P. 167 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 16601-16700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	frmdup3lem 16601*	Lemma for frmdup3 16602. (Contributed by Mario Carneiro, 18-Jul-2016.)
|- M = (freeMnd ` I )   &    |- B = ( Base ` G )   &    |- U = (varFMnd ` I )   =>    |- ( ( ( G e. Mnd /\ I e. V /\ A : I --> B ) /\ ( F e. ( M MndHom G ) /\ ( F o. U ) = A ) ) -> F = ( x e. Word I |-> ( G gsumg ( A o. x ) ) ) )
Theorem	frmdup3 16602*	Universal property of the free monoid by existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by Mario Carneiro, 18-Jul-2016.)
|- M = (freeMnd ` I )   &    |- B = ( Base ` G )   &    |- U = (varFMnd ` I )   =>    |- ( ( G e. Mnd /\ I e. V /\ A : I --> B ) -> E! m e. ( M MndHom G ) ( m o. U ) = A )
10.1.9  Examples and counterexamples for magmas, semigroups and monoids
Theorem	mgm2nsgrplem1 16603*	Lemma 1 for mgm2nsgrp 16607: M is a magma, even if A = B ( M is the trivial magma in this case, see mgmb1mgm1 16450). (Contributed by AV, 27-Jan-2020.)
|- S = { A , B }   &    |- ( Base ` M ) = S   &    |- ( +g ` M ) = ( x e. S , y e. S |-> if ( ( x = A /\ y = A ) , B , A ) )   =>    |- ( ( A e. V /\ B e. W ) -> M e. Mgm )
Theorem	mgm2nsgrplem2 16604*	Lemma 2 for mgm2nsgrp 16607. (Contributed by AV, 27-Jan-2020.)
|- S = { A , B }   &    |- ( Base ` M ) = S   &    |- ( +g ` M ) = ( x e. S , y e. S |-> if ( ( x = A /\ y = A ) , B , A ) )   &    |- .o. = ( +g ` M )   =>    |- ( ( A e. V /\ B e. W ) -> ( ( A .o. A ) .o. B ) = A )
Theorem	mgm2nsgrplem3 16605*	Lemma 3 for mgm2nsgrp 16607. (Contributed by AV, 28-Jan-2020.)
|- S = { A , B }   &    |- ( Base ` M ) = S   &    |- ( +g ` M ) = ( x e. S , y e. S |-> if ( ( x = A /\ y = A ) , B , A ) )   &    |- .o. = ( +g ` M )   =>    |- ( ( A e. V /\ B e. W ) -> ( A .o. ( A .o. B ) ) = B )
Theorem	mgm2nsgrplem4 16606*	Lemma 4 for mgm2nsgrp 16607: M is not a semigroup. (Contributed by AV, 28-Jan-2020.) (Proof shortened by AV, 31-Jan-2020.)
|- S = { A , B }   &    |- ( Base ` M ) = S   &    |- ( +g ` M ) = ( x e. S , y e. S |-> if ( ( x = A /\ y = A ) , B , A ) )   =>    |- ( ( # ` S ) = 2 -> M e/ SGrp )
Theorem	mgm2nsgrp 16607*	A small magma (with two elements) which is not a semigroup. (Contributed by AV, 28-Jan-2020.)
|- S = { A , B }   &    |- ( Base ` M ) = S   &    |- ( +g ` M ) = ( x e. S , y e. S |-> if ( ( x = A /\ y = A ) , B , A ) )   =>    |- ( ( # ` S ) = 2 -> ( M e. Mgm /\ M e/ SGrp ) )
Theorem	sgrp2nmndlem1 16608*	Lemma 1 for sgrp2nmnd 16615: M is a magma, even if A = B ( M is the trivial magma in this case, see mgmb1mgm1 16450). (Contributed by AV, 29-Jan-2020.)
|- S = { A , B }   &    |- ( Base ` M ) = S   &    |- ( +g ` M ) = ( x e. S , y e. S |-> if ( x = A , A , B ) )   =>    |- ( ( A e. V /\ B e. W ) -> M e. Mgm )
Theorem	sgrp2nmndlem2 16609*	Lemma 2 for sgrp2nmnd 16615. (Contributed by AV, 29-Jan-2020.)
|- S = { A , B }   &    |- ( Base ` M ) = S   &    |- ( +g ` M ) = ( x e. S , y e. S |-> if ( x = A , A , B ) )   &    |- .o. = ( +g ` M )   =>    |- ( ( A e. S /\ C e. S ) -> ( A .o. C ) = A )
Theorem	sgrp2nmndlem3 16610*	Lemma 3 for sgrp2nmnd 16615. (Contributed by AV, 29-Jan-2020.)
