P. 138 of Theorem List - Intuitionistic Logic Explorer

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

Type	Label	Description
Statement
Theorem	mulgfng 13701	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 13702	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 13703	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 13704*	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 13705*	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 13706	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 13707	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 13708	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 13709	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 13710	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 13711*	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 13712*	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 13713*	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 13714	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 13715	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 13716	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 13717	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 13718	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 13719	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 13720	Closure of the group multiple (exponentiation) operation for a nonnegative multiplier in a monoid. Deduction associated with mulgnn0cl 13715. (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 13721	Deduction associated with mulgcl 13716. (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 13722	Lemma for mulgaddcom 13723. (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 13723	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 ) ) )
Theorem	mulginvcom 13724	The group multiple operator commutes with the group inverse function. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 31-Aug-2021.)
|- B = ( Base ` G )   &    |- .x. = (.g ` G )   &    |- I = ( invg ` G )   =>    |- ( ( G e. Grp /\ N e. ZZ /\ X e. B ) -> ( N .x. ( I ` X ) ) = ( I ` ( N .x. X ) ) )
Theorem	mulginvinv 13725	The group multiple operator commutes with the group inverse function. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 31-Aug-2021.)
|- B = ( Base ` G )   &    |- .x. = (.g ` G )   &    |- I = ( invg ` G )   =>    |- ( ( G e. Grp /\ N e. ZZ /\ X e. B ) -> ( I ` ( N .x. ( I ` X ) ) ) = ( N .x. X ) )
Theorem	mulgnn0z 13726	A group multiple of the identity, for nonnegative multiple. (Contributed by Mario Carneiro, 13-Dec-2014.)
|- B = ( Base ` G )   &    |- .x. = (.g ` G )   &    |- .0. = ( 0g ` G )   =>    |- ( ( G e. Mnd /\ N e. NN0 ) -> ( N .x. .0. ) = .0. )
Theorem	mulgz 13727	A group multiple of the identity, for integer multiple. (Contributed by Mario Carneiro, 13-Dec-2014.)
|- B = ( Base ` G )   &    |- .x. = (.g ` G )   &    |- .0. = ( 0g ` G )   =>    |- ( ( G e. Grp /\ N e. ZZ ) -> ( N .x. .0. ) = .0. )
Theorem	mulgnndir 13728	Sum of group multiples, for positive multiples. (Contributed by Mario Carneiro, 11-Dec-2014.) (Revised by AV, 29-Aug-2021.)
|- B = ( Base ` G )   &    |- .x. = (.g ` G )   &    |- .+ = ( +g ` G )   =>    |- ( ( G e. Smgrp /\ ( M e. NN /\ N e. NN /\ X e. B ) ) -> ( ( M + N ) .x. X ) = ( ( M .x. X ) .+ ( N .x. X ) ) )
Theorem	mulgnn0dir 13729	Sum of group multiples, generalized to NN0. (Contributed by Mario Carneiro, 11-Dec-2014.)
|- B = ( Base ` G )   &    |- .x. = (.g ` G )   &    |- .+ = ( +g ` G )   =>    |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> ( ( M + N ) .x. X ) = ( ( M .x. X ) .+ ( N .x. X ) ) )
Theorem	mulgdirlem 13730	Lemma for mulgdir 13731. (Contributed by Mario Carneiro, 13-Dec-2014.)
|- B = ( Base ` G )   &    |- .x. = (.g ` G )   &    |- .+ = ( +g ` G )   =>    |- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) /\ ( M + N ) e. NN0 ) -> ( ( M + N ) .x. X ) = ( ( M .x. X ) .+ ( N .x. X ) ) )
Theorem	mulgdir 13731	Sum of group multiples, generalized to ZZ. (Contributed by Mario Carneiro, 13-Dec-2014.)
|- B = ( Base ` G )   &    |- .x. = (.g ` G )   &    |- .+ = ( +g ` G )   =>    |- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> ( ( M + N ) .x. X ) = ( ( M .x. X ) .+ ( N .x. X ) ) )
Theorem	mulgp1 13732	Group multiple (exponentiation) operation at a successor, extended to ZZ. (Contributed by Mario Carneiro, 11-Dec-2014.)
|- B = ( Base ` G )   &    |- .x. = (.g ` G )   &    |- .+ = ( +g ` G )   =>    |- ( ( G e. Grp /\ N e. ZZ /\ X e. B ) -> ( ( N + 1 ) .x. X ) = ( ( N .x. X ) .+ X ) )
Theorem	mulgneg2 13733	Group multiple (exponentiation) operation at a negative integer. (Contributed by Mario Carneiro, 13-Dec-2014.)
