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Theorem List for Metamath Proof Explorer - 41901-42000   *Has distinct variable group(s)
TypeLabelDescription
Statement

TheoremrhmsubcALTV 41901 According to df-subc 16295, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 ( see subcssc 16323 and subcss2 16326). Therefore, the set of unital ring homomorphisms is a "subcategory" of the category of non-unital rings. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCatALTV‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       (𝜑𝐻 ∈ (Subcat‘(RngCatALTV‘𝑈)))

TheoremrhmsubcALTVcat 41902 The restriction of the category of non-unital rings to the set of unital ring homomorphisms is a category. (Contributed by AV, 4-Mar-2020.) (New usage is discouraged.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCatALTV‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       (𝜑 → ((RngCatALTV‘𝑈) ↾cat 𝐻) ∈ Cat)

21.34.13  Basic algebraic structures (extension)

21.34.13.1  Auxiliary theorems

Theoremxpprsng 41903 The Cartesian product of an unordered pair and a singleton. (Contributed by AV, 20-May-2019.)
((𝐴𝑉𝐵𝑊𝐶𝑈) → ({𝐴, 𝐵} × {𝐶}) = {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐶⟩})

Theoremopeliun2xp 41904 Membership of an ordered pair in a union of Cartesian products over its second component, analogous to opeliunxp 5093. (Contributed by AV, 30-Mar-2019.)
(⟨𝐶, 𝑦⟩ ∈ 𝑦𝐵 (𝐴 × {𝑦}) ↔ (𝑦𝐵𝐶𝐴))

Theoremeliunxp2 41905* Membership in a union of Cartesian products over its second component, analogous to eliunxp 5181. (Contributed by AV, 30-Mar-2019.)
(𝐶 𝑦𝐵 (𝐴 × {𝑦}) ↔ ∃𝑥𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))

Theoremmpt2mptx2 41906* Express a two-argument function as a one-argument function, or vice-versa. In this version 𝐴(𝑦) is not assumed to be constant w.r.t 𝑦, analogous to mpt2mptx 6649. (Contributed by AV, 30-Mar-2019.)
(𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)       (𝑧 𝑦𝐵 (𝐴 × {𝑦}) ↦ 𝐶) = (𝑥𝐴, 𝑦𝐵𝐷)

Theoremcbvmpt2x2 41907* Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpt2 6632 allows 𝐴 to be a function of 𝑦, analogous to cbvmpt2x 6631. (Contributed by AV, 30-Mar-2019.)
𝑧𝐴    &   𝑦𝐷    &   𝑧𝐶    &   𝑤𝐶    &   𝑥𝐸    &   𝑦𝐸    &   (𝑦 = 𝑧𝐴 = 𝐷)    &   ((𝑦 = 𝑧𝑥 = 𝑤) → 𝐶 = 𝐸)       (𝑥𝐴, 𝑦𝐵𝐶) = (𝑤𝐷, 𝑧𝐵𝐸)

Theoremdmmpt2ssx2 41908* The domain of a mapping is a subset of its base classes expressed as union of Cartesian products over its second component, analogous to dmmpt2ssx 7124. (Contributed by AV, 30-Mar-2019.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       dom 𝐹 𝑦𝐵 (𝐴 × {𝑦})

Theoremmpt2exxg2 41909* Existence of an operation class abstraction (version for dependent domains, i.e. the first base class may depend on the second base class), analogous to mpt2exxg 7133. (Contributed by AV, 30-Mar-2019.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       ((𝐵𝑅 ∧ ∀𝑦𝐵 𝐴𝑆) → 𝐹 ∈ V)

Theoremovmpt2rdxf 41910* Value of an operation given by a maps-to rule, deduction form, with substitution of second argument, analogous to ovmpt2dxf 6684. (Contributed by AV, 30-Mar-2019.)
(𝜑𝐹 = (𝑥𝐶, 𝑦𝐷𝑅))    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)    &   ((𝜑𝑦 = 𝐵) → 𝐶 = 𝐿)    &   (𝜑𝐴𝐿)    &   (𝜑𝐵𝐷)    &   (𝜑𝑆𝑋)    &   𝑥𝜑    &   𝑦𝜑    &   𝑦𝐴    &   𝑥𝐵    &   𝑥𝑆    &   𝑦𝑆       (𝜑 → (𝐴𝐹𝐵) = 𝑆)

Theoremovmpt2rdx 41911* Value of an operation given by a maps-to rule, deduction form, with substitution of second argument, analogous to ovmpt2dxf 6684. (Contributed by AV, 30-Mar-2019.)
(𝜑𝐹 = (𝑥𝐶, 𝑦𝐷𝑅))    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)    &   ((𝜑𝑦 = 𝐵) → 𝐶 = 𝐿)    &   (𝜑𝐴𝐿)    &   (𝜑𝐵𝐷)    &   (𝜑𝑆𝑋)       (𝜑 → (𝐴𝐹𝐵) = 𝑆)

