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Theorem List for Metamath Proof Explorer - 26701-26800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdivrngpr 26701 A division ring is a prime ring. (Contributed by Jeff Madsen, 6-Jan-2011.)

Theoremisdmn 26702 The predicate "is a domain". (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremisdmn2 26703 The predicate "is a domain". (Contributed by Jeff Madsen, 10-Jun-2010.)
CRingOps

Theoremdmncrng 26704 A domain is a commutative ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
CRingOps

Theoremdmnrngo 26705 A domain is a ring. (Contributed by Jeff Madsen, 6-Jan-2011.)

Theoremflddmn 26706 A field is a domain. (Contributed by Jeff Madsen, 10-Jun-2010.)

19.14.19  Ideal generators

Syntaxcigen 26707 Extend class notation with the ideal generation function.

Definitiondf-igen 26708* Define the ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremigenval 26709* The ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof shortened by Mario Carneiro, 20-Dec-2013.)

Theoremigenss 26710 A set is a subset of the ideal it generates. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremigenidl 26711 The ideal generated by a set is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremigenmin 26712 The ideal generated by a set is the minimal ideal containing that set. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremigenidl2 26713 The ideal generated by an ideal is that ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremigenval2 26714* The ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremprnc 26715* A principal ideal (an ideal generated by one element) in a commutative ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
CRingOps

Theoremisfldidl 26716 Determine if a ring is a field based on its ideals. (Contributed by Jeff Madsen, 10-Jun-2010.)
GId       GId       CRingOps

Theoremisfldidl2 26717 Determine if a ring is a field based on its ideals. (Contributed by Jeff Madsen, 6-Jan-2011.)
GId       CRingOps

Theoremispridlc 26718* The predicate "is a prime ideal". Alternate definition for commutative rings. (Contributed by Jeff Madsen, 19-Jun-2010.)
CRingOps

Theorempridlc 26719 Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.)
CRingOps

Theorempridlc2 26720 Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.)
CRingOps

Theorempridlc3 26721 Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.)
CRingOps

Theoremisdmn3 26722* The predicate "is a domain", alternate expression. (Contributed by Jeff Madsen, 19-Jun-2010.)
GId       GId       CRingOps

Theoremdmnnzd 26723 A domain has no zero-divisors (besides zero). (Contributed by Jeff Madsen, 19-Jun-2010.)
GId

Theoremdmncan1 26724 Cancellation law for domains. (Contributed by Jeff Madsen, 6-Jan-2011.)
GId

Theoremdmncan2 26725 Cancellation law for domains. (Contributed by Jeff Madsen, 6-Jan-2011.)
GId

19.15  Mathbox for Rodolfo Medina

19.15.1  Partitions

Theoremprtlem60 26726 Lemma for prter3 26769. (Contributed by Rodolfo Medina, 9-Oct-2010.)

Theorembicomdd 26727 Commute two sides of a biconditional in a deduction. (Contributed by Rodolfo Medina, 19-Oct-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremprtlem1 26728 Add a disjunct in the antecedent. (Contributed by Rodolfo Medina, 24-Sep-2010.)

Theoremjca2 26729 Inference conjoining the consequents of two implications. (Contributed by Rodolfo Medina, 12-Oct-2010.)

Theoremjca2r 26730 Inference conjoining the consequents of two implications. (Contributed by Rodolfo Medina, 17-Oct-2010.)

Theoremjca3 26731 Inference conjoining the consequents of two implications. (Contributed by Rodolfo Medina, 14-Oct-2010.)

Theoremprtlem50 26732 Lemma for prter3 26769. (Contributed by Rodolfo Medina, 12-Oct-2010.)

Theoreman43 26733 Rearrangement of 4 conjuncts. (Contributed by Rodolfo Medina, 24-Sep-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoreman43OLD 26734 Rearrangement of 4 conjuncts. (Contributed by Rodolfo Medina, 24-Sep-2010.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoreman3 26735 A rearrangement of conjuncts. (Contributed by Rodolfo Medina, 25-Sep-2010.)

Theoremprtlem70 26736 Lemma for prter3 26769: a rearrangement of conjuncts. (Contributed by Rodolfo Medina, 20-Oct-2010.)

Theoremibdr 26737 Reverse of ibd. (Contributed by Rodolfo Medina, 30-Sep-2010.)

Theorempm5.31r 26738 Variant of pm5.31 573. (Contributed by Rodolfo Medina, 15-Oct-2010.)

Theoremexan3 26739 Cancel a conjunct from the scope of an existential quantifier. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremexan3OLD 26740 Cancel a conjunct from the scope of an existential quantifier. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorem2r19.29 26741 Double the quantifiers of theorem r19.29. (Contributed by Rodolfo Medina, 25-Sep-2010.)

