P. 142 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 14101-14200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	lmodvaddsub4 14101	Vector addition/subtraction law. (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- V = ( Base ` W )   &    |- .+ = ( +g ` W )   &    |- .- = ( -g ` W )   =>    |- ( ( W e. LMod /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> ( ( A .+ B ) = ( C .+ D ) <-> ( A .- C ) = ( D .- B ) ) )
Theorem	lmodvpncan 14102	Addition/subtraction cancellation law for vectors. (Contributed by NM, 16-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- V = ( Base ` W )   &    |- .+ = ( +g ` W )   &    |- .- = ( -g ` W )   =>    |- ( ( W e. LMod /\ A e. V /\ B e. V ) -> ( ( A .+ B ) .- B ) = A )
Theorem	lmodvnpcan 14103	Cancellation law for vector subtraction (Contributed by NM, 19-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- V = ( Base ` W )   &    |- .+ = ( +g ` W )   &    |- .- = ( -g ` W )   =>    |- ( ( W e. LMod /\ A e. V /\ B e. V ) -> ( ( A .- B ) .+ B ) = A )
Theorem	lmodvsubval2 14104	Value of vector subtraction in terms of addition. (Contributed by NM, 31-Mar-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
|- V = ( Base ` W )   &    |- .+ = ( +g ` W )   &    |- .- = ( -g ` W )   &    |- F = (Scalar ` W )   &    |- .x. = ( .s ` W )   &    |- N = ( invg ` F )   &    |- .1. = ( 1r ` F )   =>    |- ( ( W e. LMod /\ A e. V /\ B e. V ) -> ( A .- B ) = ( A .+ ( ( N ` .1. ) .x. B ) ) )
Theorem	lmodsubvs 14105	Subtraction of a scalar product in terms of addition. (Contributed by NM, 9-Apr-2015.)
|- V = ( Base ` W )   &    |- .+ = ( +g ` W )   &    |- .- = ( -g ` W )   &    |- .x. = ( .s ` W )   &    |- F = (Scalar ` W )   &    |- K = ( Base ` F )   &    |- N = ( invg ` F )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> A e. K )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   =>    |- ( ph -> ( X .- ( A .x. Y ) ) = ( X .+ ( ( N ` A ) .x. Y ) ) )
Theorem	lmodsubdi 14106	Scalar multiplication distributive law for subtraction. (Contributed by NM, 2-Jul-2014.)
|- V = ( Base ` W )   &    |- .x. = ( .s ` W )   &    |- F = (Scalar ` W )   &    |- K = ( Base ` F )   &    |- .- = ( -g ` W )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> A e. K )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   =>    |- ( ph -> ( A .x. ( X .- Y ) ) = ( ( A .x. X ) .- ( A .x. Y ) ) )
Theorem	lmodsubdir 14107	Scalar multiplication distributive law for subtraction. (Contributed by NM, 2-Jul-2014.)
|- V = ( Base ` W )   &    |- .x. = ( .s ` W )   &    |- F = (Scalar ` W )   &    |- K = ( Base ` F )   &    |- .- = ( -g ` W )   &    |- S = ( -g ` F )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> A e. K )   &    |- ( ph -> B e. K )   &    |- ( ph -> X e. V )   =>    |- ( ph -> ( ( A S B ) .x. X ) = ( ( A .x. X ) .- ( B .x. X ) ) )
Theorem	lmodsubeq0 14108	If the difference between two vectors is zero, they are equal. (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- V = ( Base ` W )   &    |- .0. = ( 0g ` W )   &    |- .- = ( -g ` W )   =>    |- ( ( W e. LMod /\ A e. V /\ B e. V ) -> ( ( A .- B ) = .0. <-> A = B ) )
Theorem	lmodsubid 14109	Subtraction of a vector from itself. (Contributed by NM, 16-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- V = ( Base ` W )   &    |- .0. = ( 0g ` W )   &    |- .- = ( -g ` W )   =>    |- ( ( W e. LMod /\ A e. V ) -> ( A .- A ) = .0. )
Theorem	lmodprop2d 14110*	If two structures have the same components (properties), one is a left module iff the other one is. This version of lmodpropd 14111 also breaks up the components of the scalar ring. (Contributed by Mario Carneiro, 27-Jun-2015.)