Theoremovmpt2x2 41912* The value of an operation class abstraction. Variant of ovmpt2ga 6688 which does not require 𝐷 and 𝑥 to be distinct. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 20-Dec-2013.)
((𝑥 = 𝐴𝑦 = 𝐵) → 𝑅 = 𝑆)    &   (𝑦 = 𝐵𝐶 = 𝐿)    &   𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)       ((𝐴𝐿𝐵𝐷𝑆𝐻) → (𝐴𝐹𝐵) = 𝑆)

Theoremfdmdifeqresdif 41913* The restriction of a conditional mapping to function values of a function having a domain which is a difference with a singleton equals this function. (Contributed by AV, 23-Apr-2019.)
𝐹 = (𝑥𝐷 ↦ if(𝑥 = 𝑌, 𝑋, (𝐺𝑥)))       (𝐺:(𝐷 ∖ {𝑌})⟶𝑅𝐺 = (𝐹 ↾ (𝐷 ∖ {𝑌})))

Theoremoffvalfv 41914* The function operation expressed as a mapping with function values. (Contributed by AV, 6-Apr-2019.)
(𝜑𝐴𝑉)    &   (𝜑𝐹 Fn 𝐴)    &   (𝜑𝐺 Fn 𝐴)       (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))

Theoremofaddmndmap 41915 The function operation applied to the addition for functions (with the same domain) into a monoid is a function (with the same domain) into the monoid. (Contributed by AV, 6-Apr-2019.)
𝑅 = (Base‘𝑀)    &    + = (+g𝑀)       ((𝑀 ∈ Mnd ∧ 𝑉𝑌 ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (𝐴𝑓 + 𝐵) ∈ (𝑅𝑚 𝑉))

Theoremmapsnop 41916 A singleton of an ordered pair as an element of the mapping operation. (Contributed by AV, 12-Apr-2019.)
𝐹 = {⟨𝑋, 𝑌⟩}       ((𝑋𝑉𝑌𝑅𝑅𝑊) → 𝐹 ∈ (𝑅𝑚 {𝑋}))

Theoremmapprop 41917 An unordered pair containing two ordered pairs as an element of the mapping operation. (Contributed by AV, 16-Apr-2019.)
𝐹 = {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩}       (((𝑋𝑉𝐴𝑅) ∧ (𝑌𝑉𝐵𝑅) ∧ (𝑋𝑌𝑅𝑊)) → 𝐹 ∈ (𝑅𝑚 {𝑋, 𝑌}))

Theoremztprmneprm 41918 A prime is not an integer multiple of another prime. (Contributed by AV, 23-May-2019.)
((𝑍 ∈ ℤ ∧ 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵𝐴 = 𝐵))

Theorem2t6m3t4e0 41919 2 times 6 minus 3 times 4 equals 0. (Contributed by AV, 24-May-2019.)
((2 · 6) − (3 · 4)) = 0

Theoremssnn0ssfz 41920* For any finite subset of 0, find a superset in the form of a set of sequential integers, analogous to ssnnssfz 28937. (Contributed by AV, 30-Sep-2019.)
(𝐴 ∈ (𝒫 ℕ0 ∩ Fin) → ∃𝑛 ∈ ℕ0 𝐴 ⊆ (0...𝑛))

Theoremnn0sumltlt 41921 If the sum of two nonnegative integers is less than a third integer, then one of the summands is already less than this third integer. (Contributed by AV, 19-Oct-2019.)
((𝑎 ∈ ℕ0𝑏 ∈ ℕ0𝑐 ∈ ℕ0) → ((𝑎 + 𝑏) < 𝑐𝑏 < 𝑐))

21.34.13.2  The binomial coefficient operation (extension)

Theorembcpascm1 41922 Pascal's rule for the binomial coefficient, generalized to all integers 𝐾, shifted down by 1. (Contributed by AV, 8-Sep-2019.)
((𝑁 ∈ ℕ ∧ 𝐾 ∈ ℤ) → (((𝑁 − 1)C𝐾) + ((𝑁 − 1)C(𝐾 − 1))) = (𝑁C𝐾))

Theoremaltgsumbc 41923* The sum of binomial coefficients for a fixed positive 𝑁 with alternating signs is zero. Notice that this is not valid for 𝑁 = 0 (since ((-1↑0) · (0C0)) = (1 · 1) = 1). For a proof using Pascal's rule (bcpascm1 41922) instead of the binomial theorem (binom 14401) , see altgsumbcALT 41924. (Contributed by AV, 13-Sep-2019.)
(𝑁 ∈ ℕ → Σ𝑘 ∈ (0...𝑁)((-1↑𝑘) · (𝑁C𝑘)) = 0)

TheoremaltgsumbcALT 41924* Alternate proof of altgsumbc 41923, using Pascal's rule (bcpascm1 41922) instead of the binomial theorem (binom 14401). (Contributed by AV, 8-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑁 ∈ ℕ → Σ𝑘 ∈ (0...𝑁)((-1↑𝑘) · (𝑁C𝑘)) = 0)

21.34.13.3  The ` ZZ `-module ` ZZ X. ZZ `

Theoremzlmodzxzlmod 41925 The -module ℤ × ℤ is a (left) module with the ring of integers as base set. (Contributed by AV, 20-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (ℤring freeLMod {0, 1})       (𝑍 ∈ LMod ∧ ℤring = (Scalar‘𝑍))

