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Theorem List for Metamath Proof Explorer - 14901-15000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmulgnn0p1 14901 Group multiple (exponentiation) operation at a successor, extended to . (Contributed by Mario Carneiro, 11-Dec-2014.)
.g

Theoremmulgnnsubcl 14902* Closure of the group multiple (exponentiation) operation in a subsemigroup. (Contributed by Mario Carneiro, 10-Jan-2015.)
.g

Theoremmulgnn0subcl 14903* Closure of the group multiple (exponentiation) operation in a submonoid. (Contributed by Mario Carneiro, 10-Jan-2015.)
.g

Theoremmulgsubcl 14904* Closure of the group multiple (exponentiation) operation in a subgroup. (Contributed by Mario Carneiro, 10-Jan-2015.)
.g

Theoremmulgnncl 14905 Closure of the group multiple (exponentiation) operation. TODO: This can be generalized to a magma if/when we introduce them. (Contributed by Mario Carneiro, 11-Dec-2014.)
.g

Theoremmulgnn0cl 14906 Closure of the group multiple (exponentiation) operation. (Contributed by Mario Carneiro, 11-Dec-2014.)
.g

Theoremmulgcl 14907 Closure of the group multiple (exponentiation) operation. (Contributed by Mario Carneiro, 11-Dec-2014.)
.g

Theoremmulgneg 14908 Group multiple (exponentiation) operation at a negative integer. (Contributed by Mario Carneiro, 11-Dec-2014.)
.g

Theoremmulgm1 14909 Group multiple (exponentiation) operation at negative one. (Contributed by Mario Carneiro, 20-Dec-2014.)
.g

Theoremmulgnn0z 14910 A group multiple of the identity, for nonnegative multiple. (Contributed by Mario Carneiro, 13-Dec-2014.)
.g

Theoremmulgz 14911 A group multiple of the identity, for integer multiple. (Contributed by Mario Carneiro, 13-Dec-2014.)
.g

Theoremmulgnndir 14912 Sum of group multiples, for positive multiples. TODO: This can be generalized to a semigroup if/when we introduce them. (Contributed by Mario Carneiro, 11-Dec-2014.)
.g

Theoremmulgnn0dir 14913 Sum of group multiples, generalized to . (Contributed by Mario Carneiro, 11-Dec-2014.)
.g

Theoremmulgdirlem 14914 Lemma for mulgdir 14915. (Contributed by Mario Carneiro, 13-Dec-2014.)
.g

Theoremmulgdir 14915 Sum of group multiples, generalized to . (Contributed by Mario Carneiro, 13-Dec-2014.)
.g

Theoremmulgp1 14916 Group multiple (exponentiation) operation at a successor, extended to . (Contributed by Mario Carneiro, 11-Dec-2014.)
.g

Theoremmulgneg2 14917 Group multiple (exponentiation) operation at a negative integer. (Contributed by Mario Carneiro, 13-Dec-2014.)
.g

Theoremmulgnnass 14918 Product of group multiples, for positive multiples. TODO: This can be generalized to a semigroup if/when we introduce them. (Contributed by Mario Carneiro, 13-Dec-2014.)
.g

Theoremmulgnn0ass 14919 Product of group multiples, generalized to . (Contributed by Mario Carneiro, 13-Dec-2014.)
.g

Theoremmulgass 14920 Product of group multiples, generalized to . (Contributed by Mario Carneiro, 13-Dec-2014.)
.g

Theoremmulgsubdir 14921 Subtraction of a group element from itself. (Contributed by Mario Carneiro, 13-Dec-2014.)
.g

Theoremmhmmulg 14922 A homomorphism of monoids preserves group multiples. (Contributed by Mario Carneiro, 14-Jun-2015.)
.g       .g       MndHom

Theoremmulgpropd 14923* Two structures with the same group-nature have the same group multiple function. is expected to either be (when strong equality is available) or (when closure is available). (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
.g       .g