|- ( ph -> B = ( Base ` K ) )   &    |- ( ph -> B = ( Base ` L ) )   &    |- F = (Scalar ` K )   &    |- G = (Scalar ` L )   &    |- ( ph -> P = ( Base ` F ) )   &    |- ( ph -> P = ( Base ` G ) )   &    |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )   &    |- ( ( ph /\ ( x e. P /\ y e. P ) ) -> ( x ( +g ` F ) y ) = ( x ( +g ` G ) y ) )   &    |- ( ( ph /\ ( x e. P /\ y e. P ) ) -> ( x ( .r ` F ) y ) = ( x ( .r ` G ) y ) )   &    |- ( ( ph /\ ( x e. P /\ y e. B ) ) -> ( x ( .s ` K ) y ) = ( x ( .s ` L ) y ) )   =>    |- ( ph -> ( K e. LMod <-> L e. LMod ) )
Theorem	lmodpropd 14111*	If two structures have the same components (properties), one is a left module iff the other one is. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 27-Jun-2015.)
|- ( ph -> B = ( Base ` K ) )   &    |- ( ph -> B = ( Base ` L ) )   &    |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )   &    |- ( ph -> F = (Scalar ` K ) )   &    |- ( ph -> F = (Scalar ` L ) )   &    |- P = ( Base ` F )   &    |- ( ( ph /\ ( x e. P /\ y e. B ) ) -> ( x ( .s ` K ) y ) = ( x ( .s ` L ) y ) )   =>    |- ( ph -> ( K e. LMod <-> L e. LMod ) )
Theorem	rmodislmodlem 14112*	Lemma for rmodislmod 14113. This is the part of the proof of rmodislmod 14113 which requires the scalar ring to be commutative. (Contributed by AV, 3-Dec-2021.)
|- V = ( Base ` R )   &    |- .+ = ( +g ` R )   &    |- .x. = ( .s ` R )   &    |- F = (Scalar ` R )   &    |- K = ( Base ` F )   &    |- .+^ = ( +g ` F )   &    |- .X. = ( .r ` F )   &    |- .1. = ( 1r ` F )   &    |- ( R e. Grp /\ F e. Ring /\ A. q e. K A. r e. K A. x e. V A. w e. V ( ( ( w .x. r ) e. V /\ ( ( w .+ x ) .x. r ) = ( ( w .x. r ) .+ ( x .x. r ) ) /\ ( w .x. ( q .+^ r ) ) = ( ( w .x. q ) .+ ( w .x. r ) ) ) /\ ( ( w .x. ( q .X. r ) ) = ( ( w .x. q ) .x. r ) /\ ( w .x. .1. ) = w ) ) )   &    |- .* = ( s e. K , v e. V |-> ( v .x. s ) )   &    |- L = ( R sSet <. ( .s ` ndx ) , .* >. )   =>    |- ( ( F e. CRing /\ ( a e. K /\ b e. K /\ c e. V ) ) -> ( ( a .X. b ) .* c ) = ( a .* ( b .* c ) ) )
Theorem	rmodislmod 14113*	The right module R induces a left module L by replacing the scalar multiplication with a reversed multiplication if the scalar ring is commutative. The hypothesis "rmodislmod.r" is a definition of a right module analogous to Definition df-lmod 14051 of a left module, see also islmod 14053. (Contributed by AV, 3-Dec-2021.) (Proof shortened by AV, 18-Oct-2024.)
