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Theorem List for Metamath Proof Explorer - 14501-14600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlatnlej2l 14501 An idiom to express that a lattice element differs from two others. (Contributed by NM, 19-Jul-2012.)

Theoremlatnlej2r 14502 An idiom to express that a lattice element differs from two others. (Contributed by NM, 19-Jul-2012.)

Theoremlatjidm 14503 Lattice join is idempotent. (chjidm 23022 analog.) (Contributed by NM, 8-Oct-2011.)

Theoremlatmcom 14504 The join of a lattice commutes. (incom 3533 analog.) (Contributed by NM, 6-Nov-2011.)

Theoremlatmle1 14505 A meet is less than or equal to its first argument. (inss1 3561 analog.) (Contributed by NM, 21-Oct-2011.)

Theoremlatmle2 14506 A meet is less than or equal to its second argument. (inss2 3562 analog.) (Contributed by NM, 21-Oct-2011.)

Theoremlatlem12 14507 An element is less than or equal to a meet iff the element is less than or equal to each argument of the meet. (ssin 3563 analog.) (Contributed by NM, 21-Oct-2011.)

Theoremlatleeqm1 14508 Less-than-or-equal-to in terms of meet. (df-ss 3334 analog.) (Contributed by NM, 7-Nov-2011.)

Theoremlatleeqm2 14509 Less-than-or-equal-to in terms of meet. (sseqin2 3560 analog.) (Contributed by NM, 7-Nov-2011.)

Theoremlatmlem1 14510 Add meet to both sides of a lattice ordering. (ssrin 3566 analog.) (Contributed by NM, 10-Nov-2011.)

Theoremlatmlem2 14511 Add meet to both sides of a lattice ordering. (sslin 3567 analog.) (Contributed by NM, 10-Nov-2011.)

Theoremlatmlem12 14512 Add join to both sides of a lattice ordering. (ss2in 3568 analog.) (Contributed by NM, 10-Nov-2011.)

Theoremlatnlemlt 14513 Negation of less-than-or-equal-to expressed in terms of meet and less-than. (nssinpss 3573 analog.) (Contributed by NM, 5-Feb-2012.)

Theoremlatnle 14514 Equivalent expressions for "not less than" in a lattice. (chnle 23016 analog.) (Contributed by NM, 16-Nov-2011.)

Theoremlatmidm 14515 Lattice join is idempotent. (inidm 3550 analog.) (Contributed by NM, 8-Nov-2011.)

Theoremlatabs1 14516 Lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (chabs1 23018 analog.) (Contributed by NM, 8-Nov-2011.)

Theoremlatabs2 14517 Lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (chabs2 23019 analog.) (Contributed by NM, 8-Nov-2011.)

Theoremlatledi 14518 An ortholattice is distributive in one ordering direction. (ledi 23042 analog.) (Contributed by NM, 7-Nov-2011.)

Theoremlatmlej11 14519 Ordering of a meet and join with a common variable. (Contributed by NM, 4-Oct-2012.)

Theoremlatmlej12 14520 Ordering of a meet and join with a common variable. (Contributed by NM, 4-Oct-2012.)

Theoremlatmlej21 14521 Ordering of a meet and join with a common variable. (Contributed by NM, 4-Oct-2012.)

Theoremlatmlej22 14522 Ordering of a meet and join with a common variable. (Contributed by NM, 4-Oct-2012.)

Theoremlubsn 14523 The least upper bound of a singleton. (chsupsn 22915 analog.) (Contributed by NM, 20-Oct-2011.)

Theoremlatjass 14524 Lattice join is associative. Lemma 2.2 in [MegPav2002] p. 362. (chjass 23035 analog.) (Contributed by NM, 17-Sep-2011.)

Theoremlatj12 14525 Swap 1st and 2nd members of lattice join. (chj12 23036 analog.) (Contributed by NM, 4-Jun-2012.)

Theoremlatj32 14526 Swap 2nd and 3rd members of lattice join. Lemma 2.2 in [MegPav2002] p. 362. (Contributed by NM, 2-Dec-2011.)

Theoremlatj13 14527 Swap 1sd and 3rd members of lattice join. (Contributed by NM, 4-Jun-2012.)

Theoremlatj31 14528 Swap 2nd and 3rd members of lattice join. Lemma 2.2 in [MegPav2002] p. 362. (Contributed by NM, 23-Jun-2012.)

Theoremlatjrot 14529 Rotate lattice join of 3 classes. (Contributed by NM, 23-Jul-2012.)

Theoremlatj4 14530 Rearrangement of lattice join of 4 classes. (chj4 23037 analog.) (Contributed by NM, 14-Jun-2012.)

Theoremlatj4rot 14531 Rotate lattice join of 4 classes. (Contributed by NM, 11-Jul-2012.)

Theoremlatjjdi 14532 Lattice join distributes over itself. (Contributed by NM, 30-Jul-2012.)