Theoremzlmodzxzel 41926 An element of the (base set of the) -module ℤ × ℤ. (Contributed by AV, 21-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (ℤring freeLMod {0, 1})       ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∈ (Base‘𝑍))

Theoremzlmodzxz0 41927 The 0 of the -module ℤ × ℤ. (Contributed by AV, 20-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (ℤring freeLMod {0, 1})    &    0 = {⟨0, 0⟩, ⟨1, 0⟩}        0 = (0g𝑍)

Theoremzlmodzxzscm 41928 The scalar multiplication of the -module ℤ × ℤ. (Contributed by AV, 20-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (ℤring freeLMod {0, 1})    &    = ( ·𝑠𝑍)       ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 {⟨0, 𝐵⟩, ⟨1, 𝐶⟩}) = {⟨0, (𝐴 · 𝐵)⟩, ⟨1, (𝐴 · 𝐶)⟩})

Theoremzlmodzxzadd 41929 The addition of the -module ℤ × ℤ. (Contributed by AV, 22-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (ℤring freeLMod {0, 1})    &    + = (+g𝑍)       (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → ({⟨0, 𝐴⟩, ⟨1, 𝐶⟩} + {⟨0, 𝐵⟩, ⟨1, 𝐷⟩}) = {⟨0, (𝐴 + 𝐵)⟩, ⟨1, (𝐶 + 𝐷)⟩})

Theoremzlmodzxzsubm 41930 The subtraction of the -module ℤ × ℤ expressed as addition. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (ℤring freeLMod {0, 1})    &    = (-g𝑍)       (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → ({⟨0, 𝐴⟩, ⟨1, 𝐶⟩} {⟨0, 𝐵⟩, ⟨1, 𝐷⟩}) = ({⟨0, 𝐴⟩, ⟨1, 𝐶⟩} (+g𝑍)(-1( ·𝑠𝑍){⟨0, 𝐵⟩, ⟨1, 𝐷⟩})))

Theoremzlmodzxzsub 41931 The subtraction of the -module ℤ × ℤ. (Contributed by AV, 22-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (ℤring freeLMod {0, 1})    &    = (-g𝑍)       (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → ({⟨0, 𝐴⟩, ⟨1, 𝐶⟩} {⟨0, 𝐵⟩, ⟨1, 𝐷⟩}) = {⟨0, (𝐴𝐵)⟩, ⟨1, (𝐶𝐷)⟩})

21.34.13.4  Ordered group sum operation (extension)

Theoremgsumpr 41932* Group sum of a pair. (Contributed by AV, 6-Dec-2018.) (Proof shortened by AV, 28-Jul-2019.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   (𝑘 = 𝑀𝐴 = 𝐶)    &   (𝑘 = 𝑁𝐴 = 𝐷)       ((𝐺 ∈ CMnd ∧ (𝑀𝑉𝑁𝑊𝑀𝑁) ∧ (𝐶𝐵𝐷𝐵)) → (𝐺 Σg (𝑘 ∈ {𝑀, 𝑁} ↦ 𝐴)) = (𝐶 + 𝐷))

Theoremmgpsumunsn 41933* Extract a summand/factor from the group sum for the multiplicative group of a unital ring. (Contributed by AV, 29-Dec-2018.)
𝑀 = (mulGrp‘𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝐼𝑁)    &   ((𝜑𝑘𝑁) → 𝐴 ∈ (Base‘𝑅))    &   (𝜑𝑋 ∈ (Base‘𝑅))    &   (𝑘 = 𝐼𝐴 = 𝑋)       (𝜑 → (𝑀 Σg (𝑘𝑁𝐴)) = ((𝑀 Σg (𝑘 ∈ (𝑁 ∖ {𝐼}) ↦ 𝐴)) · 𝑋))

Theoremmgpsumz 41934* If the group sum for the multiplicative group of a unital ring contains a summand/factor that is the zero of the ring, the group sum itself is zero. (Contributed by AV, 29-Dec-2018.)
𝑀 = (mulGrp‘𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝐼𝑁)    &   ((𝜑𝑘𝑁) → 𝐴 ∈ (Base‘𝑅))    &    0 = (0g𝑅)    &   (𝑘 = 𝐼𝐴 = 0 )       (𝜑 → (𝑀 Σg (𝑘𝑁𝐴)) = 0 )

Theoremmgpsumn 41935* If the group sum for the multiplicative group of a unital ring contains a summand/factor that is the one of the ring, this summand/ factor can be removed from the group sum. (Contributed by AV, 29-Dec-2018.)
𝑀 = (mulGrp‘𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝐼𝑁)    &   ((𝜑𝑘𝑁) → 𝐴 ∈ (Base‘𝑅))    &    1 = (1r𝑅)    &   (𝑘 = 𝐼𝐴 = 1 )       (𝜑 → (𝑀 Σg (𝑘𝑁𝐴)) = (𝑀 Σg (𝑘 ∈ (𝑁 ∖ {𝐼}) ↦ 𝐴)))