Theoremsubmmulgcl 14924 Closure of the group multiple (exponentiation) operation in a submonoid. (Contributed by Mario Carneiro, 13-Jan-2015.)
.g       SubMnd

Theoremsubmmulg 14925 A group multiple is the same if evaluated in a submonoid. (Contributed by Mario Carneiro, 15-Jun-2015.)
.g       s        .g       SubMnd

Theoremprdsinvlem 14926* Characterization of inverses in a structure product. (Contributed by Mario Carneiro, 10-Jan-2015.)
s

Theoremprdsgrpd 14927 The product of a family of groups is a group. (Contributed by Stefan O'Rear, 10-Jan-2015.)
s

Theoremprdsinvgd 14928* Negation in a product of groups. (Contributed by Stefan O'Rear, 10-Jan-2015.)
s

Theorempwsgrp 14929 The product of a family of groups is a group. (Contributed by Mario Carneiro, 11-Jan-2015.)
s

Theorempwsinvg 14930 Negation in a group power. (Contributed by Mario Carneiro, 11-Jan-2015.)
s

Theorempwssub 14931 Subtraction in a group power. (Contributed by Mario Carneiro, 12-Jan-2015.)
s

Theorempwsmulg 14932 Value of a group multiple in a structure power. (Contributed by Mario Carneiro, 15-Jun-2015.)
s               .g       .g

Theoremimasgrp2 14933* The image structure of a group is a group. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-2015.)
s

Theoremimasgrp 14934* The image structure of a group is a group. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-2015.)
s

Theoremimasgrpf1 14935 The image of a group under an injection is a group. (Contributed by Mario Carneiro, 20-Aug-2015.)
s

Theoremdivsgrp2 14936* Prove that a quotient structure is a group. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)
s

Theoremxpsgrp 14937 The binary product of groups is a group. (Contributed by Mario Carneiro, 20-Aug-2015.)
s

10.2.2  Subgroups and Quotient groups

Syntaxcsubg 14938 Extend class notation with all subgroups of a group.
SubGrp

Syntaxcnsg 14939 Extend class notation with all normal subgroups of a group.
NrmSGrp

Syntaxcqg 14940 Quotient group equivalence class.
~QG

Definitiondf-subg 14941* Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 14959), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 14954), contains the neutral element of the group (see subg0 14950) and contains the inverses for all of its elements (see subginvcl 14953). (Contributed by Mario Carneiro, 2-Dec-2014.)
SubGrp s

Definitiondf-nsg 14942* Define the equivalence relation in a quotient ring or quotient group (where is a two-sided ideal or a normal subgroup). For non-normal subgroups this generates the left cosets. (Contributed by Mario Carneiro, 15-Jun-2015.)
NrmSGrp SubGrp

Definitiondf-eqg 14943* Define the equivalence relation in a quotient ring or quotient group (where is a two-sided ideal or a normal subgroup). For non-normal subgroups this generates the left cosets. (Contributed by Mario Carneiro, 15-Jun-2015.)
~QG

Theoremissubg 14944 The subgroup predicate. (Contributed by Mario Carneiro, 2-Dec-2014.)
SubGrp s

Theoremsubgss 14945 A subgroup is a subset. (Contributed by Mario Carneiro, 2-Dec-2014.)
SubGrp

Theoremsubgid 14946 A group is a subgroup of itself. (Contributed by Mario Carneiro, 7-Dec-2014.)
SubGrp

Theoremsubggrp 14947 A subgroup is a group. (Contributed by Mario Carneiro, 2-Dec-2014.)
s        SubGrp

Theoremsubgbas 14948 The base of the restricted group in a subgroup. (Contributed by Mario Carneiro, 2-Dec-2014.)
s        SubGrp

Theoremsubgrcl 14949 Reverse closure for the subgroup predicate. (Contributed by Mario Carneiro, 2-Dec-2014.)
SubGrp

Theoremsubg0 14950 A subgroup of a group must have the same identity as the group. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
s               SubGrp

