P. 140 of Theorem List - Intuitionistic Logic Explorer

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

Type	Label	Description
Statement
Theorem	lspsnel3 13901	A member of the span of the singleton of a vector is a member of a subspace containing the vector. (Contributed by NM, 4-Jul-2014.)
|- S = ( LSubSp ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> U e. S )   &    |- ( ph -> X e. U )   &    |- ( ph -> Y e. ( N ` { X } ) )   =>    |- ( ph -> Y e. U )
Theorem	lspprss 13902	The span of a pair of vectors in a subspace belongs to the subspace. (Contributed by NM, 12-Jan-2015.)
|- S = ( LSubSp ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> U e. S )   &    |- ( ph -> X e. U )   &    |- ( ph -> Y e. U )   =>    |- ( ph -> ( N ` { X , Y } ) C_ U )
Theorem	lspsnid 13903	A vector belongs to the span of its singleton. (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- V = ( Base ` W )   &    |- N = ( LSpan ` W )   =>    |- ( ( W e. LMod /\ X e. V ) -> X e. ( N ` { X } ) )
Theorem	lspsnel6 13904	Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 8-Aug-2014.) (Revised by Mario Carneiro, 8-Jan-2015.)
|- V = ( Base ` W )   &    |- S = ( LSubSp ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> U e. S )   =>    |- ( ph -> ( X e. U <-> ( X e. V /\ ( N ` { X } ) C_ U ) ) )
Theorem	lspsnel5 13905	Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 8-Aug-2014.)
|- V = ( Base ` W )   &    |- S = ( LSubSp ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> U e. S )   &    |- ( ph -> X e. V )   =>    |- ( ph -> ( X e. U <-> ( N ` { X } ) C_ U ) )
Theorem	lspsnel5a 13906	Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 20-Feb-2015.)
|- S = ( LSubSp ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> U e. S )   &    |- ( ph -> X e. U )   =>    |- ( ph -> ( N ` { X } ) C_ U )
Theorem	lspprid1 13907	A member of a pair of vectors belongs to their span. (Contributed by NM, 14-May-2015.)
|- V = ( Base ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   =>    |- ( ph -> X e. ( N ` { X , Y } ) )
Theorem	lspprid2 13908	A member of a pair of vectors belongs to their span. (Contributed by NM, 14-May-2015.)
|- V = ( Base ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   =>    |- ( ph -> Y e. ( N ` { X , Y } ) )
Theorem	lspprvacl 13909	The sum of two vectors belongs to their span. (Contributed by NM, 20-May-2015.)
|- V = ( Base ` W )   &    |- .+ = ( +g ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   =>    |- ( ph -> ( X .+ Y ) e. ( N ` { X , Y } ) )
Theorem	lssats2 13910*	A way to express atomisticity (a subspace is the union of its atoms). (Contributed by NM, 3-Feb-2015.)
|- S = ( LSubSp ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> U e. S )   =>    |- ( ph -> U = U_ x e. U ( N ` { x } ) )
Theorem	lspsneli 13911	A scalar product with a vector belongs to the span of its singleton. (Contributed by NM, 2-Jul-2014.)
|- V = ( Base ` W )   &    |- .x. = ( .s ` W )   &    |- F = (Scalar ` W )   &    |- K = ( Base ` F )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> A e. K )   &    |- ( ph -> X e. V )   =>    |- ( ph -> ( A .x. X ) e. ( N ` { X } ) )
Theorem	lspsn 13912*	Span of the singleton of a vector. (Contributed by NM, 14-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
|- F = (Scalar ` W )   &    |- K = ( Base ` F )   &    |- V = ( Base ` W )   &    |- .x. = ( .s ` W )   &    |- N = ( LSpan ` W )   =>    |- ( ( W e. LMod /\ X e. V ) -> ( N ` { X } ) = { v | E. k e. K v = ( k .x. X ) } )
Theorem	ellspsn 13913*	Member of span of the singleton of a vector. (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- F = (Scalar ` W )   &    |- K = ( Base ` F )   &    |- V = ( Base ` W )   &    |- .x. = ( .s ` W )   &    |- N = ( LSpan ` W )   =>    |- ( ( W e. LMod /\ X e. V ) -> ( U e. ( N ` { X } ) <-> E. k e. K U = ( k .x. X ) ) )
Theorem	lspsnvsi 13914	Span of a scalar product of a singleton. (Contributed by NM, 23-Apr-2014.) (Proof shortened by Mario Carneiro, 4-Sep-2014.)
