mmtheorems81 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 8001-8100 - Page 81 of 107

Type	Label	Description
Statement
Theorem	grpidinvlem2 8001	Lemma for grpidinv 8004.
Theorem	grpidinvlem3 8002	Lemma for grpidinv 8004.
Theorem	grpidinvlem4 8003	Lemma for grpidinv 8004.
Theorem	grpidinv 8004	A group has a left and right identity element, and every member has a left and right inverse.
|- X = ran G   =>   |- (G e. Grp -> E.u e. X A.x e. X (((uGx) = x /\ (xGu) = x) /\ E.y e. X ((yGx) = u /\ (xGy) = u)))
Theorem	grpideu 8005	The left identity element of a group is unique. Lemma 2.2.1(a) of [Herstein] p. 55.
|- X = ran G   =>   |- (G e. Grp -> E!u e. X A.x e. X (uGx) = x)
Theorem	grprndm 8006	A group's range in terms of its domain.
|- (G e. Grp -> ran G = dom dom G)
Theorem	0ngrp 8007	The empty set is not a group.
|- -. (/) e. Grp
Theorem	grprn 8008	The range of a group operation. Useful for satisfying group base set hypotheses of the form X = ran G.
|- G e. Grp   &   |- dom G = (X X. X)   =>   |- X = ran G
Theorem	grprnOLD 8009	The range of a group operation. Useful for satisfying X = ran G hypothesis for specific groups.
|- G e. Grp   &   |- G:(X X. X)-->X   =>   |- X = ran G
Theorem	grpidval 8010	The value of the identity element of a group.
|- X = ran G   &   |- U = (Id` G)   =>   |- (G e. Grp -> U = U.{u e. X | A.x e. X (uGx) = x})
Theorem	grpidcl 8011	The identity element of a group belongs to the group.
|- X = ran G   &   |- U = (Id` G)   =>   |- (G e. Grp -> U e. X)
Theorem	grpidinv2 8012	A group's properties using the explicit identity element.
|- X = ran G   &   |- U = (Id` G)   =>   |- ((G e. Grp /\ A e. X) -> (((UGA) = A /\ (AGU) = A) /\ E.y e. X ((yGA) = U /\ (AGy) = U)))
Theorem	grplid 8013	The identity element of a group is a left identity.
|- X = ran G   &   |- U = (Id` G)   =>   |- ((G e. Grp /\ A e. X) -> (UGA) = A)
Theorem	grprid 8014	The identity element of a group is a right identity.
|- X = ran G   &   |- U = (Id` G)   =>   |- ((G e. Grp /\ A e. X) -> (AGU) = A)
Theorem	grprcan 8015	Right cancellation law for groups.
|- X = ran G   =>   |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> ((AGC) = (BGC) <-> A = B))
Theorem	grpinveu 8016	The left inverse element of a group is unique. Lemma 2.2.1(b) of [Herstein] p. 55.
|- X = ran G   &   |- U = (Id` G)   =>   |- ((G e. Grp /\ A e. X) -> E!y e. X (yGA) = U)
Theorem	grpid 8017	Two ways of saying that an element of a group is the identity element. (Contributed by Paul Chapman, 25-Feb-2008.)
|- X = ran G   &   |- U = (Id` G)   =>   |- ((G e. Grp /\ A e. X) -> (A = U <-> (AGA) = A))
Theorem	grpinvfval 8018	The inverse function of a group.
|- X = ran G   &   |- U = (Id` G)   &   |- N = (inv` G)   =>   |- (G e. Grp -> N = {<.x, n>. | (x e. X /\ n = U.{y e. X | (yGx) = U})})
Theorem	grpinvval 8019	The inverse of a group element.
|- X = ran G   &   |- U = (Id` G)   &   |- N = (inv` G)   =>   |- ((G e. Grp /\ A e. X) -> (N` A) = U.{y e. X | (yGA) = U})
Theorem	grpinvcl 8020	A group element's inverse is a group element.
|- X = ran G   &   |- N = (inv` G)   =>   |- ((G e. Grp /\ A e. X) -> (N` A) e. X)
Theorem	grpinv 8021	The properties of a group element's inverse.
|- X = ran G   &   |- U = (Id` G)   &   |- N = (inv` G)   =>   |- ((G e. Grp /\ A e. X) -> (((N` A)GA) = U /\ (AG(N` A)) = U))
Theorem	grplinv 8022	The left inverse of a group element.
|- X = ran G   &   |- U = (Id` G)   &   |- N = (inv` G)   =>   |- ((G e. Grp /\ A e. X) -> ((N` A)GA) = U)
Theorem	grprinv 8023	The right inverse of a group element.
