P. 488 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 48701-48800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	lincval1 48701	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 48702	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 48703*	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 48704*	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 48705*	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 48706	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 48707*	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 48708	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 48709	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 48710	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 48711	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 48712*	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 48713	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 48714	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 48715	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 48716	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 48717*	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 48718	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 48719	Lemma for lspeqlco 48721. (Contributed by AV, 17-Apr-2019.)
⊢ 𝐵 = (Base‘𝑀)    ⇒   ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → ((LSpan‘𝑀)‘𝑉) ⊆ (𝑀 LinCo 𝑉))
Theorem	lcosslsp 48720	Lemma for lspeqlco 48721. (Contributed by AV, 20-Apr-2019.)
⊢ 𝐵 = (Base‘𝑀)    ⇒   ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 LinCo 𝑉) ⊆ ((LSpan‘𝑀)‘𝑉))
Theorem	lspeqlco 48721	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 48724 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 48724): "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 21765) and (linearly) independent sets (df-linds 21766). 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 21766 and df-lininds 48724 for (linear) independence for (left) modules is shown in lindslininds 48746. Remark 2: Subsets of the base set of a (left) module are linearly dependent if they are not linearly independent (see df-lindeps 48726) 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 48765. 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 48767) and not for (left) modules in general.
Syntax	clininds 48722	Extend class notation with the relation between a module and its linearly independent subsets.
class linIndS
Syntax	clindeps 48723	Extend class notation with the relation between a module and its linearly dependent subsets.
class linDepS
Definition	df-lininds 48724*	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 48725	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 48726*	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 48727	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 48728*	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 48729*	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 48730*	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 48731*	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 48732*	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 48733	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 48734	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 48735*	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 48736*	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 48737*	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 48738*	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 48739*	Implication 1 for lindslininds 48746. (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 48740*	Lemma 1 for lindslinindsimp2 48745. (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 48741*	Lemma 2 for lindslinindsimp2 48745. (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 48742*	Lemma 3 for lindslinindsimp2 48745. (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 48743*	Lemma 4 for lindslinindsimp2 48745. (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 48744*	Lemma 5 for lindslinindsimp2 48745. (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 48745*	Implication 2 for lindslininds 48746. (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 48746	Equivalence of definitions df-linds 21766 and df-lininds 48724 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 48747	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 48748	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 48749	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 48750	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 48751	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 48752*	Lemma for snlindsntor 48753. (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 48753*	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 48754	Lemma for ldepspr 48755. (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 48755	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 48756	Lemma 3 for lincresunit3 48763. (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 48757	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 48758	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 48759*	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 48760*	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 48761*	Lemma 1 for lincresunit3 48763. (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 48762*	Lemma 2 for lincresunit3 48763. (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 48763*	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 48764*	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 48765*	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))
Theorem	islininds2 48766*	Implication of being a linearly independent 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 linIndS 𝑀 → ∀𝑠 ∈ 𝑆 ∀𝑓 ∈ (𝐸 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑓 finSupp 0 ∨ (𝑓( linC ‘𝑀)(𝑆 ∖ {𝑠})) ≠ 𝑠)))
Theorem	isldepslvec2 48767*	Alternative definition of being a linearly dependent subset of a (left) vector space. In this case, the reverse implication of islindeps2 48765 holds, so that both definitions are equivalent (see theorem 1.6 in [Roman] p. 46 and the note in [Roman] p. 112: if a nontrivial linear combination of elements (where not all of the coefficients are 0) in an R-vector space is 0, then and only then each of the elements is a linear combination of the others. (Contributed by AV, 30-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 ∈ LVec ∧ 𝑆 ∈ 𝒫 𝐵) → (∃𝑠 ∈ 𝑆 ∃𝑓 ∈ (𝐸 ↑m (𝑆 ∖ {𝑠}))(𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)(𝑆 ∖ {𝑠})) = 𝑠) ↔ 𝑆 linDepS 𝑀))
Theorem	lindssnlvec 48768	A singleton not containing the zero element of a vector space is always linearly independent. (Contributed by AV, 16-Apr-2019.) (Revised by AV, 28-Apr-2019.)
⊢ ((𝑀 ∈ LVec ∧ 𝑆 ∈ (Base‘𝑀) ∧ 𝑆 ≠ (0g‘𝑀)) → {𝑆} linIndS 𝑀)
21.48.20.4  Simple left modules and the ` ZZ `-module
Theorem	lmod1lem1 48769*	Lemma 1 for lmod1 48774. (Contributed by AV, 28-Apr-2019.)
⊢ 𝑀 = ({⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)⟩})    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) → (𝑟( ·𝑠 ‘𝑀)𝐼) ∈ {𝐼})
Theorem	lmod1lem2 48770*	Lemma 2 for lmod1 48774. (Contributed by AV, 28-Apr-2019.)
