝑋 ≀ (π‘Œ ∨ 𝑍)) β†’ (Β¬ 𝑋 ≀ π‘Œ ∧ Β¬ 𝑋 ≀ 𝑍))
 
Theoremlatnlej2l 18417 An idiom to express that a lattice element differs from two others. (Contributed by NM, 19-Jul-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡) ∧ Β¬ 𝑋 ≀ (π‘Œ ∨ 𝑍)) β†’ Β¬ 𝑋 ≀ π‘Œ)
 
Theoremlatnlej2r 18418 An idiom to express that a lattice element differs from two others. (Contributed by NM, 19-Jul-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡) ∧ Β¬ 𝑋 ≀ (π‘Œ ∨ 𝑍)) β†’ Β¬ 𝑋 ≀ 𝑍)
 
Theoremlatjidm 18419 Lattice join is idempotent. Analogue of unidm 4151. (Contributed by NM, 8-Oct-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡) β†’ (𝑋 ∨ 𝑋) = 𝑋)
 
Theoremlatmcom 18420 The join of a lattice commutes. (Contributed by NM, 6-Nov-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∧ π‘Œ) = (π‘Œ ∧ 𝑋))
 
Theoremlatmle1 18421 A meet is less than or equal to its first argument. (Contributed by NM, 21-Oct-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∧ π‘Œ) ≀ 𝑋)
 
Theoremlatmle2 18422 A meet is less than or equal to its second argument. (Contributed by NM, 21-Oct-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∧ π‘Œ) ≀ π‘Œ)
 
Theoremlatlem12 18423 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.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ ((𝑋 ≀ π‘Œ ∧ 𝑋 ≀ 𝑍) ↔ 𝑋 ≀ (π‘Œ ∧ 𝑍)))
 
Theoremlatleeqm1 18424 "Less than or equal to" in terms of meet. (Contributed by NM, 7-Nov-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ≀ π‘Œ ↔ (𝑋 ∧ π‘Œ) = 𝑋))
 
Theoremlatleeqm2 18425 "Less than or equal to" in terms of meet. (Contributed by NM, 7-Nov-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ≀ π‘Œ ↔ (π‘Œ ∧ 𝑋) = 𝑋))
 
Theoremlatmlem1 18426 Add meet to both sides of a lattice ordering. (Contributed by NM, 10-Nov-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ (𝑋 ≀ π‘Œ β†’ (𝑋 ∧ 𝑍) ≀ (π‘Œ ∧ 𝑍)))
 
Theoremlatmlem2 18427 Add meet to both sides of a lattice ordering. (sslin 4233 analog.) (Contributed by NM, 10-Nov-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ (𝑋 ≀ π‘Œ β†’ (𝑍 ∧ 𝑋) ≀ (𝑍 ∧ π‘Œ)))
 
Theoremlatmlem12 18428 Add join to both sides of a lattice ordering. (ss2in 4235 analog.) (Contributed by NM, 10-Nov-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ (𝑍 ∈ 𝐡 ∧ π‘Š ∈ 𝐡)) β†’ ((𝑋 ≀ π‘Œ ∧ 𝑍 ≀ π‘Š) β†’ (𝑋 ∧ 𝑍) ≀ (π‘Œ ∧ π‘Š)))
 
Theoremlatnlemlt 18429 Negation of "less than or equal to" expressed in terms of meet and less-than. (nssinpss 4255 analog.) (Contributed by NM, 5-Feb-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    < = (ltβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (Β¬ 𝑋 ≀ π‘Œ ↔ (𝑋 ∧ π‘Œ) < 𝑋))
 
Theoremlatnle 18430 Equivalent expressions for "not less than" in a lattice. (chnle 31034 analog.) (Contributed by NM, 16-Nov-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    < = (ltβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (Β¬ π‘Œ ≀ 𝑋 ↔ 𝑋 < (𝑋 ∨ π‘Œ)))
 