|- X = ran G   &   |- U = (Id` G)   &   |- N = (inv` G)   =>   |- ((G e. Grp /\ A e. X) -> (AG(N` A)) = U)
Theorem	grpinvid1 8024	The inverse of a group element expressed in terms of the identity element.
|- X = ran G   &   |- U = (Id` G)   &   |- N = (inv` G)   =>   |- ((G e. Grp /\ A e. X /\ B e. X) -> ((N` A) = B <-> (AGB) = U))
Theorem	grpinvid2 8025	The inverse of a group element expressed in terms of the identity element.
|- X = ran G   &   |- U = (Id` G)   &   |- N = (inv` G)   =>   |- ((G e. Grp /\ A e. X /\ B e. X) -> ((N` A) = B <-> (BGA) = U))
Theorem	grpinvid 8026	The inverse of the identity element of a group.
|- U = (Id` G)   &   |- N = (inv` G)   =>   |- (G e. Grp -> (N` U) = U)
Theorem	grplcan 8027	Left cancellation law for groups.
|- X = ran G   =>   |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> ((CGA) = (CGB) <-> A = B))
Theorem	isgrp2i 8028	An alternate way to show a group operation. Exercise 1 of [Herstein] p. 57.
|- X e. V   &   |- X =/= (/)   &   |- G:(X X. X)-->X   &   |- ((x e. X /\ y e. X /\ z e. X) -> ((xGy)Gz) = (xG(yGz)))   &   |- ((x e. X /\ y e. X) -> E.z e. X (zGx) = y)   &   |- ((x e. X /\ y e. X) -> E.z e. X (xGz) = y)   =>   |- G e. Grp
Theorem	grpasscan1 8029	An associative cancellation law for groups. (Contributed by Paul Chapman, 25-Feb-2008.)
|- X = ran G   &   |- N = (inv` G)   =>   |- ((G e. Grp /\ A e. X /\ B e. X) -> (AG((N` A)GB)) = B)
Theorem	grp2inv 8030	Double inverse law for groups. Lemma 2.2.1(c) of [Herstein] p. 55.
|- X = ran G   &   |- N = (inv` G)   =>   |- ((G e. Grp /\ A e. X) -> (N` (N` A)) = A)
Theorem	grpinvf 8031	Mapping of the inverse function of a group.
|- X = ran G   &   |- N = (inv` G)   =>   |- (G e. Grp -> N:X-1-1-onto->X)
Theorem	grpinvop 8032	The inverse of the group operation reverses the arguments. Lemma 2.2.1(d) of [Herstein] p. 55.
|- X = ran G   &   |- N = (inv` G)   =>   |- ((G e. Grp /\ A e. X /\ B e. X) -> (N` (AGB)) = ((N` B)G(N` A)))
Theorem	grpdivfval 8033	Group division (or subtraction) operation.
|- X = ran G   &   |- N = (inv` G)   &   |- D = ( /g ` G)   =>   |- (G e. Grp -> D = {<.<.x, y>., z>. | ((x e. X /\ y e. X) /\ z = (xG(N` y)))})
Theorem	grpdivval 8034	Group division (or subtraction) operation value.
|- X = ran G   &   |- N = (inv` G)   &   |- D = ( /g ` G)   =>   |- ((G e. Grp /\ A e. X /\ B e. X) -> (ADB) = (AG(N` B)))
Theorem	grpdivinv 8035	Group division by an inverse.
|- X = ran G   &   |- N = (inv` G)   &   |- D = ( /g ` G)   =>   |- ((G e. Grp /\ A e. X /\ B e. X) -> (AD(N` B)) = (AGB))
Theorem	grpinvdiv 8036	Inverse of a group division.
|- X = ran G   &   |- N = (inv` G)   &   |- D = ( /g ` G)   =>   |- ((G e. Grp /\ A e. X /\ B e. X) -> (N` (ADB)) = (BDA))
Theorem	grpdivf 8037	Mapping for group division.
|- X = ran G   &   |- D = ( /g ` G)   =>   |- (G e. Grp -> D:(X X. X)-->X)
Theorem	grpdivcl 8038	Closure of group division (or subtraction) operation.
|- X = ran G   &   |- D = ( /g ` G)   =>   |- ((G e. Grp /\ A e. X /\ B e. X) -> (ADB) e. X)
Theorem	grpdivdiv 8039	Double group division.
|- X = ran G   &   |- D = ( /g ` G)   =>   |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> (AD(BDC)) = (AG(CDB)))
Theorem	grpmuldivass 8040	Associative-type law for multiplication and division.
|- X = ran G   &   |- D = ( /g ` G)   =>   |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> ((AGB)DC) = (AG(BDC)))
Theorem	grpdivid 8041	Division of a group member by itself.
|- X = ran G   &   |- D = ( /g ` G)   &   |- U = (Id` G)   =>   |- ((G