P. 192 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 19101-19200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	pj1lmhm2 19101	The left projection function is a linear operator. (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
|- L = ( LSubSp ` W )   &    |- .(+) = ( LSSum ` W )   &    |- .0. = ( 0g ` W )   &    |- P = ( proj1 ` W )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> T e. L )   &    |- ( ph -> U e. L )   &    |- ( ph -> ( T i^i U ) = { .0. } )   =>    |- ( ph -> ( T P U ) e. ( ( W ↾s ( T .(+) U ) ) LMHom ( W ↾s T ) ) )
10.7  Vector spaces
10.7.1  Definition and basic properties
Syntax	clvec 19102	Extend class notation with class of all left vector spaces.
class LVec
Definition	df-lvec 19103	Define the class of all left vector spaces. A left vector space over a division ring is an Abelian group (vectors) together with a division ring (scalars) and a left scalar product connecting them. Some authors call this a "left module over a division ring", reserving "vector space" for those where the division ring multiplication is commutative i.e. a field. (Contributed by NM, 11-Nov-2013.)
|- LVec = { f e. LMod | (Scalar ` f ) e. DivRing }
Theorem	islvec 19104	The predicate "is a left vector space". (Contributed by NM, 11-Nov-2013.)
|- F = (Scalar ` W )   =>    |- ( W e. LVec <-> ( W e. LMod /\ F e. DivRing ) )
Theorem	lvecdrng 19105	The set of scalars of a left vector space is a division ring. (Contributed by NM, 17-Apr-2014.)
|- F = (Scalar ` W )   =>    |- ( W e. LVec -> F e. DivRing )
Theorem	lveclmod 19106	A left vector space is a left module. (Contributed by NM, 9-Dec-2013.)
|- ( W e. LVec -> W e. LMod )
Theorem	lsslvec 19107	A vector subspace is a vector space. (Contributed by NM, 14-Mar-2015.)
|- X = ( W ↾s U )   &    |- S = ( LSubSp ` W )   =>    |- ( ( W e. LVec /\ U e. S ) -> X e. LVec )
Theorem	lvecvs0or 19108	If a scalar product is zero, one of its factors must be zero. (hvmul0or 27882 analog.) (Contributed by NM, 2-Jul-2014.)
|- V = ( Base ` W )   &    |- .x. = ( .s ` W )   &    |- F = (Scalar ` W )   &    |- K = ( Base ` F )   &    |- O = ( 0g ` F )   &    |- .0. = ( 0g ` W )   &    |- ( ph -> W e. LVec )   &    |- ( ph -> A e. K )   &    |- ( ph -> X e. V )   =>    |- ( ph -> ( ( A .x. X ) = .0. <-> ( A = O \/ X = .0. ) ) )
Theorem	lvecvsn0 19109	A scalar product is nonzero iff both of its factors are nonzero. (Contributed by NM, 3-Jan-2015.)
|- V = ( Base ` W )   &    |- .x. = ( .s ` W )   &    |- F = (Scalar ` W )   &    |- K = ( Base ` F )   &    |- O = ( 0g ` F )   &    |- .0. = ( 0g ` W )   &    |- ( ph -> W e. LVec )   &    |- ( ph -> A e. K )   &    |- ( ph -> X e. V )   =>    |- ( ph -> ( ( A .x. X ) =/= .0. <-> ( A =/= O /\ X =/= .0. ) ) )
Theorem	lssvs0or 19110	If a scalar product belongs to a subspace, either the scalar component is zero or the vector component also belongs to the subspace. (Contributed by NM, 5-Apr-2015.)
|- V = ( Base ` W )   &    |- .x. = ( .s ` W )   &    |- F = (Scalar ` W )   &    |- K = ( Base ` F )   &    |- .0. = ( 0g ` F )   &    |- S = ( LSubSp ` W )   &    |- ( ph -> W e. LVec )   &    |- ( ph -> U e. S )   &    |- ( ph -> X e. V )   &    |- ( ph -> A e. K )   =>    |- ( ph -> ( ( A .x. X ) e. U <-> ( A = .0. \/ X e. U ) ) )
Theorem	lvecvscan 19111	Cancellation law for scalar multiplication. (hvmulcan 27929 analog.) (Contributed by NM, 2-Jul-2014.)
|- V = ( Base ` W )   &    |- .x. = ( .s ` W )   &    |- F = (Scalar ` W )   &    |- K = ( Base ` F )   &    |- .0. = ( 0g ` F )   &    |- ( ph -> W e. LVec )   &    |- ( ph -> A e. K )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   &    |- ( ph -> A =/= .0. )   =>    |- ( ph -> ( ( A .x. X ) = ( A .x. Y ) <-> X = Y ) )
Theorem	lvecvscan2 19112	Cancellation law for scalar multiplication. (hvmulcan2 27930 analog.) (Contributed by NM, 2-Jul-2014.)
|- V = ( Base ` W )   &    |- .x. = ( .s ` W )   &    |- F = (Scalar ` W )   &    |- K = ( Base ` F )   &    |- .0. = ( 0g ` W )   &    |- ( ph -> W e. LVec )   &    |- ( ph -> A e. K )   &    |- ( ph -> B e. K )   &    |- ( ph -> X e. V )   &    |- ( ph -> X =/= .0. )   =>    |- ( ph -> ( ( A .x. X ) = ( B .x. X ) <-> A = B ) )
Theorem	lvecinv 19113	Invert coefficient of scalar product. (Contributed by NM, 11-Apr-2015.)
