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Theorem List for Metamath Proof Explorer - 25801-25900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-idl 25801* Define the class of (two-sided) ideals of a ring . A subset of is an ideal if it contains , is closed under addition, and is closed under multiplication on either side by any element of . (Contributed by Jeff Madsen, 10-Jun-2010.)
GId

Definitiondf-pridl 25802* Define the class of prime ideals of a ring . A proper ideal of is prime if whenever for ideals and , either or . The more familiar definition using elements rather than ideals is equivalent provided is commutative; see ispridl2 25829 and ispridlc 25861. (Contributed by Jeff Madsen, 10-Jun-2010.)

Definitiondf-maxidl 25803* Define the class of maximal ideals of a ring . A proper ideal is called maximal if it is maximal with respect to inclusion among proper ideals. (Contributed by Jeff Madsen, 5-Jan-2011.)

Theoremidlval 25804* The class of ideals of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
GId

Theoremisidl 25805* The predicate "is an ideal of the ring ." (Contributed by Jeff Madsen, 10-Jun-2010.)
GId

Theoremisidlc 25806* The predicate "is an ideal of the commutative ring ." (Contributed by Jeff Madsen, 10-Jun-2010.)
GId       CRingOps

Theoremidlss 25807 An ideal of is a subset of . (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremidlcl 25808 An element of an ideal is an element of the ring. (Contributed by Jeff Madsen, 19-Jun-2010.)

Theoremidl0cl 25809 An ideal contains . (Contributed by Jeff Madsen, 10-Jun-2010.)
GId

Theoremidladdcl 25810 An ideal is closed under addition. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremidllmulcl 25811 An ideal is closed under multiplication on the left. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremidlrmulcl 25812 An ideal is closed under multiplication on the right. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremidlnegcl 25813 An ideal is closed under negation. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremidlsubcl 25814 An ideal is closed under subtraction. (Contributed by Jeff Madsen, 19-Jun-2010.)

Theoremrngoidl 25815 A ring is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theorem0idl 25816 The set containing only is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
GId

Theorem1idl 25817 Two ways of expressing the unit ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
GId

Theorem0rngo 25818 In a ring, iff the ring contains only . (Contributed by Jeff Madsen, 6-Jan-2011.)
GId       GId

Theoremdivrngidl 25819 The only ideals in a division ring are the zero ideal and the unit ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
GId

Theoremintidl 25820 The intersection of a nonempty collection of ideals is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoreminidl 25821 The intersection of two ideals is an ideal. (Contributed by Jeff Madsen, 16-Jun-2011.)

Theoremunichnidl 25822* The union of a nonempty chain of ideals is an ideal. (Contributed by Jeff Madsen, 5-Jan-2011.)

Theoremkeridl 25823 The kernel of a ring homomorphism is an ideal. (Contributed by Jeff Madsen, 3-Jan-2011.)
GId

Theorempridlval 25824* The class of prime ideals of a ring . (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremispridl 25825* The predicate "is a prime ideal". (Contributed by Jeff Madsen, 10-Jun-2010.)

Theorempridlidl 25826 A prime ideal is an ideal. (Contributed by Jeff Madsen, 19-Jun-2010.)

Theorempridlnr 25827 A prime ideal is a proper ideal. (Contributed by Jeff Madsen, 19-Jun-2010.)

Theorempridl 25828* The main property of a prime ideal. (Contributed by Jeff Madsen, 19-Jun-2010.)

Theoremispridl2 25829* A condition that shows an ideal is prime. For commutative rings, this is often taken to be the definition. See ispridlc 25861 for the equivalence in the commutative case. (Contributed by Jeff Madsen, 19-Jun-2010.)

Theoremmaxidlval 25830* The set of maximal ideals of a ring. (Contributed by Jeff Madsen, 5-Jan-2011.)

Theoremismaxidl 25831* The predicate "is a maximal ideal". (Contributed by Jeff Madsen, 5-Jan-2011.)

Theoremmaxidlidl 25832 A maximal ideal is an ideal. (Contributed by Jeff Madsen, 5-Jan-2011.)

Theoremmaxidlnr 25833 A maximal ideal is proper. (Contributed by Jeff Madsen, 16-Jun-2011.)