|- V = ( Base ` R )   &    |- .+ = ( +g ` R )   &    |- .x. = ( .s ` R )   &    |- F = (Scalar ` R )   &    |- K = ( Base ` F )   &    |- .+^ = ( +g ` F )   &    |- .X. = ( .r ` F )   &    |- .1. = ( 1r ` F )   &    |- ( R e. Grp /\ F e. Ring /\ A. q e. K A. r e. K A. x e. V A. w e. V ( ( ( w .x. r ) e. V /\ ( ( w .+ x ) .x. r ) = ( ( w .x. r ) .+ ( x .x. r ) ) /\ ( w .x. ( q .+^ r ) ) = ( ( w .x. q ) .+ ( w .x. r ) ) ) /\ ( ( w .x. ( q .X. r ) ) = ( ( w .x. q ) .x. r ) /\ ( w .x. .1. ) = w ) ) )   &    |- .* = ( s e. K , v e. V |-> ( v .x. s ) )   &    |- L = ( R sSet <. ( .s ` ndx ) , .* >. )   =>    |- ( F e. CRing -> L e. LMod )
7.5.2  Subspaces and spans in a left module
Syntax	clss 14114	Extend class notation with linear subspaces of a left module or left vector space.
class LSubSp
Definition	df-lssm 14115*	A linear subspace of a left module or left vector space is an inhabited (in contrast to non-empty for non-intuitionistic logic) subset of the base set of the left-module/vector space with a closure condition on vector addition and scalar multiplication. (Contributed by NM, 8-Dec-2013.)
|- LSubSp = ( w e. _V |-> { s e. ~P ( Base ` w ) | ( E. j j e. s /\ A. x e. ( Base ` (Scalar ` w ) ) A. a e. s A. b e. s ( ( x ( .s ` w ) a ) ( +g ` w ) b ) e. s ) } )
Theorem	lssex 14116	Existence of a linear subspace. (Contributed by Jim Kingdon, 27-Apr-2025.)
|- ( W e. V -> ( LSubSp ` W ) e. _V )
Theorem	lssmex 14117	If a linear subspace is inhabited, the class it is built from is a set. (Contributed by Jim Kingdon, 28-Apr-2025.)
|- S = ( LSubSp ` W )   =>    |- ( U e. S -> W e. _V )
Theorem	lsssetm 14118*	The set of all (not necessarily closed) linear subspaces of a left module or left vector space. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 15-Jul-2014.)
|- F = (Scalar ` W )   &    |- B = ( Base ` F )   &    |- V = ( Base ` W )   &    |- .+ = ( +g ` W )   &    |- .x. = ( .s ` W )   &    |- S = ( LSubSp ` W )   =>    |- ( W e. X -> S = { s e. ~P V | ( E. j j e. s /\ A. x e. B A. a e. s A. b e. s ( ( x .x. a ) .+ b ) e. s ) } )
Theorem	islssm 14119*	The predicate "is a subspace" (of a left module or left vector space). (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)
|- F = (Scalar ` W )   &    |- B = ( Base ` F )   &    |- V = ( Base ` W )   &    |- .+ = ( +g ` W )   &    |- .x. = ( .s ` W )   &    |- S = ( LSubSp ` W )   =>    |- ( U e. S <-> ( U C_ V /\ E. j j e. U /\ A. x e. B A. a e. U A. b e. U ( ( x .x. a ) .+ b ) e. U ) )
Theorem	islssmg 14120*	The predicate "is a subspace" (of a left module or left vector space). (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.) Use islssm 14119 instead. (New usage is discouraged.)