|- B = ( Base ` G )   &    |- .x. = (.g ` G )   &    |- I = ( invg ` G )   =>    |- ( ( G e. Grp /\ N e. ZZ /\ X e. B ) -> ( -u N .x. X ) = ( N .x. ( I ` X ) ) )
Theorem	mulgnnass 13734	Product of group multiples, for positive multiples in a semigroup. (Contributed by Mario Carneiro, 13-Dec-2014.) (Revised by AV, 29-Aug-2021.)
|- B = ( Base ` G )   &    |- .x. = (.g ` G )   =>    |- ( ( G e. Smgrp /\ ( M e. NN /\ N e. NN /\ X e. B ) ) -> ( ( M x. N ) .x. X ) = ( M .x. ( N .x. X ) ) )
Theorem	mulgnn0ass 13735	Product of group multiples, generalized to NN0. (Contributed by Mario Carneiro, 13-Dec-2014.)
|- B = ( Base ` G )   &    |- .x. = (.g ` G )   =>    |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> ( ( M x. N ) .x. X ) = ( M .x. ( N .x. X ) ) )
Theorem	mulgass 13736	Product of group multiples, generalized to ZZ. (Contributed by Mario Carneiro, 13-Dec-2014.)
|- B = ( Base ` G )   &    |- .x. = (.g ` G )   =>    |- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> ( ( M x. N ) .x. X ) = ( M .x. ( N .x. X ) ) )
Theorem	mulgassr 13737	Reversed product of group multiples. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 30-Aug-2021.)
|- B = ( Base ` G )   &    |- .x. = (.g ` G )   =>    |- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> ( ( N x. M ) .x. X ) = ( M .x. ( N .x. X ) ) )
Theorem	mulgmodid 13738	Casting out multiples of the identity element leaves the group multiple unchanged. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 30-Aug-2021.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- .x. = (.g ` G )   =>    |- ( ( G e. Grp /\ ( N e. ZZ /\ M e. NN ) /\ ( X e. B /\ ( M .x. X ) = .0. ) ) -> ( ( N mod M ) .x. X ) = ( N .x. X ) )
Theorem	mulgsubdir 13739	Distribution of group multiples over subtraction for group elements, subdir 8555 analog. (Contributed by Mario Carneiro, 13-Dec-2014.)
|- B = ( Base ` G )   &    |- .x. = (.g ` G )   &    |- .- = ( -g ` G )   =>    |- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> ( ( M - N ) .x. X ) = ( ( M .x. X ) .- ( N .x. X ) ) )
Theorem	mhmmulg 13740	A homomorphism of monoids preserves group multiples. (Contributed by Mario Carneiro, 14-Jun-2015.)
|- B = ( Base ` G )   &    |- .x. = (.g ` G )   &    |- .X. = (.g ` H )   =>    |- ( ( F e. ( G MndHom H ) /\ N e. NN0 /\ X e. B ) -> ( F ` ( N .x. X ) ) = ( N .X. ( F ` X ) ) )
Theorem	mulgpropdg 13741*	Two structures with the same group-nature have the same group multiple function. K is expected to either be _V (when strong equality is available) or B (when closure is available). (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
|- ( ph -> .x. = (.g ` G ) )   &    |- ( ph -> .X. = (.g ` H ) )   &    |- ( ph -> G e. V )   &    |- ( ph -> H e. W )   &    |- ( ph -> B = ( Base ` G ) )   &    |- ( ph -> B = ( Base ` H ) )   &    |- ( ph -> B C_ K )   &    |- ( ( ph /\ ( x e. K /\ y e. K ) ) -> ( x ( +g ` G ) y ) e. K )   &    |- ( ( ph /\ ( x e. K /\ y e. K ) ) -> ( x ( +g ` G ) y ) = ( x ( +g ` H ) y ) )   =>    |- ( ph -> .x. = .X. )
Theorem	submmulgcl 13742	Closure of the group multiple (exponentiation) operation in a submonoid. (Contributed by Mario Carneiro, 13-Jan-2015.)
|- .xb = (.g ` G )   =>    |- ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) -> ( N .xb X ) e. S )
Theorem	submmulg 13743	A group multiple is the same if evaluated in a submonoid. (Contributed by Mario Carneiro, 15-Jun-2015.)