Theoremlatmidm 18431 Lattice meet is idempotent. Analogue of inidm 4217. (Contributed by NM, 8-Nov-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡) β†’ (𝑋 ∧ 𝑋) = 𝑋)
 
Theoremlatabs1 18432 Lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (chabs1 31036 analog.) (Contributed by NM, 8-Nov-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∨ (𝑋 ∧ π‘Œ)) = 𝑋)
 
Theoremlatabs2 18433 Lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (chabs2 31037 analog.) (Contributed by NM, 8-Nov-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∧ (𝑋 ∨ π‘Œ)) = 𝑋)
 
Theoremlatledi 18434 An ortholattice is distributive in one ordering direction. (ledi 31060 analog.) (Contributed by NM, 7-Nov-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ ((𝑋 ∧ π‘Œ) ∨ (𝑋 ∧ 𝑍)) ≀ (𝑋 ∧ (π‘Œ ∨ 𝑍)))
 
Theoremlatmlej11 18435 Ordering of a meet and join with a common variable. (Contributed by NM, 4-Oct-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ (𝑋 ∧ π‘Œ) ≀ (𝑋 ∨ 𝑍))
 
Theoremlatmlej12 18436 Ordering of a meet and join with a common variable. (Contributed by NM, 4-Oct-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ (𝑋 ∧ π‘Œ) ≀ (𝑍 ∨ 𝑋))
 
Theoremlatmlej21 18437 Ordering of a meet and join with a common variable. (Contributed by NM, 4-Oct-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ (π‘Œ ∧ 𝑋) ≀ (𝑋 ∨ 𝑍))
 
Theoremlatmlej22 18438 Ordering of a meet and join with a common variable. (Contributed by NM, 4-Oct-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ (π‘Œ ∧ 𝑋) ≀ (𝑍 ∨ 𝑋))
 
Theoremlubsn 18439 The least upper bound of a singleton. (chsupsn 30933 analog.) (Contributed by NM, 20-Oct-2011.)
𝐡 = (Baseβ€˜πΎ)    &   π‘ˆ = (lubβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡) β†’ (π‘ˆβ€˜{𝑋}) = 𝑋)
 
Theoremlatjass 18440 Lattice join is associative. Lemma 2.2 in [MegPav2002] p. 362. (chjass 31053 analog.) (Contributed by NM, 17-Sep-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ ((𝑋 ∨ π‘Œ) ∨ 𝑍) = (𝑋 ∨ (π‘Œ ∨ 𝑍)))
 
Theoremlatj12 18441 Swap 1st and 2nd members of lattice join. (chj12 31054 analog.) (Contributed by NM, 4-Jun-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ (𝑋 ∨ (π‘Œ ∨ 𝑍)) = (π‘Œ ∨ (𝑋 ∨ 𝑍)))
 
Theoremlatj32 18442 Swap 2nd and 3rd members of lattice join. Lemma 2.2 in [MegPav2002] p. 362. (Contributed by NM, 2-Dec-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ ((𝑋 ∨ π‘Œ) ∨ 𝑍) = ((𝑋 ∨ 𝑍) ∨ π‘Œ))
 
Theoremlatj13 18443 Swap 1st and 3rd members of lattice join. (Contributed by NM, 4-Jun-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ (𝑋 ∨ (π‘Œ ∨ 𝑍)) = (𝑍 ∨ (π‘Œ ∨ 𝑋)))
 
Theoremlatj31 18444 Swap 2nd and 3rd members of lattice join. Lemma 2.2 in [MegPav2002] p. 362. (Contributed by NM, 23-Jun-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ ((𝑋 ∨ π‘Œ) ∨ 𝑍) = ((𝑍 ∨ π‘Œ) ∨ 𝑋))
 
Theoremlatjrot 18445 Rotate lattice join of 3 classes. (Contributed by NM, 23-Jul-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ ((𝑋 ∨ π‘Œ) ∨ 𝑍) = ((𝑍 ∨ 𝑋) ∨ π‘Œ))
 