Theoremsubginv 14951 The inverse of an element in a subgroup is the same as the inverse in the larger group. (Contributed by Mario Carneiro, 2-Dec-2014.)
s                      SubGrp

Theoremsubg0cl 14952 The group identity is an element of any subgroup. (Contributed by Mario Carneiro, 2-Dec-2014.)
SubGrp

Theoremsubginvcl 14953 The inverse of an element is closed in a subgroup. (Contributed by Mario Carneiro, 2-Dec-2014.)
SubGrp

Theoremsubgcl 14954 A subgroup is closed under group operation. (Contributed by Mario Carneiro, 2-Dec-2014.)
SubGrp

Theoremsubgsubcl 14955 A subgroup is closed under group subtraction. (Contributed by Mario Carneiro, 18-Jan-2015.)
SubGrp

Theoremsubgsub 14956 The subtraction of elements in a subgroup is the same as subtraction in the group. (Contributed by Mario Carneiro, 15-Jun-2015.)
s               SubGrp

Theoremsubgmulgcl 14957 Closure of the group multiple (exponentiation) operation in a subgroup. (Contributed by Mario Carneiro, 13-Jan-2015.)
.g       SubGrp

Theoremsubgmulg 14958 A group multiple is the same if evaluated in a subgroup. (Contributed by Mario Carneiro, 15-Jan-2015.)
.g       s        .g       SubGrp

Theoremissubg2 14959* Characterize the subgroups of a group by closure properties. (Contributed by Mario Carneiro, 2-Dec-2014.)
SubGrp

Theoremissubg3 14960* A subgroup is a symmetric submonoid. (Contributed by Mario Carneiro, 7-Mar-2015.)
SubGrp SubMnd

Theoremissubg4 14961* A subgroup is a nonempty subset of the group closed under subtraction. (Contributed by Mario Carneiro, 17-Sep-2015.)
SubGrp

Theoremsubgsubm 14962 A subgroup is a submonoid. (Contributed by Mario Carneiro, 18-Jun-2015.)
SubGrp SubMnd

Theoremsubsubg 14963 A subgroup of a subgroup is a subgroup. (Contributed by Mario Carneiro, 19-Jan-2015.)
s        SubGrp SubGrp SubGrp

Theoremsubgint 14964 The intersection of a nonempty collection of subgroups is a subgroup. (Contributed by Mario Carneiro, 7-Dec-2014.)
SubGrp SubGrp

Theorem0subg 14965 The zero subgroup of an arbitrary group. (Contributed by Stefan O'Rear, 10-Dec-2014.)
SubGrp

Theoremcycsubgcl 14966* The set of integer powers of an element of a group forms a subgroup containing , called the cyclic group generated by the element . (Contributed by Mario Carneiro, 13-Jan-2015.)
.g              SubGrp

Theoremcycsubgss 14967* The cyclic subgroup generated by an element is a subset of any subgroup containing . (Contributed by Mario Carneiro, 13-Jan-2015.)
.g              SubGrp

Theoremcycsubg 14968* The cyclic group generated by is the smallest subgroup containing . (Contributed by Mario Carneiro, 13-Jan-2015.)
.g              SubGrp

Theoremisnsg 14969* Property of being a normal subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
NrmSGrp SubGrp

Theoremisnsg2 14970* Weaken the condition of isnsg 14969 to only one side of the implication. (Contributed by Mario Carneiro, 18-Jan-2015.)
NrmSGrp SubGrp

Theoremnsgbi 14971 Defining property of a normal subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
NrmSGrp

Theoremnsgsubg 14972 A normal subgroup is a subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
NrmSGrp SubGrp

Theoremnsgconj 14973 The conjugation of an element of a normal subgroup is in the subgroup. (Contributed by Mario Carneiro, 4-Feb-2015.)
NrmSGrp

Theoremisnsg3 14974* A subgroup is normal iff the conjugation of all the elements of the subgroup is in the subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
NrmSGrp SubGrp

