P. 138 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 13701-13800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	resmhm2 13701	One direction of resmhm2b 13702. (Contributed by Mario Carneiro, 18-Jun-2015.)
|- U = ( T ↾s X )   =>    |- ( ( F e. ( S MndHom U ) /\ X e. (SubMnd ` T ) ) -> F e. ( S MndHom T ) )
Theorem	resmhm2b 13702	Restriction of the codomain of a homomorphism. (Contributed by Mario Carneiro, 18-Jun-2015.)
|- U = ( T ↾s X )   =>    |- ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) -> ( F e. ( S MndHom T ) <-> F e. ( S MndHom U ) ) )
Theorem	mhmco 13703	The composition of monoid homomorphisms is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
|- ( ( F e. ( T MndHom U ) /\ G e. ( S MndHom T ) ) -> ( F o. G ) e. ( S MndHom U ) )
Theorem	mhmima 13704	The homomorphic image of a submonoid is a submonoid. (Contributed by Mario Carneiro, 10-Mar-2015.)
|- ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) -> ( F " X ) e. (SubMnd ` N ) )
Theorem	mhmeql 13705	The equalizer of two monoid homomorphisms is a submonoid. (Contributed by Stefan O'Rear, 7-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
|- ( ( F e. ( S MndHom T ) /\ G e. ( S MndHom T ) ) -> dom ( F i^i G ) e. (SubMnd ` S ) )
7.1.7  Iterated sums in a monoid One important use of words is as formal composites in cases where order is significant, using the general sum operator df-igsum 13472. If order is not significant, it is simpler to use families instead.
Theorem	gsumvallem2 13706*	Lemma for properties of the set of identities of G. The set of identities of a monoid is exactly the unique identity element. (Contributed by Mario Carneiro, 7-Dec-2014.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- .+ = ( +g ` G )   &    |- O = { x e. B | A. y e. B ( ( x .+ y ) = y /\ ( y .+ x ) = y ) }   =>    |- ( G e. Mnd -> O = { .0. } )
Theorem	gsumsubm 13707	Evaluate a group sum in a submonoid. (Contributed by Mario Carneiro, 19-Dec-2014.)
|- ( ph -> A e. V )   &    |- ( ph -> S e. (SubMnd ` G ) )   &    |- ( ph -> F : A --> S )   &    |- H = ( G ↾s S )   =>    |- ( ph -> ( G gsumg F ) = ( H gsumg F ) )
Theorem	gsumfzz 13708*	Value of a group sum over the zero element. (Contributed by Mario Carneiro, 7-Dec-2014.) (Revised by Jim Kingdon, 15-Aug-2025.)
|- .0. = ( 0g ` G )   =>    |- ( ( G e. Mnd /\ M e. ZZ /\ N e. ZZ ) -> ( G gsumg ( k e. ( M ... N ) |-> .0. ) ) = .0. )
Theorem	gsumwsubmcl 13709	Closure of the composite in any submonoid. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 1-Oct-2015.)
|- ( ( S e. (SubMnd ` G ) /\ W e. Word S ) -> ( G gsumg W ) e. S )
Theorem	gsumwcl 13710	Closure of the composite of a word in a structure G. (Contributed by Stefan O'Rear, 15-Aug-2015.)
|- B = ( Base ` G )   =>    |- ( ( G e. Mnd /\ W e. Word B ) -> ( G gsumg W ) e. B )
Theorem	gsumwmhm 13711	Behavior of homomorphisms on finite monoidal sums. (Contributed by Stefan O'Rear, 27-Aug-2015.)
|- B = ( Base ` M )   =>    |- ( ( H e. ( M MndHom N ) /\ W e. Word B ) -> ( H ` ( M gsumg W ) ) = ( N gsumg ( H o. W ) ) )
Theorem	gsumfzcl 13712	Closure of a finite group sum. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by AV, 3-Jun-2019.) (Revised by Jim Kingdon, 16-Aug-2025.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. Mnd )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> F : ( M ... N ) --> B )   =>    |- ( ph -> ( G gsumg F ) e. B )
7.2  Groups
7.2.1  Definition and basic properties
Syntax	cgrp 13713	Extend class notation with class of all groups.
class Grp
Syntax	cminusg 13714	Extend class notation with inverse of group element.
class invg
Syntax	csg 13715	Extend class notation with group subtraction (or division) operation.