|- F = (Scalar ` W )   &    |- K = ( Base ` F )   &    |- V = ( Base ` W )   &    |- .x. = ( .s ` W )   &    |- N = ( LSpan ` W )   =>    |- ( ( W e. LMod /\ R e. K /\ X e. V ) -> ( N ` { ( R .x. X ) } ) C_ ( N ` { X } ) )
Theorem	lspsnss2 13915*	Comparable spans of singletons must have proportional vectors. (Contributed by NM, 7-Jun-2015.)
|- V = ( Base ` W )   &    |- S = (Scalar ` W )   &    |- K = ( Base ` S )   &    |- .x. = ( .s ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   =>    |- ( ph -> ( ( N ` { X } ) C_ ( N ` { Y } ) <-> E. k e. K X = ( k .x. Y ) ) )
Theorem	lspsnneg 13916	Negation does not change the span of a singleton. (Contributed by NM, 24-Apr-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
|- V = ( Base ` W )   &    |- M = ( invg ` W )   &    |- N = ( LSpan ` W )   =>    |- ( ( W e. LMod /\ X e. V ) -> ( N ` { ( M ` X ) } ) = ( N ` { X } ) )
Theorem	lspsnsub 13917	Swapping subtraction order does not change the span of a singleton. (Contributed by NM, 4-Apr-2015.)
|- V = ( Base ` W )   &    |- .- = ( -g ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   =>    |- ( ph -> ( N ` { ( X .- Y ) } ) = ( N ` { ( Y .- X ) } ) )
Theorem	lspsn0 13918	Span of the singleton of the zero vector. (Contributed by NM, 15-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
|- .0. = ( 0g ` W )   &    |- N = ( LSpan ` W )   =>    |- ( W e. LMod -> ( N ` { .0. } ) = { .0. } )
Theorem	lsp0 13919	Span of the empty set. (Contributed by Mario Carneiro, 5-Sep-2014.)
|- .0. = ( 0g ` W )   &    |- N = ( LSpan ` W )   =>    |- ( W e. LMod -> ( N ` (/) ) = { .0. } )
Theorem	lspuni0 13920	Union of the span of the empty set. (Contributed by NM, 14-Mar-2015.)
|- .0. = ( 0g ` W )   &    |- N = ( LSpan ` W )   =>    |- ( W e. LMod -> U. ( N ` (/) ) = .0. )
Theorem	lspun0 13921	The span of a union with the zero subspace. (Contributed by NM, 22-May-2015.)
|- V = ( Base ` W )   &    |- .0. = ( 0g ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> X C_ V )   =>    |- ( ph -> ( N ` ( X u. { .0. } ) ) = ( N ` X ) )
Theorem	lspsneq0 13922	Span of the singleton is the zero subspace iff the vector is zero. (Contributed by NM, 27-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- V = ( Base ` W )   &    |- .0. = ( 0g ` W )   &    |- N = ( LSpan ` W )   =>    |- ( ( W e. LMod /\ X e. V ) -> ( ( N ` { X } ) = { .0. } <-> X = .0. ) )
Theorem	lspsneq0b 13923	Equal singleton spans imply both arguments are zero or both are nonzero. (Contributed by NM, 21-Mar-2015.)
|- V = ( Base ` W )   &    |- .0. = ( 0g ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   &    |- ( ph -> ( N ` { X } ) = ( N ` { Y } ) )   =>    |- ( ph -> ( X = .0. <-> Y = .0. ) )
Theorem	lmodindp1 13924	Two independent (non-colinear) vectors have nonzero sum. (Contributed by NM, 22-Apr-2015.)
|- V = ( Base ` W )   &    |- .+ = ( +g ` W )   &    |- .0. = ( 0g ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   &    |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )   =>    |- ( ph -> ( X .+ Y ) =/= .0. )
Theorem	lsslsp 13925	Spans in submodules correspond to spans in the containing module. (Contributed by Stefan O'Rear, 12-Dec-2014.) Terms in the equation were swapped as proposed by NM on 15-Mar-2015. (Revised by AV, 18-Apr-2025.)