Theoremlatjjdir 14533 Lattice join distributes over itself. (Contributed by NM, 2-Aug-2012.)

Theoremmod1ile 14534 The weak direction of the modular law (e.g. pmod1i 30645, atmod1i1 30654) that holds in any lattice. (Contributed by NM, 11-May-2012.)

Theoremmod2ile 14535 The weak direction of the modular law (e.g. pmod2iN 30646) that holds in any lattice. (Contributed by NM, 11-May-2012.)

Syntaxccla 14536 Extend class notation with complete lattices.

Definitiondf-clat 14537* Define the class of all complete lattices. (Contributed by NM, 18-Oct-2012.)

Theoremisclat 14538* The predicate "is a complete lattice." (Contributed by NM, 18-Oct-2012.)

Theoremclatlem 14539 Lemma for properties of a complete lattice. (Contributed by NM, 14-Sep-2011.)

Theoremclatlubcl 14540 LUB always exists in a complete lattice. (chsupcl 22842 analog.) (Contributed by NM, 14-Sep-2011.)

Theoremclatglbcl 14541 GLB always exists in a complete lattice. (chintcl 22834 analog.) (Contributed by NM, 14-Sep-2011.)

Theoremisclati 14542* Properties that determine a complete lattice. (Contributed by NM, 12-Sep-2011.)

Theoremclatl 14543 A complete lattice is a lattice. (Contributed by NM, 18-Sep-2011.)

Theoremisglbd 14544* Properties that determine the greatest lower bound of a complete lattice. (Contributed by Mario Carneiro, 19-Mar-2014.)

Theoremlublem 14545* Lemma for the least upper bound properties in a complete lattice. (Contributed by NM, 19-Oct-2011.)

Theoremlubub 14546 The LUB of a complete lattice subset is an upper bound. (Contributed by NM, 19-Oct-2011.)

Theoremlubl 14547* The LUB of a complete lattice subset is the least bound. (Contributed by NM, 19-Oct-2011.)

Theoremlubss 14548 Subset law for least upper bounds. (chsupss 22844 analog.) (Contributed by NM, 20-Oct-2011.)

Theoremlubel 14549 An element of a set is less than or equal to the least upper bound of the set. (Contributed by NM, 21-Oct-2011.)

Theoremlubun 14550 The LUB of a union. (Contributed by NM, 5-Mar-2012.)

Theoremclatglb 14551* Properties of greatest lower bound of a complete lattice. (Contributed by NM, 5-Dec-2011.)

Theoremclatglble 14552 The greatest lower bound is the least element. (Contributed by NM, 5-Dec-2011.)

Theoremclatleglb 14553* Two ways of expressing "less than or equal to the greatest lower bound." (Contributed by NM, 5-Dec-2011.)

Theoremclatglbss 14554 Subset law for greatest lower bound. (Contributed by Mario Carneiro, 16-Apr-2014.)

9.2.3  The dual of an ordered set

Syntaxcodu 14555 Class function defining dual orders.
ODual

Definitiondf-odu 14556 Define the dual of an ordered structure, which replaces the order component of the structure with its reverse. See odubas 14560, oduleval 14558, and oduleg 14559 for its principal properties.

EDITORIAL: likely usable to simplify many lattice proofs, as it allows for duality arguments to be formalized; for instance latmass 14614. (Contributed by Stefan O'Rear, 29-Jan-2015.)

ODual sSet

Theoremoduval 14557 Value of an order dual structure. (Contributed by Stefan O'Rear, 29-Jan-2015.)
ODual              sSet

Theoremoduleval 14558 Value of the less-equal relation in an order dual structure. (Contributed by Stefan O'Rear, 29-Jan-2015.)
ODual

Theoremoduleg 14559 Truth of the less-equal relation in an order dual structure. (Contributed by Stefan O'Rear, 29-Jan-2015.)
ODual

Theoremodubas 14560 Base set of an order dual structure. (Contributed by Stefan O'Rear, 29-Jan-2015.)
ODual

Theorempospropd 14561* 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.)

Theoremodupos 14562 Being a poset is a self-dual property. (Contributed by Stefan O'Rear, 29-Jan-2015.)
ODual

Theoremoduposb 14563 Being a poset is a self-dual property. (Contributed by Stefan O'Rear, 29-Jan-2015.)
ODual

Theoremmeet0 14564 Lemma for odujoin 14569. (Contributed by Stefan O'Rear, 29-Jan-2015.)

Theoremjoin0 14565 Lemma for odumeet 14567. (Contributed by Stefan O'Rear, 29-Jan-2015.)