Theoremprtlem100 26742 Lemma for prter3 26769. (Contributed by Rodolfo Medina, 19-Oct-2010.)

Theoremprtlem5 26743* Lemma for prter1 26766, prter2 26768, prter3 26769 and prtex 26767. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)

Theoremprtlem90 26744 Lemma for prter2 26768. (Contributed by Rodolfo Medina, 17-Oct-2010.)

Theoremprtlem80 26745 Lemma for prter2 26768. (Contributed by Rodolfo Medina, 17-Oct-2010.)

Theoremn0el 26746* Negated membership of the empty set in another class. (Contributed by Rodolfo Medina, 25-Sep-2010.)

Theoremceqsex3OLD 26747* Version of ceqsex 2996 with an antecedent instead of a hypothesis. (Use ceqsexg 3073 instead of this one. --NM 13-Aug-11) (Contributed by Rodolfo Medina, 19-Oct-2010.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremceqsex3vOLD 26748* Version of ceqsexv 2997 with an antecedent instead of a hypothesis. (Use ceqsexgv 3074 instead of this one. --NM 13-Aug-11) (Contributed by Rodolfo Medina, 19-Oct-2010.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorembrabsb2 26749* Closed form of brabsbOLD 4493. (Contributed by Rodolfo Medina, 13-Oct-2010.)

Theoremeqbrrdv2 26750* Other version of eqbrrdiv 5003. (Contributed by Rodolfo Medina, 30-Sep-2010.)

Theoremprtlem9 26751* Lemma for prter3 26769. (Contributed by Rodolfo Medina, 25-Sep-2010.)

Theoremprtlem10 26752* Lemma for prter3 26769. (Contributed by Rodolfo Medina, 14-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremprtlem11 26753 Lemma for prter2 26768. (Contributed by Rodolfo Medina, 12-Oct-2010.)

Theoremprtlem12 26754* Lemma for prtex 26767 and prter3 26769. (Contributed by Rodolfo Medina, 13-Oct-2010.)

Theoremprtlem13 26755* Lemma for prter1 26766, prter2 26768, prter3 26769 and prtex 26767. (Contributed by Rodolfo Medina, 13-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremprtlem16 26756* Lemma for prtex 26767, prter2 26768 and prter3 26769. (Contributed by Rodolfo Medina, 14-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremprtlem400 26757* Lemma for prter2 26768 and also a property of partitions . (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)

Syntaxwprt 26758 Extend the definition of a wff to include the partition predicate.

Definitiondf-prt 26759* Define the partition predicate. (Contributed by Rodolfo Medina, 13-Oct-2010.)

Theoremerprt 26760 The quotient set of an equivalence relation is a partition. (Contributed by Rodolfo Medina, 13-Oct-2010.)

Theoremprtlem14 26761* Lemma for prter1 26766, prter2 26768 and prtex 26767. (Contributed by Rodolfo Medina, 13-Oct-2010.)

Theoremprtlem15 26762* Lemma for prter1 26766 and prtex 26767. (Contributed by Rodolfo Medina, 13-Oct-2010.)

Theoremprtlem17 26763* Lemma for prter2 26768. (Contributed by Rodolfo Medina, 15-Oct-2010.)

Theoremprtlem18 26764* Lemma for prter2 26768. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremprtlem19 26765* Lemma for prter2 26768. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremprter1 26766* Every partition generates an equivalence relation. (Contributed by Rodolfo Medina, 13-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremprtex 26767* The equivalence relation generated by a partition is a set if and only if the partition itself is a set. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremprter2 26768* The quotient set of the equivalence relation generated by a partition equals the partition itself. (Contributed by Rodolfo Medina, 17-Oct-2010.)

Theoremprter3 26769* For every partition there exists a unique equivalence relation whose quotient set equals the partition. (Contributed by Rodolfo Medina, 19-Oct-2010.) (Proof shortened by Mario Carneiro, 12-Aug-2015.)

19.16  Mathbox for Stefan O'Rear

19.16.1  Additional elementary logic and set theory

Theoremnelss 26770 Demonstrate by witnesses that two classes lack a subclass relation. (Contributed by Stefan O'Rear, 5-Feb-2015.)

Theoremmoxfr 26771* Transfer at-most-one between related expressions. (Contributed by Stefan O'Rear, 12-Feb-2015.)

Theoremraldifsni 26772 Rearrangement of a property of a singleton difference. (Contributed by Stefan O'Rear, 27-Feb-2015.)

19.16.2  Additional theory of functions

Theoremfninfp 26773* Express the class of fixed points of a function. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Theoremfnelfp 26774 Property of a fixed point of a function. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Theoremfndifnfp 26775* Express the class of non-fixed points of a function. (Contributed by Stefan O'Rear, 14-Aug-2015.)