Theoremgsumsplit2f 41936* Split a group sum into two parts. (Contributed by AV, 4-Sep-2019.)
𝑘𝜑    &   𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝑋𝐵)    &   (𝜑 → (𝑘𝐴𝑋) finSupp 0 )    &   (𝜑 → (𝐶𝐷) = ∅)    &   (𝜑𝐴 = (𝐶𝐷))       (𝜑 → (𝐺 Σg (𝑘𝐴𝑋)) = ((𝐺 Σg (𝑘𝐶𝑋)) + (𝐺 Σg (𝑘𝐷𝑋))))

Theoremgsumdifsndf 41937* Extract a summand from a finitely supported group sum. (Contributed by AV, 4-Sep-2019.)
𝑘𝑌    &   𝑘𝜑    &   𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴𝑊)    &   (𝜑 → (𝑘𝐴𝑋) finSupp (0g𝐺))    &   ((𝜑𝑘𝐴) → 𝑋𝐵)    &   (𝜑𝑀𝐴)    &   (𝜑𝑌𝐵)    &   ((𝜑𝑘 = 𝑀) → 𝑋 = 𝑌)       (𝜑 → (𝐺 Σg (𝑘𝐴𝑋)) = ((𝐺 Σg (𝑘 ∈ (𝐴 ∖ {𝑀}) ↦ 𝑋)) + 𝑌))

21.34.13.5  Symmetric groups (extension)

Theoremnn0le2is012 41938 A nonnegative integer which is less than or equal to 2 is either 0 or 1 or 2. (Contributed by AV, 16-Mar-2019.)
((𝑁 ∈ ℕ0𝑁 ≤ 2) → (𝑁 = 0 ∨ 𝑁 = 1 ∨ 𝑁 = 2))

Theoremexple2lt6 41939 A nonnegative integer to the power of itself is less than 6 if it is less than or equal to 2. (Contributed by AV, 16-Mar-2019.)
((𝑁 ∈ ℕ0𝑁 ≤ 2) → (𝑁𝑁) < 6)

Theorempgrple2abl 41940 Every symmetric group on a set with at most 2 elements is abelian. (Contributed by AV, 16-Mar-2019.)
𝐺 = (SymGrp‘𝐴)       ((𝐴𝑉 ∧ (#‘𝐴) ≤ 2) → 𝐺 ∈ Abel)

Theorempgrpgt2nabl 41941 Every symmetric group on a set with more than 2 elements is not abelian, see also the remark in [Rotman] p. 28. (Contributed by AV, 21-Mar-2019.)
𝐺 = (SymGrp‘𝐴)       ((𝐴𝑉 ∧ 2 < (#‘𝐴)) → 𝐺 ∉ Abel)

21.34.13.6  Divisibility (extension)

Theoreminvginvrid 41942 Identity for a multiplication with additive and multiplicative inverses in a ring. (Contributed by AV, 18-May-2018.)
𝐵 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &   𝑁 = (invg𝑅)    &   𝐼 = (invr𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵𝑌𝑈) → ((𝑁𝑌) · ((𝐼‘(𝑁𝑌)) · 𝑋)) = 𝑋)

21.34.13.7  The support of functions (extension)

Theoremrmsupp0 41943* The support of a mapping of a multiplication of zero with a function into a ring is empty. (Contributed by AV, 10-Apr-2019.)
𝑅 = (Base‘𝑀)       (((𝑀 ∈ Ring ∧ 𝑉𝑋𝐶 = (0g𝑀)) ∧ 𝐴 ∈ (𝑅𝑚 𝑉)) → ((𝑣𝑉 ↦ (𝐶(.r𝑀)(𝐴𝑣))) supp (0g𝑀)) = ∅)

Theoremdomnmsuppn0 41944* The support of a mapping of a multiplication of a nonzero constant with a function into a (ring theoretic) domain equals the support of the function. (Contributed by AV, 11-Apr-2019.)
𝑅 = (Base‘𝑀)       (((𝑀 ∈ Domn ∧ 𝑉𝑋) ∧ (𝐶𝑅𝐶 ≠ (0g𝑀)) ∧ 𝐴 ∈ (𝑅𝑚 𝑉)) → ((𝑣𝑉 ↦ (𝐶(.r𝑀)(𝐴𝑣))) supp (0g𝑀)) = (𝐴 supp (0g𝑀)))

Theoremrmsuppss 41945* The support of a mapping of a multiplication of a constant with a function into a ring is a subset of the support of the function. (Contributed by AV, 11-Apr-2019.)
𝑅 = (Base‘𝑀)       (((𝑀 ∈ Ring ∧ 𝑉𝑋𝐶𝑅) ∧ 𝐴 ∈ (𝑅𝑚 𝑉)) → ((𝑣𝑉 ↦ (𝐶(.r𝑀)(𝐴𝑣))) supp (0g𝑀)) ⊆ (𝐴 supp (0g𝑀)))

Theoremmndpsuppss 41946 The support of a mapping of a scalar multiplication with a function of scalars is a subset of the support of the function of scalars. (Contributed by AV, 5-Apr-2019.)
𝑅 = (Base‘𝑀)       (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → ((𝐴𝑓 (+g𝑀)𝐵) supp (0g𝑀)) ⊆ ((𝐴 supp (0g𝑀)) ∪ (𝐵 supp (0g𝑀))))