|- X = ( W ↾s U )   &    |- M = ( LSpan ` W )   &    |- N = ( LSpan ` X )   &    |- L = ( LSubSp ` W )   =>    |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( N ` G ) = ( M ` G ) )
Theorem	lss0v 13926	The zero vector in a submodule equals the zero vector in the including module. (Contributed by NM, 15-Mar-2015.)
|- X = ( W ↾s U )   &    |- .0. = ( 0g ` W )   &    |- Z = ( 0g ` X )   &    |- L = ( LSubSp ` W )   =>    |- ( ( W e. LMod /\ U e. L ) -> Z = .0. )
Theorem	lsspropdg 13927*	If two structures have the same components (properties), they have the same subspace structure. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.)
|- ( ph -> B = ( Base ` K ) )   &    |- ( ph -> B = ( Base ` L ) )   &    |- ( ph -> B C_ W )   &    |- ( ( ph /\ ( x e. W /\ y e. W ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )   &    |- ( ( ph /\ ( x e. P /\ y e. B ) ) -> ( x ( .s ` K ) y ) e. W )   &    |- ( ( ph /\ ( x e. P /\ y e. B ) ) -> ( x ( .s ` K ) y ) = ( x ( .s ` L ) y ) )   &    |- ( ph -> P = ( Base ` (Scalar ` K ) ) )   &    |- ( ph -> P = ( Base ` (Scalar ` L ) ) )   &    |- ( ph -> K e. X )   &    |- ( ph -> L e. Y )   =>    |- ( ph -> ( LSubSp ` K ) = ( LSubSp ` L ) )
Theorem	lsppropd 13928*	If two structures have the same components (properties), they have the same span function. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 24-Apr-2024.)
|- ( ph -> B = ( Base ` K ) )   &    |- ( ph -> B = ( Base ` L ) )   &    |- ( ph -> B C_ W )   &    |- ( ( ph /\ ( x e. W /\ y e. W ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )   &    |- ( ( ph /\ ( x e. P /\ y e. B ) ) -> ( x ( .s ` K ) y ) e. W )   &    |- ( ( ph /\ ( x e. P /\ y e. B ) ) -> ( x ( .s ` K ) y ) = ( x ( .s ` L ) y ) )   &    |- ( ph -> P = ( Base ` (Scalar ` K ) ) )   &    |- ( ph -> P = ( Base ` (Scalar ` L ) ) )   &    |- ( ph -> K e. X )   &    |- ( ph -> L e. Y )   =>    |- ( ph -> ( LSpan ` K ) = ( LSpan ` L ) )
7.6  Subring algebras and ideals
7.6.1  Subring algebras
Syntax	csra 13929	Extend class notation with the subring algebra generator.
class subringAlg
Syntax	crglmod 13930	Extend class notation with the left module induced by a ring over itself.
class ringLMod
Definition	df-sra 13931*	Any ring can be regarded as a left algebra over any of its subrings. The function subringAlg associates with any ring and any of its subrings the left algebra consisting in the ring itself regarded as a left algebra over the subring. It has an inner product which is simply the ring product. (Contributed by Mario Carneiro, 27-Nov-2014.) (Revised by Thierry Arnoux, 16-Jun-2019.)
|- subringAlg = ( w e. _V |-> ( s e. ~P ( Base ` w ) |-> ( ( ( w sSet <. (Scalar ` ndx ) , ( w ↾s s ) >. ) sSet <. ( .s ` ndx ) , ( .r ` w ) >. ) sSet <. ( .i ` ndx ) , ( .r ` w ) >. ) ) )
Definition	df-rgmod 13932	Any ring can be regarded as a left algebra over itself. The function ringLMod associates with any ring the left algebra consisting in the ring itself regarded as a left algebra over itself. It has an inner product which is simply the ring product. (Contributed by Stefan O'Rear, 6-Dec-2014.)
|- ringLMod = ( w e. _V |-> ( (subringAlg ` w ) ` ( Base ` w ) ) )
Theorem	sraval 13933	Lemma for srabaseg 13935 through sravscag 13939. (Contributed by Mario Carneiro, 27-Nov-2014.) (Revised by Thierry Arnoux, 16-Jun-2019.)