|- .xb = (.g ` G )   &    |- H = ( G ↾s S )   &    |- .x. = (.g ` H )   =>    |- ( ( S e. (SubMnd ` G ) /\ N e. NN0 /\ X e. S ) -> ( N .xb X ) = ( N .x. X ) )
7.2.3  Subgroups and Quotient groups
Syntax	csubg 13744	Extend class notation with all subgroups of a group.
class SubGrp
Syntax	cnsg 13745	Extend class notation with all normal subgroups of a group.
class NrmSGrp
Syntax	cqg 13746	Quotient group equivalence class.
class ~QG
Definition	df-subg 13747*	Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2m 13766), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 13761), contains the neutral element of the group (see subg0 13757) and contains the inverses for all of its elements (see subginvcl 13760). (Contributed by Mario Carneiro, 2-Dec-2014.)
|- SubGrp = ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )
Definition	df-nsg 13748*	Define the equivalence relation in a quotient ring or quotient group (where i is a two-sided ideal or a normal subgroup). For non-normal subgroups this generates the left cosets. (Contributed by Mario Carneiro, 15-Jun-2015.)
|- NrmSGrp = ( w e. Grp |-> { s e. (SubGrp ` w ) | [. ( Base ` w ) / b ]. [. ( +g ` w ) / p ]. A. x e. b A. y e. b ( ( x p y ) e. s <-> ( y p x ) e. s ) } )
Definition	df-eqg 13749*	Define the equivalence relation in a group generated by a subgroup. More precisely, if G is a group and H is a subgroup, then G ~QG H is the equivalence relation on G associated with the left cosets of H. A typical application of this definition is the construction of the quotient group (resp. ring) of a group (resp. ring) by a normal subgroup (resp. two-sided ideal). (Contributed by Mario Carneiro, 15-Jun-2015.)
|- ~QG = ( r e. _V , i e. _V |-> { <. x , y >. | ( { x , y } C_ ( Base ` r ) /\ ( ( ( invg ` r ) ` x ) ( +g ` r ) y ) e. i ) } )
Theorem	issubg 13750	The subgroup predicate. (Contributed by Mario Carneiro, 2-Dec-2014.)
|- B = ( Base ` G )   =>    |- ( S e. (SubGrp ` G ) <-> ( G e. Grp /\ S C_ B /\ ( G ↾s S ) e. Grp ) )
Theorem	subgss 13751	A subgroup is a subset. (Contributed by Mario Carneiro, 2-Dec-2014.)
|- B = ( Base ` G )   =>    |- ( S e. (SubGrp ` G ) -> S C_ B )
Theorem	subgid 13752	A group is a subgroup of itself. (Contributed by Mario Carneiro, 7-Dec-2014.)
|- B = ( Base ` G )   =>    |- ( G e. Grp -> B e. (SubGrp ` G ) )
Theorem	subgex 13753	The class of subgroups of a group is a set. (Contributed by Jim Kingdon, 8-Mar-2025.)
|- ( G e. Grp -> (SubGrp ` G ) e. _V )
Theorem	subggrp 13754	A subgroup is a group. (Contributed by Mario Carneiro, 2-Dec-2014.)
|- H = ( G ↾s S )   =>    |- ( S e. (SubGrp ` G ) -> H e. Grp )
Theorem	subgbas 13755	The base of the restricted group in a subgroup. (Contributed by Mario Carneiro, 2-Dec-2014.)
|- H = ( G ↾s S )   =>    |- ( S e. (SubGrp ` G ) -> S = ( Base ` H ) )
Theorem	subgrcl 13756	Reverse closure for the subgroup predicate. (Contributed by Mario Carneiro, 2-Dec-2014.)
|- ( S e. (SubGrp ` G ) -> G e. Grp )
Theorem	subg0 13757	A subgroup of a group must have the same identity as the group. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
|- H = ( G ↾s S )   &    |- .0. = ( 0g ` G )   =>    |- ( S e. (SubGrp ` G ) -> .0. = ( 0g ` H ) )
Theorem	subginv 13758	The inverse of an element in a subgroup is the same as the inverse in the larger group. (Contributed by Mario Carneiro, 2-Dec-2014.)
|- H = ( G ↾s S )   &    |- I = ( invg ` G )   &    |- J = ( invg ` H )   =>    |- ( ( S e. (SubGrp ` G ) /\ X e. S ) -> ( I ` X ) = ( J ` X ) )
Theorem	subg0cl 13759	The group identity is an element of any subgroup. (Contributed by Mario Carneiro, 2-Dec-2014.)