Theoremscmsuppss 41947* The support of a mapping of a scalar multiplication with a function of scalars is a subset of the support of the function of scalars. (Contributed by AV, 5-Apr-2019.)
𝑆 = (Scalar‘𝑀)    &   𝑅 = (Base‘𝑆)       ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅𝑚 𝑉)) → ((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) supp (0g𝑀)) ⊆ (𝐴 supp (0g𝑆)))

21.34.13.8  Finitely supported functions (extension)

Theoremrmsuppfi 41948* The support of a mapping of a multiplication of a constant with a function into a ring is finite if the support of the function is finite. (Contributed by AV, 11-Apr-2019.)
𝑅 = (Base‘𝑀)       (((𝑀 ∈ Ring ∧ 𝑉𝑋𝐶𝑅) ∧ 𝐴 ∈ (𝑅𝑚 𝑉) ∧ (𝐴 supp (0g𝑀)) ∈ Fin) → ((𝑣𝑉 ↦ (𝐶(.r𝑀)(𝐴𝑣))) supp (0g𝑀)) ∈ Fin)

Theoremrmfsupp 41949* A mapping of a multiplication of a constant with a function into a ring is finitely supported if the function is finitely supported. (Contributed by AV, 9-Jun-2019.)
𝑅 = (Base‘𝑀)       (((𝑀 ∈ Ring ∧ 𝑉𝑋𝐶𝑅) ∧ 𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐴 finSupp (0g𝑀)) → (𝑣𝑉 ↦ (𝐶(.r𝑀)(𝐴𝑣))) finSupp (0g𝑀))

Theoremmndpsuppfi 41950 The support of a mapping of a scalar multiplication with a function of scalars is finite if the support of the function of scalars is finite. (Contributed by AV, 5-Apr-2019.)
𝑅 = (Base‘𝑀)       (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉)) ∧ ((𝐴 supp (0g𝑀)) ∈ Fin ∧ (𝐵 supp (0g𝑀)) ∈ Fin)) → ((𝐴𝑓 (+g𝑀)𝐵) supp (0g𝑀)) ∈ Fin)

Theoremmndpfsupp 41951 A mapping of a scalar multiplication with a function of scalars is finitely supported if the function of scalars is finitely supported. (Contributed by AV, 9-Jun-2019.)
𝑅 = (Base‘𝑀)       (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉)) ∧ (𝐴 finSupp (0g𝑀) ∧ 𝐵 finSupp (0g𝑀))) → (𝐴𝑓 (+g𝑀)𝐵) finSupp (0g𝑀))

Theoremscmsuppfi 41952* The support of a mapping of a scalar multiplication with a function of scalars is finite if the support of the function of scalars is finite. (Contributed by AV, 5-Apr-2019.)
𝑆 = (Scalar‘𝑀)    &   𝑅 = (Base‘𝑆)       (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅𝑚 𝑉) ∧ (𝐴 supp (0g𝑆)) ∈ Fin) → ((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) supp (0g𝑀)) ∈ Fin)

Theoremscmfsupp 41953* A mapping of a scalar multiplication with a function of scalars is finitely supported if the function of scalars is finitely supported. (Contributed by AV, 9-Jun-2019.)
𝑆 = (Scalar‘𝑀)    &   𝑅 = (Base‘𝑆)       (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐴 finSupp (0g𝑆)) → (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) finSupp (0g𝑀))

Theoremsuppmptcfin 41954* The support of a mapping with value 0 except of one is finite. (Contributed by AV, 27-Apr-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   𝐹 = (𝑥𝑉 ↦ if(𝑥 = 𝑋, 1 , 0 ))       ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) → (𝐹 supp 0 ) ∈ Fin)

Theoremmptcfsupp 41955* A mapping with value 0 except of one is finitely supported. (Contributed by AV, 9-Jun-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   𝐹 = (𝑥𝑉 ↦ if(𝑥 = 𝑋, 1 , 0 ))       ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) → 𝐹 finSupp 0 )

Theoremfsuppmptdmf 41956* A mapping with a finite domain is finitely supported. (Contributed by AV, 4-Sep-2019.)
𝑥𝜑    &   𝐹 = (𝑥𝐴𝑌)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑥𝐴) → 𝑌𝑉)    &   (𝜑𝑍𝑊)       (𝜑𝐹 finSupp 𝑍)

21.34.13.9  Left modules (extension)

Theoremlmodvsmdi 41957 Multiple distributive law for scalar product (left-distributivity). (Contributed by AV, 5-Sep-2019.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)    &    = (.g𝑊)    &   𝐸 = (.g𝐹)       ((𝑊 ∈ LMod ∧ (𝑅𝐾𝑁 ∈ ℕ0𝑋𝑉)) → (𝑅 · (𝑁 𝑋)) = ((𝑁𝐸𝑅) · 𝑋))

