P. 484 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 48301-48400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	coe1id 48301*	Coefficient vector of the unit polynomial. (Contributed by AV, 9-Aug-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐼 = (1r‘𝑃)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (coe1‘𝐼) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, 1 , 0 )))
Theorem	coe1sclmulval 48302	The value of the coefficient vector of a polynomial multiplied on the left by a scalar. (Contributed by AV, 14-Aug-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ 𝑆 = ( ·𝑠 ‘𝑃)    &   ⊢ ∙ = (.r‘𝑃)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ (𝑌 ∈ 𝐾 ∧ 𝑍 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) → ((coe1‘(𝑌𝑆𝑍))‘𝑁) = (𝑌 · ((coe1‘𝑍)‘𝑁)))
Theorem	ply1mulgsumlem1 48303*	Lemma 1 for ply1mulgsum 48307. (Contributed by AV, 19-Oct-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐴 = (coe1‘𝐾)    &   ⊢ 𝐶 = (coe1‘𝐿)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ × = (.r‘𝑃)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ ∗ = (.r‘𝑅)    &   ⊢ 𝑀 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝑀)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅))))
Theorem	ply1mulgsumlem2 48304*	Lemma 2 for ply1mulgsum 48307. (Contributed by AV, 19-Oct-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐴 = (coe1‘𝐾)    &   ⊢ 𝐶 = (coe1‘𝐿)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ × = (.r‘𝑃)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ ∗ = (.r‘𝑅)    &   ⊢ 𝑀 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝑀)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙))))) = (0g‘𝑅)))
Theorem	ply1mulgsumlem3 48305*	Lemma 3 for ply1mulgsum 48307. (Contributed by AV, 20-Oct-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐴 = (coe1‘𝐾)    &   ⊢ 𝐶 = (coe1‘𝐿)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ × = (.r‘𝑃)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ ∗ = (.r‘𝑅)    &   ⊢ 𝑀 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝑀)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ (𝑅 Σg (𝑙 ∈ (0...𝑘) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙)))))) finSupp (0g‘𝑅))
Theorem	ply1mulgsumlem4 48306*	Lemma 4 for ply1mulgsum 48307. (Contributed by AV, 19-Oct-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐴 = (coe1‘𝐾)    &   ⊢ 𝐶 = (coe1‘𝐿)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ × = (.r‘𝑃)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ ∗ = (.r‘𝑅)    &   ⊢ 𝑀 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝑀)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝑅 Σg (𝑙 ∈ (0...𝑘) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙))))) · (𝑘 ↑ 𝑋))) finSupp (0g‘𝑃))
Theorem	ply1mulgsum 48307*	The product of two polynomials expressed as group sum of scaled monomials. (Contributed by AV, 20-Oct-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐴 = (coe1‘𝐾)    &   ⊢ 𝐶 = (coe1‘𝐿)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ × = (.r‘𝑃)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ ∗ = (.r‘𝑅)    &   ⊢ 𝑀 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝑀)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → (𝐾 × 𝐿) = (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑅 Σg (𝑙 ∈ (0...𝑘) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙))))) · (𝑘 ↑ 𝑋)))))
Theorem	evl1at0 48308	Polynomial evaluation for the 0 scalar. (Contributed by AV, 10-Aug-2019.)
⊢ 𝑂 = (eval1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑍 = (0g‘𝑃)    ⇒   ⊢ (𝑅 ∈ CRing → ((𝑂‘𝑍)‘ 0 ) = 0 )
Theorem	evl1at1 48309	Polynomial evaluation for the 1 scalar. (Contributed by AV, 10-Aug-2019.)
⊢ 𝑂 = (eval1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝐼 = (1r‘𝑃)    ⇒   ⊢ (𝑅 ∈ CRing → ((𝑂‘𝐼)‘ 1 ) = 1 )
21.48.19.12  Univariate polynomials (examples)
Theorem	linply1 48310	A term of the form 𝑥 − 𝐶 is a (univariate) polynomial, also called "linear polynomial". (Part of ply1remlem 26204). (Contributed by AV, 3-Jul-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ − = (-g‘𝑃)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ 𝐺 = (𝑋 − (𝐴‘𝐶))    &   ⊢ (𝜑 → 𝐶 ∈ 𝐾)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    ⇒   ⊢ (𝜑 → 𝐺 ∈ 𝐵)
Theorem	lineval 48311	A term of the form 𝑥 − 𝐶 evaluated for 𝑥 = 𝑉 results in 𝑉 − 𝐶 (part of ply1remlem 26204). (Contributed by AV, 3-Jul-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ − = (-g‘𝑃)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ 𝐺 = (𝑋 − (𝐴‘𝐶))    &   ⊢ (𝜑 → 𝐶 ∈ 𝐾)    &   ⊢ 𝑂 = (eval1‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑉 ∈ 𝐾)    ⇒   ⊢ (𝜑 → ((𝑂‘𝐺)‘𝑉) = (𝑉(-g‘𝑅)𝐶))
Theorem	linevalexample 48312	The polynomial 𝑥 − 3 over ℤ evaluated for 𝑥 = 5 results in 2. (Contributed by AV, 3-Jul-2019.)
⊢ 𝑃 = (Poly1‘ℤring)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝑋 = (var1‘ℤring)    &   ⊢ − = (-g‘𝑃)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ 𝐺 = (𝑋 − (𝐴‘3))    &   ⊢ 𝑂 = (eval1‘ℤring)    ⇒   ⊢ ((𝑂‘(𝑋 − (𝐴‘3)))‘5) = 2
21.48.20  Linear algebra (extension)
21.48.20.1  The subalgebras of diagonal and scalar matrices (extension) In the following, alternative definitions for diagonal and scalar matrices are provided. These definitions define diagonal and scalar matrices as extensible structures, whereas Definitions df-dmat 22496 and df-scmat 22497 define diagonal and scalar matrices as sets.
Syntax	cdmatalt 48313	Alternative notation for the algebra of diagonal matrices.
class DMatALT
Syntax	cscmatalt 48314	Alternative notation for the algebra of scalar matrices.
class ScMatALT
Definition	df-dmatalt 48315*	Define the set of n x n diagonal (square) matrices over a set (usually a ring) r, see definition in [Roman] p. 4 or Definition 3.12 in [Hefferon] p. 240. (Contributed by AV, 8-Dec-2019.)