|- V = ( Base ` W )   &    |- .x. = ( .s ` W )   &    |- F = (Scalar ` W )   &    |- K = ( Base ` F )   &    |- .0. = ( 0g ` F )   &    |- I = ( invr ` F )   &    |- ( ph -> W e. LVec )   &    |- ( ph -> A e. ( K \ { .0. } ) )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   =>    |- ( ph -> ( X = ( A .x. Y ) <-> Y = ( ( I ` A ) .x. X ) ) )
Theorem	lspsnvs 19114	A nonzero scalar product does not change the span of a singleton. (spansncol 28427 analog.) (Contributed by NM, 23-Apr-2014.)
|- V = ( Base ` W )   &    |- F = (Scalar ` W )   &    |- .x. = ( .s ` W )   &    |- K = ( Base ` F )   &    |- .0. = ( 0g ` F )   &    |- N = ( LSpan ` W )   =>    |- ( ( W e. LVec /\ ( R e. K /\ R =/= .0. ) /\ X e. V ) -> ( N ` { ( R .x. X ) } ) = ( N ` { X } ) )
Theorem	lspsneleq 19115	Membership relation that implies equality of spans. (spansneleq 28429 analog.) (Contributed by NM, 4-Jul-2014.)
|- V = ( Base ` W )   &    |- .0. = ( 0g ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LVec )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. ( N ` { X } ) )   &    |- ( ph -> Y =/= .0. )   =>    |- ( ph -> ( N ` { Y } ) = ( N ` { X } ) )
Theorem	lspsncmp 19116	Comparable spans of nonzero singletons are equal. (Contributed by NM, 27-Apr-2015.)
|- V = ( Base ` W )   &    |- .0. = ( 0g ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LVec )   &    |- ( ph -> X e. ( V \ { .0. } ) )   &    |- ( ph -> Y e. V )   =>    |- ( ph -> ( ( N ` { X } ) C_ ( N ` { Y } ) <-> ( N ` { X } ) = ( N ` { Y } ) ) )
Theorem	lspsnne1 19117	Two ways to express that vectors have different spans. (Contributed by NM, 28-May-2015.)
|- V = ( Base ` W )   &    |- .0. = ( 0g ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LVec )   &    |- ( ph -> X e. ( V \ { .0. } ) )   &    |- ( ph -> Y e. V )   &    |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )   =>    |- ( ph -> -. X e. ( N ` { Y } ) )
Theorem	lspsnne2 19118	Two ways to express that vectors have different spans. (Contributed by NM, 20-May-2015.)
|- V = ( Base ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   &    |- ( ph -> -. X e. ( N ` { Y } ) )   =>    |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
Theorem	lspsnnecom 19119	Swap two vectors with different spans. (Contributed by NM, 20-May-2015.)
|- V = ( Base ` W )   &    |- .0. = ( 0g ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LVec )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. ( V \ { .0. } ) )   &    |- ( ph -> -. X e. ( N ` { Y } ) )   =>    |- ( ph -> -. Y e. ( N ` { X } ) )
Theorem	lspabs2 19120	Absorption law for span of vector sum. (Contributed by NM, 30-Apr-2015.)
|- V = ( Base ` W )   &    |- .+ = ( +g ` W )   &    |- .0. = ( 0g ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LVec )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. ( V \ { .0. } ) )   &    |- ( ph -> ( N ` { X } ) = ( N ` { ( X .+ Y ) } ) )   =>    |- ( ph -> ( N ` { X } ) = ( N ` { Y } ) )
Theorem	lspabs3 19121	Absorption law for span of vector sum. (Contributed by NM, 30-Apr-2015.)
|- V = ( Base ` W )   &    |- .+ = ( +g ` W )   &    |- .0. = ( 0g ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LVec )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   &    |- ( ph -> ( X .+ Y ) =/= .0. )   &    |- ( ph -> ( N ` { X } ) = ( N ` { Y } ) )   =>    |- ( ph -> ( N ` { X } ) = ( N ` { ( X .+ Y ) } ) )
Theorem	lspsneq 19122*	Equal spans of singletons must have proportional vectors. See lspsnss2 19005 for comparable span version. TODO: can proof be shortened? (Contributed by NM, 21-Mar-2015.)
|- V = ( Base ` W )   &    |- S = (Scalar ` W )   &    |- K = ( Base ` S )   &    |- .0. = ( 0g ` S )   &    |- .x. = ( .s ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LVec )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   =>    |- ( ph -> ( ( N ` { X } ) = ( N ` { Y } ) <-> E. k e. ( K \ { .0. } ) X = ( k .x. Y ) ) )
Theorem	lspsneu 19123*	Nonzero vectors with equal singleton spans have a unique proportionality constant. (Contributed by NM, 31-May-2015.)
|- V = ( Base ` W )   &    |- S = (Scalar ` W )   &    |- K = ( Base ` S )   &    |- O = ( 0g ` S )   &    |- .x. = ( .s ` W )   &    |- .0. = ( 0g ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LVec )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. ( V \ { .0. } ) )   =>    |- ( ph -> ( ( N ` { X } ) = ( N ` { Y } ) <-> E! k e. ( K \ { O } ) X = ( k .x. Y ) ) )
Theorem	lspsnel4 19124	A member of the span of the singleton of a vector is a member of a subspace containing the vector. (elspansn4 28432 analog.) (Contributed by NM, 4-Jul-2014.)