Theoremgsumlsscl 41958* 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 ∧ 𝑍𝑆𝑉𝑍) → ((𝐹 ∈ (𝐵𝑚 𝑉) ∧ 𝐹 finSupp (0g𝑅)) → (𝑀 Σg (𝑣𝑉 ↦ ((𝐹𝑣)( ·𝑠𝑀)𝑣))) ∈ 𝑍))

21.34.13.10  Associative algebras (extension)

Theoremascl0 41959 The scalar 0 embedded into a left module corresponds to the 0 of the left module if the left module is also a ring. (Contributed by AV, 31-Jul-2019.)
𝐴 = (algSc‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑊 ∈ Ring)       (𝜑 → (𝐴‘(0g𝐹)) = (0g𝑊))

Theoremascl1 41960 The scalar 1 embedded into a left module corresponds to the 1 of the left module if the left module is also a ring. (Contributed by AV, 31-Jul-2019.)
𝐴 = (algSc‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑊 ∈ Ring)       (𝜑 → (𝐴‘(1r𝐹)) = (1r𝑊))

Theoremassaascl0 41961 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)    &   (𝜑𝑊 ∈ AssAlg)       (𝜑 → (𝐴‘(0g𝐹)) = (0g𝑊))

Theoremassaascl1 41962 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)    &   (𝜑𝑊 ∈ AssAlg)       (𝜑 → (𝐴‘(1r𝐹)) = (1r𝑊))

21.34.13.11  Univariate polynomials (extension)

Theoremply1vr1smo 41963 The variable in a polynomial expressed as scaled monomial. (Contributed by AV, 12-Aug-2019.)
𝑃 = (Poly1𝑅)    &    1 = (1r𝑅)    &    · = ( ·𝑠𝑃)    &   𝐺 = (mulGrp‘𝑃)    &    = (.g𝐺)    &   𝑋 = (var1𝑅)       (𝑅 ∈ Ring → ( 1 · (1 𝑋)) = 𝑋)

Theoremply1ass23l 41964 Associative identity with scalar and ring multiplication for the polynomial ring. (Contributed by AV, 14-Aug-2019.)
𝑃 = (Poly1𝑅)    &    × = (.r𝑃)    &   𝐵 = (Base‘𝑃)    &   𝐾 = (Base‘𝑅)    &    · = ( ·𝑠𝑃)       ((𝑅 ∈ Ring ∧ (𝐴𝐾𝑋𝐵𝑌𝐵)) → ((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)))

Theoremply1sclrmsm 41965 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 = (Poly1𝑅)    &   𝐸 = (Base‘𝑃)    &   𝑋 = (var1𝑅)    &    · = ( ·𝑠𝑃)    &    × = (.r𝑃)    &   𝑁 = (mulGrp‘𝑃)    &    = (.g𝑁)    &   𝐴 = (algSc‘𝑃)       ((𝑅 ∈ Ring ∧ 𝐹𝐾𝑍𝐸) → ((𝐴𝐹) × 𝑍) = (𝐹 · 𝑍))

Theoremcoe1id 41966* Coefficient vector of the unit polynomial. (Contributed by AV, 9-Aug-2019.)
𝑃 = (Poly1𝑅)    &   𝐼 = (1r𝑃)    &    0 = (0g𝑅)    &    1 = (1r𝑅)       (𝑅 ∈ Ring → (coe1𝐼) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, 1 , 0 )))

Theoremcoe1sclmulval 41967 The value of the coefficient vector of a polynomial multiplied on the left by a scalar. (Contributed by AV, 14-Aug-2019.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐾 = (Base‘𝑅)    &   𝐴 = (algSc‘𝑃)    &   𝑆 = ( ·𝑠𝑃)    &    = (.r𝑃)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ (𝑌𝐾𝑍𝐵) ∧ 𝑁 ∈ ℕ0) → ((coe1‘(𝑌𝑆𝑍))‘𝑁) = (𝑌 · ((coe1𝑍)‘𝑁)))

Theoremply1mulgsumlem1 41968* Lemma 1 for ply1mulgsum 41972. (Contributed by AV, 19-Oct-2019.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐴 = (coe1𝐾)    &   𝐶 = (coe1𝐿)    &   𝑋 = (var1𝑅)    &    × = (.r𝑃)    &    · = ( ·𝑠𝑃)    &    = (.r𝑅)    &   𝑀 = (mulGrp‘𝑃)    &    = (.g𝑀)       ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → ∃𝑠 ∈ ℕ0𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴𝑛) = (0g𝑅) ∧ (𝐶𝑛) = (0g𝑅))))

Theoremply1mulgsumlem2 41969* Lemma 2 for ply1mulgsum 41972. (Contributed by AV, 19-Oct-2019.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐴 = (coe1𝐾)    &   𝐶 = (coe1𝐿)    &   𝑋 = (var1𝑅)    &    × = (.r𝑃)    &    · = ( ·𝑠𝑃)    &    = (.r𝑅)    &   𝑀 = (mulGrp‘𝑃)    &    = (.g𝑀)       ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → ∃𝑠 ∈ ℕ0𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴𝑙) (𝐶‘(𝑛𝑙))))) = (0g𝑅)))

