P. 140 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 13901-14000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	grpinveu 13901*	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 13902	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 13903	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 13904*	Deduce the identity element of a group from its properties. Useful in conjunction with isgrpd 13886. (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 13905*	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 13906*	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 13907	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 13908*	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 13909	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 13910	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 13911	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 13912	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 13913	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 13914	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 13915	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 13916	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 13917*	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 13918	The left inverse of a group element. Deduction associated with grplinv 13913. (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 13919	The right inverse of a group element. Deduction associated with grprinv 13914. (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 13920*	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 13921*	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 13922*	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 13923	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 13924	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 13483. (Contributed by Jim Kingdon, 28-Feb-2025.)
|- B = ( Base ` G )   =>    |- ( G e. Grp -> ( G ↾s B ) e. Grp )
Theorem	grplcan 13925	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 13926	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 13927	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 13928	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 13929	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 13930	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 13931	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 13932	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 13933	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 13934	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 13935	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 13936	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 13937*	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 13938*	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 13939*	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 13940*	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 13941	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 13942	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 13943	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 13944	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 13945	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 13946	A df-neg 8507-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 13947	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 13948	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 13949	If the difference between two group elements is zero, they are equal. (subeq0 8559 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 13950	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 13951	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 13952	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 13953	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 13954	Cancellation law for subtraction (pncan 8539 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 13955	Cancellation law for subtraction (npcan 8542 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 13956	Double group subtraction (subsub4 8566 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 13957	Cancellation law for mixed addition and subtraction. (pnpcan2 8573 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 13958	Cancellation law for group subtraction. (npncan 8554 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 ) )
Theorem	grpnpncan0 13959	Cancellation law for group subtraction (npncan2 8560 analog). (Contributed by AV, 24-Nov-2019.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .- = ( -g ` G )   &    |- .0. = ( 0g ` G )   =>    |- ( ( G e. Grp /\ ( X e. B /\ Y e. B ) ) -> ( ( X .- Y ) .+ ( Y .- X ) ) = .0. )
Theorem	grpnnncan2 13960	Cancellation law for group subtraction. (nnncan2 8570 analog.) (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 2-Dec-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	dfgrp3mlem 13961*	Lemma for dfgrp3m 13962. (Contributed by AV, 28-Aug-2021.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( ( G e. Smgrp /\ E. w w e. B /\ A. x e. B A. y e. B ( E. l e. B ( l .+ x ) = y /\ E. r e. B ( x .+ r ) = y ) ) -> E. u e. B A. a e. B ( ( u .+ a ) = a /\ E. i e. B ( i .+ a ) = u ) )
Theorem	dfgrp3m 13962*	Alternate definition of a group as semigroup (with at least one element) which is also a quasigroup, i.e. a magma in which solutions x and y of the equations ( a .+ x ) = b and ( x .+ a ) = b exist. Theorem 3.2 of [Bruck] p. 28. (Contributed by AV, 28-Aug-2021.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( G e. Grp <-> ( G e. Smgrp /\ E. w w e. B /\ A. x e. B A. y e. B ( E. l e. B ( l .+ x ) = y /\ E. r e. B ( x .+ r ) = y ) ) )
Theorem	dfgrp3me 13963*	Alternate definition of a group as a set with a closed, associative operation, for which solutions x and y of the equations ( a .+ x ) = b and ( x .+ a ) = b exist. Exercise 1 of [Herstein] p. 57. (Contributed by NM, 5-Dec-2006.) (Revised by AV, 28-Aug-2021.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( G e. Grp <-> ( E. w w e. B /\ A. x e. B A. y e. B ( ( x .+ y ) e. B /\ A. z e. B ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) /\ ( E. l e. B ( l .+ x ) = y /\ E. r e. B ( x .+ r ) = y ) ) ) )
Theorem	grplactfval 13964*	The left group action of element A of group G. (Contributed by Paul Chapman, 18-Mar-2008.)
|- F = ( g e. X |-> ( a e. X |-> ( g .+ a ) ) )   &    |- X = ( Base ` G )   =>    |- ( A e. X -> ( F ` A ) = ( a e. X |-> ( A .+ a ) ) )
Theorem	grplactcnv 13965*	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.) (Proof shortened by Mario Carneiro, 14-Aug-2015.)