|- V = ( Base ` W )   &    |- .0. = ( 0g ` W )   &    |- S = ( LSubSp ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LVec )   &    |- ( ph -> U e. S )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. ( N ` { X } ) )   &    |- ( ph -> Y =/= .0. )   =>    |- ( ph -> ( X e. U <-> Y e. U ) )
Theorem	lspdisj 19125	The span of a vector not in a subspace is disjoint with the subspace. (Contributed by NM, 6-Apr-2015.)
|- V = ( Base ` W )   &    |- .0. = ( 0g ` W )   &    |- N = ( LSpan ` W )   &    |- S = ( LSubSp ` W )   &    |- ( ph -> W e. LVec )   &    |- ( ph -> U e. S )   &    |- ( ph -> X e. V )   &    |- ( ph -> -. X e. U )   =>    |- ( ph -> ( ( N ` { X } ) i^i U ) = { .0. } )
Theorem	lspdisjb 19126	A nonzero vector is not in a subspace iff its span is disjoint with the subspace. (Contributed by NM, 23-Apr-2015.)
|- V = ( Base ` W )   &    |- .0. = ( 0g ` W )   &    |- N = ( LSpan ` W )   &    |- S = ( LSubSp ` W )   &    |- ( ph -> W e. LVec )   &    |- ( ph -> U e. S )   &    |- ( ph -> X e. ( V \ { .0. } ) )   =>    |- ( ph -> ( -. X e. U <-> ( ( N ` { X } ) i^i U ) = { .0. } ) )
Theorem	lspdisj2 19127	Unequal spans are disjoint (share only the zero vector). (Contributed by NM, 22-Mar-2015.)
|- V = ( Base ` W )   &    |- .0. = ( 0g ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LVec )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   &    |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )   =>    |- ( ph -> ( ( N ` { X } ) i^i ( N ` { Y } ) ) = { .0. } )
Theorem	lspfixed 19128*	Show membership in the span of the sum of two vectors, one of which ( Y) is fixed in advance. (Contributed by NM, 27-May-2015.)
|- V = ( Base ` W )   &    |- .+ = ( +g ` W )   &    |- .0. = ( 0g ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LVec )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   &    |- ( ph -> Z e. V )   &    |- ( ph -> -. X e. ( N ` { Y } ) )   &    |- ( ph -> -. X e. ( N ` { Z } ) )   &    |- ( ph -> X e. ( N ` { Y , Z } ) )   =>    |- ( ph -> E. z e. ( ( N ` { Z } ) \ { .0. } ) X e. ( N ` { ( Y .+ z ) } ) )
Theorem	lspexch 19129	Exchange property for span of a pair. TODO: see if a version with Y,Z and X,Z reversed will shorten proofs (analogous to lspexchn1 19130 vs. lspexchn2 19131); look for lspexch 19129 and prcom 4267 in same proof. TODO: would a hypothesis of -. X e. ( N ` { Z } ) instead of ( N ` { X } ) =/= ( N { Z } ) ` be better overall? This would be shorter and also satisfy the X =/= .0. condition. Here and also lspindp* and all proofs affected by them (all in NM's mathbox); there are 58 hypotheses with the =/= pattern as of 24-May-2015. (Contributed by NM, 11-Apr-2015.)
|- V = ( Base ` W )   &    |- .0. = ( 0g ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LVec )   &    |- ( ph -> X e. ( V \ { .0. } ) )   &    |- ( ph -> Y e. V )   &    |- ( ph -> Z e. V )   &    |- ( ph -> ( N ` { X } ) =/= ( N ` { Z } ) )   &    |- ( ph -> X e. ( N ` { Y , Z } ) )   =>    |- ( ph -> Y e. ( N ` { X , Z } ) )
Theorem	lspexchn1 19130	Exchange property for span of a pair with negated membership. TODO: look at uses of lspexch 19129 to see if this will shorten proofs. (Contributed by NM, 20-May-2015.)
|- V = ( Base ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LVec )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   &    |- ( ph -> Z e. V )   &    |- ( ph -> -. Y e. ( N ` { Z } ) )   &    |- ( ph -> -. X e. ( N ` { Y , Z } ) )   =>    |- ( ph -> -. Y e. ( N ` { X , Z } ) )
Theorem	lspexchn2 19131	Exchange property for span of a pair with negated membership. TODO: look at uses of lspexch 19129 to see if this will shorten proofs. (Contributed by NM, 24-May-2015.)
|- V = ( Base ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LVec )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   &    |- ( ph -> Z e. V )   &    |- ( ph -> -. Y e. ( N ` { Z } ) )   &    |- ( ph -> -. X e. ( N ` { Z , Y } ) )   =>    |- ( ph -> -. Y e. ( N ` { Z , X } ) )
Theorem	lspindpi 19132	Partial independence property. (Contributed by NM, 23-Apr-2015.)
|- V = ( Base ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LVec )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   &    |- ( ph -> Z e. V )   &    |- ( ph -> -. X e. ( N ` { Y , Z } ) )   =>    |- ( ph -> ( ( N ` { X } ) =/= ( N ` { Y } ) /\ ( N ` { X } ) =/= ( N ` { Z } ) ) )
Theorem	lspindp1 19133	Alternate way to say 3 vectors are mutually independent (swap 1st and 2nd). (Contributed by NM, 11-Apr-2015.)