⊢ 𝑀 = ({⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)⟩})    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) → (𝑟( ·𝑠 ‘𝑀)(𝐼(+g‘𝑀)𝐼)) = ((𝑟( ·𝑠 ‘𝑀)𝐼)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼)))
Theorem	lmod1lem3 48771*	Lemma 3 for lmod1 48774. (Contributed by AV, 29-Apr-2019.)
⊢ 𝑀 = ({⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)⟩})    ⇒   ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝐼) = ((𝑞( ·𝑠 ‘𝑀)𝐼)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼)))
Theorem	lmod1lem4 48772*	Lemma 4 for lmod1 48774. (Contributed by AV, 29-Apr-2019.)
⊢ 𝑀 = ({⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)⟩})    ⇒   ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → ((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝐼) = (𝑞( ·𝑠 ‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼)))
Theorem	lmod1lem5 48773*	Lemma 5 for lmod1 48774. (Contributed by AV, 28-Apr-2019.)
⊢ 𝑀 = ({⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)⟩})    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝐼) = 𝐼)
Theorem	lmod1 48774*	The (smallest) structure representing a zero module over an arbitrary ring. (Contributed by AV, 29-Apr-2019.)
⊢ 𝑀 = ({⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)⟩})    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → 𝑀 ∈ LMod)
Theorem	lmod1zr 48775	The (smallest) structure representing a zero module over a zero ring. (Contributed by AV, 29-Apr-2019.)
⊢ 𝑅 = {⟨(Base‘ndx), {𝑍}⟩, ⟨(+g‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩, ⟨(.r‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩}    &   ⊢ 𝑀 = ({⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), {⟨⟨𝑍, 𝐼⟩, 𝐼⟩}⟩})    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → 𝑀 ∈ LMod)
Theorem	lmod1zrnlvec 48776	There is a (left) module (a zero module) which is not a (left) vector space. (Contributed by AV, 29-Apr-2019.)
⊢ 𝑅 = {⟨(Base‘ndx), {𝑍}⟩, ⟨(+g‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩, ⟨(.r‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩}    &   ⊢ 𝑀 = ({⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), {⟨⟨𝑍, 𝐼⟩, 𝐼⟩}⟩})    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → 𝑀 ∉ LVec)
Theorem	lmodn0 48777	Left modules exist. (Contributed by AV, 29-Apr-2019.)
⊢ LMod ≠ ∅
Theorem	zlmodzxzequa 48778	Example of an equation within the ℤ-module ℤ × ℤ (see example in [Roman] p. 112 for a linearly dependent set). (Contributed by AV, 22-May-2019.) (Revised by AV, 10-Jun-2019.)
⊢ 𝑍 = (ℤring freeLMod {0, 1})    &   ⊢ 0 = {⟨0, 0⟩, ⟨1, 0⟩}    &   ⊢ ∙ = ( ·𝑠 ‘𝑍)    &   ⊢ − = (-g‘𝑍)    &   ⊢ 𝐴 = {⟨0, 3⟩, ⟨1, 6⟩}    &   ⊢ 𝐵 = {⟨0, 2⟩, ⟨1, 4⟩}    ⇒   ⊢ ((2 ∙ 𝐴) − (3 ∙ 𝐵)) = 0
Theorem	zlmodzxznm 48779	Example of a linearly dependent set whose elements are not linear combinations of the others, see note in [Roman] p. 112). (Contributed by AV, 23-May-2019.) (Revised by AV, 10-Jun-2019.)
⊢ 𝑍 = (ℤring freeLMod {0, 1})    &   ⊢ 0 = {⟨0, 0⟩, ⟨1, 0⟩}    &   ⊢ ∙ = ( ·𝑠 ‘𝑍)    &   ⊢ − = (-g‘𝑍)    &   ⊢ 𝐴 = {⟨0, 3⟩, ⟨1, 6⟩}    &   ⊢ 𝐵 = {⟨0, 2⟩, ⟨1, 4⟩}    ⇒   ⊢ ∀𝑖 ∈ ℤ ((𝑖 ∙ 𝐴) ≠ 𝐵 ∧ (𝑖 ∙ 𝐵) ≠ 𝐴)
Theorem	zlmodzxzldeplem 48780	A and B are not equal. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.)
⊢ 𝑍 = (ℤring freeLMod {0, 1})    &   ⊢ 𝐴 = {⟨0, 3⟩, ⟨1, 6⟩}    &   ⊢ 𝐵 = {⟨0, 2⟩, ⟨1, 4⟩}    ⇒   ⊢ 𝐴 ≠ 𝐵
Theorem	zlmodzxzequap 48781	Example of an equation within the ℤ-module ℤ × ℤ (see example in [Roman] p. 112 for a linearly dependent set), written as a sum. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.)