|- .0. = ( 0g ` G )   =>    |- ( S e. (SubGrp ` G ) -> .0. e. S )
Theorem	subginvcl 13760	The inverse of an element is closed in a subgroup. (Contributed by Mario Carneiro, 2-Dec-2014.)
|- I = ( invg ` G )   =>    |- ( ( S e. (SubGrp ` G ) /\ X e. S ) -> ( I ` X ) e. S )
Theorem	subgcl 13761	A subgroup is closed under group operation. (Contributed by Mario Carneiro, 2-Dec-2014.)
|- .+ = ( +g ` G )   =>    |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> ( X .+ Y ) e. S )
Theorem	subgsubcl 13762	A subgroup is closed under group subtraction. (Contributed by Mario Carneiro, 18-Jan-2015.)
|- .- = ( -g ` G )   =>    |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> ( X .- Y ) e. S )
Theorem	subgsub 13763	The subtraction of elements in a subgroup is the same as subtraction in the group. (Contributed by Mario Carneiro, 15-Jun-2015.)
|- .- = ( -g ` G )   &    |- H = ( G ↾s S )   &    |- N = ( -g ` H )   =>    |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> ( X .- Y ) = ( X N Y ) )
Theorem	subgmulgcl 13764	Closure of the group multiple (exponentiation) operation in a subgroup. (Contributed by Mario Carneiro, 13-Jan-2015.)
|- .x. = (.g ` G )   =>    |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> ( N .x. X ) e. S )
Theorem	subgmulg 13765	A group multiple is the same if evaluated in a subgroup. (Contributed by Mario Carneiro, 15-Jan-2015.)
|- .x. = (.g ` G )   &    |- H = ( G ↾s S )   &    |- .xb = (.g ` H )   =>    |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> ( N .x. X ) = ( N .xb X ) )
Theorem	issubg2m 13766*	Characterize the subgroups of a group by closure properties. (Contributed by Mario Carneiro, 2-Dec-2014.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- I = ( invg ` G )   =>    |- ( G e. Grp -> ( S e. (SubGrp ` G ) <-> ( S C_ B /\ E. u u e. S /\ A. x e. S ( A. y e. S ( x .+ y ) e. S /\ ( I ` x ) e. S ) ) ) )
Theorem	issubgrpd2 13767*	Prove a subgroup by closure (definition version). (Contributed by Stefan O'Rear, 7-Dec-2014.)
|- ( ph -> S = ( I ↾s D ) )   &    |- ( ph -> .0. = ( 0g ` I ) )   &    |- ( ph -> .+ = ( +g ` I ) )   &    |- ( ph -> D C_ ( Base ` I ) )   &    |- ( ph -> .0. e. D )   &    |- ( ( ph /\ x e. D /\ y e. D ) -> ( x .+ y ) e. D )   &    |- ( ( ph /\ x e. D ) -> ( ( invg ` I ) ` x ) e. D )   &    |- ( ph -> I e. Grp )   =>    |- ( ph -> D e. (SubGrp ` I ) )
Theorem	issubgrpd 13768*	Prove a subgroup by closure. (Contributed by Stefan O'Rear, 7-Dec-2014.)
|- ( ph -> S = ( I ↾s D ) )   &    |- ( ph -> .0. = ( 0g ` I ) )   &    |- ( ph -> .+ = ( +g ` I ) )   &    |- ( ph -> D C_ ( Base ` I ) )   &    |- ( ph -> .0. e. D )   &    |- ( ( ph /\ x e. D /\ y e. D ) -> ( x .+ y ) e. D )   &    |- ( ( ph /\ x e. D ) -> ( ( invg ` I ) ` x ) e. D )   &    |- ( ph -> I e. Grp )   =>    |- ( ph -> S e. Grp )
Theorem	issubg3 13769*	A subgroup is a symmetric submonoid. (Contributed by Mario Carneiro, 7-Mar-2015.)
|- I = ( invg ` G )   =>    |- ( G e. Grp -> ( S e. (SubGrp ` G ) <-> ( S e. (SubMnd ` G ) /\ A. x e. S ( I ` x ) e. S ) ) )
Theorem	issubg4m 13770*	A subgroup is an inhabited subset of the group closed under subtraction. (Contributed by Mario Carneiro, 17-Sep-2015.)
|- B = ( Base ` G )   &    |- .- = ( -g ` G )   =>    |- ( G e. Grp -> ( S e. (SubGrp ` G ) <-> ( S C_ B /\ E. w w e. S /\ A. x e. S A. y e. S ( x .- y ) e. S ) ) )
Theorem	grpissubg 13771	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 (base set of the) group is subgroup of the other group. (Contributed by AV, 14-Mar-2019.)