|- S = { A , B }   &    |- ( Base ` M ) = S   &    |- ( +g ` M ) = ( x e. S , y e. S |-> if ( x = A , A , B ) )   &    |- .o. = ( +g ` M )   =>    |- ( ( C e. S /\ B e. S /\ A =/= B ) -> ( B .o. C ) = B )
Theorem	sgrp2rid2 16611*	A small semigroup (with two elements) with two right identities which are different if A =/= B. (Contributed by AV, 10-Feb-2020.)
|- S = { A , B }   &    |- ( Base ` M ) = S   &    |- ( +g ` M ) = ( x e. S , y e. S |-> if ( x = A , A , B ) )   &    |- .o. = ( +g ` M )   =>    |- ( ( A e. V /\ B e. W ) -> A. x e. S A. y e. S ( y .o. x ) = y )
Theorem	sgrp2rid2ex 16612*	A small semigroup (with two elements) with two right identities which are different. (Contributed by AV, 10-Feb-2020.)
|- S = { A , B }   &    |- ( Base ` M ) = S   &    |- ( +g ` M ) = ( x e. S , y e. S |-> if ( x = A , A , B ) )   &    |- .o. = ( +g ` M )   =>    |- ( ( # ` S ) = 2 -> E. x e. S E. z e. S A. y e. S ( x =/= z /\ ( y .o. x ) = y /\ ( y .o. z ) = y ) )
Theorem	sgrp2nmndlem4 16613*	Lemma 4 for sgrp2nmnd 16615: M is a semigroup. (Contributed by AV, 29-Jan-2020.)
|- S = { A , B }   &    |- ( Base ` M ) = S   &    |- ( +g ` M ) = ( x e. S , y e. S |-> if ( x = A , A , B ) )   =>    |- ( ( # ` S ) = 2 -> M e. SGrp )
Theorem	sgrp2nmndlem5 16614*	Lemma 5 for sgrp2nmnd 16615: M is not a monoid. (Contributed by AV, 29-Jan-2020.)
|- S = { A , B }   &    |- ( Base ` M ) = S   &    |- ( +g ` M ) = ( x e. S , y e. S |-> if ( x = A , A , B ) )   =>    |- ( ( # ` S ) = 2 -> M e/ Mnd )
Theorem	sgrp2nmnd 16615*	A small semigroup (with two elements) which is not a monoid. (Contributed by AV, 26-Jan-2020.)
|- S = { A , B }   &    |- ( Base ` M ) = S   &    |- ( +g ` M ) = ( x e. S , y e. S |-> if ( x = A , A , B ) )   =>    |- ( ( # ` S ) = 2 -> ( M e. SGrp /\ M e/ Mnd ) )
Theorem	mgmnsgrpex 16616	There is a magma which is not a semigroup. (Contributed by AV, 29-Jan-2020.)
|- E. m e. Mgm m e/ SGrp
Theorem	sgrpnmndex 16617	There is a semigroup which is not a monoid. (Contributed by AV, 29-Jan-2020.)
|- E. m e. SGrp m e/ Mnd
Theorem	sgrpssmgm 16618	The class of all semigroups is a proper subclass of the class of all semigroups. (Contributed by AV, 29-Jan-2020.)
|- SGrp C. Mgm
Theorem	mndsssgrp 16619	The class of all monoids is a proper subclass of the class of all monoids. (Contributed by AV, 29-Jan-2020.)
|- Mnd C. SGrp
10.2  Groups
10.2.1  Definition and basic properties
Syntax	cgrp 16620	Extend class notation with class of all groups.
class Grp
Syntax	cminusg 16621	Extend class notation with inverse of group element.
class invg
Syntax	csg 16622	Extend class notation with group subtraction (or division) operation.
class -g
Syntax	cmg 16623	Extend class notation with a function mapping a group operation to the power operation for the group.