|- F = ( g e. X |-> ( a e. X |-> ( g .+ a ) ) )   &    |- X = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- I = ( invg ` G )   =>    |- ( ( G e. Grp /\ A e. X ) -> ( ( F ` A ) : X -1-1-onto-> X /\ `' ( F ` A ) = ( F ` ( I ` A ) ) ) )
Theorem	grplactf1o 13966*	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.) (Proof shortened by Mario Carneiro, 14-Aug-2015.)
|- F = ( g e. X |-> ( a e. X |-> ( g .+ a ) ) )   &    |- X = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( ( G e. Grp /\ A e. X ) -> ( F ` A ) : X -1-1-onto-> X )
Theorem	grpsubpropdg 13967	Weak property deduction for the group subtraction operation. (Contributed by Mario Carneiro, 27-Mar-2015.)
|- ( ph -> ( Base ` G ) = ( Base ` H ) )   &    |- ( ph -> ( +g ` G ) = ( +g ` H ) )   &    |- ( ph -> G e. V )   &    |- ( ph -> H e. W )   =>    |- ( ph -> ( -g ` G ) = ( -g ` H ) )
Theorem	grpsubpropd2 13968*	Strong property deduction for the group subtraction operation. (Contributed by Mario Carneiro, 4-Oct-2015.)
|- ( ph -> B = ( Base ` G ) )   &    |- ( ph -> B = ( Base ` H ) )   &    |- ( ph -> G e. Grp )   &    |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` G ) y ) = ( x ( +g ` H ) y ) )   =>    |- ( ph -> ( -g ` G ) = ( -g ` H ) )
Theorem	grp1 13969	The (smallest) structure representing a trivial group. According to Wikipedia ("Trivial group", 28-Apr-2019, https://en.wikipedia.org/wiki/Trivial_group) "In mathematics, a trivial group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element". (Contributed by AV, 28-Apr-2019.)
|- M = { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. }   =>    |- ( I e. V -> M e. Grp )
Theorem	grp1inv 13970	The inverse function of the trivial group. (Contributed by FL, 21-Jun-2010.) (Revised by AV, 26-Aug-2021.)
|- M = { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. }   =>    |- ( I e. V -> ( invg ` M ) = ( _I |` { I } ) )
Theorem	imasgrp2 13971*	The image structure of a group is a group. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-2015.)
|- ( ph -> U = ( F "s R ) )   &    |- ( ph -> V = ( Base ` R ) )   &    |- ( ph -> .+ = ( +g ` R ) )   &    |- ( ph -> F : V -onto-> B )   &    |- ( ( ph /\ ( a e. V /\ b e. V ) /\ ( p e. V /\ q e. V ) ) -> ( ( ( F ` a ) = ( F ` p ) /\ ( F ` b ) = ( F ` q ) ) -> ( F ` ( a .+ b ) ) = ( F ` ( p .+ q ) ) ) )   &    |- ( ph -> R e. W )   &    |- ( ( ph /\ x e. V /\ y e. V ) -> ( x .+ y ) e. V )   &    |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> ( F ` ( ( x .+ y ) .+ z ) ) = ( F ` ( x .+ ( y .+ z ) ) ) )   &    |- ( ph -> .0. e. V )   &    |- ( ( ph /\ x e. V ) -> ( F ` ( .0. .+ x ) ) = ( F ` x ) )   &    |- ( ( ph /\ x e. V ) -> N e. V )   &    |- ( ( ph /\ x e. V ) -> ( F ` ( N .+ x ) ) = ( F ` .0. ) )   =>    |- ( ph -> ( U e. Grp /\ ( F ` .0. ) = ( 0g ` U ) ) )
Theorem	imasgrp 13972*	The image structure of a group is a group. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-2015.)