⊢ DMatALT = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ⦋(𝑛 Mat 𝑟) / 𝑎⦌(𝑎 ↾s {𝑚 ∈ (Base‘𝑎) ∣ ∀𝑖 ∈ 𝑛 ∀𝑗 ∈ 𝑛 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = (0g‘𝑟))}))
Definition	df-scmatalt 48316*	Define the algebra of n x n scalar matrices over a set (usually a ring) r, see definition in [Connell] p. 57: "A scalar matrix is a diagonal matrix for which all the diagonal terms are equal, i.e., a matrix of the form cIn";. (Contributed by AV, 8-Dec-2019.)
⊢ ScMatALT = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ⦋(𝑛 Mat 𝑟) / 𝑎⦌(𝑎 ↾s {𝑚 ∈ (Base‘𝑎) ∣ ∃𝑐 ∈ (Base‘𝑟)∀𝑖 ∈ 𝑛 ∀𝑗 ∈ 𝑛 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑟))}))
Theorem	dmatALTval 48317*	The algebra of 𝑁 x 𝑁 diagonal matrices over a ring 𝑅. (Contributed by AV, 8-Dec-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐷 = (𝑁 DMatALT 𝑅)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → 𝐷 = (𝐴 ↾s {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )}))
Theorem	dmatALTbas 48318*	The base set of the algebra of 𝑁 x 𝑁 diagonal matrices over a ring 𝑅, i.e. the set of all 𝑁 x 𝑁 diagonal matrices over the ring 𝑅. (Contributed by AV, 8-Dec-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐷 = (𝑁 DMatALT 𝑅)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (Base‘𝐷) = {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )})
Theorem	dmatALTbasel 48319*	An element of the base set of the algebra of 𝑁 x 𝑁 diagonal matrices over a ring 𝑅, i.e. an 𝑁 x 𝑁 diagonal matrix over the ring 𝑅. (Contributed by AV, 8-Dec-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐷 = (𝑁 DMatALT 𝑅)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (𝑀 ∈ (Base‘𝐷) ↔ (𝑀 ∈ 𝐵 ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 ))))
Theorem	dmatbas 48320	The set of all 𝑁 x 𝑁 diagonal matrices over (the ring) 𝑅 is the base set of the algebra of 𝑁 x 𝑁 diagonal matrices over (the ring) 𝑅. (Contributed by AV, 8-Dec-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐷 = (𝑁 DMat 𝑅)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐷 = (Base‘(𝑁 DMatALT 𝑅)))
21.48.20.2  Linear combinations According to Wikipedia ("Linear combination", 29-Mar-2019, https://en.wikipedia.org/wiki/Linear_combination) "In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g., a linear combination of x and y would be any expression of the form ax + by, where a and b are constants). The concept of linear combinations is central to linear algebra and related fields of mathematics." In linear algebra, these "terms" are "vectors" (elements from vector spaces or left modules), and the constants are elements of the underlying field resp. ring. This corresponds to the definition in [Lang] p. 129: "Let M be a module over a ring A and let S be a subset of M. By a linear combination of elements of S (with coefficients in A) one means a sum ∑x ∈S axx where {ax} is a set of elements of A, ...". In the definition in [Lang] p. 129, it is additionally claimed that "..., almost all of which [elements of A] are equal to 0.". This is not necessarily required in the following definition df-linc 48323, but it is essential if additions and scalar multiplications of linear combinations are considered. Therefore, we define the set of all linear combinations with finite support in df-lco 48324, so that we can show that such sets are submodules of the corresponding modules, see lincolss 48351. Remark:According to Wikipedia ("Linear span", 28-Apr-2019, https://en.wikipedia.org/wiki/Linear_span 48351) "In linear algebra, the linear span (also called the linear hull or just span) of a set of vectors in a vector space [or module] is the intersection of all linear subspaces which each contain every vector in that set.", and "Alternately, the span of [a set] S may be defined as the set of all finite linear combinations of elements (vectors) of S". Whereas spans are defined according to the first approach in df-lsp 20970, the set of all linear combinations as defined by df-lco 48324 follows the alternative approach. That both definitions are equivalent is shown by lspeqlco 48356.
Syntax	clinc 48321	Extend class notation with the operation constructing a linear combination (of vectors from a left module).
class linC
Syntax	clinco 48322	Extend class notation with the operation constructing a set of linear combinations (of vectors from a left module) with finite support.
class LinCo
Definition	df-linc 48323*	Define the operation constructing a linear combination. Although this definition is taylored for linear combinations of vectors from left modules, it can be used for any structure having a Base, Scalar s and a scalar multiplication ·𝑠. (Contributed by AV, 29-Mar-2019.)
⊢ linC = (𝑚 ∈ V ↦ (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ (𝑚 Σg (𝑥 ∈ 𝑣 ↦ ((𝑠‘𝑥)( ·𝑠 ‘𝑚)𝑥)))))
Definition	df-lco 48324*	Define the operation constructing the set of all linear combinations for a set of vectors. (Contributed by AV, 31-Mar-2019.) (Revised by AV, 28-Jul-2019.)
⊢ LinCo = (𝑚 ∈ V, 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ {𝑐 ∈ (Base‘𝑚) ∣ ∃𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣)(𝑠 finSupp (0g‘(Scalar‘𝑚)) ∧ 𝑐 = (𝑠( linC ‘𝑚)𝑣))})
Theorem	lincop 48325*	A linear combination as operation. (Contributed by AV, 30-Mar-2019.)
⊢ (𝑀 ∈ 𝑋 → ( linC ‘𝑀) = (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥 ∈ 𝑣 ↦ ((𝑠‘𝑥)( ·𝑠 ‘𝑀)𝑥)))))
Theorem	lincval 48326*	The value of a linear combination. (Contributed by AV, 30-Mar-2019.)
⊢ ((𝑀 ∈ 𝑋 ∧ 𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑆( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑥 ∈ 𝑉 ↦ ((𝑆‘𝑥)( ·𝑠 ‘𝑀)𝑥))))
Theorem	dflinc2 48327*	Alternative definition of linear combinations using the function operation. (Contributed by AV, 1-Apr-2019.)