Theoremlatj4 18446 Rearrangement of lattice join of 4 classes. (chj4 31055 analog.) (Contributed by NM, 14-Jun-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ (𝑍 ∈ 𝐡 ∧ π‘Š ∈ 𝐡)) β†’ ((𝑋 ∨ π‘Œ) ∨ (𝑍 ∨ π‘Š)) = ((𝑋 ∨ 𝑍) ∨ (π‘Œ ∨ π‘Š)))
 
Theoremlatj4rot 18447 Rotate lattice join of 4 classes. (Contributed by NM, 11-Jul-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ (𝑍 ∈ 𝐡 ∧ π‘Š ∈ 𝐡)) β†’ ((𝑋 ∨ π‘Œ) ∨ (𝑍 ∨ π‘Š)) = ((π‘Š ∨ 𝑋) ∨ (π‘Œ ∨ 𝑍)))
 
Theoremlatjjdi 18448 Lattice join distributes over itself. (Contributed by NM, 30-Jul-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ (𝑋 ∨ (π‘Œ ∨ 𝑍)) = ((𝑋 ∨ π‘Œ) ∨ (𝑋 ∨ 𝑍)))
 
Theoremlatjjdir 18449 Lattice join distributes over itself. (Contributed by NM, 2-Aug-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ ((𝑋 ∨ π‘Œ) ∨ 𝑍) = ((𝑋 ∨ 𝑍) ∨ (π‘Œ ∨ 𝑍)))
 
Theoremmod1ile 18450 The weak direction of the modular law (e.g., pmod1i 39022, atmod1i1 39031) that holds in any lattice. (Contributed by NM, 11-May-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ (𝑋 ≀ 𝑍 β†’ (𝑋 ∨ (π‘Œ ∧ 𝑍)) ≀ ((𝑋 ∨ π‘Œ) ∧ 𝑍)))
 
Theoremmod2ile 18451 The weak direction of the modular law (e.g., pmod2iN 39023) that holds in any lattice. (Contributed by NM, 11-May-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ (𝑍 ≀ 𝑋 β†’ ((𝑋 ∧ π‘Œ) ∨ 𝑍) ≀ (𝑋 ∧ (π‘Œ ∨ 𝑍))))
 
Theoremlatmass 18452 Lattice meet is associative. (Contributed by Stefan O'Rear, 30-Jan-2015.)
𝐡 = (Baseβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ ((𝑋 ∧ π‘Œ) ∧ 𝑍) = (𝑋 ∧ (π‘Œ ∧ 𝑍)))
 
Theoremlatdisdlem 18453* Lemma for latdisd 18454. (Contributed by Stefan O'Rear, 30-Jan-2015.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    β‡’   (𝐾 ∈ Lat β†’ (βˆ€π‘’ ∈ 𝐡 βˆ€π‘£ ∈ 𝐡 βˆ€π‘€ ∈ 𝐡 (𝑒 ∨ (𝑣 ∧ 𝑀)) = ((𝑒 ∨ 𝑣) ∧ (𝑒 ∨ 𝑀)) β†’ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ 𝐡 (π‘₯ ∧ (𝑦 ∨ 𝑧)) = ((π‘₯ ∧ 𝑦) ∨ (π‘₯ ∧ 𝑧))))
 
Theoremlatdisd 18454* 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.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    β‡’   (𝐾 ∈ Lat β†’ (βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ 𝐡 (π‘₯ ∨ (𝑦 ∧ 𝑧)) = ((π‘₯ ∨ 𝑦) ∧ (π‘₯ ∨ 𝑧)) ↔ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ 𝐡 (π‘₯ ∧ (𝑦 ∨ 𝑧)) = ((π‘₯ ∧ 𝑦) ∨ (π‘₯ ∧ 𝑧))))
 