|- ( ph -> U = ( F "s R ) )   &    |- ( ph -> V = ( Base ` R ) )   &    |- ( ph -> .+ = ( +g ` R ) )   &    |- ( ph -> F : V -onto-> B )   &    |- ( ( ph /\ ( a e. V /\ b e. V ) /\ ( p e. V /\ q e. V ) ) -> ( ( ( F ` a ) = ( F ` p ) /\ ( F ` b ) = ( F ` q ) ) -> ( F ` ( a .+ b ) ) = ( F ` ( p .+ q ) ) ) )   &    |- ( ph -> R e. Grp )   &    |- .0. = ( 0g ` R )   =>    |- ( ph -> ( U e. Grp /\ ( F ` .0. ) = ( 0g ` U ) ) )
Theorem	imasgrpf1 13973	The image of a group under an injection is a group. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- U = ( F "s R )   &    |- V = ( Base ` R )   =>    |- ( ( F : V -1-1-> B /\ R e. Grp ) -> U e. Grp )
Theorem	qusgrp2 13974*	Prove that a quotient structure is a group. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)
|- ( ph -> U = ( R /.s .~ ) )   &    |- ( ph -> V = ( Base ` R ) )   &    |- ( ph -> .+ = ( +g ` R ) )   &    |- ( ph -> .~ Er V )   &    |- ( ph -> R e. X )   &    |- ( ph -> ( ( a .~ p /\ b .~ q ) -> ( a .+ b ) .~ ( p .+ q ) ) )   &    |- ( ( ph /\ x e. V /\ y e. V ) -> ( x .+ y ) e. V )   &    |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> ( ( x .+ y ) .+ z ) .~ ( x .+ ( y .+ z ) ) )   &    |- ( ph -> .0. e. V )   &    |- ( ( ph /\ x e. V ) -> ( .0. .+ x ) .~ x )   &    |- ( ( ph /\ x e. V ) -> N e. V )   &    |- ( ( ph /\ x e. V ) -> ( N .+ x ) .~ .0. )   =>    |- ( ph -> ( U e. Grp /\ [ .0. ] .~ = ( 0g ` U ) ) )
Theorem	mhmlem 13975*	Lemma for mhmmnd 13977 and ghmgrp 13979. (Contributed by Paul Chapman, 25-Apr-2008.) (Revised by Mario Carneiro, 12-May-2014.) (Revised by Thierry Arnoux, 25-Jan-2020.)
|- ( ( ph /\ x e. X /\ y e. X ) -> ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) )   &    |- ( ph -> A e. X )   &    |- ( ph -> B e. X )   =>    |- ( ph -> ( F ` ( A .+ B ) ) = ( ( F ` A ) .+^ ( F ` B ) ) )
Theorem	mhmid 13976*	A surjective monoid morphism preserves identity element. (Contributed by Thierry Arnoux, 25-Jan-2020.)
|- ( ( ph /\ x e. X /\ y e. X ) -> ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) )   &    |- X = ( Base ` G )   &    |- Y = ( Base ` H )   &    |- .+ = ( +g ` G )   &    |- .+^ = ( +g ` H )   &    |- ( ph -> F : X -onto-> Y )   &    |- ( ph -> G e. Mnd )   &    |- .0. = ( 0g ` G )   =>    |- ( ph -> ( F ` .0. ) = ( 0g ` H ) )
Theorem	mhmmnd 13977*	The image of a monoid G under a monoid homomorphism F is a monoid. (Contributed by Thierry Arnoux, 25-Jan-2020.)
|- ( ( ph /\ x e. X /\ y e. X ) -> ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) )   &    |- X = ( Base ` G )   &    |- Y = ( Base ` H )   &    |- .+ = ( +g ` G )   &    |- .+^ = ( +g ` H )   &    |- ( ph -> F : X -onto-> Y )   &    |- ( ph -> G e. Mnd )   =>    |- ( ph -> H e. Mnd )
Theorem	mhmfmhm 13978*	The function fulfilling the conditions of mhmmnd 13977 is a monoid homomorphism. (Contributed by Thierry Arnoux, 26-Jan-2020.)
|- ( ( ph /\ x e. X /\ y e. X ) -> ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) )   &    |- X = ( Base ` G )   &    |- Y = ( Base ` H )   &    |- .+ = ( +g ` G )   &    |- .+^ = ( +g ` H )   &    |- ( ph -> F : X -onto-> Y )   &    |- ( ph -> G e. Mnd )   =>    |- ( ph -> F e. ( G MndHom H ) )
Theorem	ghmgrp 13979*	The image of a group G under a group homomorphism F is a group. 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.) (Revised by Mario Carneiro, 12-May-2014.) (Revised by Thierry Arnoux, 25-Jan-2020.)
|- ( ( ph /\ x e. X /\ y e. X ) -> ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) )   &    |- X = ( Base ` G )   &    |- Y = ( Base ` H )   &    |- .+ = ( +g ` G )   &    |- .+^ = ( +g ` H )   &    |- ( ph -> F : X -onto-> Y )   &    |- ( ph -> G e. Grp )   =>    |- ( ph -> H e. Grp )
7.2.2  Group multiple operation The "group multiple" operation (if the group is multiplicative, also called "group power" or "group exponentiation" operation), can be defined for arbitrary magmas, if the multiplier/exponent is a nonnegative integer. See also the definition in [Lang] p. 6, where an element x(of a monoid) to the power of a nonnegative integer n is defined and denoted by x ^ n. Definition df-mulg 13981, however, defines the group multiple for arbitrary (i.e. also negative) integers. This is meaningful for groups only, and requires Definition df-minusg 13867 of the inverse operation invg.