|- F = (Scalar ` W )   &    |- B = ( Base ` F )   &    |- V = ( Base ` W )   &    |- .+ = ( +g ` W )   &    |- .x. = ( .s ` W )   &    |- S = ( LSubSp ` W )   =>    |- ( W e. X -> ( U e. S <-> ( U C_ V /\ E. j j e. U /\ A. x e. B A. a e. U A. b e. U ( ( x .x. a ) .+ b ) e. U ) ) )
Theorem	islssmd 14121*	Properties that determine a subspace of a left module or left vector space. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)
|- ( ph -> F = (Scalar ` W ) )   &    |- ( ph -> B = ( Base ` F ) )   &    |- ( ph -> V = ( Base ` W ) )   &    |- ( ph -> .+ = ( +g ` W ) )   &    |- ( ph -> .x. = ( .s ` W ) )   &    |- ( ph -> S = ( LSubSp ` W ) )   &    |- ( ph -> U C_ V )   &    |- ( ph -> E. j j e. U )   &    |- ( ( ph /\ ( x e. B /\ a e. U /\ b e. U ) ) -> ( ( x .x. a ) .+ b ) e. U )   &    |- ( ph -> W e. X )   =>    |- ( ph -> U e. S )
Theorem	lssssg 14122	A subspace is a set of vectors. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)
|- V = ( Base ` W )   &    |- S = ( LSubSp ` W )   =>    |- ( ( W e. X /\ U e. S ) -> U C_ V )
Theorem	lsselg 14123	A subspace member is a vector. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 8-Jan-2015.)
|- V = ( Base ` W )   &    |- S = ( LSubSp ` W )   =>    |- ( ( W e. C /\ U e. S /\ X e. U ) -> X e. V )
Theorem	lss1 14124	The set of vectors in a left module is a subspace. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- V = ( Base ` W )   &    |- S = ( LSubSp ` W )   =>    |- ( W e. LMod -> V e. S )
Theorem	lssuni 14125	The union of all subspaces is the vector space. (Contributed by NM, 13-Mar-2015.)
|- V = ( Base ` W )   &    |- S = ( LSubSp ` W )   &    |- ( ph -> W e. LMod )   =>    |- ( ph -> U. S = V )
Theorem	lssclg 14126	Closure property of a subspace. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)
|- F = (Scalar ` W )   &    |- B = ( Base ` F )   &    |- .+ = ( +g ` W )   &    |- .x. = ( .s ` W )   &    |- S = ( LSubSp ` W )   =>    |- ( ( W e. C /\ U e. S /\ ( Z e. B /\ X e. U /\ Y e. U ) ) -> ( ( Z .x. X ) .+ Y ) e. U )
Theorem	lssvacl 14127	Closure of vector addition in a subspace. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- .+ = ( +g ` W )   &    |- S = ( LSubSp ` W )   =>    |- ( ( ( W e. LMod /\ U e. S ) /\ ( X e. U /\ Y e. U ) ) -> ( X .+ Y ) e. U )
Theorem	lssvsubcl 14128	Closure of vector subtraction in a subspace. (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- .- = ( -g ` W )   &    |- S = ( LSubSp ` W )   =>    |- ( ( ( W e. LMod /\ U e. S ) /\ ( X e. U /\ Y e. U ) ) -> ( X .- Y ) e. U )
Theorem	lssvancl1 14129	Non-closure: if one vector belongs to a subspace but another does not, their sum does not belong. Useful for obtaining a new vector not in a subspace. (Contributed by NM, 14-May-2015.)
|- V = ( Base ` W )   &    |- .+ = ( +g ` W )   &    |- S = ( LSubSp ` W )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> U e. S )   &    |- ( ph -> X e. U )   &    |- ( ph -> Y e. V )   &    |- ( ph -> -. Y e. U )   =>    |- ( ph -> -. ( X .+ Y ) e. U )
Theorem	lssvancl2 14130	Non-closure: if one vector belongs to a subspace but another does not, their sum does not belong. Useful for obtaining a new vector not in a subspace. (Contributed by NM, 20-May-2015.)