9.5.2  Complete lattices
 
Syntaxccla 18455 Extend class notation with complete lattices.
class CLat
 
Definitiondf-clat 18456 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 (lubβ€˜π‘) = 𝒫 (Baseβ€˜π‘) ∧ dom (glbβ€˜π‘) = 𝒫 (Baseβ€˜π‘))}
 
Theoremisclat 18457 The predicate "is a complete lattice". (Contributed by NM, 18-Oct-2012.) (Revised by NM, 12-Sep-2018.)
𝐡 = (Baseβ€˜πΎ)    &   π‘ˆ = (lubβ€˜πΎ)    &   πΊ = (glbβ€˜πΎ)    β‡’   (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom π‘ˆ = 𝒫 𝐡 ∧ dom 𝐺 = 𝒫 𝐡)))
 
Theoremclatpos 18458 A complete lattice is a poset. (Contributed by NM, 8-Sep-2018.)
(𝐾 ∈ CLat β†’ 𝐾 ∈ Poset)
 
Theoremclatlem 18459 Lemma for properties of a complete lattice. (Contributed by NM, 14-Sep-2011.)
𝐡 = (Baseβ€˜πΎ)    &   π‘ˆ = (lubβ€˜πΎ)    &   πΊ = (glbβ€˜πΎ)    β‡’   ((𝐾 ∈ CLat ∧ 𝑆 βŠ† 𝐡) β†’ ((π‘ˆβ€˜π‘†) ∈ 𝐡 ∧ (πΊβ€˜π‘†) ∈ 𝐡))
 
Theoremclatlubcl 18460 Any subset of the base set has an LUB in a complete lattice. (Contributed by NM, 14-Sep-2011.)
𝐡 = (Baseβ€˜πΎ)    &   π‘ˆ = (lubβ€˜πΎ)    β‡’   ((𝐾 ∈ CLat ∧ 𝑆 βŠ† 𝐡) β†’ (π‘ˆβ€˜π‘†) ∈ 𝐡)
 
Theoremclatlubcl2 18461 Any subset of the base set has an LUB in a complete lattice. (Contributed by NM, 13-Sep-2018.)
𝐡 = (Baseβ€˜πΎ)    &   π‘ˆ = (lubβ€˜πΎ)    β‡’   ((𝐾 ∈ CLat ∧ 𝑆 βŠ† 𝐡) β†’ 𝑆 ∈ dom π‘ˆ)
 
Theoremclatglbcl 18462 Any subset of the base set has a GLB in a complete lattice. (Contributed by NM, 14-Sep-2011.)
𝐡 = (Baseβ€˜πΎ)    &   πΊ = (glbβ€˜πΎ)    β‡’   ((𝐾 ∈ CLat ∧ 𝑆 βŠ† 𝐡) β†’ (πΊβ€˜π‘†) ∈ 𝐡)
 
Theoremclatglbcl2 18463 Any subset of the base set has a GLB in a complete lattice. (Contributed by NM, 13-Sep-2018.)
𝐡 = (Baseβ€˜πΎ)    &   πΊ = (glbβ€˜πΎ)    β‡’   ((𝐾 ∈ CLat ∧ 𝑆 βŠ† 𝐡) β†’ 𝑆 ∈ dom 𝐺)
 
Theoremoduclatb 18464 Being a complete lattice is self-dual. (Contributed by Stefan O'Rear, 29-Jan-2015.)
𝐷 = (ODualβ€˜π‘‚)    β‡’   (𝑂 ∈ CLat ↔ 𝐷 ∈ CLat)
 
Theoremclatl 18465 A complete lattice is a lattice. (Contributed by NM, 18-Sep-2011.) TODO: use eqrelrdv2 5794 to shorten proof and eliminate joindmss 18336 and meetdmss 18350?
(𝐾 ∈ CLat β†’ 𝐾 ∈ Lat)
 