|- V = ( Base ` W )   &    |- .0. = ( 0g ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LVec )   &    |- ( ph -> X e. ( V \ { .0. } ) )   &    |- ( ph -> Y e. V )   &    |- ( ph -> Z e. V )   &    |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )   &    |- ( ph -> -. Z e. ( N ` { X , Y } ) )   =>    |- ( ph -> ( ( N ` { Z } ) =/= ( N ` { Y } ) /\ -. X e. ( N ` { Z , Y } ) ) )
Theorem	lspindp2l 19134	Alternate way to say 3 vectors are mutually independent (rotate left). (Contributed by NM, 10-May-2015.)
|- V = ( Base ` W )   &    |- .0. = ( 0g ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LVec )   &    |- ( ph -> X e. ( V \ { .0. } ) )   &    |- ( ph -> Y e. V )   &    |- ( ph -> Z e. V )   &    |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )   &    |- ( ph -> -. Z e. ( N ` { X , Y } ) )   =>    |- ( ph -> ( ( N ` { Y } ) =/= ( N ` { Z } ) /\ -. X e. ( N ` { Y , Z } ) ) )
Theorem	lspindp2 19135	Alternate way to say 3 vectors are mutually independent (rotate right). (Contributed by NM, 12-Apr-2015.)
|- V = ( Base ` W )   &    |- .0. = ( 0g ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LVec )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. ( V \ { .0. } ) )   &    |- ( ph -> Z e. V )   &    |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )   &    |- ( ph -> -. Z e. ( N ` { X , Y } ) )   =>    |- ( ph -> ( ( N ` { Z } ) =/= ( N ` { X } ) /\ -. Y e. ( N ` { Z , X } ) ) )
Theorem	lspindp3 19136	Independence of 2 vectors is preserved by vector sum. (Contributed by NM, 26-Apr-2015.)
|- V = ( Base ` W )   &    |- .+ = ( +g ` W )   &    |- .0. = ( 0g ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LVec )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. ( V \ { .0. } ) )   &    |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )   =>    |- ( ph -> ( N ` { X } ) =/= ( N ` { ( X .+ Y ) } ) )
Theorem	lspindp4 19137	(Partial) independence of 3 vectors is preserved by vector sum. (Contributed by NM, 26-Apr-2015.)
|- V = ( Base ` W )   &    |- .+ = ( +g ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   &    |- ( ph -> Z e. V )   &    |- ( ph -> -. Z e. ( N ` { X , Y } ) )   =>    |- ( ph -> -. Z e. ( N ` { X , ( X .+ Y ) } ) )
Theorem	lvecindp 19138	Compute the X coefficient in a sum with an independent vector X (first conjunct), which can then be removed to continue with the remaining vectors summed in expressions Y and Z (second conjunct). Typically, U is the span of the remaining vectors. (Contributed by NM, 5-Apr-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
|- V = ( Base ` W )   &    |- .+ = ( +g ` W )   &    |- F = (Scalar ` W )   &    |- K = ( Base ` F )   &    |- .x. = ( .s ` W )   &    |- S = ( LSubSp ` W )   &    |- ( ph -> W e. LVec )   &    |- ( ph -> U e. S )   &    |- ( ph -> X e. V )   &    |- ( ph -> -. X e. U )   &    |- ( ph -> Y e. U )   &    |- ( ph -> Z e. U )   &    |- ( ph -> A e. K )   &    |- ( ph -> B e. K )   &    |- ( ph -> ( ( A .x. X ) .+ Y ) = ( ( B .x. X ) .+ Z ) )   =>    |- ( ph -> ( A = B /\ Y = Z ) )
Theorem	lvecindp2 19139	Sums of independent vectors must have equal coefficients. (Contributed by NM, 22-Mar-2015.)
|- V = ( Base ` W )   &    |- .+ = ( +g ` W )   &    |- F = (Scalar ` W )   &    |- K = ( Base ` F )   &    |- .x. = ( .s ` W )   &    |- .0. = ( 0g ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LVec )   &    |- ( ph -> X e. ( V \ { .0. } ) )   &    |- ( ph -> Y e. ( V \ { .0. } ) )   &    |- ( ph -> A e. K )   &    |- ( ph -> B e. K )   &    |- ( ph -> C e. K )   &    |- ( ph -> D e. K )   &    |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )   &    |- ( ph -> ( ( A .x. X ) .+ ( B .x. Y ) ) = ( ( C .x. X ) .+ ( D .x. Y ) ) )   =>    |- ( ph -> ( A = C /\ B = D ) )
Theorem	lspsnsubn0 19140	Unequal singleton spans imply nonzero vector subtraction. (Contributed by NM, 19-Mar-2015.)
|- V = ( Base ` W )   &    |- .0. = ( 0g ` W )   &    |- .- = ( -g ` W )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   &    |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )   =>    |- ( ph -> ( X .- Y ) =/= .0. )
Theorem	lsmcv 19141	Subspace sum has the covering property (using spans of singletons to represent atoms). Similar to Exercise 5 of [Kalmbach] p. 153. (spansncvi 28511 analog.) TODO: ugly proof; can it be shortened? (Contributed by NM, 2-Oct-2014.)
|- V = ( Base ` W )   &    |- S = ( LSubSp ` W )   &    |- N = ( LSpan ` W )   &    |- .(+) = ( LSSum ` W )   &    |- ( ph -> W e. LVec )   &    |- ( ph -> T e. S )   &    |- ( ph -> U e. S )   &    |- ( ph -> X e. V )   =>    |- ( ( ph /\ T C. U /\ U C_ ( T .(+) ( N ` { X } ) ) ) -> U = ( T .(+) ( N ` { X } ) ) )
Theorem	lspsolvlem 19142*	Lemma for lspsolv 19143. (Contributed by Mario Carneiro, 25-Jun-2014.)