⊢ 𝑍 = (ℤring freeLMod {0, 1})    &   ⊢ 𝐴 = {⟨0, 3⟩, ⟨1, 6⟩}    &   ⊢ 𝐵 = {⟨0, 2⟩, ⟨1, 4⟩}    &   ⊢ 0 = {⟨0, 0⟩, ⟨1, 0⟩}    &   ⊢ + = (+g‘𝑍)    &   ⊢ ∙ = ( ·𝑠 ‘𝑍)    ⇒   ⊢ ((2 ∙ 𝐴) + (-3 ∙ 𝐵)) = 0
Theorem	zlmodzxzldeplem1 48782	Lemma 1 for zlmodzxzldep 48786. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.)
⊢ 𝑍 = (ℤring freeLMod {0, 1})    &   ⊢ 𝐴 = {⟨0, 3⟩, ⟨1, 6⟩}    &   ⊢ 𝐵 = {⟨0, 2⟩, ⟨1, 4⟩}    &   ⊢ 𝐹 = {⟨𝐴, 2⟩, ⟨𝐵, -3⟩}    ⇒   ⊢ 𝐹 ∈ (ℤ ↑m {𝐴, 𝐵})
Theorem	zlmodzxzldeplem2 48783	Lemma 2 for zlmodzxzldep 48786. (Contributed by AV, 24-May-2019.) (Revised by AV, 30-Jul-2019.)
⊢ 𝑍 = (ℤring freeLMod {0, 1})    &   ⊢ 𝐴 = {⟨0, 3⟩, ⟨1, 6⟩}    &   ⊢ 𝐵 = {⟨0, 2⟩, ⟨1, 4⟩}    &   ⊢ 𝐹 = {⟨𝐴, 2⟩, ⟨𝐵, -3⟩}    ⇒   ⊢ 𝐹 finSupp 0
Theorem	zlmodzxzldeplem3 48784	Lemma 3 for zlmodzxzldep 48786. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.)
⊢ 𝑍 = (ℤring freeLMod {0, 1})    &   ⊢ 𝐴 = {⟨0, 3⟩, ⟨1, 6⟩}    &   ⊢ 𝐵 = {⟨0, 2⟩, ⟨1, 4⟩}    &   ⊢ 𝐹 = {⟨𝐴, 2⟩, ⟨𝐵, -3⟩}    ⇒   ⊢ (𝐹( linC ‘𝑍){𝐴, 𝐵}) = (0g‘𝑍)
Theorem	zlmodzxzldeplem4 48785*	Lemma 4 for zlmodzxzldep 48786. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.)
⊢ 𝑍 = (ℤring freeLMod {0, 1})    &   ⊢ 𝐴 = {⟨0, 3⟩, ⟨1, 6⟩}    &   ⊢ 𝐵 = {⟨0, 2⟩, ⟨1, 4⟩}    &   ⊢ 𝐹 = {⟨𝐴, 2⟩, ⟨𝐵, -3⟩}    ⇒   ⊢ ∃𝑦 ∈ {𝐴, 𝐵} (𝐹‘𝑦) ≠ 0
Theorem	zlmodzxzldep 48786	{ A , B } is a linearly dependent set within the ℤ-module ℤ × ℤ (see example in [Roman] p. 112). (Contributed by AV, 22-May-2019.) (Revised by AV, 10-Jun-2019.)
⊢ 𝑍 = (ℤring freeLMod {0, 1})    &   ⊢ 𝐴 = {⟨0, 3⟩, ⟨1, 6⟩}    &   ⊢ 𝐵 = {⟨0, 2⟩, ⟨1, 4⟩}    ⇒   ⊢ {𝐴, 𝐵} linDepS 𝑍
Theorem	ldepsnlinclem1 48787	Lemma 1 for ldepsnlinc 48790. (Contributed by AV, 25-May-2019.) (Revised by AV, 10-Jun-2019.)
⊢ 𝑍 = (ℤring freeLMod {0, 1})    &   ⊢ 𝐴 = {⟨0, 3⟩, ⟨1, 6⟩}    &   ⊢ 𝐵 = {⟨0, 2⟩, ⟨1, 4⟩}    ⇒   ⊢ (𝐹 ∈ ((Base‘ℤring) ↑m {𝐵}) → (𝐹( linC ‘𝑍){𝐵}) ≠ 𝐴)
Theorem	ldepsnlinclem2 48788	Lemma 2 for ldepsnlinc 48790. (Contributed by AV, 25-May-2019.) (Revised by AV, 10-Jun-2019.)