class .g
Definition	df-grp 16624*	Define class of all groups. A group is a monoid (df-mnd 16488) whose internal operation is such that every element admits a left inverse (which can be proven to be a two-sided inverse). Thus, a group G is an algebraic structure formed from a base set of elements (notated ( Base ` G ) per df-base 15089) and an internal group operation (notated ( +g ` G ) per df-plusg 15165). The operation combines any two elements of the group base set and must satisfy the 4 group axioms: closure (the result of the group operation must always be a member of the base set, see grpcl 16630), associativity (so ( ( a +g b ) +g c ) = ( a +g ( b +g c ) ) for any a, b, c, see grpass 16631), identity (there must be an element e = ( 0g ` G ) such that e +g a = a +g e = a for any a), and inverse (for each element a in the base set, there must be an element b = invg a in the base set such that a +g b = b +g a = e). It can be proven that the identity element is unique (grpideu 16633). Groups need not be commutative; a commutative group is an Abelian group (see df-abl 17368). Subgroups can often be formed from groups, see df-subg 16765. An example of an (Abelian) group is the set of complex numbers CC over the group operation + (addition), as proven in cnaddablx 17441; an Abelian group is a group as proven in ablgrp 17370. Other structures include groups, including unital rings (df-ring 17717) and fields (df-field 17913). (Contributed by NM, 17-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.)
|- Grp = { g e. Mnd | A. a e. ( Base ` g ) E. m e. ( Base ` g ) ( m ( +g ` g ) a ) = ( 0g ` g ) }
Definition	df-minusg 16625*	Define inverse of group element. (Contributed by NM, 24-Aug-2011.)
|- invg = ( g e. _V |-> ( x e. ( Base ` g ) |-> ( iota_ w e. ( Base ` g ) ( w ( +g ` g ) x ) = ( 0g ` g ) ) ) )
Definition	df-sbg 16626*	Define group subtraction (also called division for multiplicative groups). (Contributed by NM, 31-Mar-2014.)
|- -g = ( g e. _V |-> ( x e. ( Base ` g ) , y e. ( Base ` g ) |-> ( x ( +g ` g ) ( ( invg ` g ) ` y ) ) ) )
Definition	df-mulg 16627*	Define the group multiple function, also known as group exponentiation when viewed multiplicatively. (Contributed by Mario Carneiro, 11-Dec-2014.)
|- .g = ( g e. _V |-> ( n e. ZZ , x e. ( Base ` g ) |-> if ( n = 0 , ( 0g ` g ) , [_ seq 1 ( ( +g ` g ) , ( NN X. { x } ) ) / s ]_ if ( 0 < n , ( s ` n ) , ( ( invg ` g ) ` ( s ` -u n ) ) ) ) ) )
Theorem	isgrp 16628*	The predicate "is a group." (This theorem demonstrates the use of symbols as variable names, first proposed by FL in 2010.) (Contributed by NM, 17-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .0. = ( 0g ` G )   =>    |- ( G e. Grp <-> ( G e. Mnd /\ A. a e. B E. m e. B ( m .+ a ) = .0. ) )
Theorem	grpmnd 16629	A group is a monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)
|- ( G e. Grp -> G e. Mnd )
Theorem	grpcl 16630	Closure of the operation of a group. (Contributed by NM, 14-Aug-2011.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( X .+ Y ) e. B )
Theorem	grpass 16631	A group operation is associative. (Contributed by NM, 14-Aug-2011.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .+ Y ) .+ Z ) = ( X .+ ( Y .+ Z ) ) )
Theorem	grpinvex 16632*	Every member of a group has a left inverse. (Contributed by NM, 16-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .0. = ( 0g ` G )   =>    |- ( ( G e. Grp /\ X e. B ) -> E. y e. B ( y .+ X ) = .0. )
Theorem	grpideu 16633*	The two-sided identity element of a group is unique. Lemma 2.2.1(a) of [Herstein] p. 55. (Contributed by NM, 16-Aug-2011.) (Revised by Mario Carneiro, 8-Dec-2014.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .0. = ( 0g ` G )   =>    |- ( G e. Grp -> E! u e. B A. x e. B ( ( u .+ x ) = x /\ ( x .+ u ) = x ) )
Theorem	grpplusf 16634	The group addition operation is a function. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- B = ( Base ` G )   &    |- F = ( +f ` G )   =>    |- ( G e. Grp -> F : ( B X. B ) --> B )
Theorem	grppropd 16635*	If two structures have the same group components (properties), one is a group iff the other one is. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
|- ( ph -> B = ( Base ` K ) )   &    |- ( ph -> B = ( Base ` L ) )   &    |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )   =>    |- ( ph -> ( K e. Grp <-> L e. Grp ) )
Theorem	grpprop 16636	If two structures have the same group components (properties), one is a group iff the other one is. (Contributed by NM, 11-Oct-2013.)
