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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | lincvalpr 42001 | 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 42002 | 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 42003 | 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 42004* | 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 42005* | 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 42006* | 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 42007 | 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 42008* | 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 42009 | 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 42010 | 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 42011 | 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 42012 | 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 42013* | 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 42014 | 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 42015 | 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 42016 | 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 42017 | 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 42018* | 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 42019 | 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 42020 | Lemma for lspeqlco 42022. (Contributed by AV, 17-Apr-2019.) |
⊢ 𝐵 = (Base‘𝑀) ⇒ ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → ((LSpan‘𝑀)‘𝑉) ⊆ (𝑀 LinCo 𝑉)) | ||
Theorem | lcosslsp 42021 | Lemma for lspeqlco 42022. (Contributed by AV, 20-Apr-2019.) |
⊢ 𝐵 = (Base‘𝑀) ⇒ ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 LinCo 𝑉) ⊆ ((LSpan‘𝑀)‘𝑉)) | ||
Theorem | lspeqlco 42022 | 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 18793) 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)) | ||
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 42025 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. | ||
Syntax | clininds 42023 | Extend class notation with the relation between a module and its linearly independent subsets. |
class linIndS | ||
Syntax | clindeps 42024 | Extend class notation with the relation between a module and its linearly dependent subsets. |
class linDepS | ||
Definition | df-lininds 42025* | 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 42026 | 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 42027* | 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 42028 | 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 42029* | 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 42030* | 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 42031* | 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 42032* | 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 42033* | 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 42034 | 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 42035 | 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 42036* | 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 42037* | 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 42038* | 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 42039* | 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 42040* | Implication 1 for lindslininds 42047. (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 ∈ 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 42041* | Lemma 1 for lindslinindsimp2 42046. (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 42042* | Lemma 2 for lindslinindsimp2 42046. (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 42043* | Lemma 3 for lindslinindsimp2 42046. (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 42044* | Lemma 4 for lindslinindsimp2 42046. (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 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (𝑀 Σg (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓‘𝑦)( ·𝑠 ‘𝑀)𝑦))) = (𝑌( ·𝑠 ‘𝑀)𝑥)) | ||
Theorem | lindslinindsimp2lem5 42045* | Lemma 5 for lindslinindsimp2 42046. (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 42046* | Implication 2 for lindslininds 42047. (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 42047 | Equivalence of definitions df-linds 19965 and df-lininds 42025 for (linear) independency 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 42048 | The empty set is always a linearly independet 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 42049 | 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 42050 | 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 42051 | 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 18699), 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 42052 | 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 42053* | Lemma for snlindsntor 42054. (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 42054* | 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 42055 | Lemma for ldepspr 42056. (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 42056 | 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 42057 | Lemma 3 for lincresunit3 42064. (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 42058 | 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 42059 | 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 42060* | 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 42061* | 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 42062* | Lemma 1 for lincresunit3 42064. (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 42063* | Lemma 2 for lincresunit3 42064. (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 42064* | 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 42065* | 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 42066* | 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 42067* | 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 42068* | Alternative definition of being a linearly dependent subset of a (left) vector space. In this case, the reverse implication of islindeps2 42066 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 42069 | 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) | ||
Theorem | lmod1lem1 42070* | Lemma 1 for lmod1 42075. (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 42071* | Lemma 2 for lmod1 42075. (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 42072* | Lemma 3 for lmod1 42075. (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 42073* | Lemma 4 for lmod1 42075. (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 42074* | Lemma 5 for lmod1 42075. (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 42075* | 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 42076 | 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 42077 | 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 42078 | Left modules exist. (Contributed by AV, 29-Apr-2019.) |
⊢ LMod ≠ ∅ | ||
Theorem | zlmodzxzequa 42079 | 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 42080 | 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 42081 | 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 42082 | 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 42083 | Lemma 1 for zlmodzxzldep 42087. (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 42084 | Lemma 2 for zlmodzxzldep 42087. (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 42085 | Lemma 3 for zlmodzxzldep 42087. (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 42086* | Lemma 4 for zlmodzxzldep 42087. (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 42087 | { 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 42088 | Lemma 1 for ldepsnlinc 42091. (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 42089 | Lemma 2 for ldepsnlinc 42091. (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) | ||
Theorem | lvecpsslmod 42090 | 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 18927) 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 42077. (Contributed by AV, 29-Apr-2019.) |
⊢ LVec ⊊ LMod | ||
Theorem | ldepsnlinc 42091* | The reverse implication of islindeps2 42066 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 combinantion of the other elements of this subset. Such a left module can be constructed by using zlmodzxzequa 42079 and zlmodzxznm 42080. (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 42092* | For (left) vector spaces, isldepslvec2 42068 provides an alternative definition of being a linearly dependent subset, whereas ldepsnlinc 42091 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))) | ||
Theorem | offval0 42093* | Value of an operation applied to two functions. (Contributed by AV, 15-May-2020.) |
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐹 ∘𝑓 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) | ||
Theorem | suppdm 42094 | 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 42095 | 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 42096 | 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 42097 | 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 42098 | 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 42099 | 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 42100 | 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))) |
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