P. 440 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 43901-44000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	altgsumbcALT 43901*	Alternate proof of altgsumbc 43900, using Pascal's rule (bcpascm1 43899) instead of the binomial theorem (binom 15022). (Contributed by AV, 8-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (0...𝑁)((-1↑𝑘) · (𝑁C𝑘)) = 0)
20.38.19.3  The ` ZZ `-module ` ZZ X. ZZ `
Theorem	zlmodzxzlmod 43902	The ℤ-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))
Theorem	zlmodzxzel 43903	An element of the (base set of the) ℤ-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))
Theorem	zlmodzxz0 43904	The 0 of the ℤ-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 = (0g‘𝑍)
Theorem	zlmodzxzscm 43905	The scalar multiplication of the ℤ-module ℤ × ℤ. (Contributed by AV, 20-May-2019.) (Revised by AV, 10-Jun-2019.)
⊢ 𝑍 = (ℤring freeLMod {0, 1})    &   ⊢ ∙ = ( ·𝑠 ‘𝑍)    ⇒   ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 ∙ {⟨0, 𝐵⟩, ⟨1, 𝐶⟩}) = {⟨0, (𝐴 · 𝐵)⟩, ⟨1, (𝐴 · 𝐶)⟩})
Theorem	zlmodzxzadd 43906	The addition of the ℤ-module ℤ × ℤ. (Contributed by AV, 22-May-2019.) (Revised by AV, 10-Jun-2019.)
⊢ 𝑍 = (ℤring freeLMod {0, 1})    &   ⊢ + = (+g‘𝑍)    ⇒   ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → ({⟨0, 𝐴⟩, ⟨1, 𝐶⟩} + {⟨0, 𝐵⟩, ⟨1, 𝐷⟩}) = {⟨0, (𝐴 + 𝐵)⟩, ⟨1, (𝐶 + 𝐷)⟩})
Theorem	zlmodzxzsubm 43907	The subtraction of the ℤ-module ℤ × ℤ expressed as addition. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.)
⊢ 𝑍 = (ℤring freeLMod {0, 1})    &   ⊢ − = (-g‘𝑍)    ⇒   ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → ({⟨0, 𝐴⟩, ⟨1, 𝐶⟩} − {⟨0, 𝐵⟩, ⟨1, 𝐷⟩}) = ({⟨0, 𝐴⟩, ⟨1, 𝐶⟩} (+g‘𝑍)(-1( ·𝑠 ‘𝑍){⟨0, 𝐵⟩, ⟨1, 𝐷⟩})))
Theorem	zlmodzxzsub 43908	The subtraction of the ℤ-module ℤ × ℤ. (Contributed by AV, 22-May-2019.) (Revised by AV, 10-Jun-2019.)
⊢ 𝑍 = (ℤring freeLMod {0, 1})    &   ⊢ − = (-g‘𝑍)    ⇒   ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → ({⟨0, 𝐴⟩, ⟨1, 𝐶⟩} − {⟨0, 𝐵⟩, ⟨1, 𝐷⟩}) = {⟨0, (𝐴 − 𝐵)⟩, ⟨1, (𝐶 − 𝐷)⟩})
20.38.19.4  Ordered group sum operation (extension)
Theorem	mgpsumunsn 43909*	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)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑁)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑁) → 𝐴 ∈ (Base‘𝑅))    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑅))    &   ⊢ (𝑘 = 𝐼 → 𝐴 = 𝑋)    ⇒   ⊢ (𝜑 → (𝑀 Σg (𝑘 ∈ 𝑁 ↦ 𝐴)) = ((𝑀 Σg (𝑘 ∈ (𝑁 ∖ {𝐼}) ↦ 𝐴)) · 𝑋))
Theorem	mgpsumz 43910*	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)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑁)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑁) → 𝐴 ∈ (Base‘𝑅))    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝑘 = 𝐼 → 𝐴 = 0 )    ⇒   ⊢ (𝜑 → (𝑀 Σg (𝑘 ∈ 𝑁 ↦ 𝐴)) = 0 )
Theorem	mgpsumn 43911*	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)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑁)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑁) → 𝐴 ∈ (Base‘𝑅))    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ (𝑘 = 𝐼 → 𝐴 = 1 )    ⇒   ⊢ (𝜑 → (𝑀 Σg (𝑘 ∈ 𝑁 ↦ 𝐴)) = (𝑀 Σg (𝑘 ∈ (𝑁 ∖ {𝐼}) ↦ 𝐴)))
Theorem	gsumsplit2f 43912*	Split a group sum into two parts. (Contributed by AV, 4-Sep-2019.)