Theoremoduglb 14566 Greatest lower bounds in a dual order are least upper bounds in the original order. (Contributed by Stefan O'Rear, 29-Jan-2015.)
ODual

Theoremodumeet 14567 Meets in a dual order are joins in the original. (Contributed by Stefan O'Rear, 29-Jan-2015.)
ODual

Theoremodulub 14568 Least upper bounds in a dual order are greatest lower bounds in the original order. (Contributed by Stefan O'Rear, 29-Jan-2015.)
ODual

Theoremodujoin 14569 Joins in a dual order are meets in the original. (Contributed by Stefan O'Rear, 29-Jan-2015.)
ODual

Theoremodulatb 14570 Being a lattice is self-dual. (Contributed by Stefan O'Rear, 29-Jan-2015.)
ODual

Theoremoduclatb 14571 Being a complete lattice is self-dual. (Contributed by Stefan O'Rear, 29-Jan-2015.)
ODual

Theoremodulat 14572 Being a lattice is self-dual. (Contributed by Stefan O'Rear, 29-Jan-2015.)
ODual

Theoremposlubmo 14573* 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.)

Theoremposlubd 14574* Properties which determine the least upper bound in a poset. (Contributed by Stefan O'Rear, 31-Jan-2015.)

Theoremposlubdg 14575* Properties which determine the least upper bound in a poset. (Contributed by Stefan O'Rear, 31-Jan-2015.)

Theoremposglbd 14576* Properties which determine the greatest lower bound in a poset. (Contributed by Stefan O'Rear, 31-Jan-2015.)

9.2.4  Subset order structures

Syntaxcipo 14577 Class function defining inclusion posets.
toInc

Definitiondf-ipo 14578* For any family of sets, define the poset of that family ordered by inclusion. See ipobas 14581, ipolerval 14582, and ipole 14584 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 TopSet ordTop

Theoremipostr 14579 The structure of df-ipo 14578 is a structure defining indexes up to 11. (Contributed by Mario Carneiro, 25-Oct-2015.)
TopSet Struct ;

Theoremipoval 14580* Value of the inclusion poset. (Contributed by Stefan O'Rear, 30-Jan-2015.)
toInc              TopSet ordTop

Theoremipobas 14581 Base set of the inclusion poset. (Contributed by Stefan O'Rear, 30-Jan-2015.) (Revised by Mario Carneiro, 25-Oct-2015.)
toInc

Theoremipolerval 14582* Relation of the inclusion poset. (Contributed by Stefan O'Rear, 30-Jan-2015.)
toInc

Theoremipotset 14583 Topology of the inclusion poset. (Contributed by Mario Carneiro, 24-Oct-2015.)
toInc              ordTop TopSet

Theoremipole 14584 Weak order condition of the inclusion poset. (Contributed by Stefan O'Rear, 30-Jan-2015.)
toInc

Theoremipolt 14585 Strict order condition of the inclusion poset. (Contributed by Stefan O'Rear, 30-Jan-2015.)
toInc

Theoremipopos 14586 The inclusion poset on a family of sets is actually a poset. (Contributed by Stefan O'Rear, 30-Jan-2015.)
toInc

Theoremisipodrs 14587* Condition for a family of sets to be directed by inclusion. (Contributed by Stefan O'Rear, 2-Apr-2015.)
toInc Dirset

Theoremipodrscl 14588 Direction by inclusion as used here implies sethood. (Contributed by Stefan O'Rear, 2-Apr-2015.)
toInc Dirset

Theoremipodrsfi 14589* Finite upper bound property for directed collections of sets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
toInc Dirset

Theoremfpwipodrs 14590 The finite subsets of any set are directed by inclusion. (Contributed by Stefan O'Rear, 2-Apr-2015.)
toInc Dirset

Theoremipodrsima 14591* The monotone image of a directed set. (Contributed by Stefan O'Rear, 2-Apr-2015.)
toInc Dirset                     toInc Dirset

Theoremisacs3lem 14592* An algebraic closure system satisfies isacs3 14600. (Contributed by Stefan O'Rear, 2-Apr-2015.)
ACS Moore toInc Dirset

Theoremacsdrsel 14593 An algebraic closure system contains all directed unions of closed sets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
ACS toInc Dirset

Theoremisacs4lem 14594* 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.)
mrCls       Moore toInc Dirset Moore toInc Dirset

Theoremisacs5lem 14595* 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.)
mrCls       Moore toInc Dirset Moore

Theoremacsdrscl 14596 In an algebraic closure system, closure commutes with directed unions. (Contributed by Stefan O'Rear, 2-Apr-2015.)
mrCls       ACS toInc Dirset

Theoremacsficl 14597 A closure in an algebraic closure system is the union of the closures of finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
mrCls       ACS

Theoremisacs5 14598* 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.)
mrCls       ACS Moore

Theoremisacs4 14599* A closure system is algebraic iff closure commutes with directed unions. (Contributed by Stefan O'Rear, 2-Apr-2015.)
mrCls       ACS Moore toInc Dirset

Theoremisacs3 14600* A closure system is algebraic iff directed unions of closed sets are closed. (Contributed by Stefan O'Rear, 2-Apr-2015.)
ACS Moore toInc Dirset

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