|- V = ( Base ` W )   &    |- .+ = ( +g ` W )   &    |- S = ( LSubSp ` W )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> U e. S )   &    |- ( ph -> X e. U )   &    |- ( ph -> Y e. V )   &    |- ( ph -> -. Y e. U )   =>    |- ( ph -> -. ( Y .+ X ) e. U )
Theorem	lss0cl 14131	The zero vector belongs to every subspace. (Contributed by NM, 12-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
|- .0. = ( 0g ` W )   &    |- S = ( LSubSp ` W )   =>    |- ( ( W e. LMod /\ U e. S ) -> .0. e. U )
Theorem	lsssn0 14132	The singleton of the zero vector is a subspace. (Contributed by NM, 13-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- .0. = ( 0g ` W )   &    |- S = ( LSubSp ` W )   =>    |- ( W e. LMod -> { .0. } e. S )
Theorem	lss0ss 14133	The zero subspace is included in every subspace. (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- .0. = ( 0g ` W )   &    |- S = ( LSubSp ` W )   =>    |- ( ( W e. LMod /\ X e. S ) -> { .0. } C_ X )
Theorem	lssle0 14134	No subspace is smaller than the zero subspace. (Contributed by NM, 20-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- .0. = ( 0g ` W )   &    |- S = ( LSubSp ` W )   =>    |- ( ( W e. LMod /\ X e. S ) -> ( X C_ { .0. } <-> X = { .0. } ) )
Theorem	lssvneln0 14135	A vector X which doesn't belong to a subspace U is nonzero. (Contributed by NM, 14-May-2015.) (Revised by AV, 19-Jul-2022.)
|- .0. = ( 0g ` W )   &    |- S = ( LSubSp ` W )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> U e. S )   &    |- ( ph -> -. X e. U )   =>    |- ( ph -> X =/= .0. )
Theorem	lssneln0 14136	A vector X which doesn't belong to a subspace U is nonzero. (Contributed by NM, 14-May-2015.) (Revised by AV, 17-Jul-2022.) (Proof shortened by AV, 19-Jul-2022.)
|- .0. = ( 0g ` W )   &    |- S = ( LSubSp ` W )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> U e. S )   &    |- ( ph -> X e. V )   &    |- ( ph -> -. X e. U )   =>    |- ( ph -> X e. ( V \ { .0. } ) )
Theorem	lssvscl 14137	Closure of scalar product in a subspace. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- F = (Scalar ` W )   &    |- .x. = ( .s ` W )   &    |- B = ( Base ` F )   &    |- S = ( LSubSp ` W )   =>    |- ( ( ( W e. LMod /\ U e. S ) /\ ( X e. B /\ Y e. U ) ) -> ( X .x. Y ) e. U )
Theorem	lssvnegcl 14138	Closure of negative vectors in a subspace. (Contributed by Stefan O'Rear, 11-Dec-2014.)
|- S = ( LSubSp ` W )   &    |- N = ( invg ` W )   =>    |- ( ( W e. LMod /\ U e. S /\ X e. U ) -> ( N ` X ) e. U )
Theorem	lsssubg 14139	All subspaces are subgroups. (Contributed by Stefan O'Rear, 11-Dec-2014.)
|- S = ( LSubSp ` W )   =>    |- ( ( W e. LMod /\ U e. S ) -> U e. (SubGrp ` W ) )
Theorem	lsssssubg 14140	All subspaces are subgroups. (Contributed by Mario Carneiro, 19-Apr-2016.)
|- S = ( LSubSp ` W )   =>    |- ( W e. LMod -> S C_ (SubGrp ` W ) )
Theorem	islss3 14141	A linear subspace of a module is a subset which is a module in its own right. (Contributed by Stefan O'Rear, 6-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
|- X = ( W ↾s U )   &    |- V = ( Base ` W )   &    |- S = ( LSubSp ` W )   =>    |- ( W e. LMod -> ( U e. S <-> ( U C_ V /\ X e. LMod ) ) )
Theorem	lsslmod 14142	A submodule is a module. (Contributed by Stefan O'Rear, 12-Dec-2014.)
|- X = ( W ↾s U )   &    |- S = ( LSubSp ` W )   =>    |- ( ( W e. LMod /\ U e. S ) -> X e. LMod )
Theorem	lsslss 14143	The subspaces of a subspace are the smaller subspaces. (Contributed by Stefan O'Rear, 12-Dec-2014.)