Syntax	cmg 13980	Extend class notation with a function mapping a group operation to the multiple/power operation for the magma/group.
class .g
Definition	df-mulg 13981*	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	mulgfvalg 13982*	Group multiple (exponentiation) operation. (Contributed by Mario Carneiro, 11-Dec-2014.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .0. = ( 0g ` G )   &    |- I = ( invg ` G )   &    |- .x. = (.g ` G )   =>    |- ( G e. V -> .x. = ( n e. ZZ , x e. B |-> if ( n = 0 , .0. , if ( 0 < n , ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) , ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) ) ) ) )
Theorem	mulgval 13983	Value of the group multiple (exponentiation) operation. (Contributed by Mario Carneiro, 11-Dec-2014.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .0. = ( 0g ` G )   &    |- I = ( invg ` G )   &    |- .x. = (.g ` G )   &    |- S = seq 1 ( .+ , ( NN X. { X } ) )   =>    |- ( ( N e. ZZ /\ X e. B ) -> ( N .x. X ) = if ( N = 0 , .0. , if ( 0 < N , ( S ` N ) , ( I ` ( S ` -u N ) ) ) ) )
Theorem	mulgex 13984	Existence of the group multiple operation. (Contributed by Jim Kingdon, 22-Apr-2025.)
|- ( G e. V -> (.g ` G ) e. _V )
Theorem	mulgfng 13985	Functionality of the group multiple operation. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
|- B = ( Base ` G )   &    |- .x. = (.g ` G )   =>    |- ( G e. V -> .x. Fn ( ZZ X. B ) )
Theorem	mulg0 13986	Group multiple (exponentiation) operation at zero. (Contributed by Mario Carneiro, 11-Dec-2014.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- .x. = (.g ` G )   =>    |- ( X e. B -> ( 0 .x. X ) = .0. )
Theorem	mulgnn 13987	Group multiple (exponentiation) operation at a positive integer. (Contributed by Mario Carneiro, 11-Dec-2014.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .x. = (.g ` G )   &    |- S = seq 1 ( .+ , ( NN X. { X } ) )   =>    |- ( ( N e. NN /\ X e. B ) -> ( N .x. X ) = ( S ` N ) )
Theorem	mulgnngzsum 13988*	Group multiple (exponentiation) operation at a positive integer expressed by a group sum. (Contributed by AV, 28-Dec-2023.)
|- B = ( Base ` G )   &    |- .x. = (.g ` G )   &    |- F = ( x e. ( 1 ... N ) |-> X )   =>    |- ( ( N e. NN /\ X e. B ) -> ( N .x. X ) = ( G gzsumgz F ) )
Theorem	mulgnn0gzsum 13989*	Group multiple (exponentiation) operation at a nonnegative integer expressed by a group sum. This corresponds to the definition in [Lang] p. 6, second formula. (Contributed by AV, 28-Dec-2023.)
|- B = ( Base ` G )   &    |- .x. = (.g ` G )   &    |- F = ( x e. ( 1 ... N ) |-> X )   =>    |- ( ( N e. NN0 /\ X e. B ) -> ( N .x. X ) = ( G gzsumgz F ) )
Theorem	mulg1 13990	Group multiple (exponentiation) operation at one. (Contributed by Mario Carneiro, 11-Dec-2014.)
|- B = ( Base ` G )   &    |- .x. = (.g ` G )   =>    |- ( X e. B -> ( 1 .x. X ) = X )
Theorem	mulgnnp1 13991	Group multiple (exponentiation) operation at a successor. (Contributed by Mario Carneiro, 11-Dec-2014.)
|- B = ( Base ` G )   &    |- .x. = (.g ` G )   &    |- .+ = ( +g ` G )   =>    |- ( ( N e. NN /\ X e. B ) -> ( ( N + 1 ) .x. X ) = ( ( N .x. X ) .+ X ) )
Theorem	mulg2 13992	Group multiple (exponentiation) operation at two. (Contributed by Mario Carneiro, 15-Oct-2015.)