|- B = ( Base ` G )   &    |- S = ( Base ` H )   =>    |- ( ( G e. Grp /\ H e. Grp ) -> ( ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) -> S e. (SubGrp ` G ) ) )
Theorem	resgrpisgrp 13772	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 other group restricted to the base set of the group is a group. (Contributed by AV, 14-Mar-2019.)
|- B = ( Base ` G )   &    |- S = ( Base ` H )   =>    |- ( ( G e. Grp /\ H e. Grp ) -> ( ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) -> ( G ↾s S ) e. Grp ) )
Theorem	subgsubm 13773	A subgroup is a submonoid. (Contributed by Mario Carneiro, 18-Jun-2015.)
|- ( S e. (SubGrp ` G ) -> S e. (SubMnd ` G ) )
Theorem	subsubg 13774	A subgroup of a subgroup is a subgroup. (Contributed by Mario Carneiro, 19-Jan-2015.)
|- H = ( G ↾s S )   =>    |- ( S e. (SubGrp ` G ) -> ( A e. (SubGrp ` H ) <-> ( A e. (SubGrp ` G ) /\ A C_ S ) ) )
Theorem	subgintm 13775*	The intersection of an inhabited collection of subgroups is a subgroup. (Contributed by Mario Carneiro, 7-Dec-2014.)
|- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> |^| S e. (SubGrp ` G ) )
Theorem	0subg 13776	The zero subgroup of an arbitrary group. (Contributed by Stefan O'Rear, 10-Dec-2014.) (Proof shortened by SN, 31-Jan-2025.)
|- .0. = ( 0g ` G )   =>    |- ( G e. Grp -> { .0. } e. (SubGrp ` G ) )
Theorem	trivsubgd 13777	The only subgroup of a trivial group is itself. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. Grp )   &    |- ( ph -> B = { .0. } )   &    |- ( ph -> A e. (SubGrp ` G ) )   =>    |- ( ph -> A = B )
Theorem	trivsubgsnd 13778	The only subgroup of a trivial group is itself. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. Grp )   &    |- ( ph -> B = { .0. } )   =>    |- ( ph -> (SubGrp ` G ) = { B } )
Theorem	isnsg 13779*	Property of being a normal subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
|- X = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( S e. (NrmSGrp ` G ) <-> ( S e. (SubGrp ` G ) /\ A. x e. X A. y e. X ( ( x .+ y ) e. S <-> ( y .+ x ) e. S ) ) )
Theorem	isnsg2 13780*	Weaken the condition of isnsg 13779 to only one side of the implication. (Contributed by Mario Carneiro, 18-Jan-2015.)
|- X = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( S e. (NrmSGrp ` G ) <-> ( S e. (SubGrp ` G ) /\ A. x e. X A. y e. X ( ( x .+ y ) e. S -> ( y .+ x ) e. S ) ) )
Theorem	nsgbi 13781	Defining property of a normal subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
|- X = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. X ) -> ( ( A .+ B ) e. S <-> ( B .+ A ) e. S ) )
Theorem	nsgsubg 13782	A normal subgroup is a subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
|- ( S e. (NrmSGrp ` G ) -> S e. (SubGrp ` G ) )
Theorem	nsgconj 13783	The conjugation of an element of a normal subgroup is in the subgroup. (Contributed by Mario Carneiro, 4-Feb-2015.)
|- X = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .- = ( -g ` G )   =>    |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> ( ( A .+ B ) .- A ) e. S )
Theorem	isnsg3 13784*	A subgroup is normal iff the conjugation of all the elements of the subgroup is in the subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
|- X = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .- = ( -g ` G )   =>    |- ( S e. (NrmSGrp ` G ) <-> ( S e. (SubGrp ` G ) /\ A. x e. X A. y e. S ( ( x .+ y ) .- x ) e. S ) )
Theorem	elnmz 13785*	Elementhood in the normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)
|- N = { x e. X | A. y e. X ( ( x .+ y ) e. S <-> ( y .+ x ) e. S ) }   =>    |- ( A e. N <-> ( A e. X /\ A. z e. X ( ( A .+ z ) e. S <-> ( z .+ A ) e. S ) ) )
Theorem	nmzbi 13786*	Defining property of the normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)
|- N = { x e. X | A. y e. X ( ( x .+ y ) e. S <-> ( y .+ x ) e. S ) }   =>    |- ( ( A e. N /\ B e. X ) -> ( ( A .+ B ) e. S <-> ( B .+ A ) e. S ) )
Theorem	nmzsubg 13787*	The normalizer NG(S) of a subset S of the group is a subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
|- N = { x e. X | A. y e. X ( ( x .+ y ) e. S <-> ( y .+ x ) e. S ) }   &    |- X = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( G e. Grp -> N e. (SubGrp ` G ) )
Theorem	ssnmz 13788*	A subgroup is a subset of its normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)
|- N = { x e. X | A. y e. X ( ( x .+ y ) e. S <-> ( y .+ x ) e. S ) }   &    |- X = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( S e. (SubGrp ` G ) -> S C_ N )
Theorem	isnsg4 13789*	A subgroup is normal iff its normalizer is the entire group. (Contributed by Mario Carneiro, 18-Jan-2015.)