class -g
Definition	df-grp 13716*	Define class of all groups. A group is a monoid (df-mnd 13630) 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 13218) and an internal group operation (notated ( +g ` G ) per df-plusg 13303). 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 13721), associativity (so ( ( a +g b ) +g c ) = ( a +g ( b +g c ) ) for any a, b, c, see grpass 13722), 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 13724). Groups need not be commutative; a commutative group is an Abelian group. Subgroups can often be formed from groups. An example of an (Abelian) group is the set of complex numbers CC over the group operation + (addition). Other structures include groups, including unital rings and fields. (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 13717*	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 13718*	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 ) ) ) )
Theorem	isgrp 13719*	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 13720	A group is a monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)
|- ( G e. Grp -> G e. Mnd )
Theorem	grpcl 13721	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 13722	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 13723*	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 13724*	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	grpassd 13725	A group operation is associative. (Contributed by SN, 29-Jan-2025.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- ( ph -> G e. Grp )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   =>    |- ( ph -> ( ( X .+ Y ) .+ Z ) = ( X .+ ( Y .+ Z ) ) )
Theorem	grpmndd 13726	A group is a monoid. (Contributed by SN, 1-Jun-2024.)
|- ( ph -> G e. Grp )   =>    |- ( ph -> G e. Mnd )
Theorem	grpcld 13727	Closure of the operation of a group. (Contributed by SN, 29-Jul-2024.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- ( ph -> G e. Grp )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( X .+ Y ) e. B )
Theorem	grpplusf 13728	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	grpplusfo 13729	The group addition operation is a function onto the base set/set of group elements. (Contributed by NM, 30-Oct-2006.) (Revised by AV, 30-Aug-2021.)
|- B = ( Base ` G )   &    |- F = ( +f ` G )   =>    |- ( G e. Grp -> F : ( B X. B ) -onto-> B )
Theorem	grppropd 13730*	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 13731	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	grppropstrg 13732	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. V -> ( K e. Grp <-> L e. Grp ) )
Theorem	isgrpd2e 13733*	Deduce a group from its properties. In this version of isgrpd2 13734, 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 13734*	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 2232, but we make an exception for theorems such as isgrpd2 13734 and ismndd 13650 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 13735*	Deduce a group from its properties. In this version of isgrpd 13736, 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 13736*	Deduce a group from its properties. Unlike isgrpd2 13734, 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 13737*	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	grpsgrp 13738	A group is a semigroup. (Contributed by AV, 28-Aug-2021.)
|- ( G e. Grp -> G e. Smgrp )
Theorem	grpmgmd 13739	A group is a magma, deduction form. (Contributed by SN, 14-Apr-2025.)
|- ( ph -> G e. Grp )   =>    |- ( ph -> G e. Mgm )
Theorem	dfgrp2 13740*	Alternate definition of a group as semigroup with a left identity and a left inverse for each element. This "definition" is weaker than df-grp 13716, based on the definition of a monoid which provides a left and a right identity. (Contributed by AV, 28-Aug-2021.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( G e. Grp <-> ( G e. Smgrp /\ E. n e. B A. x e. B ( ( n .+ x ) = x /\ E. i e. B ( i .+ x ) = n ) ) )
Theorem	dfgrp2e 13741*	Alternate definition of a group as a set with a closed, associative operation, a left identity and a left inverse for each element. Alternate definition in [Lang] p. 7. (Contributed by NM, 10-Oct-2006.) (Revised by AV, 28-Aug-2021.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( G e. Grp <-> ( A. x e. B A. y e. B ( ( x .+ y ) e. B /\ A. z e. B ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) ) /\ E. n e. B A. x e. B ( ( n .+ x ) = x /\ E. i e. B ( i .+ x ) = n ) ) )
Theorem	grpidcl 13742	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 13743	The base set of a group is not empty. It is also inhabited (see grpidcl 13742). (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)
|- B = ( Base ` G )   =>    |- ( G e. Grp -> B =/= (/) )
Theorem	grplid 13744	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 13745	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	grplidd 13746	The identity element of a group is a left identity. Deduction associated with grplid 13744. (Contributed by SN, 29-Jan-2025.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. Grp )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( .0. .+ X ) = X )
Theorem	grpridd 13747	The identity element of a group is a right identity. Deduction associated with grprid 13745. (Contributed by SN, 29-Jan-2025.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. Grp )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( X .+ .0. ) = X )
Theorem	grpn0 13748	A group is not empty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) (Revised by Mario Carneiro, 2-Dec-2014.)
|- ( G e. Grp -> G =/= (/) )
Theorem	hashfingrpnn 13749	A finite group has positive integer size. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- B = ( Base ` G )   &    |- ( ph -> G e. Grp )   &    |- ( ph -> B e. Fin )   =>    |- ( ph -> (♯ ` B ) e. NN )
Theorem	grprcan 13750	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 13751*	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 13752	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 13753	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 13754*	Deduce the identity element of a group from its properties. Useful in conjunction with isgrpd 13736. (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	grpinvfvalg 13755*	The inverse function of a group. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.) (Revised by Rohan Ridenour, 13-Aug-2023.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .0. = ( 0g ` G )   &    |- N = ( invg ` G )   =>    |- ( G e. V -> N = ( x e. B |-> ( iota_ y e. B ( y .+ x ) = .0. ) ) )
Theorem	grpinvval 13756*	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	grpinvfng 13757	Functionality of the group inverse function. (Contributed by Stefan O'Rear, 21-Mar-2015.)