Theoremisglbd 18466* Properties that determine the greatest lower bound of a complete lattice. (Contributed by Mario Carneiro, 19-Mar-2014.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   πΊ = (glbβ€˜πΎ)    &   ((πœ‘ ∧ 𝑦 ∈ 𝑆) β†’ 𝐻 ≀ 𝑦)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐡 ∧ βˆ€π‘¦ ∈ 𝑆 π‘₯ ≀ 𝑦) β†’ π‘₯ ≀ 𝐻)    &   (πœ‘ β†’ 𝐾 ∈ CLat)    &   (πœ‘ β†’ 𝑆 βŠ† 𝐡)    &   (πœ‘ β†’ 𝐻 ∈ 𝐡)    β‡’   (πœ‘ β†’ (πΊβ€˜π‘†) = 𝐻)
 
Theoremlublem 18467* Lemma for the least upper bound properties in a complete lattice. (Contributed by NM, 19-Oct-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   π‘ˆ = (lubβ€˜πΎ)    β‡’   ((𝐾 ∈ CLat ∧ 𝑆 βŠ† 𝐡) β†’ (βˆ€π‘¦ ∈ 𝑆 𝑦 ≀ (π‘ˆβ€˜π‘†) ∧ βˆ€π‘§ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝑆 𝑦 ≀ 𝑧 β†’ (π‘ˆβ€˜π‘†) ≀ 𝑧)))
 
Theoremlubub 18468 The LUB of a complete lattice subset is an upper bound. (Contributed by NM, 19-Oct-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   π‘ˆ = (lubβ€˜πΎ)    β‡’   ((𝐾 ∈ CLat ∧ 𝑆 βŠ† 𝐡 ∧ 𝑋 ∈ 𝑆) β†’ 𝑋 ≀ (π‘ˆβ€˜π‘†))
 
Theoremlubl 18469* The LUB of a complete lattice subset is the least bound. (Contributed by NM, 19-Oct-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   π‘ˆ = (lubβ€˜πΎ)    β‡’   ((𝐾 ∈ CLat ∧ 𝑆 βŠ† 𝐡 ∧ 𝑋 ∈ 𝐡) β†’ (βˆ€π‘¦ ∈ 𝑆 𝑦 ≀ 𝑋 β†’ (π‘ˆβ€˜π‘†) ≀ 𝑋))
 
Theoremlubss 18470 Subset law for least upper bounds. (chsupss 30862 analog.) (Contributed by NM, 20-Oct-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   π‘ˆ = (lubβ€˜πΎ)    β‡’   ((𝐾 ∈ CLat ∧ 𝑇 βŠ† 𝐡 ∧ 𝑆 βŠ† 𝑇) β†’ (π‘ˆβ€˜π‘†) ≀ (π‘ˆβ€˜π‘‡))
 
Theoremlubel 18471 An element of a set is less than or equal to the least upper bound of the set. (Contributed by NM, 21-Oct-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   π‘ˆ = (lubβ€˜πΎ)    β‡’   ((𝐾 ∈ CLat ∧ 𝑋 ∈ 𝑆 ∧ 𝑆 βŠ† 𝐡) β†’ 𝑋 ≀ (π‘ˆβ€˜π‘†))
 
Theoremlubun 18472 The LUB of a union. (Contributed by NM, 5-Mar-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π‘ˆ = (lubβ€˜πΎ)    β‡’   ((𝐾 ∈ CLat ∧ 𝑆 βŠ† 𝐡 ∧ 𝑇 βŠ† 𝐡) β†’ (π‘ˆβ€˜(𝑆 βˆͺ 𝑇)) = ((π‘ˆβ€˜π‘†) ∨ (π‘ˆβ€˜π‘‡)))
 
Theoremclatglb 18473* Properties of greatest lower bound of a complete lattice. (Contributed by NM, 5-Dec-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   πΊ = (glbβ€˜πΎ)    β‡’   ((𝐾 ∈ CLat ∧ 𝑆 βŠ† 𝐡) β†’ (βˆ€π‘¦ ∈ 𝑆 (πΊβ€˜π‘†) ≀ 𝑦 ∧ βˆ€π‘§ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝑆 𝑧 ≀ 𝑦 β†’ 𝑧 ≀ (πΊβ€˜π‘†))))
 