|- V = ( Base ` W )   &    |- S = ( LSubSp ` W )   &    |- N = ( LSpan ` W )   &    |- F = (Scalar ` W )   &    |- B = ( Base ` F )   &    |- .+ = ( +g ` W )   &    |- .x. = ( .s ` W )   &    |- Q = { z e. V | E. r e. B ( z .+ ( r .x. Y ) ) e. ( N ` A ) }   &    |- ( ph -> W e. LMod )   &    |- ( ph -> A C_ V )   &    |- ( ph -> Y e. V )   &    |- ( ph -> X e. ( N ` ( A u. { Y } ) ) )   =>    |- ( ph -> E. r e. B ( X .+ ( r .x. Y ) ) e. ( N ` A ) )
Theorem	lspsolv 19143	If X is in the span of A u. { Y } but not A, then Y is in the span of A u. { X }. (Contributed by Mario Carneiro, 25-Jun-2014.)
|- V = ( Base ` W )   &    |- S = ( LSubSp ` W )   &    |- N = ( LSpan ` W )   =>    |- ( ( W e. LVec /\ ( A C_ V /\ Y e. V /\ X e. ( ( N ` ( A u. { Y } ) ) \ ( N ` A ) ) ) ) -> Y e. ( N ` ( A u. { X } ) ) )
Theorem	lssacsex 19144*	In a vector space, subspaces form an algebraic closure system whose closure operator has the exchange property. Strengthening of lssacs 18967 by lspsolv 19143. (Contributed by David Moews, 1-May-2017.)
|- A = ( LSubSp ` W )   &    |- N = (mrCls ` A )   &    |- X = ( Base ` W )   =>    |- ( W e. LVec -> ( A e. (ACS ` X ) /\ A. s e. ~P X A. y e. X A. z e. ( ( N ` ( s u. { y } ) ) \ ( N ` s ) ) y e. ( N ` ( s u. { z } ) ) ) )
Theorem	lspsnat 19145	There is no subspace strictly between the zero subspace and the span of a vector (i.e. a 1-dimensional subspace is an atom). (h1datomi 28440 analog.) (Contributed by NM, 20-Apr-2014.) (Proof shortened by Mario Carneiro, 22-Jun-2014.)
|- V = ( Base ` W )   &    |- .0. = ( 0g ` W )   &    |- S = ( LSubSp ` W )   &    |- N = ( LSpan ` W )   =>    |- ( ( ( W e. LVec /\ U e. S /\ X e. V ) /\ U C_ ( N ` { X } ) ) -> ( U = ( N ` { X } ) \/ U = { .0. } ) )
Theorem	lspsncv0 19146*	The span of a singleton covers the zero subspace, using Definition 3.2.18 of [PtakPulmannova] p. 68 for "covers".) (Contributed by NM, 12-Aug-2014.)
|- V = ( Base ` W )   &    |- .0. = ( 0g ` W )   &    |- S = ( LSubSp ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LVec )   &    |- ( ph -> X e. V )   &    |- ( ph -> X =/= .0. )   =>    |- ( ph -> -. E. y e. S ( { .0. } C. y /\ y C. ( N ` { X } ) ) )
Theorem	lsppratlem1 19147	Lemma for lspprat 19153. Let x e. ( U \ { 0 } ) (if there is no such x then U is the zero subspace), and let y e. ( U \ ( N ` { x } ) ) (assuming the conclusion is false). The goal is to write X, Y in terms of x, y, which would normally be done by solving the system of linear equations. The span equivalent of this process is lspsolv 19143 (hence the name), which we use extensively below. In this lemma, we show that since x e. ( N ` { X , Y } ), either x e. ( N ` { Y } ) or X e. ( N ` { x , Y } ). (Contributed by NM, 29-Aug-2014.)
|- V = ( Base ` W )   &    |- S = ( LSubSp ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LVec )   &    |- ( ph -> U e. S )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   &    |- ( ph -> U C. ( N ` { X , Y } ) )   &    |- .0. = ( 0g ` W )   &    |- ( ph -> x e. ( U \ { .0. } ) )   &    |- ( ph -> y e. ( U \ ( N ` { x } ) ) )   =>    |- ( ph -> ( x e. ( N ` { Y } ) \/ X e. ( N ` { x , Y } ) ) )
Theorem	lsppratlem2 19148	Lemma for lspprat 19153. Show that if X and Y are both in ( N ` { x , y } ) (which will be our goal for each of the two cases above), then ( N ` { X , Y } ) C_ U, contradicting the hypothesis for U. (Contributed by NM, 29-Aug-2014.) (Revised by Mario Carneiro, 5-Sep-2014.)
|- V = ( Base ` W )   &    |- S = ( LSubSp ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LVec )   &    |- ( ph -> U e. S )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   &    |- ( ph -> U C. ( N ` { X , Y } ) )   &    |- .0. = ( 0g ` W )   &    |- ( ph -> x e. ( U \ { .0. } ) )   &    |- ( ph -> y e. ( U \ ( N ` { x } ) ) )   &    |- ( ph -> X e. ( N ` { x , y } ) )   &    |- ( ph -> Y e. ( N ` { x , y } ) )   =>    |- ( ph -> ( N ` { X , Y } ) C_ U )
Theorem	lsppratlem3 19149	Lemma for lspprat 19153. In the first case of lsppratlem1 19147, since x e/ ( N ` (/) ), also Y e. ( N ` { x } ), and since y e. ( N ` { X , Y } ) C_ ( N ` { X , x } ) and y e/ ( N ` { x } ), we have X e. ( N ` { x , y } ) as desired. (Contributed by NM, 29-Aug-2014.)