|- ( ( W e. V /\ S C_ ( Base ` W ) ) -> ( (subringAlg ` W ) ` S ) = ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) )
Theorem	sralemg 13934	Lemma for srabaseg 13935 and similar theorems. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
|- ( ph -> A = ( (subringAlg ` W ) ` S ) )   &    |- ( ph -> S C_ ( Base ` W ) )   &    |- ( ph -> W e. X )   &    |- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )   &    |- (Scalar ` ndx ) =/= ( E ` ndx )   &    |- ( .s ` ndx ) =/= ( E ` ndx )   &    |- ( .i ` ndx ) =/= ( E ` ndx )   =>    |- ( ph -> ( E ` W ) = ( E ` A ) )
Theorem	srabaseg 13935	Base set of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
|- ( ph -> A = ( (subringAlg ` W ) ` S ) )   &    |- ( ph -> S C_ ( Base ` W ) )   &    |- ( ph -> W e. X )   =>    |- ( ph -> ( Base ` W ) = ( Base ` A ) )
Theorem	sraaddgg 13936	Additive operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
|- ( ph -> A = ( (subringAlg ` W ) ` S ) )   &    |- ( ph -> S C_ ( Base ` W ) )   &    |- ( ph -> W e. X )   =>    |- ( ph -> ( +g ` W ) = ( +g ` A ) )
Theorem	sramulrg 13937	Multiplicative operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
|- ( ph -> A = ( (subringAlg ` W ) ` S ) )   &    |- ( ph -> S C_ ( Base ` W ) )   &    |- ( ph -> W e. X )   =>    |- ( ph -> ( .r ` W ) = ( .r ` A ) )
Theorem	srascag 13938	The set of scalars of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Proof shortened by AV, 12-Nov-2024.)
|- ( ph -> A = ( (subringAlg ` W ) ` S ) )   &    |- ( ph -> S C_ ( Base ` W ) )   &    |- ( ph -> W e. X )   =>    |- ( ph -> ( W ↾s S ) = (Scalar ` A ) )
Theorem	sravscag 13939	The scalar product operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Proof shortened by AV, 12-Nov-2024.)
|- ( ph -> A = ( (subringAlg ` W ) ` S ) )   &    |- ( ph -> S C_ ( Base ` W ) )   &    |- ( ph -> W e. X )   =>    |- ( ph -> ( .r ` W ) = ( .s ` A ) )
Theorem	sraipg 13940	The inner product operation of a subring algebra. (Contributed by Thierry Arnoux, 16-Jun-2019.)
|- ( ph -> A = ( (subringAlg ` W ) ` S ) )   &    |- ( ph -> S C_ ( Base ` W ) )   &    |- ( ph -> W e. X )   =>    |- ( ph -> ( .r ` W ) = ( .i ` A ) )
Theorem	sratsetg 13941	Topology component of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
|- ( ph -> A = ( (subringAlg ` W ) ` S ) )   &    |- ( ph -> S C_ ( Base ` W ) )   &    |- ( ph -> W e. X )   =>    |- ( ph -> (TopSet ` W ) = (TopSet ` A ) )
Theorem	sraex 13942	Existence of a subring algebra. (Contributed by Jim Kingdon, 16-Apr-2025.)
|- ( ph -> A = ( (subringAlg ` W ) ` S ) )   &    |- ( ph -> S C_ ( Base ` W ) )   &    |- ( ph -> W e. X )   =>    |- ( ph -> A e. _V )
Theorem	sratopng 13943	Topology component of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
|- ( ph -> A = ( (subringAlg ` W ) ` S ) )   &    |- ( ph -> S C_ ( Base ` W ) )   &    |- ( ph -> W e. X )   =>    |- ( ph -> ( TopOpen ` W ) = ( TopOpen ` A ) )
Theorem	sradsg 13944	Distance function of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
|- ( ph -> A = ( (subringAlg ` W ) ` S ) )   &    |- ( ph -> S C_ ( Base ` W ) )   &    |- ( ph -> W e. X )   =>    |- ( ph -> ( dist ` W ) = ( dist ` A ) )
Theorem	sraring 13945	Condition for a subring algebra to be a ring. (Contributed by Thierry Arnoux, 24-Jul-2023.)
|- A = ( (subringAlg ` R ) ` V )   &    |- B = ( Base ` R )   =>    |- ( ( R e. Ring /\ V C_ B ) -> A e. Ring )
Theorem	sralmod 13946	The subring algebra is a left module. (Contributed by Stefan O'Rear, 27-Nov-2014.)
