P. 134 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 13301-13400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	grprinv 13301	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 13302	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 13303	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 13304*	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 13305	The left inverse of a group element. Deduction associated with grplinv 13300. (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 13306	The right inverse of a group element. Deduction associated with grprinv 13301. (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 13307*	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 13308*	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 13309*	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 13310	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 13311	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 12822. (Contributed by Jim Kingdon, 28-Feb-2025.)
|- B = ( Base ` G )   =>    |- ( G e. Grp -> ( G ↾s B ) e. Grp )
Theorem	grplcan 13312	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 13313	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 13314	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 13315	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 13316	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 13317	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 13318	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 13319	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 13320	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 13321	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 13322	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 13323	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 13324*	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 13325*	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 13326*	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 13327*	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 13328	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 13329	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 13330	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 13331	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 13332	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 13333	A df-neg 8228-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 13334	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 13335	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 13336	If the difference between two group elements is zero, they are equal. (subeq0 8280 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 13337	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 13338	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 13339	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 13340	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 13341	Cancellation law for subtraction (pncan 8260 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 13342	Cancellation law for subtraction (npcan 8263 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 13343	Double group subtraction (subsub4 8287 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 13344	Cancellation law for mixed addition and subtraction. (pnpcan2 8294 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 13345	Cancellation law for group subtraction. (npncan 8275 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 13346	Cancellation law for group subtraction (npncan2 8281 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 13347	Cancellation law for group subtraction. (nnncan2 8291 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 13348*	Lemma for dfgrp3m 13349. (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 13349*	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 13350*	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 13351*	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 13352*	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 13353*	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 13354	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 13355*	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 13356	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 13357	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 13358*	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 13359	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 13360*	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 13361	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 13362	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 13363	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 13364*	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 13365*	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 13366	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 13367*	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 13368*	Lemma for mhmmnd 13370 and ghmgrp 13372. (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 13369*	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 13370*	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 13371*	The function fulfilling the conditions of mhmmnd 13370 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 13372*	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 13374, however, defines the group multiple for arbitrary (i.e. also negative) integers. This is meaningful for groups only, and requires Definition df-minusg 13254 of the inverse operation invg.
Syntax	cmg 13373	Extend class notation with a function mapping a group operation to the multiple/power operation for the magma/group.
class .g
Definition	df-mulg 13374*	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 13375*	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 13376	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 13377	Existence of the group multiple operation. (Contributed by Jim Kingdon, 22-Apr-2025.)
|- ( G e. V -> (.g ` G ) e. _V )
Theorem	mulgfng 13378	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 13379	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 13380	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	mulgnngsum 13381*	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 gsumg F ) )
Theorem	mulgnn0gsum 13382*	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 gsumg F ) )
Theorem	mulg1 13383	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 13384	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 13385	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 13386	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 13387	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 13388*	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 13389*	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 13390*	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 13391	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 13392	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 13393	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 )
Theorem	mulgneg 13394	Group multiple (exponentiation) operation at a negative integer. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 11-Dec-2014.)
|- B = ( Base ` G )   &    |- .x. = (.g ` G )   &    |- I = ( invg ` G )   =>    |- ( ( G e. Grp /\ N e. ZZ /\ X e. B ) -> ( -u N .x. X ) = ( I ` ( N .x. X ) ) )
Theorem	mulgnegneg 13395	The inverse of a negative group multiple is the positive group multiple. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 30-Aug-2021.)
|- B = ( Base ` G )   &    |- .x. = (.g ` G )   &    |- I = ( invg ` G )   =>    |- ( ( G e. Grp /\ N e. ZZ /\ X e. B ) -> ( I ` ( -u N .x. X ) ) = ( N .x. X ) )
Theorem	mulgm1 13396	Group multiple (exponentiation) operation at negative one. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 20-Dec-2014.)
|- B = ( Base ` G )   &    |- .x. = (.g ` G )   &    |- I = ( invg ` G )   =>    |- ( ( G e. Grp /\ X e. B ) -> ( -u 1 .x. X ) = ( I ` X ) )
Theorem	mulgnn0cld 13397	Closure of the group multiple (exponentiation) operation for a nonnegative multiplier in a monoid. Deduction associated with mulgnn0cl 13392. (Contributed by SN, 1-Feb-2025.)
|- B = ( Base ` G )   &    |- .x. = (.g ` G )   &    |- ( ph -> G e. Mnd )   &    |- ( ph -> N e. NN0 )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( N .x. X ) e. B )
Theorem	mulgcld 13398	Deduction associated with mulgcl 13393. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- B = ( Base ` G )   &    |- .x. = (.g ` G )   &    |- ( ph -> G e. Grp )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( N .x. X ) e. B )
Theorem	mulgaddcomlem 13399	Lemma for mulgaddcom 13400. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 31-Aug-2021.)
|- B = ( Base ` G )   &    |- .x. = (.g ` G )   &    |- .+ = ( +g ` G )   =>    |- ( ( ( G e. Grp /\ y e. ZZ /\ X e. B ) /\ ( ( y .x. X ) .+ X ) = ( X .+ ( y .x. X ) ) ) -> ( ( -u y .x. X ) .+ X ) = ( X .+ ( -u y .x. X ) ) )
Theorem	mulgaddcom 13400	The group multiple operator commutes with the group operation. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 31-Aug-2021.)
|- B = ( Base ` G )   &    |- .x. = (.g ` G )   &    |- .+ = ( +g ` G )   =>    |- ( ( G e. Grp /\ N e. ZZ /\ X e. B ) -> ( ( N .x. X ) .+ X ) = ( X .+ ( N .x. X ) ) )