|- B = ( Base ` G )   &    |- N = ( invg ` G )   =>    |- ( G e. V -> N Fn B )
Theorem	grpsubfvalg 13758*	Group subtraction (division) operation. (Contributed by NM, 31-Mar-2014.) (Revised by Stefan O'Rear, 27-Mar-2015.) (Proof shortened by AV, 19-Feb-2024.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- I = ( invg ` G )   &    |- .- = ( -g ` G )   =>    |- ( G e. V -> .- = ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) )
Theorem	grpsubval 13759	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 13760	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 13761	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	grpinvcld 13762	A group element's inverse is a group element. (Contributed by SN, 29-Jan-2025.)
|- B = ( Base ` G )   &    |- N = ( invg ` G )   &    |- ( ph -> G e. Grp )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( N ` X ) e. B )
Theorem	grplinv 13763	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 13764	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 13765	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 13766	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 13767*	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	grplinvd 13768	The left inverse of a group element. Deduction associated with grplinv 13763. (Contributed by SN, 29-Jan-2025.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .0. = ( 0g ` G )   &    |- N = ( invg ` G )   &    |- ( ph -> G e. Grp )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( ( N ` X ) .+ X ) = .0. )
Theorem	grprinvd 13769	The right inverse of a group element. Deduction associated with grprinv 13764. (Contributed by SN, 29-Jan-2025.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .0. = ( 0g ` G )   &    |- N = ( invg ` G )   &    |- ( ph -> G e. Grp )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( X .+ ( N ` X ) ) = .0. )
Theorem	grplrinv 13770*	In a group, every member has a left and right inverse. (Contributed by AV, 1-Sep-2021.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .0. = ( 0g ` G )   =>    |- ( G e. Grp -> A. x e. B E. y e. B ( ( y .+ x ) = .0. /\ ( x .+ y ) = .0. ) )
Theorem	grpidinv2 13771*	A group's properties using the explicit identity element. (Contributed by NM, 5-Feb-2010.) (Revised by AV, 1-Sep-2021.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .0. = ( 0g ` G )   =>    |- ( ( G e. Grp /\ A e. B ) -> ( ( ( .0. .+ A ) = A /\ ( A .+ .0. ) = A ) /\ E. y e. B ( ( y .+ A ) = .0. /\ ( A .+ y ) = .0. ) ) )
Theorem	grpidinv 13772*	A group has a left and right identity element, and every member has a left and right inverse. (Contributed by NM, 14-Oct-2006.) (Revised by AV, 1-Sep-2021.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( G e. Grp -> E. u e. B A. x e. B ( ( ( u .+ x ) = x /\ ( x .+ u ) = x ) /\ E. y e. B ( ( y .+ x ) = u /\ ( x .+ y ) = u ) ) )
Theorem	grpinvid 13773	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	grpressid 13774	A group restricted to its base set is a group. It will usually be the original group exactly, of course, but to show that needs additional conditions such as those in strressid 13284. (Contributed by Jim Kingdon, 28-Feb-2025.)
|- B = ( Base ` G )   =>    |- ( G e. Grp -> ( G ↾s B ) e. Grp )
Theorem	grplcan 13775	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	grpasscan1 13776	An associative cancellation law for groups. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by AV, 30-Aug-2021.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- N = ( invg ` G )   =>    |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( X .+ ( ( N ` X ) .+ Y ) ) = Y )
Theorem	grpasscan2 13777	An associative cancellation law for groups. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 30-Aug-2021.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- N = ( invg ` G )   =>    |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( X .+ ( N ` Y ) ) .+ Y ) = X )
Theorem	grpidrcan 13778	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 13779	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 13780	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 13781	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 13782	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 13783	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 13784	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 13785	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 13786	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 13787*	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	grpinvpropdg 13788*	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 -> K e. V )   &    |- ( ph -> L e. W )   &    |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )   =>    |- ( ph -> ( invg ` K ) = ( invg ` L ) )
Theorem	grpidssd 13789*	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 13790*	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 13791	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 13792	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 13793	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 13794	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 13795	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 13796	A df-neg 8447-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 13797	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 13798	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 13799	If the difference between two group elements is zero, they are equal. (subeq0 8499 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 13800	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 ) ) )