Theoremmaxidlmax 25834 A maximal ideal is a maximal proper ideal. (Contributed by Jeff Madsen, 16-Jun-2011.)

Theoremmaxidln1 25835 One is not contained in any maximal ideal. (Contributed by Jeff Madsen, 17-Jun-2011.)
GId

Theoremmaxidln0 25836 A ring with a maximal ideal is not the zero ring. (Contributed by Jeff Madsen, 17-Jun-2011.)
GId       GId

16.13.19  Prime rings and integral domains

Syntaxcprrng 25837 Extend class notation with the class of prime rings.

Syntaxcdmn 25838 Extend class notation with the class of domains.

Definitiondf-prrngo 25839 Define the class of prime rings. A ring is prime if the zero ideal is a prime ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
GId

Definitiondf-dmn 25840 Define the class of (integral) domains. A domain is a commutative prime ring. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremisprrngo 25841 The predicate "is a prime ring". (Contributed by Jeff Madsen, 10-Jun-2010.)
GId

Theoremprrngorngo 25842 A prime ring is a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremsmprngopr 25843 A simple ring (one whose only ideals are and ) is a prime ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
GId       GId

Theoremdivrngpr 25844 A division ring is a prime ring. (Contributed by Jeff Madsen, 6-Jan-2011.)

Theoremisdmn 25845 The predicate "is a domain". (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremisdmn2 25846 The predicate "is a domain". (Contributed by Jeff Madsen, 10-Jun-2010.)
CRingOps

Theoremdmncrng 25847 A domain is a commutative ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
CRingOps

Theoremdmnrngo 25848 A domain is a ring. (Contributed by Jeff Madsen, 6-Jan-2011.)

Theoremflddmn 25849 A field is a domain. (Contributed by Jeff Madsen, 10-Jun-2010.)

16.13.20  Ideal generators

Syntaxcigen 25850 Extend class notation with the ideal generation function.

Definitiondf-igen 25851* Define the ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremigenval 25852* 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 25853 A set is a subset of the ideal it generates. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremigenidl 25854 The ideal generated by a set is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremigenmin 25855 The ideal generated by a set is the minimal ideal containing that set. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremigenidl2 25856 The ideal generated by an ideal is that ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremigenval2 25857* The ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremprnc 25858* A principal ideal (an ideal generated by one element) in a commutative ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
CRingOps

Theoremisfldidl 25859 Determine if a ring is a field based on its ideals. (Contributed by Jeff Madsen, 10-Jun-2010.)
GId       GId       CRingOps

Theoremisfldidl2 25860 Determine if a ring is a field based on its ideals. (Contributed by Jeff Madsen, 6-Jan-2011.)
GId       CRingOps

Theoremispridlc 25861* The predicate "is a prime ideal". Alternate definition for commutative rings. (Contributed by Jeff Madsen, 19-Jun-2010.)
CRingOps

Theorempridlc 25862 Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.)
CRingOps

Theorempridlc2 25863 Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.)
CRingOps

Theorempridlc3 25864 Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.)
CRingOps

Theoremisdmn3 25865* The predicate "is a domain", alternate expression. (Contributed by Jeff Madsen, 19-Jun-2010.)
GId       GId       CRingOps

Theoremdmnnzd 25866 A domain has no zero-divisors (besides zero). (Contributed by Jeff Madsen, 19-Jun-2010.)
GId

Theoremdmncan1 25867 Cancellation law for domains. (Contributed by Jeff Madsen, 6-Jan-2011.)
GId

Theoremdmncan2 25868 Cancellation law for domains. (Contributed by Jeff Madsen, 6-Jan-2011.)
GId

16.14  Mathbox for Rodolfo Medina

16.14.1  Partitions

Theoremprtlem60 25869 Lemma for prter3 25916. (Contributed by Rodolfo Medina, 9-Oct-2010.)

TheoremimpbiddOLD 25870 Lemma for prter3 25916. (Moved to impbidd 183 in main set.mm and may be deleted by mathbox owner, RM. --NM 15-May-2013.) (Contributed by Rodolfo Medina, 12-Oct-2010.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorembicomdd 25871 Commute two sides of a biconditional in a deduction. (Contributed by Rodolfo Medina, 19-Oct-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

TheorembicomddOLD 25872 Commute two sides of a biconditional in a deduction. (Contributed by Rodolfo Medina, 19-Oct-2010.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremprtlem1 25873 Add a disjunct in the antecedent. (Contributed by Rodolfo Medina, 24-Sep-2010.)