|- ( Base ` K ) = ( Base ` L )   &    |- ( +g ` K ) = ( +g ` L )   =>    |- ( K e. Grp <-> L e. Grp )
Theorem	grppropstr 16637	Generalize a specific 2-element group L to show that any set K with the same (relevant) properties is also a group. (Contributed by NM, 28-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.)
|- ( Base ` K ) = B   &    |- ( +g ` K ) = .+   &    |- L = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. }   =>    |- ( K e. Grp <-> L e. Grp )
Theorem	grpss 16638	Show that a structure extending a constructed group (e.g. a ring) is also a group. This allows us to prove that a constructed potential ring R is a group before we know that it is also a ring. (Theorem ringgrp 17720, on the other hand, requires that we know in advance that R is a ring.) (Contributed by NM, 11-Oct-2013.)
|- G = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. }   &    |- R e. _V   &    |- G C_ R   &    |- Fun R   =>    |- ( G e. Grp <-> R e. Grp )
Theorem	isgrpd2e 16639*	Deduce a group from its properties. In this version of isgrpd2 16640, we don't assume there is an expression for the inverse of x. (Contributed by NM, 10-Aug-2013.)
|- ( ph -> B = ( Base ` G ) )   &    |- ( ph -> .+ = ( +g ` G ) )   &    |- ( ph -> .0. = ( 0g ` G ) )   &    |- ( ph -> G e. Mnd )   &    |- ( ( ph /\ x e. B ) -> E. y e. B ( y .+ x ) = .0. )   =>    |- ( ph -> G e. Grp )
Theorem	isgrpd2 16640*	Deduce a group from its properties. N (negative) is normally dependent on x i.e. read it as N ( x ). Note: normally we don't use a ph antecedent on hypotheses that name structure components, since they can be eliminated with eqid 2429, but we make an exception for theorems such as isgrpd2 16640, ismndd 16510, and islmodd 18032 since theorems using them often rewrite the structure components. (Contributed by NM, 10-Aug-2013.)
|- ( ph -> B = ( Base ` G ) )   &    |- ( ph -> .+ = ( +g ` G ) )   &    |- ( ph -> .0. = ( 0g ` G ) )   &    |- ( ph -> G e. Mnd )   &    |- ( ( ph /\ x e. B ) -> N e. B )   &    |- ( ( ph /\ x e. B ) -> ( N .+ x ) = .0. )   =>    |- ( ph -> G e. Grp )
Theorem	isgrpde 16641*	Deduce a group from its properties. In this version of isgrpd 16642, we don't assume there is an expression for the inverse of x. (Contributed by NM, 6-Jan-2015.)
|- ( ph -> B = ( Base ` G ) )   &    |- ( ph -> .+ = ( +g ` G ) )   &    |- ( ( ph /\ x e. B /\ y e. B ) -> ( x .+ y ) e. B )   &    |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )   &    |- ( ph -> .0. e. B )   &    |- ( ( ph /\ x e. B ) -> ( .0. .+ x ) = x )   &    |- ( ( ph /\ x e. B ) -> E. y e. B ( y .+ x ) = .0. )   =>    |- ( ph -> G e. Grp )
Theorem	isgrpd 16642*	Deduce a group from its properties. Unlike isgrpd2 16640, this one goes straight from the base properties rather than going through Mnd. N (negative) is normally dependent on x i.e. read it as N ( x ). (Contributed by NM, 6-Jun-2013.) (Revised by Mario Carneiro, 6-Jan-2015.)
|- ( ph -> B = ( Base ` G ) )   &    |- ( ph -> .+ = ( +g ` G ) )   &    |- ( ( ph /\ x e. B /\ y e. B ) -> ( x .+ y ) e. B )   &    |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )   &    |- ( ph -> .0. e. B )   &    |- ( ( ph /\ x e. B ) -> ( .0. .+ x ) = x )   &    |- ( ( ph /\ x e. B ) -> N e. B )   &    |- ( ( ph /\ x e. B ) -> ( N .+ x ) = .0. )   =>    |- ( ph -> G e. Grp )
Theorem	isgrpi 16643*	Properties that determine a group. N (negative) is normally dependent on x i.e. read it as N ( x ). (Contributed by NM, 3-Sep-2011.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- ( ( x e. B /\ y e. B ) -> ( x .+ y ) e. B )   &    |- ( ( x e. B /\ y e. B /\ z e. B ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )   &    |- .0. e. B   &    |- ( x e. B -> ( .0. .+ x ) = x )   &    |- ( x e. B -> N e. B )   &    |- ( x e. B -> ( N .+ x ) = .0. )   =>    |- G e. Grp
Theorem	isgrpix 16644*	Properties that determine a group. Read N as N ( x ). Note: This theorem has hard-coded structure indices for demonstration purposes. It is not intended for general use. (New usage is discouraged.) (Contributed by NM, 4-Sep-2011.)