⊢ linC = (𝑚 ∈ V ↦ (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ (𝑚 Σg (𝑠 ∘f ( ·𝑠 ‘𝑚)( I ↾ 𝑣)))))
Theorem	lcoop 48328*	A linear combination as operation. (Contributed by AV, 5-Apr-2019.) (Revised by AV, 28-Jul-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑆 = (Scalar‘𝑀)    &   ⊢ 𝑅 = (Base‘𝑆)    ⇒   ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 LinCo 𝑉) = {𝑐 ∈ 𝐵 ∣ ∃𝑠 ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0g‘𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉))})
Theorem	lcoval 48329*	The value of a linear combination. (Contributed by AV, 5-Apr-2019.) (Revised by AV, 28-Jul-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑆 = (Scalar‘𝑀)    &   ⊢ 𝑅 = (Base‘𝑆)    ⇒   ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐶 ∈ (𝑀 LinCo 𝑉) ↔ (𝐶 ∈ 𝐵 ∧ ∃𝑠 ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0g‘𝑆) ∧ 𝐶 = (𝑠( linC ‘𝑀)𝑉)))))
Theorem	lincfsuppcl 48330	A linear combination of vectors (with finite support) is a vector. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 28-Jul-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑅 = (Scalar‘𝑀)    &   ⊢ 𝑆 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑀 ∈ LMod ∧ (𝑉 ∈ 𝑊 ∧ 𝑉 ⊆ 𝐵) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 )) → (𝐹( linC ‘𝑀)𝑉) ∈ 𝐵)
Theorem	linccl 48331	A linear combination of vectors is a vector. (Contributed by AV, 31-Mar-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑅 = (Base‘(Scalar‘𝑀))    ⇒   ⊢ ((𝑀 ∈ LMod ∧ (𝑉 ∈ Fin ∧ 𝑉 ⊆ 𝐵 ∧ 𝑆 ∈ (𝑅 ↑m 𝑉))) → (𝑆( linC ‘𝑀)𝑉) ∈ 𝐵)
Theorem	lincval0 48332	The value of an empty linear combination. (Contributed by AV, 12-Apr-2019.)
⊢ (𝑀 ∈ 𝑋 → (∅( linC ‘𝑀)∅) = (0g‘𝑀))
Theorem	lincvalsng 48333	The linear combination over a singleton. (Contributed by AV, 25-May-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑆 = (Scalar‘𝑀)    &   ⊢ 𝑅 = (Base‘𝑆)    &   ⊢ · = ( ·𝑠 ‘𝑀)    ⇒   ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → ({⟨𝑉, 𝑌⟩} ( linC ‘𝑀){𝑉}) = (𝑌 · 𝑉))
Theorem	lincvalsn 48334	The linear combination over a singleton. (Contributed by AV, 12-Apr-2019.) (Proof shortened by AV, 25-May-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑆 = (Scalar‘𝑀)    &   ⊢ 𝑅 = (Base‘𝑆)    &   ⊢ · = ( ·𝑠 ‘𝑀)    &   ⊢ 𝐹 = {⟨𝑉, 𝑌⟩}    ⇒   ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → (𝐹( linC ‘𝑀){𝑉}) = (𝑌 · 𝑉))
Theorem	lincvalpr 48335	The linear combination over an unordered pair. (Contributed by AV, 16-Apr-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑆 = (Scalar‘𝑀)    &   ⊢ 𝑅 = (Base‘𝑆)    &   ⊢ · = ( ·𝑠 ‘𝑀)    &   ⊢ + = (+g‘𝑀)    &   ⊢ 𝐹 = {⟨𝑉, 𝑋⟩, ⟨𝑊, 𝑌⟩}    ⇒   ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ≠ 𝑊) ∧ (𝑉 ∈ 𝐵 ∧ 𝑋 ∈ 𝑅) ∧ (𝑊 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅)) → (𝐹( linC ‘𝑀){𝑉, 𝑊}) = ((𝑋 · 𝑉) + (𝑌 · 𝑊)))
Theorem	lincval1 48336	The linear combination over a singleton mapping to 0. (Contributed by AV, 12-Apr-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑆 = (Scalar‘𝑀)    &   ⊢ 𝑅 = (Base‘𝑆)    &   ⊢ 𝐹 = {⟨𝑉, (0g‘𝑆)⟩}    ⇒   ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵) → (𝐹( linC ‘𝑀){𝑉}) = (0g‘𝑀))
Theorem	lcosn0 48337	Properties of a linear combination over a singleton mapping to 0. (Contributed by AV, 12-Apr-2019.) (Revised by AV, 28-Jul-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑆 = (Scalar‘𝑀)    &   ⊢ 𝑅 = (Base‘𝑆)    &   ⊢ 𝐹 = {⟨𝑉, (0g‘𝑆)⟩}    ⇒   ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵) → (𝐹 ∈ (𝑅 ↑m {𝑉}) ∧ 𝐹 finSupp (0g‘𝑆) ∧ (𝐹( linC ‘𝑀){𝑉}) = (0g‘𝑀)))
Theorem	lincvalsc0 48338*	The linear combination where all scalars are 0. (Contributed by AV, 12-Apr-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑆 = (Scalar‘𝑀)    &   ⊢ 0 = (0g‘𝑆)    &   ⊢ 𝑍 = (0g‘𝑀)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ 0 )    ⇒   ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹( linC ‘𝑀)𝑉) = 𝑍)
Theorem	lcoc0 48339*	Properties of a linear combination where all scalars are 0. (Contributed by AV, 12-Apr-2019.) (Revised by AV, 28-Jul-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑆 = (Scalar‘𝑀)    &   ⊢ 0 = (0g‘𝑆)    &   ⊢ 𝑍 = (0g‘𝑀)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ 0 )    &   ⊢ 𝑅 = (Base‘𝑆)    ⇒   ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 ∈ (𝑅 ↑m 𝑉) ∧ 𝐹 finSupp 0 ∧ (𝐹( linC ‘𝑀)𝑉) = 𝑍))
Theorem	linc0scn0 48340*	If a set contains the zero element of a module, there is a linear combination being 0 where not all scalars are 0. (Contributed by AV, 13-Apr-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑆 = (Scalar‘𝑀)    &   ⊢ 0 = (0g‘𝑆)    &   ⊢ 1 = (1r‘𝑆)    &   ⊢ 𝑍 = (0g‘𝑀)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑍, 1 , 0 ))    ⇒   ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹( linC ‘𝑀)𝑉) = 𝑍)
Theorem	lincdifsn 48341	A vector is a linear combination of a set containing this vector. (Contributed by AV, 21-Apr-2019.) (Revised by AV, 28-Jul-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑅 = (Scalar‘𝑀)    &   ⊢ 𝑆 = (Base‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑀)    &   ⊢ + = (+g‘𝑀)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) → (𝐹( linC ‘𝑀)𝑉) = ((𝐺( linC ‘𝑀)(𝑉 ∖ {𝑋})) + ((𝐹‘𝑋) · 𝑋)))
Theorem	linc1 48342*	A vector is a linear combination of a set containing this vector. (Contributed by AV, 18-Apr-2019.) (Proof shortened by AV, 28-Jul-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑆 = (Scalar‘𝑀)    &   ⊢ 0 = (0g‘𝑆)    &   ⊢ 1 = (1r‘𝑆)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑋, 1 , 0 ))    ⇒   ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹( linC ‘𝑀)𝑉) = 𝑋)
Theorem	lincellss 48343	A linear combination of a subset of a linear subspace is also contained in the linear subspace. (Contributed by AV, 20-Apr-2019.) (Revised by AV, 28-Jul-2019.)
⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ (LSubSp‘𝑀) ∧ 𝑉 ⊆ 𝑆) → ((𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ∧ 𝐹 finSupp (0g‘(Scalar‘𝑀))) → (𝐹( linC ‘𝑀)𝑉) ∈ 𝑆))
Theorem	lco0 48344	The set of empty linear combinations over a monoid is the singleton with the identity element of the monoid. (Contributed by AV, 12-Apr-2019.)
⊢ (𝑀 ∈ Mnd → (𝑀 LinCo ∅) = {(0g‘𝑀)})
Theorem	lcoel0 48345	The zero vector is always a linear combination. (Contributed by AV, 12-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (0g‘𝑀) ∈ (𝑀 LinCo 𝑉))
Theorem	lincsum 48346	The sum of two linear combinations is a linear combination, see also the proof in [Lang] p. 129. (Contributed by AV, 4-Apr-2019.) (Revised by AV, 28-Jul-2019.)
⊢ + = (+g‘𝑀)    &   ⊢ 𝑋 = (𝐴( linC ‘𝑀)𝑉)    &   ⊢ 𝑌 = (𝐵( linC ‘𝑀)𝑉)    &   ⊢ 𝑆 = (Scalar‘𝑀)    &   ⊢ 𝑅 = (Base‘𝑆)    &   ⊢ ✚ = (+g‘𝑆)    ⇒   ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0g‘𝑆) ∧ 𝐵 finSupp (0g‘𝑆))) → (𝑋 + 𝑌) = ((𝐴 ∘f ✚ 𝐵)( linC ‘𝑀)𝑉))
Theorem	lincscm 48347*	A linear combinations multiplied with a scalar is a linear combination, see also the proof in [Lang] p. 129. (Contributed by AV, 9-Apr-2019.) (Revised by AV, 28-Jul-2019.)
⊢ ∙ = ( ·𝑠 ‘𝑀)    &   ⊢ · = (.r‘(Scalar‘𝑀))    &   ⊢ 𝑋 = (𝐴( linC ‘𝑀)𝑉)    &   ⊢ 𝑅 = (Base‘(Scalar‘𝑀))    &   ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ (𝑆 · (𝐴‘𝑥)))    ⇒   ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → (𝑆 ∙ 𝑋) = (𝐹( linC ‘𝑀)𝑉))
Theorem	lincsumcl 48348	The sum of two linear combinations is a linear combination, see also the proof in [Lang] p. 129. (Contributed by AV, 4-Apr-2019.) (Proof shortened by AV, 28-Jul-2019.)
⊢ + = (+g‘𝑀)    ⇒   ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐶 ∈ (𝑀 LinCo 𝑉) ∧ 𝐷 ∈ (𝑀 LinCo 𝑉))) → (𝐶 + 𝐷) ∈ (𝑀 LinCo 𝑉))
Theorem	lincscmcl 48349	The multiplication of a linear combination with a scalar is a linear combination, see also the proof in [Lang] p. 129. (Contributed by AV, 11-Apr-2019.) (Proof shortened by AV, 28-Jul-2019.)
⊢ · = ( ·𝑠 ‘𝑀)    &   ⊢ 𝑅 = (Base‘(Scalar‘𝑀))    ⇒   ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅 ∧ 𝐷 ∈ (𝑀 LinCo 𝑉)) → (𝐶 · 𝐷) ∈ (𝑀 LinCo 𝑉))
Theorem	lincsumscmcl 48350	The sum of a linear combination and a multiplication of a linear combination with a scalar is a linear combination. (Contributed by AV, 11-Apr-2019.)
⊢ · = ( ·𝑠 ‘𝑀)    &   ⊢ 𝑅 = (Base‘(Scalar‘𝑀))    &   ⊢ + = (+g‘𝑀)    ⇒   ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐶 ∈ 𝑅 ∧ 𝐷 ∈ (𝑀 LinCo 𝑉) ∧ 𝐵 ∈ (𝑀 LinCo 𝑉))) → ((𝐶 · 𝐷) + 𝐵) ∈ (𝑀 LinCo 𝑉))
Theorem	lincolss 48351	According to the statement in [Lang] p. 129, the set (LSubSp‘𝑀) of all linear combinations of a set of vectors V is a submodule (generated by V) of the module M. The elements of V are called generators of (LSubSp‘𝑀). (Contributed by AV, 12-Apr-2019.)
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑀 LinCo 𝑉) ∈ (LSubSp‘𝑀))
Theorem	ellcoellss 48352*	Every linear combination of a subset of a linear subspace is also contained in the linear subspace. (Contributed by AV, 20-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ (LSubSp‘𝑀) ∧ 𝑉 ⊆ 𝑆) → ∀𝑥 ∈ (𝑀 LinCo 𝑉)𝑥 ∈ 𝑆)
Theorem	lcoss 48353	A set of vectors of a module is a subset of the set of all linear combinations of the set. (Contributed by AV, 18-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑉 ⊆ (𝑀 LinCo 𝑉))
Theorem	lspsslco 48354	Lemma for lspeqlco 48356. (Contributed by AV, 17-Apr-2019.)