|- N = { x e. X | A. y e. X ( ( x .+ y ) e. S <-> ( y .+ x ) e. S ) }   &    |- X = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( S e. (NrmSGrp ` G ) <-> ( S e. (SubGrp ` G ) /\ N = X ) )
Theorem	nmznsg 13790*	Any subgroup is a normal subgroup of its normalizer. (Contributed by Mario Carneiro, 19-Jan-2015.)
|- N = { x e. X | A. y e. X ( ( x .+ y ) e. S <-> ( y .+ x ) e. S ) }   &    |- X = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- H = ( G ↾s N )   =>    |- ( S e. (SubGrp ` G ) -> S e. (NrmSGrp ` H ) )
Theorem	0nsg 13791	The zero subgroup is normal. (Contributed by Mario Carneiro, 4-Feb-2015.)
|- .0. = ( 0g ` G )   =>    |- ( G e. Grp -> { .0. } e. (NrmSGrp ` G ) )
Theorem	nsgid 13792	The whole group is a normal subgroup of itself. (Contributed by Mario Carneiro, 4-Feb-2015.)
|- B = ( Base ` G )   =>    |- ( G e. Grp -> B e. (NrmSGrp ` G ) )
Theorem	0idnsgd 13793	The whole group and the zero subgroup are normal subgroups of a group. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. Grp )   =>    |- ( ph -> { { .0. } , B } C_ (NrmSGrp ` G ) )
Theorem	trivnsgd 13794	The only normal subgroup of a trivial group is itself. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. Grp )   &    |- ( ph -> B = { .0. } )   =>    |- ( ph -> (NrmSGrp ` G ) = { B } )
Theorem	triv1nsgd 13795	A trivial group has exactly one normal subgroup. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. Grp )   &    |- ( ph -> B = { .0. } )   =>    |- ( ph -> (NrmSGrp ` G ) ~~ 1o )
Theorem	1nsgtrivd 13796	A group with exactly one normal subgroup is trivial. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. Grp )   &    |- ( ph -> (NrmSGrp ` G ) ~~ 1o )   =>    |- ( ph -> B = { .0. } )
Theorem	releqgg 13797	The left coset equivalence relation is a relation. (Contributed by Mario Carneiro, 14-Jun-2015.)
|- R = ( G ~QG S )   =>    |- ( ( G e. V /\ S e. W ) -> Rel R )
Theorem	eqgex 13798	The left coset equivalence relation exists. (Contributed by Jim Kingdon, 25-Apr-2025.)
|- ( ( G e. V /\ S e. W ) -> ( G ~QG S ) e. _V )
Theorem	eqgfval 13799*	Value of the subgroup left coset equivalence relation. (Contributed by Mario Carneiro, 15-Jan-2015.)
|- X = ( Base ` G )   &    |- N = ( invg ` G )   &    |- .+ = ( +g ` G )   &    |- R = ( G ~QG S )   =>    |- ( ( G e. V /\ S C_ X ) -> R = { <. x , y >. | ( { x , y } C_ X /\ ( ( N ` x ) .+ y ) e. S ) } )
Theorem	eqgval 13800	Value of the subgroup left coset equivalence relation. (Contributed by Mario Carneiro, 15-Jan-2015.) (Revised by Mario Carneiro, 14-Jun-2015.)
|- X = ( Base ` G )   &    |- N = ( invg ` G )   &    |- .+ = ( +g ` G )   &    |- R = ( G ~QG S )   =>    |- ( ( G e. V /\ S C_ X ) -> ( A R B <-> ( A e. X /\ B e. X /\ ( ( N ` A ) .+ B ) e. S ) ) )