|- B e. _V   &    |- .+ e. _V   &    |- G = { <. 1 , B >. , <. 2 , .+ >. }   &    |- ( ( x e. B /\ y e. B ) -> ( x .+ y ) e. B )   &    |- ( ( x e. B /\ y e. B /\ z e. B ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )   &    |- .0. e. B   &    |- ( x e. B -> ( .0. .+ x ) = x )   &    |- ( x e. B -> N e. B )   &    |- ( x e. B -> ( N .+ x ) = .0. )   =>    |- G e. Grp
Theorem	grpidcl 16645	The identity element of a group belongs to the group. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   =>    |- ( G e. Grp -> .0. e. B )
Theorem	grpbn0 16646	The base set of a group is not empty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)
|- B = ( Base ` G )   =>    |- ( G e. Grp -> B =/= (/) )
Theorem	grplid 16647	The identity element of a group is a left identity. (Contributed by NM, 18-Aug-2011.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .0. = ( 0g ` G )   =>    |- ( ( G e. Grp /\ X e. B ) -> ( .0. .+ X ) = X )
Theorem	grprid 16648	The identity element of a group is a right identity. (Contributed by NM, 18-Aug-2011.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .0. = ( 0g ` G )   =>    |- ( ( G e. Grp /\ X e. B ) -> ( X .+ .0. ) = X )
Theorem	grpn0 16649	A group is not empty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) (Revised by Mario Carneiro, 2-Dec-2014.)
|- ( G e. Grp -> G =/= (/) )
Theorem	grprcan 16650	Right cancellation law for groups. (Contributed by NM, 24-Aug-2011.) (Proof shortened by Mario Carneiro, 6-Jan-2015.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .+ Z ) = ( Y .+ Z ) <-> X = Y ) )
Theorem	grpinveu 16651*	The left inverse element of a group is unique. Lemma 2.2.1(b) of [Herstein] p. 55. (Contributed by NM, 24-Aug-2011.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .0. = ( 0g ` G )   =>    |- ( ( G e. Grp /\ X e. B ) -> E! y e. B ( y .+ X ) = .0. )
Theorem	grpid 16652	Two ways of saying that an element of a group is the identity element. Provides a convenient way to compute the value of the identity element. (Contributed by NM, 24-Aug-2011.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .0. = ( 0g ` G )   =>    |- ( ( G e. Grp /\ X e. B ) -> ( ( X .+ X ) = X <-> .0. = X ) )
Theorem	isgrpid2 16653	Properties showing that an element Z is the identity element of a group. (Contributed by NM, 7-Aug-2013.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .0. = ( 0g ` G )   =>    |- ( G e. Grp -> ( ( Z e. B /\ ( Z .+ Z ) = Z ) <-> .0. = Z ) )
Theorem	grpidd2 16654*	Deduce the identity element of a group from its properties. Useful in conjunction with isgrpd 16642. (Contributed by Mario Carneiro, 14-Jun-2015.)
|- ( ph -> B = ( Base ` G ) )   &    |- ( ph -> .+ = ( +g ` G ) )   &    |- ( ph -> .0. e. B )   &    |- ( ( ph /\ x e. B ) -> ( .0. .+ x ) = x )   &    |- ( ph -> G e. Grp )   =>    |- ( ph -> .0. = ( 0g ` G ) )
Theorem	grpinvfval 16655*	The inverse function of a group. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .0. = ( 0g ` G )   &    |- N = ( invg ` G )   =>    |- N = ( x e. B |-> ( iota_ y e. B ( y .+ x ) = .0. ) )
Theorem	grpinvval 16656*	The inverse of a group element. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .0. = ( 0g ` G )   &    |- N = ( invg ` G )   =>    |- ( X e. B -> ( N ` X ) = ( iota_ y e. B ( y .+ X ) = .0. ) )
Theorem	grpinvfn 16657	Functionality of the group inverse function. (Contributed by Stefan O'Rear, 21-Mar-2015.)
|- B = ( Base ` G )   &    |- N = ( invg ` G )   =>    |- N Fn B
Theorem	grpinvfvi 16658	The group inverse function is compatible with identity-function protection. (Contributed by Stefan O'Rear, 21-Mar-2015.)