|- A = ( (subringAlg ` W ) ` S )   =>    |- ( S e. (SubRing ` W ) -> A e. LMod )
Theorem	sralmod0g 13947	The subring module inherits a zero from its ring. (Contributed by Stefan O'Rear, 27-Dec-2014.)
|- ( ph -> A = ( (subringAlg ` W ) ` S ) )   &    |- ( ph -> .0. = ( 0g ` W ) )   &    |- ( ph -> S C_ ( Base ` W ) )   &    |- ( ph -> W e. X )   =>    |- ( ph -> .0. = ( 0g ` A ) )
Theorem	issubrgd 13948*	Prove a subring 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 -> .1. = ( 1r ` I ) )   &    |- ( ph -> .x. = ( .r ` I ) )   &    |- ( ph -> .1. e. D )   &    |- ( ( ph /\ x e. D /\ y e. D ) -> ( x .x. y ) e. D )   &    |- ( ph -> I e. Ring )   =>    |- ( ph -> D e. (SubRing ` I ) )
Theorem	rlmfn 13949	ringLMod is a function. (Contributed by Stefan O'Rear, 6-Dec-2014.)
|- ringLMod Fn _V
Theorem	rlmvalg 13950	Value of the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
|- ( W e. V -> (ringLMod ` W ) = ( (subringAlg ` W ) ` ( Base ` W ) ) )
Theorem	rlmbasg 13951	Base set of the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
|- ( R e. V -> ( Base ` R ) = ( Base ` (ringLMod ` R ) ) )
Theorem	rlmplusgg 13952	Vector addition in the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
|- ( R e. V -> ( +g ` R ) = ( +g ` (ringLMod ` R ) ) )
Theorem	rlm0g 13953	Zero vector in the ring module. (Contributed by Stefan O'Rear, 6-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
|- ( R e. V -> ( 0g ` R ) = ( 0g ` (ringLMod ` R ) ) )
Theorem	rlmsubg 13954	Subtraction in the ring module. (Contributed by Thierry Arnoux, 30-Jun-2019.)
|- ( R e. V -> ( -g ` R ) = ( -g ` (ringLMod ` R ) ) )
Theorem	rlmmulrg 13955	Ring multiplication in the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
|- ( R e. V -> ( .r ` R ) = ( .r ` (ringLMod ` R ) ) )
Theorem	rlmscabas 13956	Scalars in the ring module have the same base set. (Contributed by Jim Kingdon, 29-Apr-2025.)
|- ( R e. X -> ( Base ` R ) = ( Base ` (Scalar ` (ringLMod ` R ) ) ) )
Theorem	rlmvscag 13957	Scalar multiplication in the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
|- ( R e. V -> ( .r ` R ) = ( .s ` (ringLMod ` R ) ) )
Theorem	rlmtopng 13958	Topology component of the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
|- ( R e. V -> ( TopOpen ` R ) = ( TopOpen ` (ringLMod ` R ) ) )
Theorem	rlmdsg 13959	Metric component of the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
|- ( R e. V -> ( dist ` R ) = ( dist ` (ringLMod ` R ) ) )
Theorem	rlmlmod 13960	The ring module is a module. (Contributed by Stefan O'Rear, 6-Dec-2014.)
|- ( R e. Ring -> (ringLMod ` R ) e. LMod )
Theorem	rlmvnegg 13961	Vector negation in the ring module. (Contributed by Stefan O'Rear, 6-Dec-2014.) (Revised by Mario Carneiro, 5-Jun-2015.)
|- ( R e. V -> ( invg ` R ) = ( invg ` (ringLMod ` R ) ) )
Theorem	ixpsnbasval 13962*	The value of an infinite Cartesian product of the base of a left module over a ring with a singleton. (Contributed by AV, 3-Dec-2018.)
|- ( ( R e. V /\ X e. W ) -> X_ x e. { X } ( Base ` ( ( { X } X. { (ringLMod ` R ) } ) ` x ) ) = { f | ( f Fn { X } /\ ( f ` X ) e. ( Base ` R ) ) } )
7.6.2  Ideals and spans
Syntax	clidl 13963	Ring left-ideal function.
class LIdeal
Syntax	crsp 13964	Ring span function.
