mmtheorems82 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 8101-8200 - Page 82 of 108

Type	Label	Description
Statement
Theorem	abl4 8101	Commutative/associative law for Abelian groups.
|- X = ran G   =>   |- ((G e. Abel /\ (A e. X /\ B e. X) /\ (C e. X /\ D e. X)) -> ((AGB)G(CGD)) = ((AGC)G(BGD)))
Theorem	isabli 8102	Properties that determine an Abelian group operation.
|- G e. Grp   &   |- dom G = (X X. X)   &   |- ((x e. X /\ y e. X) -> (xGy) = (yGx))   =>   |- G e. Abel
Theorem	ablmuldiv 8103	Law for group multiplication and division.
|- X = ran G   &   |- D = ( /g ` G)   =>   |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> ((AGB)DC) = ((ADC)GB))
Theorem	abldivdiv 8104	Law for double group division.
|- X = ran G   &   |- D = ( /g ` G)   =>   |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> (AD(BDC)) = ((ADB)GC))
Theorem	abldivdiv4 8105	Law for double group division.
|- X = ran G   &   |- D = ( /g ` G)   =>   |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> ((ADB)DC) = (AD(BGC)))
Theorem	abldiv23 8106	Swap the second and third terms in a double division.
|- X = ran G   &   |- D = ( /g ` G)   =>   |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> ((ADB)DC) = ((ADC)DB))
Theorem	ablnnncan 8107	Group theory analog of nnncant 5478.
|- X = ran G   &   |- D = ( /g ` G)   =>   |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> ((AD(BDC))DC) = (ADB))
Theorem	ablnncan 8108	Group theory analog of nncant 5481.
|- X = ran G   &   |- D = ( /g ` G)   =>   |- ((G e. Abel /\ A e. X /\ B e. X) -> (AD(ADB)) = B)
Theorem	ablnnncan1 8109	Group theory analog of nnncan1t 5479.
|- X = ran G   &   |- D = ( /g ` G)   =>   |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> ((ADB)D(ADC)) = (CDB))
Subgroups
Syntax	csubg 8110	Extend class notation to include the class of subgroups.
class SubGrp
Definition	df-subg 8111	Define the set of subgroups of g.
|- SubGrp = {<.g, s>. | (g e. Grp /\ s = {h e. Grp | h (_ g})}
Theorem	issubg 8112	The predicate "is a subgroup of G." (Contributed by Paul Chapman, 3-Mar-2008.)
|- (H e. (SubGrp` G) <-> (G e. Grp /\ H e. Grp /\ H (_ G))
Theorem	subgres 8113	A subgroup operation is the restriction of its parent group operation to its underlying set. (Contributed by Paul Chapman, 3-Mar-2008.)
|- W = ran H   =>   |- (H e. (SubGrp` G) -> H = (G |` (W X. W)))
Theorem	subgopr 8114	The result of a subgroup operation is the same as the result of its parent operation. (Contributed by Paul Chapman, 3-Mar-2008.)
|- W = ran H   =>   |- (H e. (SubGrp` G) -> ((A e. W /\ B e. W) -> (AHB) = (AGB)))
Theorem	subgrnss 8115	The underlying set of a subgroup is a subset of its parent group's underlying set. (Contributed by Paul Chapman, 3-Mar-2008.)
|- X = ran G   &   |- W = ran H   =>   |- (H e. (SubGrp` G) -> W (_ X)
Theorem	subgid 8116	The identity element of a subgroup is the same as its parent's. (Contributed by Paul Chapman, 3-Mar-2008.)
|- U = (Id` G)   &   |- T = (Id` H)   =>   |- (H e. (SubGrp` G) -> T = U)
Theorem	issubgilem 8117	Lemma for issubgi 8118.
Theorem	issubgi 8118	Properties that determine a subgroup. (Contributed by Paul Chapman, 25-Feb-2008.)
|- G e. Grp   &   |- X = ran G   &   |- U = (Id` G)   &   |- N = (inv` G)   &   |- Y (_ X   &   |- H = (G |` (Y X. Y))   &   |- ((x e. Y /\ y e. Y) -> (xGy) e. Y)   &   |- U e. Y   &   |- (x e. Y -> (N` x) e. Y)   =>   |- H e. (SubGrp` G)
Theorem	subgabl 8119	A subgroup of an Abelian group is Abelian. (Contributed by Paul Chapman, 25-Apr-2008.)
|- ((G e. Abel /\ H e. (SubGrp` G)) -> H e. Abel)
Examples of groups
Theorem	grpsn 8120	The group operation for the singleton group.
|- A e. V   =>   |- {<.<.A, A>., A>.} e. Grp
Examples of Abelian groups
Theorem	ablsn 8121	The Abelian group operation for the singleton group.
|- A e. V   =>   |- {<.<.A, A>., A>.} e. Abel
Theorem	cnaddabl 8122	Complex number addition is an Abelian group operation.
|- + e. Abel
Theorem	cnid 8123	The group identity element of complex number addition is zero. (Contributed by Steve Rodriguez, 3-Dec-2006.)
|- 0 = (Id` + )
Theorem	addinv 8124	Value of the group inverse of complex number addition. (Contributed by Steve Rodriguez, 3-Dec-2006.)