Theoremply1mulgsumlem3 41970* Lemma 3 for ply1mulgsum 41972. (Contributed by AV, 20-Oct-2019.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐴 = (coe1𝐾)    &   𝐶 = (coe1𝐿)    &   𝑋 = (var1𝑅)    &    × = (.r𝑃)    &    · = ( ·𝑠𝑃)    &    = (.r𝑅)    &   𝑀 = (mulGrp‘𝑃)    &    = (.g𝑀)       ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → (𝑘 ∈ ℕ0 ↦ (𝑅 Σg (𝑙 ∈ (0...𝑘) ↦ ((𝐴𝑙) (𝐶‘(𝑘𝑙)))))) finSupp (0g𝑅))

Theoremply1mulgsumlem4 41971* Lemma 4 for ply1mulgsum 41972. (Contributed by AV, 19-Oct-2019.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐴 = (coe1𝐾)    &   𝐶 = (coe1𝐿)    &   𝑋 = (var1𝑅)    &    × = (.r𝑃)    &    · = ( ·𝑠𝑃)    &    = (.r𝑅)    &   𝑀 = (mulGrp‘𝑃)    &    = (.g𝑀)       ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝑅 Σg (𝑙 ∈ (0...𝑘) ↦ ((𝐴𝑙) (𝐶‘(𝑘𝑙))))) · (𝑘 𝑋))) finSupp (0g𝑃))

Theoremply1mulgsum 41972* The product of two polynomials expressed as group sum of scaled monomials. (Contributed by AV, 20-Oct-2019.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐴 = (coe1𝐾)    &   𝐶 = (coe1𝐿)    &   𝑋 = (var1𝑅)    &    × = (.r𝑃)    &    · = ( ·𝑠𝑃)    &    = (.r𝑅)    &   𝑀 = (mulGrp‘𝑃)    &    = (.g𝑀)       ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → (𝐾 × 𝐿) = (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑅 Σg (𝑙 ∈ (0...𝑘) ↦ ((𝐴𝑙) (𝐶‘(𝑘𝑙))))) · (𝑘 𝑋)))))

Theoremevl1at0 41973 Polynomial evaluation for the 0 scalar. (Contributed by AV, 10-Aug-2019.)
𝑂 = (eval1𝑅)    &   𝑃 = (Poly1𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑃)       (𝑅 ∈ CRing → ((𝑂𝑍)‘ 0 ) = 0 )

Theoremevl1at1 41974 Polynomial evaluation for the 1 scalar. (Contributed by AV, 10-Aug-2019.)
𝑂 = (eval1𝑅)    &   𝑃 = (Poly1𝑅)    &    1 = (1r𝑅)    &   𝐼 = (1r𝑃)       (𝑅 ∈ CRing → ((𝑂𝐼)‘ 1 ) = 1 )

21.34.13.12  Univariate polynomials (examples)

Theoremlinply1 41975 A term of the form 𝑥𝐶 is a (univariate) polynomial, also called "linear polynomial". (Part of ply1remlem 23726). (Contributed by AV, 3-Jul-2019.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐾 = (Base‘𝑅)    &   𝑋 = (var1𝑅)    &    = (-g𝑃)    &   𝐴 = (algSc‘𝑃)    &   𝐺 = (𝑋 (𝐴𝐶))    &   (𝜑𝐶𝐾)    &   (𝜑𝑅 ∈ Ring)       (𝜑𝐺𝐵)

Theoremlineval 41976 A term of the form 𝑥𝐶 evaluated for 𝑥 = 𝑉 results in 𝑉𝐶 (part of ply1remlem 23726). (Contributed by AV, 3-Jul-2019.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐾 = (Base‘𝑅)    &   𝑋 = (var1𝑅)    &    = (-g𝑃)    &   𝐴 = (algSc‘𝑃)    &   𝐺 = (𝑋 (𝐴𝐶))    &   (𝜑𝐶𝐾)    &   𝑂 = (eval1𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑉𝐾)       (𝜑 → ((𝑂𝐺)‘𝑉) = (𝑉(-g𝑅)𝐶))

Theoremzringsubgval 41977 Subtraction in the ring of integers. (Contributed by AV, 3-Aug-2019.)
= (-g‘ℤring)       ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (𝑋𝑌) = (𝑋 𝑌))

Theoremlinevalexample 41978 The polynomial 𝑥 − 3 over evaluated for 𝑥 = 5 results in 2. (Contributed by AV, 3-Jul-2019.)
𝑃 = (Poly1‘ℤring)    &   𝐵 = (Base‘𝑃)    &   𝑋 = (var1‘ℤring)    &    = (-g𝑃)    &   𝐴 = (algSc‘𝑃)    &   𝐺 = (𝑋 (𝐴‘3))    &   𝑂 = (eval1‘ℤring)       ((𝑂‘(𝑋 (𝐴‘3)))‘5) = 2

21.34.14  Linear algebra (extension)

21.34.14.1  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 the definitions df-dmat 20115 and df-scmat 20116 define diagonal and scalar matrices as sets.