Theoremsubgacs 14975 Subgroups are an algebraic closure system. (Contributed by Stefan O'Rear, 4-Apr-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
SubGrp ACS

Theoremnsgacs 14976 Normal subgroups form an algebraic closure system. (Contributed by Stefan O'Rear, 4-Sep-2015.)
NrmSGrp ACS

Theoremcycsubg2 14977* The subgroup generated by an element is exhausted by its multiples. (Contributed by Stefan O'Rear, 6-Sep-2015.)
.g              mrClsSubGrp

Theoremcycsubg2cl 14978 Any multiple of an element is contained in the generated cyclic subgroup. (Contributed by Stefan O'Rear, 12-Sep-2015.)
.g       mrClsSubGrp

Theoremelnmz 14979* Elementhood in the normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)

Theoremnmzbi 14980* Defining property of the normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)

Theoremnmzsubg 14981* The normalizer NG(S) of a subset of the group is a subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
SubGrp

Theoremssnmz 14982* A subgroup is a subset of its normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)
SubGrp

Theoremisnsg4 14983* A subgroup is normal iff its normalizer is the entire group. (Contributed by Mario Carneiro, 18-Jan-2015.)
NrmSGrp SubGrp

Theoremnmznsg 14984* Any subgroup is a normal subgroup of its normalizer. (Contributed by Mario Carneiro, 19-Jan-2015.)
s        SubGrp NrmSGrp

Theorem0nsg 14985 The zero subgroup is normal. (Contributed by Mario Carneiro, 4-Feb-2015.)
NrmSGrp

Theoremnsgid 14986 The whole group is a normal subgroup of itself. (Contributed by Mario Carneiro, 4-Feb-2015.)
NrmSGrp

Theoremreleqg 14987 The left coset equivalence relation is a relation. (Contributed by Mario Carneiro, 14-Jun-2015.)
~QG

Theoremeqgfval 14988* Value of the subgroup left coset equivalence relation. (Contributed by Mario Carneiro, 15-Jan-2015.)
~QG

Theoremeqgval 14989 Value of the subgroup left coset equivalence relation. (Contributed by Mario Carneiro, 15-Jan-2015.) (Revised by Mario Carneiro, 14-Jun-2015.)
~QG

Theoremeqger 14990 The subgroup coset equivalence relation is an equivalence relation. (Contributed by Mario Carneiro, 13-Jan-2015.)
~QG        SubGrp

Theoremeqglact 14991* A left coset can be expressed as the image of a left action. (Contributed by Mario Carneiro, 20-Sep-2015.)
~QG

Theoremeqgid 14992 The left coset containing the identity is the original subgroup. (Contributed by Mario Carneiro, 20-Sep-2015.)
~QG               SubGrp

Theoremeqgen 14993 Each coset is equipotent to the subgroup itself (which is also the coset containing the identity). (Contributed by Mario Carneiro, 20-Sep-2015.)
~QG        SubGrp

Theoremeqgcpbl 14994 The subgroup coset equivalence relation is compatible with addition when the subgroup is normal. (Contributed by Mario Carneiro, 14-Jun-2015.)
~QG               NrmSGrp

Theoremdivsgrp 14995 If is a normal subgroup of , then is a group, called the quotient of by . (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)
s ~QG        NrmSGrp

Theoremdivseccl 14996 Closure of the quotient map for a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.)
s ~QG                      NrmSGrp ~QG

Theoremdivsadd 14997 Value of the group operation in a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.)
s ~QG                             NrmSGrp ~QG ~QG ~QG

Theoremdivs0 14998 Value of the group identity operation in a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.)
s ~QG               NrmSGrp ~QG

Theoremdivsinv 14999 Value of the group inverse operation in a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.)
s ~QG                             NrmSGrp ~QG ~QG

Theoremdivssub 15000 Value of the group subtraction operation in a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.)
s ~QG                             NrmSGrp ~QG ~QG ~QG

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