|- V = ( Base ` W )   &    |- S = ( LSubSp ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LVec )   &    |- ( ph -> U e. S )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   &    |- ( ph -> U C. ( N ` { X , Y } ) )   &    |- .0. = ( 0g ` W )   &    |- ( ph -> x e. ( U \ { .0. } ) )   &    |- ( ph -> y e. ( U \ ( N ` { x } ) ) )   &    |- ( ph -> x e. ( N ` { Y } ) )   =>    |- ( ph -> ( X e. ( N ` { x , y } ) /\ Y e. ( N ` { x , y } ) ) )
Theorem	lsppratlem4 19150	Lemma for lspprat 19153. In the second case of lsppratlem1 19147, y e. ( N ` { X , Y } ) C_ ( N ` { x , Y } ) and y e/ ( N ` { x } ) implies Y e. ( N ` { x , y } ) and thus X e. ( N ` { x , Y } ) C_ ( N ` { x , y } ) as well. (Contributed by NM, 29-Aug-2014.)
|- V = ( Base ` W )   &    |- S = ( LSubSp ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LVec )   &    |- ( ph -> U e. S )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   &    |- ( ph -> U C. ( N ` { X , Y } ) )   &    |- .0. = ( 0g ` W )   &    |- ( ph -> x e. ( U \ { .0. } ) )   &    |- ( ph -> y e. ( U \ ( N ` { x } ) ) )   &    |- ( ph -> X e. ( N ` { x , Y } ) )   =>    |- ( ph -> ( X e. ( N ` { x , y } ) /\ Y e. ( N ` { x , y } ) ) )
Theorem	lsppratlem5 19151	Lemma for lspprat 19153. Combine the two cases and show a contradiction to U C. ( N ` { X , Y } ) under the assumptions on x and y. (Contributed by NM, 29-Aug-2014.)
|- V = ( Base ` W )   &    |- S = ( LSubSp ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LVec )   &    |- ( ph -> U e. S )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   &    |- ( ph -> U C. ( N ` { X , Y } ) )   &    |- .0. = ( 0g ` W )   &    |- ( ph -> x e. ( U \ { .0. } ) )   &    |- ( ph -> y e. ( U \ ( N ` { x } ) ) )   =>    |- ( ph -> ( N ` { X , Y } ) C_ U )
Theorem	lsppratlem6 19152	Lemma for lspprat 19153. Negating the assumption on y, we arrive close to the desired conclusion. (Contributed by NM, 29-Aug-2014.)
|- V = ( Base ` W )   &    |- S = ( LSubSp ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LVec )   &    |- ( ph -> U e. S )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   &    |- ( ph -> U C. ( N ` { X , Y } ) )   &    |- .0. = ( 0g ` W )   =>    |- ( ph -> ( x e. ( U \ { .0. } ) -> U = ( N ` { x } ) ) )
Theorem	lspprat 19153*	A proper subspace of the span of a pair of vectors is the span of a singleton (an atom) or the zero subspace (if z is zero). Proof suggested by Mario Carneiro, 28-Aug-2014. (Contributed by NM, 29-Aug-2014.)
|- V = ( Base ` W )   &    |- S = ( LSubSp ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LVec )   &    |- ( ph -> U e. S )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   &    |- ( ph -> U C. ( N ` { X , Y } ) )   =>    |- ( ph -> E. z e. V U = ( N ` { z } ) )
Theorem	islbs2 19154*	An equivalent formulation of the basis predicate in a vector space: a subset is a basis iff no element is in the span of the rest of the set. (Contributed by Mario Carneiro, 14-Jan-2015.)
|- V = ( Base ` W )   &    |- J = (LBasis ` W )   &    |- N = ( LSpan ` W )   =>    |- ( W e. LVec -> ( B e. J <-> ( B C_ V /\ ( N ` B ) = V /\ A. x e. B -. x e. ( N ` ( B \ { x } ) ) ) ) )
Theorem	islbs3 19155*	An equivalent formulation of the basis predicate: a subset is a basis iff it is a minimal spanning set. (Contributed by Mario Carneiro, 25-Jun-2014.)
|- V = ( Base ` W )   &    |- J = (LBasis ` W )   &    |- N = ( LSpan ` W )   =>    |- ( W e. LVec -> ( B e. J <-> ( B C_ V /\ ( N ` B ) = V /\ A. s ( s C. B -> ( N ` s ) C. V ) ) ) )
Theorem	lbsacsbs 19156	Being a basis in a vector space is equivalent to being a basis in the associated algebraic closure system. Equivalent to islbs2 19154. (Contributed by David Moews, 1-May-2017.)
|- A = ( LSubSp ` W )   &    |- N = (mrCls ` A )   &    |- X = ( Base ` W )   &    |- I = (mrInd ` A )   &    |- J = (LBasis ` W )   =>    |- ( W e. LVec -> ( S e. J <-> ( S e. I /\ ( N ` S ) = X ) ) )
Theorem	lvecdim 19157	The dimension theorem for vector spaces: any two bases of the same vector space are equinumerous. Proven by using lssacsex 19144 and lbsacsbs 19156 to show that being a basis for a vector space is equivalent to being a basis for the associated algebraic closure system, and then using acsexdimd 17183. (Contributed by David Moews, 1-May-2017.)