Theoremfnelnfp 26776 Property of a non-fixed point of a function. (Contributed by Stefan O'Rear, 15-Aug-2015.)

Theoremfnnfpeq0 26777 A function is the identity iff it moves no points. (Contributed by Stefan O'Rear, 25-Aug-2015.)

Theoremimaiinfv 26778* Indexed intersection of an image. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theoremralxpxfr2d 26779* Transfer a universal quantifier between one variable with pair-like semantics and two. (Contributed by Stefan O'Rear, 27-Feb-2015.)

Theoremralxpmap 26780* Quantification over functions in terms of quantification over values and punctured functions. (Contributed by Stefan O'Rear, 27-Feb-2015.) (Revised by Stefan O'Rear, 5-May-2015.)

Theoremfunsnfsup 26781 Finite support for a function extended by a singleton. (Contributed by Stefan O'Rear, 27-Feb-2015.)

Theoremfvtresfn 26782* Functionality of a tuple-restriction function. (Contributed by Stefan O'Rear, 24-Jan-2015.)

19.16.3  Extensions beyond function theory

Theoremgsumvsmul 26783* Pull a scalar multiplication out of a sum of vectors. EDITORIAL: properly generalizes gsummulc2 15745, since every ring is a left module over itself. (Contributed by Stefan O'Rear, 6-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
Scalar                                                                      g g

Theoremlcomf 26784 A linear-combination sum is a function. (Contributed by Stefan O'Rear, 28-Feb-2015.)
Scalar

Theoremlcomfsup 26785 A linear-combination sum is finitely supported if the coefficients are. (Contributed by Stefan O'Rear, 28-Feb-2015.)
Scalar

19.16.4  Additional topology

Theoremelrfi 26786* Elementhood in a set of relative finite intersections. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theoremelrfirn 26787* Elementhood in a set of relative finite intersections of an indexed family of sets. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theoremelrfirn2 26788* Elementhood in a set of relative finite intersections of an indexed family of sets (implicit). (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theoremcmpfiiin 26789* In a compact topology, a system of closed sets with nonempty finite intersections has a nonempty intersection. (Contributed by Stefan O'Rear, 22-Feb-2015.)

19.16.5  Characterization of closure operators. Kuratowski closure axioms

Theoremismrcd1 26790* Any function from the subsets of a set to itself, which is extensive (satisfies mrcssid 13873), isotone (satisfies mrcss 13872), and idempotent (satisfies mrcidm 13875) has a collection of fixed points which is a Moore collection, and itself is the closure operator for that collection. This can be taken as an alternate definition for the closure operators. This is the first half, ismrcd2 26791 is the second. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Moore

Theoremismrcd2 26791* Second half of ismrcd1 26790. (Contributed by Stefan O'Rear, 1-Feb-2015.)
mrCls

Theoremistopclsd 26792* A closure function which satisfies sscls 17151, clsidm 17162, cls0 17175, and clsun 26369 defines a (unique) topology which it is the closure function on. (Contributed by Stefan O'Rear, 1-Feb-2015.)
TopOn

Theoremismrc 26793* A function is a Moore closure operator iff it satisfies mrcssid 13873, mrcss 13872, and mrcidm 13875. (Contributed by Stefan O'Rear, 1-Feb-2015.)
mrClsMoore

19.16.6  Algebraic closure systems

Syntaxcnacs 26794 Class of Noetherian closure systems.
NoeACS

Definitiondf-nacs 26795* Define a closure system of Noetherian type (not standard terminology) as an algebraic system where all closed sets are finitely generated. (Contributed by Stefan O'Rear, 4-Apr-2015.)
NoeACS ACS mrCls

Theoremisnacs 26796* Expand definition of Noetherian-type closure system. (Contributed by Stefan O'Rear, 4-Apr-2015.)
mrCls       NoeACS ACS

Theoremnacsfg 26797* In a Noetherian-type closure system, all closed sets are finitely generated. (Contributed by Stefan O'Rear, 4-Apr-2015.)
mrCls       NoeACS

Theoremisnacs2 26798 Express Noetherian-type closure system with fewer quantifiers. (Contributed by Stefan O'Rear, 4-Apr-2015.)
mrCls       NoeACS ACS

Theoremmrefg2 26799* Slight variation on finite genration for closure systems. (Contributed by Stefan O'Rear, 4-Apr-2015.)
mrCls       Moore

Theoremmrefg3 26800* Slight variation on finite genration for closure systems. (Contributed by Stefan O'Rear, 4-Apr-2015.)
mrCls       Moore

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