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	grppnpcan2 8101	Group theory analog of pnpcan2t 5503.
|- X = ran G   &   |- D = ( /g ` G)   =>   |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> ((AGC)D(BGC)) = (ADB))
Theorem	grpnnncan2 8102	Group theory analog of nnncan2t 5492.
|- X = ran G   &   |- D = ( /g ` G)   =>   |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> ((ADC)D(BDC)) = (ADB))
Theorem	grpnpncan 8103	Group theory analog of npncant 5424.
|- X = ran G   &   |- D = ( /g ` G)   =>   |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> ((ADB)G(BDC)) = (ADC))
Theorem	resgrprn 8104	The underlying set of a group operation which is a restriction of a mapping. (Contributed by Paul Chapman, 25-Mar-2008.)
|- H = (G |` (Y X. Y))   =>   |- ((dom G = (X X. X) /\ H e. Grp /\ Y (_ X) -> Y = ran H)
Theorem	grplactfval 8105	The left group action of element A of group G. (Contributed by Paul Chapman, 18-Mar-2008.)
|- F = {<.g, h>. | (g e. X /\ h = {<.a, b>. | (a e. X /\ b = (gGa))})}   &   |- X = ran G   =>   |- ((G e. Grp /\ A e. X) -> (F` A) = {<.a, b>. | (a e. X /\ b = (AGa))})
Theorem	grplactval 8106	The value of the left group action of element A of group G at B. (Contributed by Paul Chapman, 18-Mar-2008.)
|- F = {<.g, h>. | (g e. X /\ h = {<.a, b>. | (a e. X /\ b = (gGa))})}   &   |- X = ran G   =>   |- ((G e. Grp /\ A e. X /\ B e. X) -> ((F` A)` B) = (AGB))
Theorem	grplactf1o 8107	The left group action of element A of group G maps the underlying set X of G one-to-one onto itself. (Contributed by Paul Chapman, 18-Mar-2008.)
|- F = {<.g, h>. | (g e. X /\ h = {<.a, b>. | (a e. X /\ b = (gGa))})}   &   |- X = ran G   =>   |- ((G e. Grp /\ A e. X) -> (F` A):X-1-1-onto->X)
Definition and basic properties of Abelian groups
Syntax	cabl 8108	Extend class notation with the class of all Abelian group operations.
class Abel
Definition	df-abl 8109	Define the class of all Abelian group operations.
|- Abel = {g e. Grp | A.x e. ran gA.y e. ran g(xgy) = (ygx)}
Theorem	isabl 8110	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 8111	An Abelian group operation is a group operation.
|- (G e. Abel -> G e. Grp)
Theorem	ablcom 8112	An Abelian group operation is commutative.
|- X = ran G   =>   |- ((G e. Abel /\ A e. X /\ B e. X) -> (AGB) = (BGA))
Theorem	abl23 8113	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 8114	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 8115	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 8116	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 8117	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 8118	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 8119	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 8120	Group theory analog of nnncant 5490.
|- X = ran G   &   |- D = ( /g ` G)   =>   |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> ((AD(BDC))DC) = (ADB))
Theorem	ablnncan 8121	Group theory analog of nncant 5493.
|- X = ran G   &   |- D = ( /g ` G)   =>   |- ((G e. Abel /\ A e. X /\ B e. X) -> (AD(ADB)) = B)
Theorem	ablnnncan1 8122	Group theory analog of nnncan1t 5491.
|- 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 8123	Extend class notation to include the class of subgroups.
class SubGrp
Definition	df-subg 8124	Define the set of subgroups of g.
|- SubGrp = {<.g, s>. | (g e. Grp /\ s = {h e. Grp | h (_ g})}
Theorem	issubg 8125	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 8126	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 8127	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 8128	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 8129	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 8130	Lemma for issubgi 8131.
Theorem	issubgi 8131	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 8132	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 8133	The group operation for the singleton group.
|- A e. V   =>   |- {<.<.A, A>., A>.} e. Grp
Examples of Abelian groups
Theorem	ablsn 8134	The Abelian group operation for the singleton group.
|- A e. V   =>   |- {<.<.A, A>., A>.} e. Abel
Theorem	cnaddabl 8135	Complex number addition is an Abelian group operation.
|- + e. Abel
Theorem	cnid 8136	The group identity element of complex number addition is zero. (Contributed by Steve Rodriguez, 3-Dec-2006.)
|- 0 = (Id` + )
Theorem	addinv 8137	Value of the group inverse of complex number addition. (Contributed by Steve Rodriguez, 3-Dec-2006.)
|- (A e. CC -> ((inv` + )` A) = -uA)
Theorem	readdsubg 8138	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 8139	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 8140	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 8141	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 8142	Lemma for ghgrpi 8146.
Theorem	ghgrpilem2 8143	Lemma for ghgrpi 8146.
Theorem	ghgrpilem3 8144	Lemma for ghgrpi 8146.
Theorem	ghgrpilem4 8145	Lemma for ghgrpi 8146.
Theorem	ghgrpi 8146	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