|- N = ( invg ` G )   =>    |- N = ( invg ` ( _I ` G ) )
Theorem	grpsubfval 16659*	Group subtraction (division) operation. (Contributed by NM, 31-Mar-2014.) (Revised by Stefan O'Rear, 27-Mar-2015.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- I = ( invg ` G )   &    |- .- = ( -g ` G )   =>    |- .- = ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) )
Theorem	grpsubval 16660	Group subtraction (division) operation. (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 13-Dec-2014.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- I = ( invg ` G )   &    |- .- = ( -g ` G )   =>    |- ( ( X e. B /\ Y e. B ) -> ( X .- Y ) = ( X .+ ( I ` Y ) ) )
Theorem	grpinvf 16661	The group inversion operation is a function on the base set. (Contributed by Mario Carneiro, 4-May-2015.)
|- B = ( Base ` G )   &    |- N = ( invg ` G )   =>    |- ( G e. Grp -> N : B --> B )
Theorem	grpinvcl 16662	A group element's inverse is a group element. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 4-May-2015.)
|- B = ( Base ` G )   &    |- N = ( invg ` G )   =>    |- ( ( G e. Grp /\ X e. B ) -> ( N ` X ) e. B )
Theorem	grplinv 16663	The left inverse of a group element. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .0. = ( 0g ` G )   &    |- N = ( invg ` G )   =>    |- ( ( G e. Grp /\ X e. B ) -> ( ( N ` X ) .+ X ) = .0. )
Theorem	grprinv 16664	The right inverse of a group element. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .0. = ( 0g ` G )   &    |- N = ( invg ` G )   =>    |- ( ( G e. Grp /\ X e. B ) -> ( X .+ ( N ` X ) ) = .0. )
Theorem	grpinvid1 16665	The inverse of a group element expressed in terms of the identity element. (Contributed by NM, 24-Aug-2011.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .0. = ( 0g ` G )   &    |- N = ( invg ` G )   =>    |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( N ` X ) = Y <-> ( X .+ Y ) = .0. ) )
Theorem	grpinvid2 16666	The inverse of a group element expressed in terms of the identity element. (Contributed by NM, 24-Aug-2011.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .0. = ( 0g ` G )   &    |- N = ( invg ` G )   =>    |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( N ` X ) = Y <-> ( Y .+ X ) = .0. ) )
Theorem	isgrpinv 16667*	Properties showing that a function M is the inverse function of a group. (Contributed by NM, 7-Aug-2013.) (Revised by Mario Carneiro, 2-Oct-2015.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .0. = ( 0g ` G )   &    |- N = ( invg ` G )   =>    |- ( G e. Grp -> ( ( M : B --> B /\ A. x e. B ( ( M ` x ) .+ x ) = .0. ) <-> N = M ) )
Theorem	grpinvid 16668	The inverse of the identity element of a group. (Contributed by NM, 24-Aug-2011.)
|- .0. = ( 0g ` G )   &    |- N = ( invg ` G )   =>    |- ( G e. Grp -> ( N ` .0. ) = .0. )
Theorem	grplcan 16669	Left cancellation law for groups. (Contributed by NM, 25-Aug-2011.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( Z .+ X ) = ( Z .+ Y ) <-> X = Y ) )
Theorem	grpidrcan 16670	If right adding an element of a group to an arbitrary element of the group results in this element, the added element is the identity element and vice versa. (Contributed by AV, 15-Mar-2019.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .0. = ( 0g ` G )   =>    |- ( ( G e. Grp /\ X e. B /\ Z e. B ) -> ( ( X .+ Z ) = X <-> Z = .0. ) )
Theorem	grpidlcan 16671	If left adding an element of a group to an arbitrary element of the group results in this element, the added element is the identity element and vice versa. (Contributed by AV, 15-Mar-2019.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .0. = ( 0g ` G )   =>    |- ( ( G e. Grp /\ X e. B /\ Z e. B ) -> ( ( Z .+ X ) = X <-> Z = .0. ) )
Theorem	grpinvinv 16672	Double inverse law for groups. Lemma 2.2.1(c) of [Herstein] p. 55. (Contributed by NM, 31-Mar-2014.)
