P. 137 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 13601-13700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	dfgrp2e 13601*	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 13602	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 13603	The base set of a group is not empty. It is also inhabited (see grpidcl 13602). (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)
|- B = ( Base ` G )   =>    |- ( G e. Grp -> B =/= (/) )
Theorem	grplid 13604	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 13605	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 13606	The identity element of a group is a left identity. Deduction associated with grplid 13604. (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 13607	The identity element of a group is a right identity. Deduction associated with grprid 13605. (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 13608	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 13609	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 13610	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 13611*	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 13612	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 13613	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 13614*	Deduce the identity element of a group from its properties. Useful in conjunction with isgrpd 13596. (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 13615*	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 13616*	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 13617	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 13618*	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 13619	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 13620	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 13621	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 13622	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 13623	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 13624	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 13625	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 13626	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 13627*	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 13628	The left inverse of a group element. Deduction associated with grplinv 13623. (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 13629	The right inverse of a group element. Deduction associated with grprinv 13624. (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 13630*	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 13631*	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 13632*	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 13633	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 13634	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 13144. (Contributed by Jim Kingdon, 28-Feb-2025.)
|- B = ( Base ` G )   =>    |- ( G e. Grp -> ( G ↾s B ) e. Grp )
Theorem	grplcan 13635	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 13636	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 13637	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 13638	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 13639	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 13640	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 13641	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 13642	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 13643	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 13644	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 13645	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 13646	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 13647*	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 13648*	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 13649*	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 13650*	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 13651	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 13652	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 13653	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 13654	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 13655	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 13656	A df-neg 8343-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 13657	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 13658	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 13659	If the difference between two group elements is zero, they are equal. (subeq0 8395 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 13660	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 13661	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 13662	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 13663	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 13664	Cancellation law for subtraction (pncan 8375 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 13665	Cancellation law for subtraction (npcan 8378 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 13666	Double group subtraction (subsub4 8402 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 13667	Cancellation law for mixed addition and subtraction. (pnpcan2 8409 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 13668	Cancellation law for group subtraction. (npncan 8390 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 13669	Cancellation law for group subtraction (npncan2 8396 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 13670	Cancellation law for group subtraction. (nnncan2 8406 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 13671*	Lemma for dfgrp3m 13672. (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 13672*	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 13673*	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 13674*	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 13675*	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 13676*	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 13677	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 13678*	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 13679	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 13680	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	prdsinvlem 13681*	Characterization of inverses in a structure product. (Contributed by Mario Carneiro, 10-Jan-2015.)
|- Y = ( S X_s R )   &    |- B = ( Base ` Y )   &    |- .+ = ( +g ` Y )   &    |- ( ph -> S e. V )   &    |- ( ph -> I e. W )   &    |- ( ph -> R : I --> Grp )   &    |- ( ph -> F e. B )   &    |- .0. = ( 0g o. R )   &    |- N = ( y e. I |-> ( ( invg ` ( R ` y ) ) ` ( F ` y ) ) )   =>    |- ( ph -> ( N e. B /\ ( N .+ F ) = .0. ) )
Theorem	prdsgrpd 13682	The product of a family of groups is a group. (Contributed by Stefan O'Rear, 10-Jan-2015.)
|- Y = ( S X_s R )   &    |- ( ph -> I e. W )   &    |- ( ph -> S e. V )   &    |- ( ph -> R : I --> Grp )   =>    |- ( ph -> Y e. Grp )
Theorem	prdsinvgd 13683*	Negation in a product of groups. (Contributed by Stefan O'Rear, 10-Jan-2015.)
|- Y = ( S X_s R )   &    |- ( ph -> I e. W )   &    |- ( ph -> S e. V )   &    |- ( ph -> R : I --> Grp )   &    |- B = ( Base ` Y )   &    |- N = ( invg ` Y )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( N ` X ) = ( x e. I |-> ( ( invg ` ( R ` x ) ) ` ( X ` x ) ) ) )
Theorem	pwsgrp 13684	A structure power of a group is a group. (Contributed by Mario Carneiro, 11-Jan-2015.)
|- Y = ( R ^s I )   =>    |- ( ( R e. Grp /\ I e. V ) -> Y e. Grp )
Theorem	pwsinvg 13685	Negation in a group power. (Contributed by Mario Carneiro, 11-Jan-2015.)
|- Y = ( R ^s I )   &    |- B = ( Base ` Y )   &    |- M = ( invg ` R )   &    |- N = ( invg ` Y )   =>    |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> ( N ` X ) = ( M o. X ) )
Theorem	pwssub 13686	Subtraction in a group power. (Contributed by Mario Carneiro, 12-Jan-2015.)
|- Y = ( R ^s I )   &    |- B = ( Base ` Y )   &    |- M = ( -g ` R )   &    |- .- = ( -g ` Y )   =>    |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> ( F .- G ) = ( F oF M G ) )
Theorem	imasgrp2 13687*	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 13688*	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 13689	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 13690*	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 13691*	Lemma for mhmmnd 13693 and ghmgrp 13695. (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 13692*	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 13693*	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 13694*	The function fulfilling the conditions of mhmmnd 13693 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 13695*	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 13697, however, defines the group multiple for arbitrary (i.e. also negative) integers. This is meaningful for groups only, and requires Definition df-minusg 13577 of the inverse operation invg.
Syntax	cmg 13696	Extend class notation with a function mapping a group operation to the multiple/power operation for the magma/group.
class .g
Definition	df-mulg 13697*	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 13698*	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 13699	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 13700	Existence of the group multiple operation. (Contributed by Jim Kingdon, 22-Apr-2025.)
|- ( G e. V -> (.g ` G ) e. _V )