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	isabl 8101	The predicate "is an Abelian (commutative) group operation."
|- X = ran G   =>   |- (G e. Abel <-> (G e. Grp /\ A.x e. X A.y e. X (xGy) = (yGx)))
Theorem	ablgrp 8102	An Abelian group operation is a group operation.
|- (G e. Abel -> G e. Grp)
Theorem	ablcom 8103	An Abelian group operation is commutative.
|- X = ran G   =>   |- ((G e. Abel /\ A e. X /\ B e. X) -> (AGB) = (BGA))
Theorem	abl23 8104	Commutative/associative law for Abelian groups.
|- X = ran G   =>   |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> ((AGB)GC) = ((AGC)GB))
Theorem	abl4 8105	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 8106	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 8107	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 8108	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 8109	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 8110	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 8111	Group theory analog of nnncant 5466.
|- X = ran G   &   |- D = ( /g ` G)   =>   |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> ((AD(BDC))DC) = (ADB))
Theorem	ablnncan 8112	Group theory analog of nncant 5469.
|- X = ran G   &   |- D = ( /g ` G)   =>   |- ((G e. Abel /\ A e. X /\ B e. X) -> (AD(ADB)) = B)
Theorem	ablnnncan1 8113	Group theory analog of nnncan1t 5467.
|- 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 8114	Extend class notation to include the class of subgroups.
class SubGrp
Definition	df-subg 8115	Define the set of subgroups of g.
|- SubGrp = {<.g, s>. | (g e. Grp /\ s = {h e. Grp | h (_ g})}
Theorem	issubg 8116	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 8117	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 8118	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 8119	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 8120	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 8121	Lemma for issubgi 8122.
Theorem	issubgi 8122	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 8123	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 8124	The group operation for the singleton group.
|- A e. V   =>   |- {<.<.A, A>., A>.} e. Grp
Examples of Abelian groups
Theorem	ablsn 8125	The Abelian group operation for the singleton group.
|- A e. V   =>   |- {<.<.A, A>., A>.} e. Abel
Theorem	cnaddabl 8126	Complex number addition is an Abelian group operation.
|- + e. Abel
Theorem	cnid 8127	The group identity element of complex number addition is zero. (Contributed by Steve Rodriguez, 3-Dec-2006.)
|- 0 = (Id` + )
Theorem	addinv 8128	Value of the group inverse of complex number addition. (Contributed by Steve Rodriguez, 3-Dec-2006.)
|- (A e. CC -> ((inv` + )` A) = -uA)
Theorem	readdsubg 8129	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 8130	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 8131	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 8132	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 8133	Lemma for ghgrpi 8137.
Theorem	ghgrpilem2 8134	Lemma for ghgrpi 8137.
Theorem	ghgrpilem3 8135	Lemma for ghgrpi 8137.
Theorem	ghgrpilem4 8136	Lemma for ghgrpi 8137.
Theorem	ghgrpi 8137	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 8138	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 8139	Extend class notation with the class of all unital rings.
class Ring
Definition	df-ring 8140	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 8141	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 8142	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(yG