Syntaxcdmatalt 41979 Alternative notation for the algebra of diagonal matrices.
class DMatALT

Syntaxcscmatalt 41980 Alternative notation for the algebra of scalar matrices.
class ScMatALT

Definitiondf-dmatalt 41981* 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(𝑎s {𝑚 ∈ (Base‘𝑎) ∣ ∀𝑖𝑛𝑗𝑛 (𝑖𝑗 → (𝑖𝑚𝑗) = (0g𝑟))}))

Definitiondf-scmatalt 41982* 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(𝑎s {𝑚 ∈ (Base‘𝑎) ∣ ∃𝑐 ∈ (Base‘𝑟)∀𝑖𝑛𝑗𝑛 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g𝑟))}))

TheoremdmatALTval 41983* 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 = (0g𝑅)    &   𝐷 = (𝑁 DMatALT 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → 𝐷 = (𝐴s {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )}))

TheoremdmatALTbas 41984* 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 = (0g𝑅)    &   𝐷 = (𝑁 DMatALT 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (Base‘𝐷) = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )})

TheoremdmatALTbasel 41985* 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 = (0g𝑅)    &   𝐷 = (𝑁 DMatALT 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (𝑀 ∈ (Base‘𝐷) ↔ (𝑀𝐵 ∧ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑀𝑗) = 0 ))))

Theoremdmatbas 41986 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 = (0g𝑅)    &   𝐷 = (𝑁 DMat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝐷 = (Base‘(𝑁 DMatALT 𝑅)))

21.34.14.2  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 41989, 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 41990, so that we can show that such sets are submodules of the corresponding modules, see lincolss 42017.
Remark:According to Wikipedia ("Linear span", 28-Apr-2019, https://en.wikipedia.org/wiki/Linear_span) "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 "Alternatively, 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 18793, the set of all linear combinations as defined by df-lco 41990 follows the alternative approach. That both definitions are equivalent is shown by lspeqlco 42022.

Syntaxclinc 41987 Extend class notation with the operation constructing a linear combination (of vectors from a left module).
class linC

Syntaxclinco 41988 Extend class notation with the operation constructing a set of linear combinations (of vectors from a left module) with finite support.
class LinCo

Definitiondf-linc 41989* 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 (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑚)𝑥)))))

Definitiondf-lco 41990* 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))})

Theoremlincop 41991* A linear combination as operation. (Contributed by AV, 30-Mar-2019.)
(𝑀𝑋 → ( linC ‘𝑀) = (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥)))))

Theoremlincval 41992* The value of a linear combination. (Contributed by AV, 30-Mar-2019.)
((𝑀𝑋𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑆( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑥𝑉 ↦ ((𝑆𝑥)( ·𝑠𝑀)𝑥))))

Theoremdflinc2 41993* 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 (𝑠𝑓 ( ·𝑠𝑚)( I ↾ 𝑣)))))

Theoremlcoop 41994* 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)       ((𝑀𝑋𝑉 ∈ 𝒫 𝐵) → (𝑀 LinCo 𝑉) = {𝑐𝐵 ∣ ∃𝑠 ∈ (𝑅𝑚 𝑉)(𝑠 finSupp (0g𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉))})

Theoremlcoval 41995* 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)       ((𝑀𝑋𝑉 ∈ 𝒫 𝐵) → (𝐶 ∈ (𝑀 LinCo 𝑉) ↔ (𝐶𝐵 ∧ ∃𝑠 ∈ (𝑅𝑚 𝑉)(𝑠 finSupp (0g𝑆) ∧ 𝐶 = (𝑠( linC ‘𝑀)𝑉)))))

Theoremlincfsuppcl 41996 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 = (0g𝑅)       ((𝑀 ∈ LMod ∧ (𝑉𝑊𝑉𝐵) ∧ (𝐹 ∈ (𝑆𝑚 𝑉) ∧ 𝐹 finSupp 0 )) → (𝐹( linC ‘𝑀)𝑉) ∈ 𝐵)

Theoremlinccl 41997 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 ∧ 𝑉𝐵𝑆 ∈ (𝑅𝑚 𝑉))) → (𝑆( linC ‘𝑀)𝑉) ∈ 𝐵)

Theoremlincval0 41998 The value of an empty linear combination. (Contributed by AV, 12-Apr-2019.)
(𝑀𝑋 → (∅( linC ‘𝑀)∅) = (0g𝑀))

Theoremlincvalsng 41999 The linear combination over a singleton. (Contributed by AV, 25-May-2019.)
𝐵 = (Base‘𝑀)    &   𝑆 = (Scalar‘𝑀)    &   𝑅 = (Base‘𝑆)    &    · = ( ·𝑠𝑀)       ((𝑀 ∈ LMod ∧ 𝑉𝐵𝑌𝑅) → ({⟨𝑉, 𝑌⟩} ( linC ‘𝑀){𝑉}) = (𝑌 · 𝑉))

Theoremlincvalsn 42000 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)    &    · = ( ·𝑠𝑀)    &   𝐹 = {⟨𝑉, 𝑌⟩}       ((𝑀 ∈ LMod ∧ 𝑉𝐵𝑌𝑅) → (𝐹( linC ‘𝑀){𝑉}) = (𝑌 · 𝑉))

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