Theoremclatglble 18474 The greatest lower bound is the least element. (Contributed by NM, 5-Dec-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   πΊ = (glbβ€˜πΎ)    β‡’   ((𝐾 ∈ CLat ∧ 𝑆 βŠ† 𝐡 ∧ 𝑋 ∈ 𝑆) β†’ (πΊβ€˜π‘†) ≀ 𝑋)
 
Theoremclatleglb 18475* Two ways of expressing "less than or equal to the greatest lower bound." (Contributed by NM, 5-Dec-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   πΊ = (glbβ€˜πΎ)    β‡’   ((𝐾 ∈ CLat ∧ 𝑋 ∈ 𝐡 ∧ 𝑆 βŠ† 𝐡) β†’ (𝑋 ≀ (πΊβ€˜π‘†) ↔ βˆ€π‘¦ ∈ 𝑆 𝑋 ≀ 𝑦))
 
Theoremclatglbss 18476 Subset law for greatest lower bound. (Contributed by Mario Carneiro, 16-Apr-2014.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   πΊ = (glbβ€˜πΎ)    β‡’   ((𝐾 ∈ CLat ∧ 𝑇 βŠ† 𝐡 ∧ 𝑆 βŠ† 𝑇) β†’ (πΊβ€˜π‘‡) ≀ (πΊβ€˜π‘†))
 
9.5.3  Distributive lattices
 
Syntaxcdlat 18477 The class of distributive lattices.
class DLat
 
Definitiondf-dlat 18478* A distributive lattice is a lattice in which meets distribute over joins, or equivalently (latdisd 18454) joins distribute over meets. (Contributed by Stefan O'Rear, 30-Jan-2015.)
DLat = {π‘˜ ∈ Lat ∣ [(Baseβ€˜π‘˜) / 𝑏][(joinβ€˜π‘˜) / 𝑗][(meetβ€˜π‘˜) / π‘š]βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ 𝑏 βˆ€π‘§ ∈ 𝑏 (π‘₯π‘š(𝑦𝑗𝑧)) = ((π‘₯π‘šπ‘¦)𝑗(π‘₯π‘šπ‘§))}
 
Theoremisdlat 18479* Property of being a distributive lattice. (Contributed by Stefan O'Rear, 30-Jan-2015.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    β‡’   (𝐾 ∈ DLat ↔ (𝐾 ∈ Lat ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ 𝐡 (π‘₯ ∧ (𝑦 ∨ 𝑧)) = ((π‘₯ ∧ 𝑦) ∨ (π‘₯ ∧ 𝑧))))
 
Theoremdlatmjdi 18480 In a distributive lattice, meets distribute over joins. (Contributed by Stefan O'Rear, 30-Jan-2015.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    β‡’   ((𝐾 ∈ DLat ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ (𝑋 ∧ (π‘Œ ∨ 𝑍)) = ((𝑋 ∧ π‘Œ) ∨ (𝑋 ∧ 𝑍)))
 
Theoremdlatl 18481 A distributive lattice is a lattice. (Contributed by Stefan O'Rear, 30-Jan-2015.)
(𝐾 ∈ DLat β†’ 𝐾 ∈ Lat)
 
Theoremodudlatb 18482 The dual of a distributive lattice is a distributive lattice and conversely. (Contributed by Stefan O'Rear, 30-Jan-2015.)
𝐷 = (ODualβ€˜πΎ)    β‡’   (𝐾 ∈ 𝑉 β†’ (𝐾 ∈ DLat ↔ 𝐷 ∈ DLat))
 