|- B = ( Base ` G )   &    |- N = ( invg ` G )   =>    |- ( ( G e. Grp /\ X e. B ) -> ( N ` ( N ` X ) ) = X )
Theorem	grpinvcnv 16673	The group inverse is its own inverse function. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- B = ( Base ` G )   &    |- N = ( invg ` G )   =>    |- ( G e. Grp -> `' N = N )
Theorem	grpinv11 16674	The group inverse is one-to-one. (Contributed by NM, 22-Mar-2015.)
|- B = ( Base ` G )   &    |- N = ( invg ` G )   &    |- ( ph -> G e. Grp )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( ( N ` X ) = ( N ` Y ) <-> X = Y ) )
Theorem	grpinvf1o 16675	The group inverse is a one-to-one onto function. (Contributed by NM, 22-Oct-2014.) (Proof shortened by Mario Carneiro, 14-Aug-2015.)
|- B = ( Base ` G )   &    |- N = ( invg ` G )   &    |- ( ph -> G e. Grp )   =>    |- ( ph -> N : B -1-1-onto-> B )
Theorem	grpinvnz 16676	The inverse of a nonzero group element is not zero. (Contributed by Stefan O'Rear, 27-Feb-2015.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- N = ( invg ` G )   =>    |- ( ( G e. Grp /\ X e. B /\ X =/= .0. ) -> ( N ` X ) =/= .0. )
Theorem	grpinvnzcl 16677	The inverse of a nonzero group element is a nonzero group element. (Contributed by Stefan O'Rear, 27-Feb-2015.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- N = ( invg ` G )   =>    |- ( ( G e. Grp /\ X e. ( B \ { .0. } ) ) -> ( N ` X ) e. ( B \ { .0. } ) )
Theorem	grpsubinv 16678	Subtraction of an inverse. (Contributed by NM, 7-Apr-2015.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .- = ( -g ` G )   &    |- N = ( invg ` G )   &    |- ( ph -> G e. Grp )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( X .- ( N ` Y ) ) = ( X .+ Y ) )
Theorem	grplmulf1o 16679*	Left multiplication by a group element is a bijection on any group. (Contributed by Mario Carneiro, 17-Jan-2015.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- F = ( x e. B |-> ( X .+ x ) )   =>    |- ( ( G e. Grp /\ X e. B ) -> F : B -1-1-onto-> B )
Theorem	grpinvpropd 16680*	If two structures have the same group components (properties), they have the same group inversion function. (Contributed by Mario Carneiro, 27-Nov-2014.) (Revised by Stefan O'Rear, 21-Mar-2015.)
|- ( ph -> B = ( Base ` K ) )   &    |- ( ph -> B = ( Base ` L ) )   &    |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )   =>    |- ( ph -> ( invg ` K ) = ( invg ` L ) )
Theorem	grpidssd 16681*	If the base set of a group is contained in the base set of another group, and the group operation of the group is the restriction of the group operation of the other group to its base set, then both groups have the same identity element. (Contributed by AV, 15-Mar-2019.)
|- ( ph -> M e. Grp )   &    |- ( ph -> S e. Grp )   &    |- B = ( Base ` S )   &    |- ( ph -> B C_ ( Base ` M ) )   &    |- ( ph -> A. x e. B A. y e. B ( x ( +g ` M ) y ) = ( x ( +g ` S ) y ) )   =>    |- ( ph -> ( 0g ` M ) = ( 0g ` S ) )
Theorem	grpinvssd 16682*	If the base set of a group is contained in the base set of another group, and the group operation of the group is the restriction of the group operation of the other group to its base set, then the elements of the first group have the same inverses in both groups. (Contributed by AV, 15-Mar-2019.)
|- ( ph -> M e. Grp )   &    |- ( ph -> S e. Grp )   &    |- B = ( Base ` S )   &    |- ( ph -> B C_ ( Base ` M ) )   &    |- ( ph -> A. x e. B A. y e. B ( x ( +g ` M ) y ) = ( x ( +g ` S ) y ) )   =>    |- ( ph -> ( X e. B -> ( ( invg ` S ) ` X ) = ( ( invg ` M ) ` X ) ) )
Theorem	grpinvadd 16683	The inverse of the group operation reverses the arguments. Lemma 2.2.1(d) of [Herstein] p. 55. (Contributed by NM, 27-Oct-2006.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- N = ( invg ` G )   =>    |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( N ` ( X .+ Y ) ) = ( ( N ` Y ) .+ ( N ` X ) ) )
Theorem	grpsubf 16684	Functionality of group subtraction. (Contributed by Mario Carneiro, 9-Sep-2014.)