|- X = ( W ↾s U )   &    |- S = ( LSubSp ` W )   &    |- T = ( LSubSp ` X )   =>    |- ( ( W e. LMod /\ U e. S ) -> ( V e. T <-> ( V e. S /\ V C_ U ) ) )
Theorem	islss4 14144*	A linear subspace is a subgroup which respects scalar multiplication. (Contributed by Stefan O'Rear, 11-Dec-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
|- F = (Scalar ` W )   &    |- B = ( Base ` F )   &    |- V = ( Base ` W )   &    |- .x. = ( .s ` W )   &    |- S = ( LSubSp ` W )   =>    |- ( W e. LMod -> ( U e. S <-> ( U e. (SubGrp ` W ) /\ A. a e. B A. b e. U ( a .x. b ) e. U ) ) )
Theorem	lss1d 14145*	One-dimensional subspace (or zero-dimensional if X is the zero vector). (Contributed by NM, 14-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
|- V = ( Base ` W )   &    |- F = (Scalar ` W )   &    |- .x. = ( .s ` W )   &    |- K = ( Base ` F )   &    |- S = ( LSubSp ` W )   =>    |- ( ( W e. LMod /\ X e. V ) -> { v | E. k e. K v = ( k .x. X ) } e. S )
Theorem	lssintclm 14146*	The intersection of an inhabited set of subspaces is a subspace. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- S = ( LSubSp ` W )   =>    |- ( ( W e. LMod /\ A C_ S /\ E. w w e. A ) -> |^| A e. S )
Theorem	lssincl 14147	The intersection of two subspaces is a subspace. (Contributed by NM, 7-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- S = ( LSubSp ` W )   =>    |- ( ( W e. LMod /\ T e. S /\ U e. S ) -> ( T i^i U ) e. S )
Syntax	clspn 14148	Extend class notation with span of a set of vectors.
class LSpan
Definition	df-lsp 14149*	Define span of a set of vectors of a left module or left vector space. (Contributed by NM, 8-Dec-2013.)
|- LSpan = ( w e. _V |-> ( s e. ~P ( Base ` w ) |-> |^| { t e. ( LSubSp ` w ) | s C_ t } ) )
Theorem	lspfval 14150*	The span function for a left vector space (or a left module). (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- V = ( Base ` W )   &    |- S = ( LSubSp ` W )   &    |- N = ( LSpan ` W )   =>    |- ( W e. X -> N = ( s e. ~P V |-> |^| { t e. S | s C_ t } ) )
Theorem	lspf 14151	The span function on a left module maps subsets to subspaces. (Contributed by Stefan O'Rear, 12-Dec-2014.)
|- V = ( Base ` W )   &    |- S = ( LSubSp ` W )   &    |- N = ( LSpan ` W )   =>    |- ( W e. LMod -> N : ~P V --> S )
Theorem	lspval 14152*	The span of a set of vectors (in a left module). (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- V = ( Base ` W )   &    |- S = ( LSubSp ` W )   &    |- N = ( LSpan ` W )   =>    |- ( ( W e. LMod /\ U C_ V ) -> ( N ` U ) = |^| { t e. S | U C_ t } )
Theorem	lspcl 14153	The span of a set of vectors is a subspace. (Contributed by NM, 9-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- V = ( Base ` W )   &    |- S = ( LSubSp ` W )   &    |- N = ( LSpan ` W )   =>    |- ( ( W e. LMod /\ U C_ V ) -> ( N ` U ) e. S )
Theorem	lspsncl 14154	The span of a singleton is a subspace (frequently used special case of lspcl 14153). (Contributed by NM, 17-Jul-2014.)
|- V = ( Base ` W )   &    |- S = ( LSubSp ` W )   &    |- N = ( LSpan ` W )   =>    |- ( ( W e. LMod /\ X e. V ) -> ( N ` { X } ) e. S )
Theorem	lspprcl 14155	The span of a pair is a subspace (frequently used special case of lspcl 14153). (Contributed by NM, 11-Apr-2015.)
|- V = ( Base ` W )   &    |- S = ( LSubSp ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   =>    |- ( ph -> ( N ` { X , Y } ) e. S )
Theorem	lsptpcl 14156	The span of an unordered triple is a subspace (frequently used special case of lspcl 14153). (Contributed by NM, 22-May-2015.)