⊢ 𝐵 = (Base‘𝑀)    ⇒   ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → ((LSpan‘𝑀)‘𝑉) ⊆ (𝑀 LinCo 𝑉))
Theorem	lcosslsp 48355	Lemma for lspeqlco 48356. (Contributed by AV, 20-Apr-2019.)
⊢ 𝐵 = (Base‘𝑀)    ⇒   ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 LinCo 𝑉) ⊆ ((LSpan‘𝑀)‘𝑉))
Theorem	lspeqlco 48356	Equivalence of a span of a set of vectors of a left module defined as the intersection of all linear subspaces which each contain every vector in that set (see df-lsp 20970) and as the set of all linear combinations of the vectors of the set with finite support. (Contributed by AV, 20-Apr-2019.)
⊢ 𝐵 = (Base‘𝑀)    ⇒   ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 LinCo 𝑉) = ((LSpan‘𝑀)‘𝑉))
21.48.20.3  Linear independence According to the definition in [Lang] p. 129: "A subset S of a module M is said to be linearly independent (over [the ring] A) if whenever we have a linear combination ∑x ∈S axx which is equal to 0, then ax=0 for all x∈S." This definition does not care for the finiteness of the set S (because the definition of a linear combination in [Lang] p.129 does already assure that only a finite number of coefficients can be 0 in the sum). Our definition df-lininds 48359 does also neither claim that the subset must be finite, nor that almost all coefficients within the linear combination are 0. If this is required, it must be explicitly stated as precondition in the corresponding theorems. Usually, the linear independence is defined for vector spaces, see Wikipedia ("Linear independence", 15-Apr-2019, https://en.wikipedia.org/wiki/Linear_independence 48359): "In the theory of vector spaces, a set of vectors is said to be linearly dependent if at least one of the vectors in the set can be defined as a linear combination of the others; if no vector in the set can be written in this way, then the vectors are said to be linearly independent." Furthermore, "In order to allow the number of linearly independent vectors in a vector space to be countably infinite, it is useful to define linear dependence as follows. More generally, let V be a vector space over a field K, and let {vi | i∈I} be a family of elements of V. The family is linearly dependent over K if there exists a finite family {aj | j∈J} of elements of K, all nonzero, such that ∑j∈J ajvj=0. A set X of elements of V is linearly independent if the corresponding family{x}x∈X is linearly independent". Remark 1: There are already definitions of (linearly) independent families (df-lindf 21826) and (linearly) independent sets (df-linds 21827). These definitions are based on the principle "of vectors, no nonzero multiple of which can be expressed as a linear combination of other elements" or (see lbsind2 21080) "every element is not in the span of the remainder of the [set]". The equivalence of Definitions df-linds 21827 and df-lininds 48359 for (linear) independence for (left) modules is shown in lindslininds 48381. Remark 2: Subsets of the base set of a (left) module are linearly dependent if they are not linearly independent (see df-lindeps 48361) or, according to Wikipedia, "if at least one of the vectors in the set can be defined as a linear combination of the others", see islindeps2 48400. The reversed implication is not valid for arbitrary modules (but for arbitrary vector spaces), because it requires a division by a coefficient. Therefore, the definition of Wikipedia is equivalent to our definition for (left) vector spaces (see isldepslvec2 48402) and not for (left) modules in general.
Syntax	clininds 48357	Extend class notation with the relation between a module and its linearly independent subsets.
class linIndS
Syntax	clindeps 48358	Extend class notation with the relation between a module and its linearly dependent subsets.
class linDepS
Definition	df-lininds 48359*	Define the relation between a module and its linearly independent subsets. (Contributed by AV, 12-Apr-2019.) (Revised by AV, 24-Apr-2019.) (Revised by AV, 30-Jul-2019.)
⊢ linIndS = {⟨𝑠, 𝑚⟩ ∣ (𝑠 ∈ 𝒫 (Base‘𝑚) ∧ ∀𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑠)((𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)𝑠) = (0g‘𝑚)) → ∀𝑥 ∈ 𝑠 (𝑓‘𝑥) = (0g‘(Scalar‘𝑚))))}
Theorem	rellininds 48360	The class defining the relation between a module and its linearly independent subsets is a relation. (Contributed by AV, 13-Apr-2019.)
⊢ Rel linIndS
Definition	df-lindeps 48361*	Define the relation between a module and its linearly dependent subsets. (Contributed by AV, 26-Apr-2019.)
⊢ linDepS = {⟨𝑠, 𝑚⟩ ∣ ¬ 𝑠 linIndS 𝑚}
Theorem	linindsv 48362	The classes of the module and its linearly independent subsets are sets. (Contributed by AV, 13-Apr-2019.)
⊢ (𝑆 linIndS 𝑀 → (𝑆 ∈ V ∧ 𝑀 ∈ V))
Theorem	islininds 48363*	The property of being a linearly independent subset. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 30-Jul-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑍 = (0g‘𝑀)    &   ⊢ 𝑅 = (Scalar‘𝑀)    &   ⊢ 𝐸 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ 𝑊) → (𝑆 linIndS 𝑀 ↔ (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))))
Theorem	linindsi 48364*	The implications of being a linearly independent subset. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 30-Jul-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑍 = (0g‘𝑀)    &   ⊢ 𝑅 = (Scalar‘𝑀)    &   ⊢ 𝐸 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑆 linIndS 𝑀 → (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )))
Theorem	linindslinci 48365*	The implications of being a linearly independent subset and a linear combination of this subset being 0. (Contributed by AV, 24-Apr-2019.) (Revised by AV, 30-Jul-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑍 = (0g‘𝑀)    &   ⊢ 𝑅 = (Scalar‘𝑀)    &   ⊢ 𝐸 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑆 linIndS 𝑀 ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ 𝐹 finSupp 0 ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍)) → ∀𝑥 ∈ 𝑆 (𝐹‘𝑥) = 0 )
Theorem	islinindfis 48366*	The property of being a linearly independent finite subset. (Contributed by AV, 27-Apr-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑍 = (0g‘𝑀)    &   ⊢ 𝑅 = (Scalar‘𝑀)    &   ⊢ 𝐸 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑆 ∈ Fin ∧ 𝑀 ∈ 𝑊) → (𝑆 linIndS 𝑀 ↔ (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))))
Theorem	islinindfiss 48367*	The property of being a linearly independent finite subset. (Contributed by AV, 27-Apr-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑍 = (0g‘𝑀)    &   ⊢ 𝑅 = (Scalar‘𝑀)    &   ⊢ 𝐸 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑀 ∈ 𝑊 ∧ 𝑆 ∈ Fin ∧ 𝑆 ∈ 𝒫 𝐵) → (𝑆 linIndS 𝑀 ↔ ∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )))
Theorem	linindscl 48368	A linearly independent set is a subset of (the base set of) a module. (Contributed by AV, 13-Apr-2019.)