|- B = ( Base ` G )   &    |- .- = ( -g ` G )   =>    |- ( G e. Grp -> .- : ( B X. B ) --> B )
Theorem	grpsubcl 16685	Closure of group subtraction. (Contributed by NM, 31-Mar-2014.)
|- B = ( Base ` G )   &    |- .- = ( -g ` G )   =>    |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( X .- Y ) e. B )
Theorem	grpsubrcan 16686	Right cancellation law for group subtraction. (Contributed by NM, 31-Mar-2014.)
|- B = ( Base ` G )   &    |- .- = ( -g ` G )   =>    |- ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .- Z ) = ( Y .- Z ) <-> X = Y ) )
Theorem	grpinvsub 16687	Inverse of a group subtraction. (Contributed by NM, 9-Sep-2014.)
|- B = ( Base ` G )   &    |- .- = ( -g ` G )   &    |- N = ( invg ` G )   =>    |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( N ` ( X .- Y ) ) = ( Y .- X ) )
Theorem	grpinvval2 16688	A df-neg 9862-like equation for inverse in terms of group subtraction. (Contributed by Mario Carneiro, 4-Oct-2015.)
|- B = ( Base ` G )   &    |- .- = ( -g ` G )   &    |- N = ( invg ` G )   &    |- .0. = ( 0g ` G )   =>    |- ( ( G e. Grp /\ X e. B ) -> ( N ` X ) = ( .0. .- X ) )
Theorem	grpsubid 16689	Subtraction of a group element from itself. (Contributed by NM, 31-Mar-2014.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- .- = ( -g ` G )   =>    |- ( ( G e. Grp /\ X e. B ) -> ( X .- X ) = .0. )
Theorem	grpsubid1 16690	Subtraction of the identity from a group element. (Contributed by Mario Carneiro, 14-Jan-2015.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- .- = ( -g ` G )   =>    |- ( ( G e. Grp /\ X e. B ) -> ( X .- .0. ) = X )
Theorem	grpsubeq0 16691	If the difference between two group elements is zero, they are equal. (subeq0 9899 analog.) (Contributed by NM, 31-Mar-2014.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- .- = ( -g ` G )   =>    |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( X .- Y ) = .0. <-> X = Y ) )
Theorem	grpsubadd0sub 16692	Subtraction expressed as addition of the difference of the identity element and the subtrahend. (Contributed by AV, 9-Nov-2019.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- .- = ( -g ` G )   &    |- .+ = ( +g ` G )   =>    |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( X .- Y ) = ( X .+ ( .0. .- Y ) ) )
Theorem	grpsubadd 16693	Relationship between group subtraction and addition. (Contributed by NM, 31-Mar-2014.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .- = ( -g ` G )   =>    |- ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .- Y ) = Z <-> ( Z .+ Y ) = X ) )
Theorem	grpsubsub 16694	Double group subtraction. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 2-Dec-2014.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .- = ( -g ` G )   =>    |- ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .- ( Y .- Z ) ) = ( X .+ ( Z .- Y ) ) )
Theorem	grpaddsubass 16695	Associative-type law for group subtraction and addition. (Contributed by NM, 16-Apr-2014.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .- = ( -g ` G )   =>    |- ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .+ Y ) .- Z ) = ( X .+ ( Y .- Z ) ) )
Theorem	grppncan 16696	Cancellation law for subtraction (pncan 9880 analog). (Contributed by NM, 16-Apr-2014.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .- = ( -g ` G )   =>    |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( X .+ Y ) .- Y ) = X )
Theorem	grpnpcan 16697	Cancellation law for subtraction (npcan 9883 analog). (Contributed by NM, 19-Apr-2014.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .- = ( -g ` G )   =>    |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( X .- Y ) .+ Y ) = X )
Theorem	grpsubsub4 16698	Double group subtraction (subsub4 9906 analog). (Contributed by Mario Carneiro, 2-Dec-2014.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .- = ( -g ` G )   =>    |- ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .- Y ) .- Z ) = ( X .- ( Z .+ Y ) ) )
Theorem	grppnpcan2 16699	Cancellation law for mixed addition and subtraction. (pnpcan2 9913 analog.) (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 2-Dec-2014.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .- = ( -g ` G )   =>    |- ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .+ Z ) .- ( Y .+ Z ) ) = ( X .- Y ) )
Theorem	grpnpncan 16700	Cancellation law for group subtraction. (npncan 9894 analog.) (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 2-Dec-2014.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .- = ( -g ` G )   =>    |- ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .- Y ) .+ ( Y .- Z ) ) = ( X .- Z ) )