|- (A e. CC -> ((inv` + )` A) = -uA)
Theorem	readdsubg 8125	The real numbers under addition comprise a subgroup of the complex numbers under addition. (Contributed by Paul Chapman, 25-Apr-2008.)
|- ( + |` (RR X. RR)) e. (SubGrp` + )
Theorem	zaddsubg 8126	The integers under addition comprise a subgroup of the complex numbers under addition. (Contributed by Paul Chapman, 25-Apr-2008.)
|- ( + |` (ZZ X. ZZ)) e. (SubGrp` + )
Theorem	ablmul 8127	Nonzero complex number multiplication is an Abelian group operation. (Contributed by Steve Rodriguez, 12-Feb-2007.)
|- ( x. |` ((CC \ {0}) X. (CC \ {0}))) e. Abel
Theorem	mulid 8128	The group identity element of nonzero complex number multiplication is one. (Contributed by Steve Rodriguez, 23-Feb-2007.)
|- (Id` ( x. |` ((CC \ {0}) X. (CC \ {0})))) = 1
Group homomorphism
Theorem	ghgrpilem1 8129	Lemma for ghgrpi 8133.
Theorem	ghgrpilem2 8130	Lemma for ghgrpi 8133.
Theorem	ghgrpilem3 8131	Lemma for ghgrpi 8133.
Theorem	ghgrpilem4 8132	Lemma for ghgrpi 8133.
Theorem	ghgrpi 8133	The image of a group G under a group homomorphism F is a group, and furthermore is Abelian if G is Abelian. This is a stronger result than that usually found in the literature, since the target of the homomorphism (operator O in our model) need not have any of the properties of a group as a prerequisite. (Contributed by Paul Chapman, 25-Apr-2008.)
|- G e. Grp   &   |- X = ran G   &   |- F:X-onto->Y   &   |- Y (_ A   &   |- O Fn (A X. A)   &   |- ((x e. X /\ y e. X) -> (F` (xGy)) = ((F` x)O(F` y)))   &   |- H = (O |` (Y X. Y))   =>   |- (H e. Grp /\ (G e. Abel -> H e. Abel))
Theorem	ghsubgi 8134	The image of a subgroup S of group G under a group homomorphism F on G is a group, and furthermore is Abelian if S is Abelian. (Contributed by Paul Chapman, 25-Apr-2008.)
|- S e. (SubGrp` G)   &   |- X = ran G   &   |- F:X-->Y   &   |- Y (_ A   &   |- O Fn (A X. A)   &   |- ((x e. X /\ y e. X) -> (F` (xGy)) = ((F` x)O(F` y)))   &   |- Z = ran S   &   |- W = (F"Z)   &   |- H = (O |` (W X. W))   =>   |- (H e. Grp /\ (S e. Abel -> H e. Abel))
Ring theory
Definition and basic properties
Syntax	cring 8135	Extend class notation with the class of all unital rings.
class Ring
Definition	df-ring 8136	Define the class of all unital rings.
|- Ring = {<.g, h>. | ((g e. Abel /\ h:(ran g X. ran g)-->ran g) /\ (A.x e. ran gA.y e. ran gA.z e. ran g(((xhy)hz) = (xh(yhz)) /\ (xh(ygz)) = ((xhy)g(xhz)) /\ ((xgy)hz) = ((xhz)g(yhz))) /\ E.x e. ran gA.y e. ran g((yhx) = y /\ (xhy) = y)))}
Theorem	isring 8137	The predicate "is a (unital) ring." Definition of ring with unit in [Schechter] p. 187. (Contributed by Jeffrey Hankins, 21-Nov-2006.)
|- X = ran G   =>   |- (H e. A -> (<.G, H>. e. Ring <-> ((G e. Abel /\ H:(X X. X)-->X) /\ (A.x e. X A.y e. X A.z e. X (((xHy)Hz) = (xH(yHz)) /\ (xH(yGz)) = ((xHy)G(xHz)) /\ ((xGy)Hz) = ((xHz)G(yHz))) /\ E.x e. X A.y e. X ((yHx) = y /\ (xHy) = y)))))
Theorem	ringi 8138	The properties of a unital ring. (Contributed by Steve Rodriguez, 8-Sep-2007.)
|- G = (1st` R)   &   |- H = (2nd` R)   &   |- X = ran G   =>   |- (R e. Ring -> ((G e. Abel /\ H:(X X. X)-->X) /\ (A.x e. X A.y e. X A.z e. X (((xHy)Hz) = (xH(yHz)) /\ (xH(yGz)) = ((xHy)G(xHz)) /\ ((xGy)Hz) = ((xHz)G(yHz))) /\ E.x e. X A.y e. X ((yHx) = y /\ (xHy) = y))))
Theorem	ringsm 8139	Functionality of the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.)
|- G = (1st` R)   &   |- H = (2nd` R)   &   |- X = ran G   =>   |- (R e. Ring -> H:(X X. X)-->X)
Theorem	ringcl 8140	Closure of the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.)
|- G = (1st` R)   &   |- H = (2nd` R)   &   |- X = ran G   =>