⊢ (𝑆 linIndS 𝑀 → 𝑆 ∈ 𝒫 (Base‘𝑀))
Theorem	lindepsnlininds 48369	A linearly dependent subset is not a linearly independent subset. (Contributed by AV, 26-Apr-2019.)
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ 𝑊) → (𝑆 linDepS 𝑀 ↔ ¬ 𝑆 linIndS 𝑀))
Theorem	islindeps 48370*	The property of being a linearly dependent subset. (Contributed by AV, 26-Apr-2019.) (Revised by AV, 30-Jul-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑍 = (0g‘𝑀)    &   ⊢ 𝑅 = (Scalar‘𝑀)    &   ⊢ 𝐸 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑀 ∈ 𝑊 ∧ 𝑆 ∈ 𝒫 𝐵) → (𝑆 linDepS 𝑀 ↔ ∃𝑓 ∈ (𝐸 ↑m 𝑆)(𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍 ∧ ∃𝑥 ∈ 𝑆 (𝑓‘𝑥) ≠ 0 )))
Theorem	lincext1 48371*	Property 1 of an extension of a linear combination. (Contributed by AV, 20-Apr-2019.) (Revised by AV, 29-Apr-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑅 = (Scalar‘𝑀)    &   ⊢ 𝐸 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑍 = (0g‘𝑀)    &   ⊢ 𝑁 = (invg‘𝑅)    &   ⊢ 𝐹 = (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑋, (𝑁‘𝑌), (𝐺‘𝑧)))    ⇒   ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (𝑌 ∈ 𝐸 ∧ 𝑋 ∈ 𝑆 ∧ 𝐺 ∈ (𝐸 ↑m (𝑆 ∖ {𝑋})))) → 𝐹 ∈ (𝐸 ↑m 𝑆))
Theorem	lincext2 48372*	Property 2 of an extension of a linear combination. (Contributed by AV, 20-Apr-2019.) (Revised by AV, 30-Jul-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑅 = (Scalar‘𝑀)    &   ⊢ 𝐸 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑍 = (0g‘𝑀)    &   ⊢ 𝑁 = (invg‘𝑅)    &   ⊢ 𝐹 = (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑋, (𝑁‘𝑌), (𝐺‘𝑧)))    ⇒   ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (𝑌 ∈ 𝐸 ∧ 𝑋 ∈ 𝑆 ∧ 𝐺 ∈ (𝐸 ↑m (𝑆 ∖ {𝑋}))) ∧ 𝐺 finSupp 0 ) → 𝐹 finSupp 0 )
Theorem	lincext3 48373*	Property 3 of an extension of a linear combination. (Contributed by AV, 23-Apr-2019.) (Revised by AV, 30-Jul-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑅 = (Scalar‘𝑀)    &   ⊢ 𝐸 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑍 = (0g‘𝑀)    &   ⊢ 𝑁 = (invg‘𝑅)    &   ⊢ 𝐹 = (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑋, (𝑁‘𝑌), (𝐺‘𝑧)))    ⇒   ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (𝑌 ∈ 𝐸 ∧ 𝑋 ∈ 𝑆 ∧ 𝐺 ∈ (𝐸 ↑m (𝑆 ∖ {𝑋}))) ∧ (𝐺 finSupp 0 ∧ (𝑌( ·𝑠 ‘𝑀)𝑋) = (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑋})))) → (𝐹( linC ‘𝑀)𝑆) = 𝑍)
Theorem	lindslinindsimp1 48374*	Implication 1 for lindslininds 48381. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 30-Jul-2019.) (Proof shortened by II, 16-Feb-2023.)
⊢ 𝑅 = (Scalar‘𝑀)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑍 = (0g‘𝑀)    ⇒   ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) → ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )) → (𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑠 ∈ 𝑆 ∀𝑦 ∈ (𝐵 ∖ { 0 }) ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})))))
Theorem	lindslinindimp2lem1 48375*	Lemma 1 for lindslinindsimp2 48380. (Contributed by AV, 25-Apr-2019.)
⊢ 𝑅 = (Scalar‘𝑀)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑍 = (0g‘𝑀)    &   ⊢ 𝑌 = ((invg‘𝑅)‘(𝑓‘𝑥))    &   ⊢ 𝐺 = (𝑓 ↾ (𝑆 ∖ {𝑥}))    ⇒   ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆 ∧ 𝑓 ∈ (𝐵 ↑m 𝑆))) → 𝑌 ∈ 𝐵)
Theorem	lindslinindimp2lem2 48376*	Lemma 2 for lindslinindsimp2 48380. (Contributed by AV, 25-Apr-2019.)
⊢ 𝑅 = (Scalar‘𝑀)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑍 = (0g‘𝑀)    &   ⊢ 𝑌 = ((invg‘𝑅)‘(𝑓‘𝑥))    &   ⊢ 𝐺 = (𝑓 ↾ (𝑆 ∖ {𝑥}))    ⇒   ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆 ∧ 𝑓 ∈ (𝐵 ↑m 𝑆))) → 𝐺 ∈ (𝐵 ↑m (𝑆 ∖ {𝑥})))
Theorem	lindslinindimp2lem3 48377*	Lemma 3 for lindslinindsimp2 48380. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 30-Jul-2019.)