class RSpan
Definition	df-lidl 13965	Define the class of left ideals of a given ring. An ideal is a submodule of the ring viewed as a module over itself. (Contributed by Stefan O'Rear, 31-Mar-2015.)
|- LIdeal = ( LSubSp o. ringLMod )
Definition	df-rsp 13966	Define the linear span function in a ring (Ideal generator). (Contributed by Stefan O'Rear, 4-Apr-2015.)
|- RSpan = ( LSpan o. ringLMod )
Theorem	lidlvalg 13967	Value of the set of ring ideals. (Contributed by Stefan O'Rear, 31-Mar-2015.)
|- ( W e. V -> (LIdeal ` W ) = ( LSubSp ` (ringLMod ` W ) ) )
Theorem	rspvalg 13968	Value of the ring span function. (Contributed by Stefan O'Rear, 4-Apr-2015.)
|- ( W e. V -> (RSpan ` W ) = ( LSpan ` (ringLMod ` W ) ) )
Theorem	lidlex 13969	Existence of the set of left ideals. (Contributed by Jim Kingdon, 27-Apr-2025.)
|- ( W e. V -> (LIdeal ` W ) e. _V )
Theorem	rspex 13970	Existence of the ring span. (Contributed by Jim Kingdon, 25-Apr-2025.)
|- ( W e. V -> (RSpan ` W ) e. _V )
Theorem	lidlmex 13971	Existence of the set a left ideal is built from (when the ideal is inhabited). (Contributed by Jim Kingdon, 18-Apr-2025.)
|- I = (LIdeal ` W )   =>    |- ( U e. I -> W e. _V )
Theorem	lidlss 13972	An ideal is a subset of the base set. (Contributed by Stefan O'Rear, 28-Mar-2015.)
|- B = ( Base ` W )   &    |- I = (LIdeal ` W )   =>    |- ( U e. I -> U C_ B )
Theorem	lidlssbas 13973	The base set of the restriction of the ring to a (left) ideal is a subset of the base set of the ring. (Contributed by AV, 17-Feb-2020.)
|- L = (LIdeal ` R )   &    |- I = ( R ↾s U )   =>    |- ( U e. L -> ( Base ` I ) C_ ( Base ` R ) )
Theorem	lidlbas 13974	A (left) ideal of a ring is the base set of the restriction of the ring to this ideal. (Contributed by AV, 17-Feb-2020.)
|- L = (LIdeal ` R )   &    |- I = ( R ↾s U )   =>    |- ( U e. L -> ( Base ` I ) = U )
Theorem	islidlm 13975*	Predicate of being a (left) ideal. (Contributed by Stefan O'Rear, 1-Apr-2015.)
|- U = (LIdeal ` R )   &    |- B = ( Base ` R )   &    |- .+ = ( +g ` R )   &    |- .x. = ( .r ` R )   =>    |- ( I e. U <-> ( I C_ B /\ E. j j e. I /\ A. x e. B A. a e. I A. b e. I ( ( x .x. a ) .+ b ) e. I ) )
Theorem	rnglidlmcl 13976	A (left) ideal containing the zero element is closed under left-multiplication by elements of the full non-unital ring. If the ring is not a unital ring, and the ideal does not contain the zero element of the ring, then the closure cannot be proven. (Contributed by AV, 18-Feb-2025.)
|- .0. = ( 0g ` R )   &    |- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- U = (LIdeal ` R )   =>    |- ( ( ( R e. Rng /\ I e. U /\ .0. e. I ) /\ ( X e. B /\ Y e. I ) ) -> ( X .x. Y ) e. I )
Theorem	dflidl2rng 13977*	Alternate (the usual textbook) definition of a (left) ideal of a non-unital ring to be a subgroup of the additive group of the ring which is closed under left-multiplication by elements of the full ring. (Contributed by AV, 21-Mar-2025.)
|- U = (LIdeal ` R )   &    |- B = ( Base ` R )   &    |- .x. = ( .r ` R )   =>    |- ( ( R e. Rng /\ I e. (SubGrp ` R ) ) -> ( I e. U <-> A. x e. B A. y e. I ( x .x. y ) e. I ) )
Theorem	isridlrng 13978*	A right ideal is a left ideal of the opposite non-unital ring. This theorem shows that this definition corresponds to the usual textbook definition of a right ideal of a ring to be a subgroup of the additive group of the ring which is closed under right-multiplication by elements of the full ring. (Contributed by AV, 21-Mar-2025.)