Theoremdlatjmdi 18483 In a distributive lattice, joins distribute over meets. (Contributed by Stefan O'Rear, 30-Jan-2015.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    β‡’   ((𝐾 ∈ DLat ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ (𝑋 ∨ (π‘Œ ∧ 𝑍)) = ((𝑋 ∨ π‘Œ) ∧ (𝑋 ∨ 𝑍)))
 
9.5.4  Subset order structures
 
Syntaxcipo 18484 Class function defining inclusion posets.
class toInc
 
Definitiondf-ipo 18485* For any family of sets, define the poset of that family ordered by inclusion. See ipobas 18488, ipolerval 18489, and ipole 18491 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 ↦ ⦋{⟨π‘₯, π‘¦βŸ© ∣ ({π‘₯, 𝑦} βŠ† 𝑓 ∧ π‘₯ βŠ† 𝑦)} / π‘œβ¦Œ({⟨(Baseβ€˜ndx), π‘“βŸ©, ⟨(TopSetβ€˜ndx), (ordTopβ€˜π‘œ)⟩} βˆͺ {⟨(leβ€˜ndx), π‘œβŸ©, ⟨(ocβ€˜ndx), (π‘₯ ∈ 𝑓 ↦ βˆͺ {𝑦 ∈ 𝑓 ∣ (𝑦 ∩ π‘₯) = βˆ…})⟩}))
 
Theoremipostr 18486 The structure of df-ipo 18485 is a structure defining indices up to 11. (Contributed by Mario Carneiro, 25-Oct-2015.)
({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(TopSetβ€˜ndx), 𝐽⟩} βˆͺ {⟨(leβ€˜ndx), ≀ ⟩, ⟨(ocβ€˜ndx), βŠ₯ ⟩}) Struct ⟨1, 11⟩
 
Theoremipoval 18487* Value of the inclusion poset. (Contributed by Stefan O'Rear, 30-Jan-2015.)
𝐼 = (toIncβ€˜πΉ)    &    ≀ = {⟨π‘₯, π‘¦βŸ© ∣ ({π‘₯, 𝑦} βŠ† 𝐹 ∧ π‘₯ βŠ† 𝑦)}    β‡’   (𝐹 ∈ 𝑉 β†’ 𝐼 = ({⟨(Baseβ€˜ndx), 𝐹⟩, ⟨(TopSetβ€˜ndx), (ordTopβ€˜ ≀ )⟩} βˆͺ {⟨(leβ€˜ndx), ≀ ⟩, ⟨(ocβ€˜ndx), (π‘₯ ∈ 𝐹 ↦ βˆͺ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ π‘₯) = βˆ…})⟩}))
 
Theoremipobas 18488 Base set of the inclusion poset. (Contributed by Stefan O'Rear, 30-Jan-2015.) (Revised by Mario Carneiro, 25-Oct-2015.)
𝐼 = (toIncβ€˜πΉ)    β‡’   (𝐹 ∈ 𝑉 β†’ 𝐹 = (Baseβ€˜πΌ))
 
Theoremipolerval 18489* Relation of the inclusion poset. (Contributed by Stefan O'Rear, 30-Jan-2015.)
𝐼 = (toIncβ€˜πΉ)    β‡’   (𝐹 ∈ 𝑉 β†’ {⟨π‘₯, π‘¦βŸ© ∣ ({π‘₯, 𝑦} βŠ† 𝐹 ∧ π‘₯ βŠ† 𝑦)} = (leβ€˜πΌ))
 
Theoremipotset 18490 Topology of the inclusion poset. (Contributed by Mario Carneiro, 24-Oct-2015.)
𝐼 = (toIncβ€˜πΉ)    &    ≀ = (leβ€˜πΌ)    β‡’   (𝐹 ∈ 𝑉 β†’ (ordTopβ€˜ ≀ ) = (TopSetβ€˜πΌ))
 
Theoremipole 18491 Weak order condition of the inclusion poset. (Contributed by Stefan O'Rear, 30-Jan-2015.)
𝐼 = (toIncβ€˜πΉ)    &    ≀ = (leβ€˜πΌ)    β‡’   ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝐹 ∧ π‘Œ ∈ 𝐹) β†’ (𝑋 ≀ π‘Œ ↔ 𝑋 βŠ† π‘Œ))
 