⊢ 𝑅 = (Scalar‘𝑀)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑍 = (0g‘𝑀)    &   ⊢ 𝑌 = ((invg‘𝑅)‘(𝑓‘𝑥))    &   ⊢ 𝐺 = (𝑓 ↾ (𝑆 ∖ {𝑥}))    ⇒   ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 )) → 𝐺 finSupp 0 )
Theorem	lindslinindimp2lem4 48378*	Lemma 4 for lindslinindsimp2 48380. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 30-Jul-2019.) (Proof shortened by II, 16-Feb-2023.)
⊢ 𝑅 = (Scalar‘𝑀)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑍 = (0g‘𝑀)    &   ⊢ 𝑌 = ((invg‘𝑅)‘(𝑓‘𝑥))    &   ⊢ 𝐺 = (𝑓 ↾ (𝑆 ∖ {𝑥}))    ⇒   ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (𝑀 Σg (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓‘𝑦)( ·𝑠 ‘𝑀)𝑦))) = (𝑌( ·𝑠 ‘𝑀)𝑥))
Theorem	lindslinindsimp2lem5 48379*	Lemma 5 for lindslinindsimp2 48380. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 30-Jul-2019.)
⊢ 𝑅 = (Scalar‘𝑀)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑍 = (0g‘𝑀)    ⇒   ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → ((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))) → (𝑓‘𝑥) = 0 )))
Theorem	lindslinindsimp2 48380*	Implication 2 for lindslininds 48381. (Contributed by AV, 26-Apr-2019.) (Revised by AV, 30-Jul-2019.)
⊢ 𝑅 = (Scalar‘𝑀)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑍 = (0g‘𝑀)    ⇒   ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) → ((𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑠 ∈ 𝑆 ∀𝑦 ∈ (𝐵 ∖ { 0 }) ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠}))) → (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))))
Theorem	lindslininds 48381	Equivalence of definitions df-linds 21827 and df-lininds 48359 for (linear) independence for (left) modules. (Contributed by AV, 26-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) → (𝑆 linIndS 𝑀 ↔ 𝑆 ∈ (LIndS‘𝑀)))
Theorem	linds0 48382	The empty set is always a linearly independent subset. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 27-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
⊢ (𝑀 ∈ 𝑉 → ∅ linIndS 𝑀)
Theorem	el0ldep 48383	A set containing the zero element of a module is always linearly dependent, if the underlying ring has at least two elements. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 27-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
⊢ (((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) → 𝑆 linDepS 𝑀)
Theorem	el0ldepsnzr 48384	A set containing the zero element of a module over a nonzero ring is always linearly dependent. (Contributed by AV, 14-Apr-2019.) (Revised by AV, 27-Apr-2019.)
⊢ (((𝑀 ∈ LMod ∧ (Scalar‘𝑀) ∈ NzRing) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆) → 𝑆 linDepS 𝑀)
Theorem	lindsrng01 48385	Any subset of a module is always linearly independent if the underlying ring has at most one element. Since the underlying ring cannot be the empty set (see lmodsn0 20872), this means that the underlying ring has only one element, so it is a zero ring. (Contributed by AV, 14-Apr-2019.) (Revised by AV, 27-Apr-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑅 = (Scalar‘𝑀)    &   ⊢ 𝐸 = (Base‘𝑅)    ⇒   ⊢ ((𝑀 ∈ LMod ∧ ((♯‘𝐸) = 0 ∨ (♯‘𝐸) = 1) ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 linIndS 𝑀)
Theorem	lindszr 48386	Any subset of a module over a zero ring is always linearly independent. (Contributed by AV, 27-Apr-2019.)
⊢ ((𝑀 ∈ LMod ∧ ¬ (Scalar‘𝑀) ∈ NzRing ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)) → 𝑆 linIndS 𝑀)
Theorem	snlindsntorlem 48387*	Lemma for snlindsntor 48388. (Contributed by AV, 15-Apr-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑅 = (Scalar‘𝑀)    &   ⊢ 𝑆 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑍 = (0g‘𝑀)    &   ⊢ · = ( ·𝑠 ‘𝑀)    ⇒   ⊢ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) → (∀𝑓 ∈ (𝑆 ↑m {𝑋})((𝑓( linC ‘𝑀){𝑋}) = 𝑍 → (𝑓‘𝑋) = 0 ) → ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 )))
Theorem	snlindsntor 48388*	A singleton is linearly independent iff it does not contain a torsion element. According to Wikipedia ("Torsion (algebra)", 15-Apr-2019, https://en.wikipedia.org/wiki/Torsion_(algebra)): "An element m of a module M over a ring R is called a torsion element of the module if there exists a regular element r of the ring (an element that is neither a left nor a right zero divisor) that annihilates m, i.e., (𝑟 · 𝑚) = 0. In an integral domain (a commutative ring without zero divisors), every nonzero element is regular, so a torsion element of a module over an integral domain is one annihilated by a nonzero element of the integral domain." Analogously, the definition in [Lang] p. 147 states that "An element x of [a module] E [over a ring R] is called a torsion element if there exists 𝑎 ∈ 𝑅, 𝑎 ≠ 0, such that 𝑎 · 𝑥 = 0. This definition includes the zero element of the module. Some authors, however, exclude the zero element from the definition of torsion elements. (Contributed by AV, 14-Apr-2019.) (Revised by AV, 27-Apr-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑅 = (Scalar‘𝑀)    &   ⊢ 𝑆 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑍 = (0g‘𝑀)    &   ⊢ · = ( ·𝑠 ‘𝑀)    ⇒   ⊢ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) → (∀𝑠 ∈ (𝑆 ∖ { 0 })(𝑠 · 𝑋) ≠ 𝑍 ↔ {𝑋} linIndS 𝑀))
Theorem	ldepsprlem 48389	Lemma for ldepspr 48390. (Contributed by AV, 16-Apr-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑅 = (Scalar‘𝑀)    &   ⊢ 𝑆 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑍 = (0g‘𝑀)    &   ⊢ · = ( ·𝑠 ‘𝑀)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑁 = (invg‘𝑅)    ⇒   ⊢ ((𝑀 ∈ LMod ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝐴 ∈ 𝑆)) → (𝑋 = (𝐴 · 𝑌) → (( 1 · 𝑋)(+g‘𝑀)((𝑁‘𝐴) · 𝑌)) = 𝑍))