|- U = (LIdeal ` (oppr ` R ) )   &    |- B = ( Base ` R )   &    |- .x. = ( .r ` R )   =>    |- ( ( R e. Rng /\ I e. (SubGrp ` R ) ) -> ( I e. U <-> A. x e. B A. y e. I ( y .x. x ) e. I ) )
Theorem	lidl0cl 13979	An ideal contains 0. (Contributed by Stefan O'Rear, 3-Jan-2015.)
|- U = (LIdeal ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( ( R e. Ring /\ I e. U ) -> .0. e. I )
Theorem	lidlacl 13980	An ideal is closed under addition. (Contributed by Stefan O'Rear, 3-Jan-2015.)
|- U = (LIdeal ` R )   &    |- .+ = ( +g ` R )   =>    |- ( ( ( R e. Ring /\ I e. U ) /\ ( X e. I /\ Y e. I ) ) -> ( X .+ Y ) e. I )
Theorem	lidlnegcl 13981	An ideal contains negatives. (Contributed by Stefan O'Rear, 3-Jan-2015.)
|- U = (LIdeal ` R )   &    |- N = ( invg ` R )   =>    |- ( ( R e. Ring /\ I e. U /\ X e. I ) -> ( N ` X ) e. I )
Theorem	lidlsubg 13982	An ideal is a subgroup of the additive group. (Contributed by Mario Carneiro, 14-Jun-2015.)
|- U = (LIdeal ` R )   =>    |- ( ( R e. Ring /\ I e. U ) -> I e. (SubGrp ` R ) )
Theorem	lidlsubcl 13983	An ideal is closed under subtraction. (Contributed by Stefan O'Rear, 28-Mar-2015.) (Proof shortened by OpenAI, 25-Mar-2020.)
|- U = (LIdeal ` R )   &    |- .- = ( -g ` R )   =>    |- ( ( ( R e. Ring /\ I e. U ) /\ ( X e. I /\ Y e. I ) ) -> ( X .- Y ) e. I )
Theorem	dflidl2 13984*	Alternate (the usual textbook) definition of a (left) ideal of a ring to be a subgroup of the additive group of the ring which is closed under left-multiplication by elements of the full ring. (Contributed by AV, 13-Feb-2025.) (Proof shortened by AV, 18-Apr-2025.)
|- U = (LIdeal ` R )   &    |- B = ( Base ` R )   &    |- .x. = ( .r ` R )   =>    |- ( R e. Ring -> ( I e. U <-> ( I e. (SubGrp ` R ) /\ A. x e. B A. y e. I ( x .x. y ) e. I ) ) )
Theorem	lidl0 13985	Every ring contains a zero ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
|- U = (LIdeal ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( R e. Ring -> { .0. } e. U )
Theorem	lidl1 13986	Every ring contains a unit ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
|- U = (LIdeal ` R )   &    |- B = ( Base ` R )   =>    |- ( R e. Ring -> B e. U )
Theorem	rspcl 13987	The span of a set of ring elements is an ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
|- K = (RSpan ` R )   &    |- B = ( Base ` R )   &    |- U = (LIdeal ` R )   =>    |- ( ( R e. Ring /\ G C_ B ) -> ( K ` G ) e. U )
Theorem	rspssid 13988	The span of a set of ring elements contains those elements. (Contributed by Stefan O'Rear, 3-Jan-2015.)
|- K = (RSpan ` R )   &    |- B = ( Base ` R )   =>    |- ( ( R e. Ring /\ G C_ B ) -> G C_ ( K ` G ) )
Theorem	rsp0 13989	The span of the zero element is the zero ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
|- K = (RSpan ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( R e. Ring -> ( K ` { .0. } ) = { .0. } )
Theorem	rspssp 13990	The ideal span of a set of elements in a ring is contained in any subring which contains those elements. (Contributed by Stefan O'Rear, 3-Jan-2015.)
|- K = (RSpan ` R )   &    |- U = (LIdeal ` R )   =>    |- ( ( R e. Ring /\ I e. U /\ G C_ I ) -> ( K ` G ) C_ I )
Theorem	lidlrsppropdg 13991*	The left ideals and ring span of a ring depend only on the ring components. Here W is expected to be either B (when closure is available) or _V (when strong equality is available). (Contributed by Mario Carneiro, 14-Jun-2015.)
