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	fsuppmptdmf 48301*	A mapping with a finite domain is finitely supported. (Contributed by AV, 4-Sep-2019.)
⊢ Ⅎ𝑥𝜑    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝑌)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑊)    ⇒   ⊢ (𝜑 → 𝐹 finSupp 𝑍)
21.48.19.9  Left modules (extension)
Theorem	lmodvsmdi 48302	Multiple distributive law for scalar product (left-distributivity). (Contributed by AV, 5-Sep-2019.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ ↑ = (.g‘𝑊)    &   ⊢ 𝐸 = (.g‘𝐹)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑉)) → (𝑅 · (𝑁 ↑ 𝑋)) = ((𝑁𝐸𝑅) · 𝑋))
Theorem	gsumlsscl 48303*	Closure of a group sum in a linear subspace: A (finitely supported) sum of scalar multiplications of vectors 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.)
⊢ 𝑆 = (LSubSp‘𝑀)    &   ⊢ 𝑅 = (Scalar‘𝑀)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ ((𝑀 ∈ LMod ∧ 𝑍 ∈ 𝑆 ∧ 𝑉 ⊆ 𝑍) → ((𝐹 ∈ (𝐵 ↑m 𝑉) ∧ 𝐹 finSupp (0g‘𝑅)) → (𝑀 Σg (𝑣 ∈ 𝑉 ↦ ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣))) ∈ 𝑍))
21.48.19.10  Associative algebras (extension)
Theorem	assaascl0 48304	The scalar 0 embedded into an associative algebra corresponds to the 0 of the associative algebra. (Contributed by AV, 31-Jul-2019.)
⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ AssAlg)    ⇒   ⊢ (𝜑 → (𝐴‘(0g‘𝐹)) = (0g‘𝑊))
Theorem	assaascl1 48305	The scalar 1 embedded into an associative algebra corresponds to the 1 of the an associative algebra. (Contributed by AV, 31-Jul-2019.)
⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ AssAlg)    ⇒   ⊢ (𝜑 → (𝐴‘(1r‘𝐹)) = (1r‘𝑊))
21.48.19.11  Univariate polynomials (extension)
Theorem	ply1vr1smo 48306	The variable in a polynomial expressed as scaled monomial. (Contributed by AV, 12-Aug-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ 𝐺 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝐺)    &   ⊢ 𝑋 = (var1‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → ( 1 · (1 ↑ 𝑋)) = 𝑋)
Theorem	ply1sclrmsm 48307	The ring multiplication of a polynomial with a scalar polynomial is equal to the scalar multiplication of the polynomial with the corresponding scalar. (Contributed by AV, 14-Aug-2019.)
⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐸 = (Base‘𝑃)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ × = (.r‘𝑃)    &   ⊢ 𝑁 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝑁)    &   ⊢ 𝐴 = (algSc‘𝑃)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐾 ∧ 𝑍 ∈ 𝐸) → ((𝐴‘𝐹) × 𝑍) = (𝐹 · 𝑍))
Theorem	coe1id 48308*	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 48309	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 48310*	Lemma 1 for ply1mulgsum 48314. (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 48311*	Lemma 2 for ply1mulgsum 48314. (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 48312*	Lemma 3 for ply1mulgsum 48314. (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 48313*	Lemma 4 for ply1mulgsum 48314. (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 48314*	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 48315	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 48316	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 48317	A term of the form 𝑥 − 𝐶 is a (univariate) polynomial, also called "linear polynomial". (Part of ply1remlem 26120). (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 48318	A term of the form 𝑥 − 𝐶 evaluated for 𝑥 = 𝑉 results in 𝑉 − 𝐶 (part of ply1remlem 26120). (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 48319	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 22426 and df-scmat 22427 define diagonal and scalar matrices as sets.
Syntax	cdmatalt 48320	Alternative notation for the algebra of diagonal matrices.
class DMatALT
Syntax	cscmatalt 48321	Alternative notation for the algebra of scalar matrices.
class ScMatALT
Definition	df-dmatalt 48322*	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 48323*	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 48324*	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 48325*	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 48326*	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 48327	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 48330, 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 48331, so that we can show that such sets are submodules of the corresponding modules, see lincolss 48358. Remark:According to Wikipedia ("Linear span", 28-Apr-2019, https://en.wikipedia.org/wiki/Linear_span 48358) "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 20927, the set of all linear combinations as defined by df-lco 48331 follows the alternative approach. That both definitions are equivalent is shown by lspeqlco 48363.
Syntax	clinc 48328	Extend class notation with the operation constructing a linear combination (of vectors from a left module).
class linC
Syntax	clinco 48329	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 48330*	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 48331*	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 48332*	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 48333*	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 48334*	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 48335*	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 48336*	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 48337	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 48338	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 48339	The value of an empty linear combination. (Contributed by AV, 12-Apr-2019.)
⊢ (𝑀 ∈ 𝑋 → (∅( linC ‘𝑀)∅) = (0g‘𝑀))
Theorem	lincvalsng 48340	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 48341	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 48342	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 48343	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 48344	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 48345*	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 48346*	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 48347*	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 48348	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 48349*	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 48350	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 48351	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 48352	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 48353	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 48354*	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 48355	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 48356	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 48357	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 48358	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 48359*	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 48360	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 48361	Lemma for lspeqlco 48363. (Contributed by AV, 17-Apr-2019.)
⊢ 𝐵 = (Base‘𝑀)    ⇒   ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → ((LSpan‘𝑀)‘𝑉) ⊆ (𝑀 LinCo 𝑉))
Theorem	lcosslsp 48362	Lemma for lspeqlco 48363. (Contributed by AV, 20-Apr-2019.)
⊢ 𝐵 = (Base‘𝑀)    ⇒   ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 LinCo 𝑉) ⊆ ((LSpan‘𝑀)‘𝑉))
Theorem	lspeqlco 48363	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 20927) 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 48366 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 48366): "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 21764) and (linearly) independent sets (df-linds 21765). 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 21037) "every element is not in the span of the remainder of the [set]". The equivalence of Definitions df-linds 21765 and df-lininds 48366 for (linear) independence for (left) modules is shown in lindslininds 48388. Remark 2: Subsets of the base set of a (left) module are linearly dependent if they are not linearly independent (see df-lindeps 48368) 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 48407. 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 48409) and not for (left) modules in general.
Syntax	clininds 48364	Extend class notation with the relation between a module and its linearly independent subsets.
class linIndS
Syntax	clindeps 48365	Extend class notation with the relation between a module and its linearly dependent subsets.
class linDepS
Definition	df-lininds 48366*	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 48367	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 48368*	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 48369	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 48370*	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 48371*	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 48372*	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 48373*	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 48374*	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 48375	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 48376	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 48377*	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 48378*	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 48379*	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 48380*	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 48381*	Implication 1 for lindslininds 48388. (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 48382*	Lemma 1 for lindslinindsimp2 48387. (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 48383*	Lemma 2 for lindslinindsimp2 48387. (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 48384*	Lemma 3 for lindslinindsimp2 48387. (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 48385*	Lemma 4 for lindslinindsimp2 48387. (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 48386*	Lemma 5 for lindslinindsimp2 48387. (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 48387*	Implication 2 for lindslininds 48388. (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 48388	Equivalence of definitions df-linds 21765 and df-lininds 48366 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 48389	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 48390	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 48391	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 48392	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 20829), 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 48393	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 48394*	Lemma for snlindsntor 48395. (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 48395*	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 48396	Lemma for ldepspr 48397. (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 48397	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 48398	Lemma 3 for lincresunit3 48405. (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 48399	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 48400	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)