Theorem	ldepspr 48390	If a vector is a scalar multiple of another vector, the (unordered pair containing the) two vectors are linearly dependent. (Contributed by AV, 16-Apr-2019.) (Revised by AV, 27-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑅 = (Scalar‘𝑀)    &   ⊢ 𝑆 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑍 = (0g‘𝑀)    &   ⊢ · = ( ·𝑠 ‘𝑀)    ⇒   ⊢ ((𝑀 ∈ LMod ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ≠ 𝑌)) → ((𝐴 ∈ 𝑆 ∧ 𝑋 = (𝐴 · 𝑌)) → {𝑋, 𝑌} linDepS 𝑀))
Theorem	lincresunit3lem3 48391	Lemma 3 for lincresunit3 48398. (Contributed by AV, 18-May-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑅 = (Scalar‘𝑀)    &   ⊢ 𝐸 = (Base‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 𝑁 = (invg‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑀)    ⇒   ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝐴 ∈ 𝑈) → (((𝑁‘𝐴) · 𝑋) = ((𝑁‘𝐴) · 𝑌) ↔ 𝑋 = 𝑌))
Theorem	lincresunitlem1 48392	Lemma 1 for properties of a specially modified restriction of a linear combination containing a unit as scalar. (Contributed by AV, 18-May-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑅 = (Scalar‘𝑀)    &   ⊢ 𝐸 = (Base‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑍 = (0g‘𝑀)    &   ⊢ 𝑁 = (invg‘𝑅)    &   ⊢ 𝐼 = (invr‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑠)))    ⇒   ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈)) → (𝐼‘(𝑁‘(𝐹‘𝑋))) ∈ 𝐸)
Theorem	lincresunitlem2 48393	Lemma for properties of a specially modified restriction of a linear combination containing a unit as scalar. (Contributed by AV, 18-May-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑅 = (Scalar‘𝑀)    &   ⊢ 𝐸 = (Base‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑍 = (0g‘𝑀)    &   ⊢ 𝑁 = (invg‘𝑅)    &   ⊢ 𝐼 = (invr‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑠)))    ⇒   ⊢ ((((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈)) ∧ 𝑌 ∈ 𝑆) → ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑌)) ∈ 𝐸)
Theorem	lincresunit1 48394*	Property 1 of a specially modified restriction of a linear combination containing a unit as scalar. (Contributed by AV, 18-May-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑅 = (Scalar‘𝑀)    &   ⊢ 𝐸 = (Base‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑍 = (0g‘𝑀)    &   ⊢ 𝑁 = (invg‘𝑅)    &   ⊢ 𝐼 = (invr‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑠)))    ⇒   ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈)) → 𝐺 ∈ (𝐸 ↑m (𝑆 ∖ {𝑋})))
Theorem	lincresunit2 48395*	Property 2 of a specially modified restriction of a linear combination containing a unit as scalar. (Contributed by AV, 18-May-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑅 = (Scalar‘𝑀)    &   ⊢ 𝐸 = (Base‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑍 = (0g‘𝑀)    &   ⊢ 𝑁 = (invg‘𝑅)    &   ⊢ 𝐼 = (invr‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑠)))    ⇒   ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → 𝐺 finSupp 0 )
Theorem	lincresunit3lem1 48396*	Lemma 1 for lincresunit3 48398. (Contributed by AV, 17-May-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑅 = (Scalar‘𝑀)    &   ⊢ 𝐸 = (Base‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑍 = (0g‘𝑀)    &   ⊢ 𝑁 = (invg‘𝑅)    &   ⊢ 𝐼 = (invr‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑠)))    ⇒   ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝑧 ∈ (𝑆 ∖ {𝑋}))) → ((𝑁‘(𝐹‘𝑋))( ·𝑠 ‘𝑀)((𝐺‘𝑧)( ·𝑠 ‘𝑀)𝑧)) = ((𝐹‘𝑧)( ·𝑠 ‘𝑀)𝑧))
Theorem	lincresunit3lem2 48397*	Lemma 2 for lincresunit3 48398. (Contributed by AV, 18-May-2019.) (Proof shortened by AV, 30-Jul-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑅 = (Scalar‘𝑀)    &   ⊢ 𝐸 = (Base‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑍 = (0g‘𝑀)    &   ⊢ 𝑁 = (invg‘𝑅)    &   ⊢ 𝐼 = (invr‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑠)))    ⇒   ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → ((𝑁‘(𝐹‘𝑋))( ·𝑠 ‘𝑀)(𝑀 Σg (𝑧 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑧)( ·𝑠 ‘𝑀)𝑧)))) = ((𝐹 ↾ (𝑆 ∖ {𝑋}))( linC ‘𝑀)(𝑆 ∖ {𝑋})))
Theorem	lincresunit3 48398*	Property 3 of a specially modified restriction of a linear combination in a vector space. (Contributed by AV, 18-May-2019.) (Proof shortened by AV, 30-Jul-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑅 = (Scalar‘𝑀)    &   ⊢ 𝐸 = (Base‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑍 = (0g‘𝑀)    &   ⊢ 𝑁 = (invg‘𝑅)    &   ⊢ 𝐼 = (invr‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑠)))    ⇒   ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑋})) = 𝑋)
Theorem	lincreslvec3 48399*	Property 3 of a specially modified restriction of a linear combination in a vector space. (Contributed by AV, 18-May-2019.) (Proof shortened by AV, 30-Jul-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑅 = (Scalar‘𝑀)    &   ⊢ 𝐸 = (Base‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑍 = (0g‘𝑀)    &   ⊢ 𝑁 = (invg‘𝑅)    &   ⊢ 𝐼 = (invr‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑠)))    ⇒   ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LVec ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ (𝐹‘𝑋) ≠ 0 ∧ 𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑋})) = 𝑋)
Theorem	islindeps2 48400*	Conditions for being a linearly dependent subset of a (left) module over a nonzero ring. (Contributed by AV, 29-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑍 = (0g‘𝑀)    &   ⊢ 𝑅 = (Scalar‘𝑀)    &   ⊢ 𝐸 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵 ∧ 𝑅 ∈ NzRing) → (∃𝑠 ∈ 𝑆 ∃𝑓 ∈ (𝐸 ↑m (𝑆 ∖ {𝑠}))(𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)(𝑆 ∖ {𝑠})) = 𝑠) → 𝑆 linDepS 𝑀))