|- ( ph -> B = ( Base ` K ) )   &    |- ( ph -> B = ( Base ` L ) )   &    |- ( ph -> B C_ W )   &    |- ( ( ph /\ ( x e. W /\ y e. W ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )   &    |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) e. W )   &    |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` L ) y ) )   &    |- ( ph -> K e. X )   &    |- ( ph -> L e. Y )   =>    |- ( ph -> ( (LIdeal ` K ) = (LIdeal ` L ) /\ (RSpan ` K ) = (RSpan ` L ) ) )
Theorem	rnglidlmmgm 13992	The multiplicative group of a (left) ideal of a non-unital ring is a magma. (Contributed by AV, 17-Feb-2020.) Generalization for non-unital rings. The assumption .0. e. U is required because a left ideal of a non-unital ring does not have to be a subgroup. (Revised by AV, 11-Mar-2025.)
|- L = (LIdeal ` R )   &    |- I = ( R ↾s U )   &    |- .0. = ( 0g ` R )   =>    |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> (mulGrp ` I ) e. Mgm )
Theorem	rnglidlmsgrp 13993	The multiplicative group of a (left) ideal of a non-unital ring is a semigroup. (Contributed by AV, 17-Feb-2020.) Generalization for non-unital rings. The assumption .0. e. U is required because a left ideal of a non-unital ring does not have to be a subgroup. (Revised by AV, 11-Mar-2025.)
|- L = (LIdeal ` R )   &    |- I = ( R ↾s U )   &    |- .0. = ( 0g ` R )   =>    |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> (mulGrp ` I ) e. Smgrp )
Theorem	rnglidlrng 13994	A (left) ideal of a non-unital ring is a non-unital ring. (Contributed by AV, 17-Feb-2020.) Generalization for non-unital rings. The assumption U e. (SubGrp ` R ) is required because a left ideal of a non-unital ring does not have to be a subgroup. (Revised by AV, 11-Mar-2025.)
|- L = (LIdeal ` R )   &    |- I = ( R ↾s U )   =>    |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> I e. Rng )
7.6.3  Two-sided ideals and quotient rings
Syntax	c2idl 13995	Ring two-sided ideal function.
class 2Ideal
Definition	df-2idl 13996	Define the class of two-sided ideals of a ring. A two-sided ideal is a left ideal which is also a right ideal (or a left ideal over the opposite ring). (Contributed by Mario Carneiro, 14-Jun-2015.)
|- 2Ideal = ( r e. _V |-> ( (LIdeal ` r ) i^i (LIdeal ` (oppr ` r ) ) ) )
Theorem	2idlmex 13997	Existence of the set a two-sided ideal is built from (when the ideal is inhabited). (Contributed by Jim Kingdon, 18-Apr-2025.)
|- T = (2Ideal ` W )   =>    |- ( U e. T -> W e. _V )
Theorem	2idlval 13998	Definition of a two-sided ideal. (Contributed by Mario Carneiro, 14-Jun-2015.)
|- I = (LIdeal ` R )   &    |- O = (oppr ` R )   &    |- J = (LIdeal ` O )   &    |- T = (2Ideal ` R )   =>    |- T = ( I i^i J )
Theorem	2idlvalg 13999	Definition of a two-sided ideal. (Contributed by Mario Carneiro, 14-Jun-2015.)
|- I = (LIdeal ` R )   &    |- O = (oppr ` R )   &    |- J = (LIdeal ` O )   &    |- T = (2Ideal ` R )   =>    |- ( R e. V -> T = ( I i^i J ) )
Theorem	isridl 14000*	A right ideal is a left ideal of the opposite ring. This theorem shows that this definition corresponds to the usual textbook definition of a right ideal of a ring to be a subgroup of the additive group of the ring which is closed under right-multiplication by elements of the full ring. (Contributed by AV, 13-Feb-2025.)
|- U = (LIdeal ` (oppr ` R ) )   &    |- B = ( Base ` R )   &    |- .x. = ( .r ` R )   =>    |- ( R e. Ring -> ( I e. U <-> ( I e. (SubGrp ` R ) /\ A. x e. B A. y e. I ( y .x. x ) e. I ) ) )