|- B = ( Base ` G )   &    |- .x. = (.g ` G )   &    |- .+ = ( +g ` G )   =>    |- ( X e. B -> ( 2 .x. X ) = ( X .+ X ) )
Theorem	mulgnegnn 13993	Group multiple (exponentiation) operation at a negative integer. (Contributed by Mario Carneiro, 11-Dec-2014.)
|- B = ( Base ` G )   &    |- .x. = (.g ` G )   &    |- I = ( invg ` G )   =>    |- ( ( N e. NN /\ X e. B ) -> ( -u N .x. X ) = ( I ` ( N .x. X ) ) )
Theorem	mulgnn0p1 13994	Group multiple (exponentiation) operation at a successor, extended to NN0. (Contributed by Mario Carneiro, 11-Dec-2014.)
|- B = ( Base ` G )   &    |- .x. = (.g ` G )   &    |- .+ = ( +g ` G )   =>    |- ( ( G e. Mnd /\ N e. NN0 /\ X e. B ) -> ( ( N + 1 ) .x. X ) = ( ( N .x. X ) .+ X ) )
Theorem	mulgnnsubcl 13995*	Closure of the group multiple (exponentiation) operation in a subsemigroup. (Contributed by Mario Carneiro, 10-Jan-2015.)
|- B = ( Base ` G )   &    |- .x. = (.g ` G )   &    |- .+ = ( +g ` G )   &    |- ( ph -> G e. V )   &    |- ( ph -> S C_ B )   &    |- ( ( ph /\ x e. S /\ y e. S ) -> ( x .+ y ) e. S )   =>    |- ( ( ph /\ N e. NN /\ X e. S ) -> ( N .x. X ) e. S )
Theorem	mulgnn0subcl 13996*	Closure of the group multiple (exponentiation) operation in a submonoid. (Contributed by Mario Carneiro, 10-Jan-2015.)
|- B = ( Base ` G )   &    |- .x. = (.g ` G )   &    |- .+ = ( +g ` G )   &    |- ( ph -> G e. V )   &    |- ( ph -> S C_ B )   &    |- ( ( ph /\ x e. S /\ y e. S ) -> ( x .+ y ) e. S )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> .0. e. S )   =>    |- ( ( ph /\ N e. NN0 /\ X e. S ) -> ( N .x. X ) e. S )
Theorem	mulgsubcl 13997*	Closure of the group multiple (exponentiation) operation in a subgroup. (Contributed by Mario Carneiro, 10-Jan-2015.)
|- B = ( Base ` G )   &    |- .x. = (.g ` G )   &    |- .+ = ( +g ` G )   &    |- ( ph -> G e. V )   &    |- ( ph -> S C_ B )   &    |- ( ( ph /\ x e. S /\ y e. S ) -> ( x .+ y ) e. S )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> .0. e. S )   &    |- I = ( invg ` G )   &    |- ( ( ph /\ x e. S ) -> ( I ` x ) e. S )   =>    |- ( ( ph /\ N e. ZZ /\ X e. S ) -> ( N .x. X ) e. S )
Theorem	mulgnncl 13998	Closure of the group multiple (exponentiation) operation for a positive multiplier in a magma. (Contributed by Mario Carneiro, 11-Dec-2014.) (Revised by AV, 29-Aug-2021.)
|- B = ( Base ` G )   &    |- .x. = (.g ` G )   =>    |- ( ( G e. Mgm /\ N e. NN /\ X e. B ) -> ( N .x. X ) e. B )
Theorem	mulgnn0cl 13999	Closure of the group multiple (exponentiation) operation for a nonnegative multiplier in a monoid. (Contributed by Mario Carneiro, 11-Dec-2014.)
|- B = ( Base ` G )   &    |- .x. = (.g ` G )   =>    |- ( ( G e. Mnd /\ N e. NN0 /\ X e. B ) -> ( N .x. X ) e. B )
Theorem	mulgcl 14000	Closure of the group multiple (exponentiation) operation. (Contributed by Mario Carneiro, 11-Dec-2014.)
|- B = ( Base ` G )   &    |- .x. = (.g ` G )   =>    |- ( ( G e. Grp /\ N e. ZZ /\ X e. B ) -> ( N .x. X ) e. B )