⊢ 𝑍 = (ℤring freeLMod {0, 1})    &   ⊢ 𝐴 = {⟨0, 3⟩, ⟨1, 6⟩}    &   ⊢ 𝐵 = {⟨0, 2⟩, ⟨1, 4⟩}    ⇒   ⊢ (𝐹 ∈ ((Base‘ℤring) ↑m {𝐴}) → (𝐹( linC ‘𝑍){𝐴}) ≠ 𝐵)
21.48.20.5  Differences between (left) modules and (left) vector spaces
Theorem	lvecpsslmod 48789	The class of all (left) vector spaces is a proper subclass of the class of all (left) modules. Although it is obvious (and proven by lveclmod 21062) that every left vector space is a left module, there is (at least) one left module which is no left vector space, for example the zero module over the zero ring, see lmod1zrnlvec 48776. (Contributed by AV, 29-Apr-2019.)
⊢ LVec ⊊ LMod
Theorem	ldepsnlinc 48790*	The reverse implication of islindeps2 48765 does not hold for arbitrary (left) modules, see note in [Roman] p. 112: "... if a nontrivial linear combination of the elements ... in an R-module M is 0, ... where not all of the coefficients are 0, then we cannot conclude ... that one of the elements ... is a linear combination of the others." This means that there is at least one left module having a linearly dependent subset in which there is at least one element which is not a linear combination of the other elements of this subset. Such a left module can be constructed by using zlmodzxzequa 48778 and zlmodzxznm 48779. (Contributed by AV, 25-May-2019.) (Revised by AV, 30-Jul-2019.)
⊢ ∃𝑚 ∈ LMod ∃𝑠 ∈ 𝒫 (Base‘𝑚)(𝑠 linDepS 𝑚 ∧ ∀𝑣 ∈ 𝑠 ∀𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑m (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) → (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) ≠ 𝑣))
Theorem	ldepslinc 48791*	For (left) vector spaces, isldepslvec2 48767 provides an alternative definition of being a linearly dependent subset, whereas ldepsnlinc 48790 indicates that there is not an analogous alternative definition for arbitrary (left) modules. (Contributed by AV, 25-May-2019.) (Revised by AV, 30-Jul-2019.)
⊢ (∀𝑚 ∈ LVec ∀𝑠 ∈ 𝒫 (Base‘𝑚)(𝑠 linDepS 𝑚 ↔ ∃𝑣 ∈ 𝑠 ∃𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑m (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣)) ∧ ¬ ∀𝑚 ∈ LMod ∀𝑠 ∈ 𝒫 (Base‘𝑚)(𝑠 linDepS 𝑚 ↔ ∃𝑣 ∈ 𝑠 ∃𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑m (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣)))
21.48.21  Complexity theory
21.48.21.1  Auxiliary theorems
Theorem	suppdm 48792	If the range of a function does not contain the zero, the support of the function equals its domain. (Contributed by AV, 20-May-2020.)
⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑍 ∉ ran 𝐹) → (𝐹 supp 𝑍) = dom 𝐹)
Theorem	eluz2cnn0n1 48793	An integer greater than 1 is a complex number not equal to 0 or 1. (Contributed by AV, 23-May-2020.)
⊢ (𝐵 ∈ (ℤ≥‘2) → 𝐵 ∈ (ℂ ∖ {0, 1}))
Theorem	divge1b 48794	The ratio of a real number to a positive real number is greater than or equal to 1 iff the divisor (the positive real number) is less than or equal to the dividend (the real number). (Contributed by AV, 26-May-2020.)
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ 1 ≤ (𝐵 / 𝐴)))
Theorem	divgt1b 48795	The ratio of a real number to a positive real number is greater than 1 iff the divisor (the positive real number) is less than the dividend (the real number). (Contributed by AV, 30-May-2020.)
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 1 < (𝐵 / 𝐴)))
Theorem	ltsubaddb 48796	Equivalence for the "less than" relation between differences and sums. (Contributed by AV, 6-Jun-2020.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 − 𝐶) < (𝐵 − 𝐷) ↔ (𝐴 + 𝐷) < (𝐵 + 𝐶)))
Theorem	ltsubsubb 48797	Equivalence for the "less than" relation between differences. (Contributed by AV, 6-Jun-2020.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 − 𝐶) < (𝐵 − 𝐷) ↔ (𝐴 − 𝐵) < (𝐶 − 𝐷)))
Theorem	ltsubadd2b 48798	Equivalence for the "less than" relation between differences and sums. (Contributed by AV, 6-Jun-2020.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐷 − 𝐶) < (𝐵 − 𝐴) ↔ (𝐴 + 𝐷) < (𝐵 + 𝐶)))
Theorem	divsub1dir 48799	Distribution of division over subtraction by 1. (Contributed by AV, 6-Jun-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → ((𝐴 / 𝐵) − 1) = ((𝐴 − 𝐵) / 𝐵))
Theorem	expnegico01 48800	An integer greater than 1 to the power of a negative integer is in the closed-below, open-above interval between 0 and 1. (Contributed by AV, 24-May-2020.)
⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝑁 < 0) → (𝐵↑𝑁) ∈ (0[,)1))