Theoremipolt 18492 Strict order condition of the inclusion poset. (Contributed by Stefan O'Rear, 30-Jan-2015.)
𝐼 = (toIncβ€˜πΉ)    &    < = (ltβ€˜πΌ)    β‡’   ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝐹 ∧ π‘Œ ∈ 𝐹) β†’ (𝑋 < π‘Œ ↔ 𝑋 ⊊ π‘Œ))
 
Theoremipopos 18493 The inclusion poset on a family of sets is actually a poset. (Contributed by Stefan O'Rear, 30-Jan-2015.)
𝐼 = (toIncβ€˜πΉ)    β‡’   πΌ ∈ Poset
 
Theoremisipodrs 18494* Condition for a family of sets to be directed by inclusion. (Contributed by Stefan O'Rear, 2-Apr-2015.)
((toIncβ€˜π΄) ∈ Dirset ↔ (𝐴 ∈ V ∧ 𝐴 β‰  βˆ… ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 βˆƒπ‘§ ∈ 𝐴 (π‘₯ βˆͺ 𝑦) βŠ† 𝑧))
 
Theoremipodrscl 18495 Direction by inclusion as used here implies sethood. (Contributed by Stefan O'Rear, 2-Apr-2015.)
((toIncβ€˜π΄) ∈ Dirset β†’ 𝐴 ∈ V)
 
Theoremipodrsfi 18496* Finite upper bound property for directed collections of sets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
(((toIncβ€˜π΄) ∈ Dirset ∧ 𝑋 βŠ† 𝐴 ∧ 𝑋 ∈ Fin) β†’ βˆƒπ‘§ ∈ 𝐴 βˆͺ 𝑋 βŠ† 𝑧)
 
Theoremfpwipodrs 18497 The finite subsets of any set are directed by inclusion. (Contributed by Stefan O'Rear, 2-Apr-2015.)
(𝐴 ∈ 𝑉 β†’ (toIncβ€˜(𝒫 𝐴 ∩ Fin)) ∈ Dirset)
 
Theoremipodrsima 18498* The monotone image of a directed set. (Contributed by Stefan O'Rear, 2-Apr-2015.)
(πœ‘ β†’ 𝐹 Fn 𝒫 𝐴)    &   ((πœ‘ ∧ (𝑒 βŠ† 𝑣 ∧ 𝑣 βŠ† 𝐴)) β†’ (πΉβ€˜π‘’) βŠ† (πΉβ€˜π‘£))    &   (πœ‘ β†’ (toIncβ€˜π΅) ∈ Dirset)    &   (πœ‘ β†’ 𝐡 βŠ† 𝒫 𝐴)    &   (πœ‘ β†’ (𝐹 β€œ 𝐡) ∈ 𝑉)    β‡’   (πœ‘ β†’ (toIncβ€˜(𝐹 β€œ 𝐡)) ∈ Dirset)
 
Theoremisacs3lem 18499* An algebraic closure system satisfies isacs3 18507. (Contributed by Stefan O'Rear, 2-Apr-2015.)
(𝐢 ∈ (ACSβ€˜π‘‹) β†’ (𝐢 ∈ (Mooreβ€˜π‘‹) ∧ βˆ€π‘  ∈ 𝒫 𝐢((toIncβ€˜π‘ ) ∈ Dirset β†’ βˆͺ 𝑠 ∈ 𝐢)))
 
Theoremacsdrsel 18500 An algebraic closure system contains all directed unions of closed sets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
((𝐢 ∈ (ACSβ€˜π‘‹) ∧ π‘Œ βŠ† 𝐢 ∧ (toIncβ€˜π‘Œ) ∈ Dirset) β†’ βˆͺ π‘Œ ∈ 𝐢)
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