|- J = (LBasis ` W )   =>    |- ( ( W e. LVec /\ S e. J /\ T e. J ) -> S ~~ T )
Theorem	lbsextlem1 19158*	Lemma for lbsext 19163. The set S is the set of all linearly independent sets containing C; we show here that it is nonempty. (Contributed by Mario Carneiro, 25-Jun-2014.)
|- V = ( Base ` W )   &    |- J = (LBasis ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LVec )   &    |- ( ph -> C C_ V )   &    |- ( ph -> A. x e. C -. x e. ( N ` ( C \ { x } ) ) )   &    |- S = { z e. ~P V | ( C C_ z /\ A. x e. z -. x e. ( N ` ( z \ { x } ) ) ) }   =>    |- ( ph -> S =/= (/) )
Theorem	lbsextlem2 19159*	Lemma for lbsext 19163. Since A is a chain (actually, we only need it to be closed under binary union), the union T of the spans of each individual element of A is a subspace, and it contains all of U. A (except for our target vector x- we are trying to make x a linear combination of all the other vectors in some set from A). (Contributed by Mario Carneiro, 25-Jun-2014.)
|- V = ( Base ` W )   &    |- J = (LBasis ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LVec )   &    |- ( ph -> C C_ V )   &    |- ( ph -> A. x e. C -. x e. ( N ` ( C \ { x } ) ) )   &    |- S = { z e. ~P V | ( C C_ z /\ A. x e. z -. x e. ( N ` ( z \ { x } ) ) ) }   &    |- P = ( LSubSp ` W )   &    |- ( ph -> A C_ S )   &    |- ( ph -> A =/= (/) )   &    |- ( ph -> [ C.] Or A )   &    |- T = U_ u e. A ( N ` ( u \ { x } ) )   =>    |- ( ph -> ( T e. P /\ ( U. A \ { x } ) C_ T ) )
Theorem	lbsextlem3 19160*	Lemma for lbsext 19163. A chain in S has an upper bound in S. (Contributed by Mario Carneiro, 25-Jun-2014.)
|- V = ( Base ` W )   &    |- J = (LBasis ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LVec )   &    |- ( ph -> C C_ V )   &    |- ( ph -> A. x e. C -. x e. ( N ` ( C \ { x } ) ) )   &    |- S = { z e. ~P V | ( C C_ z /\ A. x e. z -. x e. ( N ` ( z \ { x } ) ) ) }   &    |- P = ( LSubSp ` W )   &    |- ( ph -> A C_ S )   &    |- ( ph -> A =/= (/) )   &    |- ( ph -> [ C.] Or A )   &    |- T = U_ u e. A ( N ` ( u \ { x } ) )   =>    |- ( ph -> U. A e. S )
Theorem	lbsextlem4 19161*	Lemma for lbsext 19163. lbsextlem3 19160 satisfies the conditions for the application of Zorn's lemma zorn 9329 (thus invoking AC), and so there is a maximal linearly independent set extending C. Here we prove that such a set is a basis. (Contributed by Mario Carneiro, 25-Jun-2014.)
|- V = ( Base ` W )   &    |- J = (LBasis ` W )   &    |- N = ( LSpan ` W )   &    |- ( ph -> W e. LVec )   &    |- ( ph -> C C_ V )   &    |- ( ph -> A. x e. C -. x e. ( N ` ( C \ { x } ) ) )   &    |- S = { z e. ~P V | ( C C_ z /\ A. x e. z -. x e. ( N ` ( z \ { x } ) ) ) }   &    |- ( ph -> ~P V e. dom card )   =>    |- ( ph -> E. s e. J C C_ s )
Theorem	lbsextg 19162*	For any linearly independent subset C of V, there is a basis containing the vectors in C. (Contributed by Mario Carneiro, 17-May-2015.)
|- J = (LBasis ` W )   &    |- V = ( Base ` W )   &    |- N = ( LSpan ` W )   =>    |- ( ( ( W e. LVec /\ ~P V e. dom card ) /\ C C_ V /\ A. x e. C -. x e. ( N ` ( C \ { x } ) ) ) -> E. s e. J C C_ s )
Theorem	lbsext 19163*	For any linearly independent subset C of V, there is a basis containing the vectors in C. (Contributed by Mario Carneiro, 25-Jun-2014.) (Revised by Mario Carneiro, 17-May-2015.)
|- J = (LBasis ` W )   &    |- V = ( Base ` W )   &    |- N = ( LSpan ` W )   =>    |- ( ( W e. LVec /\ C C_ V /\ A. x e. C -. x e. ( N ` ( C \ { x } ) ) ) -> E. s e. J C C_ s )
Theorem	lbsexg 19164	Every vector space has a basis. This theorem is an AC equivalent; this is the forward implication. (Contributed by Mario Carneiro, 17-May-2015.)
|- J = (LBasis ` W )   =>    |- ( (CHOICE /\ W e. LVec ) -> J =/= (/) )
Theorem	lbsex 19165	Every vector space has a basis. This theorem is an AC equivalent. (Contributed by Mario Carneiro, 25-Jun-2014.)
|- J = (LBasis ` W )   =>    |- ( W e. LVec -> J =/= (/) )
Theorem	lvecprop2d 19166*	If two structures have the same components (properties), one is a left vector space iff the other one is. This version of lvecpropd 19167 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. LVec <-> L e. LVec ) )
Theorem	lvecpropd 19167*	If two structures have the same components (properties), one is a left vector space iff the other one is. (Contributed 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. LVec <-> L e. LVec ) )
10.8  Ideals
10.8.1  The subring algebra; ideals
Syntax	csra 19168	Extend class notation with the subring algebra generator.