|- V = ( Base ` W )   &    |- S = ( LSubSp ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   &    |- ( ph -> Z e. V )   =>    |- ( ph -> ( N ` { X , Y , Z } ) e. S )
Theorem	lspex 14157	Existence of the span of a set of vectors. (Contributed by Jim Kingdon, 25-Apr-2025.)
|- ( W e. X -> ( LSpan ` W ) e. _V )
Theorem	lspsnsubg 14158	The span of a singleton is an additive subgroup (frequently used special case of lspcl 14153). (Contributed by Mario Carneiro, 21-Apr-2016.)
|- V = ( Base ` W )   &    |- N = ( LSpan ` W )   =>    |- ( ( W e. LMod /\ X e. V ) -> ( N ` { X } ) e. (SubGrp ` W ) )
Theorem	lspid 14159	The span of a subspace is itself. (Contributed by NM, 15-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- S = ( LSubSp ` W )   &    |- N = ( LSpan ` W )   =>    |- ( ( W e. LMod /\ U e. S ) -> ( N ` U ) = U )
Theorem	lspssv 14160	A span is a set of vectors. (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- V = ( Base ` W )   &    |- N = ( LSpan ` W )   =>    |- ( ( W e. LMod /\ U C_ V ) -> ( N ` U ) C_ V )
Theorem	lspss 14161	Span preserves subset ordering. (Contributed by NM, 11-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- V = ( Base ` W )   &    |- N = ( LSpan ` W )   =>    |- ( ( W e. LMod /\ U C_ V /\ T C_ U ) -> ( N ` T ) C_ ( N ` U ) )
Theorem	lspssid 14162	A set of vectors is a subset of its span. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- V = ( Base ` W )   &    |- N = ( LSpan ` W )   =>    |- ( ( W e. LMod /\ U C_ V ) -> U C_ ( N ` U ) )
Theorem	lspidm 14163	The span of a set of vectors is idempotent. (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- V = ( Base ` W )   &    |- N = ( LSpan ` W )   =>    |- ( ( W e. LMod /\ U C_ V ) -> ( N ` ( N ` U ) ) = ( N ` U ) )
Theorem	lspun 14164	The span of union is the span of the union of spans. (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- V = ( Base ` W )   &    |- N = ( LSpan ` W )   =>    |- ( ( W e. LMod /\ T C_ V /\ U C_ V ) -> ( N ` ( T u. U ) ) = ( N ` ( ( N ` T ) u. ( N ` U ) ) ) )
Theorem	lspssp 14165	If a set of vectors is a subset of a subspace, then the span of those vectors is also contained in the subspace. (Contributed by Mario Carneiro, 4-Sep-2014.)
|- S = ( LSubSp ` W )   &    |- N = ( LSpan ` W )   =>    |- ( ( W e. LMod /\ U e. S /\ T C_ U ) -> ( N ` T ) C_ U )
Theorem	lspsnss 14166	The span of the singleton of a subspace member is included in the subspace. (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 4-Sep-2014.)
|- S = ( LSubSp ` W )   &    |- N = ( LSpan ` W )   =>    |- ( ( W e. LMod /\ U e. S /\ X e. U ) -> ( N ` { X } ) C_ U )
Theorem	lspsnel3 14167	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 14168	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 14169	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 14170	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 14171	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 14172	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 14173	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 14174	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 14175	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 14176*	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 14177	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 14178*	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 14179*	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 14180	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 14181*	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 14182	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 14183	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 14184	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 14185	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 14186	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 14187	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 14188	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 14189	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 14190	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 14191	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 14192	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 14193*	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 14194*	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 14195	Extend class notation with the subring algebra generator.
class subringAlg
Syntax	crglmod 14196	Extend class notation with the left module induced by a ring over itself.
class ringLMod
Definition	df-sra 14197*	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 14198	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 14199	Lemma for srabaseg 14201 through sravscag 14205. (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 14200	Lemma for srabaseg 14201 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 ) )