Theoremjca2 25874 Inference conjoining the consequents of two implications. (Contributed by Rodolfo Medina, 12-Oct-2010.)

Theoremjca2r 25875 Inference conjoining the consequents of two implications. (Contributed by Rodolfo Medina, 17-Oct-2010.)

Theoremjca3 25876 Inference conjoining the consequents of two implications. (Contributed by Rodolfo Medina, 14-Oct-2010.)

Theoremprtlem50 25877 Lemma for prter3 25916. (Contributed by Rodolfo Medina, 12-Oct-2010.)

Theoreman43 25878 Rearrangement of 4 conjuncts. (Contributed by Rodolfo Medina, 24-Sep-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoreman43OLD 25879 Rearrangement of 4 conjuncts. (Contributed by Rodolfo Medina, 24-Sep-2010.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoreman3 25880 A rearrangement of conjuncts. (Contributed by Rodolfo Medina, 25-Sep-2010.)

Theoremprtlem70 25881 Lemma for prter3 25916: a rearrangement of conjuncts. (Contributed by Rodolfo Medina, 20-Oct-2010.)

Theoremibdr 25882 Reverse of ibd. (Contributed by Rodolfo Medina, 30-Sep-2010.)

Theorempm5.31r 25883 Variant of pm5.31 574. (Contributed by Rodolfo Medina, 15-Oct-2010.)

Theoremexan3 25884 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 25885 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 25886 Double the quantifiers of theorem r19.29. (Contributed by Rodolfo Medina, 25-Sep-2010.)

Theoremprtlem100 25887 Lemma for prter3 25916. (Contributed by Rodolfo Medina, 19-Oct-2010.)

Theoremprtlem5 25888* Lemma for prter1 25913, prter2 25915, prter3 25916 and prtex 25914. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)

Theoremprtlem90 25889 Lemma for prter2 25915. (Contributed by Rodolfo Medina, 17-Oct-2010.)

Theoremprtlem80 25890 Lemma for prter2 25915. (Contributed by Rodolfo Medina, 17-Oct-2010.)

Theoremn0el 25891* Negated membership of the empty set in another class. (Contributed by Rodolfo Medina, 25-Sep-2010.)

Theoremceqsex3OLD 25892* Version of ceqsex 2760 with an antecedent instead of a hypothesis. (Use ceqsexg 2836 instead of this one. --NM 13-Aug-11) (Contributed by Rodolfo Medina, 19-Oct-2010.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremceqsex3vOLD 25893* Version of ceqsexv 2761 with an antecedent instead of a hypothesis. (Use ceqsexgv 2837 instead of this one. --NM 13-Aug-11) (Contributed by Rodolfo Medina, 19-Oct-2010.) (Proof modification is discouraged.) (New usage is discouraged.)

TheoremeqrelrdvOLD 25894* Deduce equality of relations from equivalence of membership. (Moved to eqrelrdv 4690 in main set.mm and may be deleted by mathbox owner, RM. --NM 20-Feb-2014.) (Contributed by Rodolfo Medina, 10-Oct-2010.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremeqrelrdv2OLD 25895* Another version of eqrelrdv 4690. (Moved to eqrelrdv2 4693 in main set.mm and may be deleted by mathbox owner, RM. --NM 20-Feb-2014.) (Contributed by Rodolfo Medina, 30-Sep-2010.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorembrabsb2 25896* Closed form of brabsbOLD 4167. (Contributed by Rodolfo Medina, 13-Oct-2010.)

Theoremeqbrrdv2 25897* Other version of eqbrrdiv 4692. (Contributed by Rodolfo Medina, 30-Sep-2010.)

Theoremprtlem9 25898* Lemma for prter3 25916. (Contributed by Rodolfo Medina, 25-Sep-2010.)

Theoremprtlem10 25899* Lemma for prter3 25916. (Contributed by Rodolfo Medina, 14-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremprtlem11 25900 Lemma for prter2 25915. (Contributed by Rodolfo Medina, 12-Oct-2010.)

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