class subringAlg
Syntax	crglmod 19169	Extend class notation with the left module induced by a ring over itself.
class ringLMod
Syntax	clidl 19170	Ring left-ideal function.
class LIdeal
Syntax	crsp 19171	Ring span function.
class RSpan
Definition	df-sra 19172*	Given any subring of a ring, we can construct a left-algebra by regarding the elements of the subring as scalars and the ring itself as a set of vectors. (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 19173	Every ring can be viewed as a left module over itself. (Contributed by Stefan O'Rear, 6-Dec-2014.)
|- ringLMod = ( w e. _V |-> ( (subringAlg ` w ) ` ( Base ` w ) ) )
Definition	df-lidl 19174	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 19175	Define the linear span function in a ring (Ideal generator). (Contributed by Stefan O'Rear, 4-Apr-2015.)
|- RSpan = ( LSpan o. ringLMod )
Theorem	sraval 19176	Lemma for srabase 19178 through sravsca 19182. (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	sralem 19177	Lemma for srabase 19178 and similar theorems. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
|- ( ph -> A = ( (subringAlg ` W ) ` S ) )   &    |- ( ph -> S C_ ( Base ` W ) )   &    |- E = Slot N   &    |- N e. NN   &    |- ( N < 5 \/ 8 < N )   =>    |- ( ph -> ( E ` W ) = ( E ` A ) )
Theorem	srabase 19178	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.)
|- ( ph -> A = ( (subringAlg ` W ) ` S ) )   &    |- ( ph -> S C_ ( Base ` W ) )   =>    |- ( ph -> ( Base ` W ) = ( Base ` A ) )
Theorem	sraaddg 19179	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.)
|- ( ph -> A = ( (subringAlg ` W ) ` S ) )   &    |- ( ph -> S C_ ( Base ` W ) )   =>    |- ( ph -> ( +g ` W ) = ( +g ` A ) )
Theorem	sramulr 19180	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.)
|- ( ph -> A = ( (subringAlg ` W ) ` S ) )   &    |- ( ph -> S C_ ( Base ` W ) )   =>    |- ( ph -> ( .r ` W ) = ( .r ` A ) )
Theorem	srasca 19181	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.)
|- ( ph -> A = ( (subringAlg ` W ) ` S ) )   &    |- ( ph -> S C_ ( Base ` W ) )   =>    |- ( ph -> ( W ↾s S ) = (Scalar ` A ) )
Theorem	sravsca 19182	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.)
|- ( ph -> A = ( (subringAlg ` W ) ` S ) )   &    |- ( ph -> S C_ ( Base ` W ) )   =>    |- ( ph -> ( .r ` W ) = ( .s ` A ) )
Theorem	sraip 19183	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 -> ( .r ` W ) = ( .i ` A ) )
Theorem	sratset 19184	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 -> (TopSet ` W ) = (TopSet ` A ) )
Theorem	sratopn 19185	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 -> ( TopOpen ` W ) = ( TopOpen ` A ) )
Theorem	srads 19186	Distance function 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 -> ( dist ` W ) = ( dist ` A ) )
Theorem	sralmod 19187	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	sralmod0 19188	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 -> .0. = ( 0g ` A ) )
Theorem	issubrngd2 19189*	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 19190	ringLMod is a function. (Contributed by Stefan O'Rear, 6-Dec-2014.)
|- ringLMod Fn _V
Theorem	rlmval 19191	Value of the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
|- (ringLMod ` W ) = ( (subringAlg ` W ) ` ( Base ` W ) )
Theorem	lidlval 19192	Value of the set of ring ideals. (Contributed by Stefan O'Rear, 31-Mar-2015.)
|- (LIdeal ` W ) = ( LSubSp ` (ringLMod ` W ) )
Theorem	rspval 19193	Value of the ring span function. (Contributed by Stefan O'Rear, 4-Apr-2015.)
|- (RSpan ` W ) = ( LSpan ` (ringLMod ` W ) )
Theorem	rlmval2 19194	Value of the ring module extended. (Contributed by AV, 2-Dec-2018.) (Revised by Thierry Arnoux, 16-Jun-2019.)
|- ( W e. X -> (ringLMod ` W ) = ( ( ( W sSet <. (Scalar ` ndx ) , W >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) )
Theorem	rlmbas 19195	Base set of the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
|- ( Base ` R ) = ( Base ` (ringLMod ` R ) )
Theorem	rlmplusg 19196	Vector addition in the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
|- ( +g ` R ) = ( +g ` (ringLMod ` R ) )
Theorem	rlm0 19197	Zero vector in the ring module. (Contributed by Stefan O'Rear, 6-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
|- ( 0g ` R ) = ( 0g ` (ringLMod ` R ) )
Theorem	rlmsub 19198	Subtraction in the ring module. (Contributed by Thierry Arnoux, 30-Jun-2019.)
|- ( -g ` R ) = ( -g ` (ringLMod ` R ) )
Theorem	rlmmulr 19199	Ring multiplication in the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
|- ( .r ` R ) = ( .r ` (ringLMod ` R ) )
Theorem	rlmsca 19200	Scalars in the ring module. (Contributed by Stefan O'Rear, 6-Dec-2014.)
|- ( R e. X -> R = (Scalar ` (ringLMod ` R ) ) )