⊢ Ⅎ𝑘𝜑    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp 0 )    &   ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅)    &   ⊢ (𝜑 → 𝐴 = (𝐶 ∪ 𝐷))    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((𝐺 Σg (𝑘 ∈ 𝐶 ↦ 𝑋)) + (𝐺 Σg (𝑘 ∈ 𝐷 ↦ 𝑋))))
Theorem	gsumdifsndf 43913*	Extract a summand from a finitely supported group sum. (Contributed by AV, 4-Sep-2019.)
⊢ Ⅎ𝑘𝑌    &   ⊢ Ⅎ𝑘𝜑    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑊)    &   ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp (0g‘𝐺))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑀 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝑋 = 𝑌)    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((𝐺 Σg (𝑘 ∈ (𝐴 ∖ {𝑀}) ↦ 𝑋)) + 𝑌))
20.38.19.5  Symmetric groups (extension)
Theorem	exple2lt6 43914	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 ∈ ℕ0 ∧ 𝑁 ≤ 2) → (𝑁↑𝑁) < 6)
Theorem	pgrple2abl 43915	Every symmetric group on a set with at most 2 elements is abelian. (Contributed by AV, 16-Mar-2019.)
⊢ 𝐺 = (SymGrp‘𝐴)    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ≤ 2) → 𝐺 ∈ Abel)
Theorem	pgrpgt2nabl 43916	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)    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 2 < (♯‘𝐴)) → 𝐺 ∉ Abel)
20.38.19.6  Divisibility (extension)
Theorem	invginvrid 43917	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 = (invg‘𝑅)    &   ⊢ 𝐼 = (invr‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → ((𝑁‘𝑌) · ((𝐼‘(𝑁‘𝑌)) · 𝑋)) = 𝑋)
20.38.19.7  The support of functions (extension)
Theorem	rmsupp0 43918*	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 ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 = (0g‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑𝑚 𝑉)) → ((𝑣 ∈ 𝑉 ↦ (𝐶(.r‘𝑀)(𝐴‘𝑣))) supp (0g‘𝑀)) = ∅)
Theorem	domnmsuppn0 43919*	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 ∧ 𝑉 ∈ 𝑋) ∧ (𝐶 ∈ 𝑅 ∧ 𝐶 ≠ (0g‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑𝑚 𝑉)) → ((𝑣 ∈ 𝑉 ↦ (𝐶(.r‘𝑀)(𝐴‘𝑣))) supp (0g‘𝑀)) = (𝐴 supp (0g‘𝑀)))
Theorem	rmsuppss 43920*	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 ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 ∈ 𝑅) ∧ 𝐴 ∈ (𝑅 ↑𝑚 𝑉)) → ((𝑣 ∈ 𝑉 ↦ (𝐶(.r‘𝑀)(𝐴‘𝑣))) supp (0g‘𝑀)) ⊆ (𝐴 supp (0g‘𝑀)))
Theorem	mndpsuppss 43921	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 ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ 𝐵 ∈ (𝑅 ↑𝑚 𝑉))) → ((𝐴 ∘𝑓 (+g‘𝑀)𝐵) supp (0g‘𝑀)) ⊆ ((𝐴 supp (0g‘𝑀)) ∪ (𝐵 supp (0g‘𝑀))))
Theorem	scmsuppss 43922*	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 ∈ (𝑅 ↑𝑚 𝑉)) → ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) supp (0g‘𝑀)) ⊆ (𝐴 supp (0g‘𝑆)))
20.38.19.8  Finitely supported functions (extension)
Theorem	rmsuppfi 43923*	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 ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 ∈ 𝑅) ∧ 𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ (𝐴 supp (0g‘𝑀)) ∈ Fin) → ((𝑣 ∈ 𝑉 ↦ (𝐶(.r‘𝑀)(𝐴‘𝑣))) supp (0g‘𝑀)) ∈ Fin)
Theorem	rmfsupp 43924*	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 ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 ∈ 𝑅) ∧ 𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ 𝐴 finSupp (0g‘𝑀)) → (𝑣 ∈ 𝑉 ↦ (𝐶(.r‘𝑀)(𝐴‘𝑣))) finSupp (0g‘𝑀))
Theorem	mndpsuppfi 43925	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 ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ 𝐵 ∈ (𝑅 ↑𝑚 𝑉)) ∧ ((𝐴 supp (0g‘𝑀)) ∈ Fin ∧ (𝐵 supp (0g‘𝑀)) ∈ Fin)) → ((𝐴 ∘𝑓 (+g‘𝑀)𝐵) supp (0g‘𝑀)) ∈ Fin)
Theorem	mndpfsupp 43926	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 ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ 𝐵 ∈ (𝑅 ↑𝑚 𝑉)) ∧ (𝐴 finSupp (0g‘𝑀) ∧ 𝐵 finSupp (0g‘𝑀))) → (𝐴 ∘𝑓 (+g‘𝑀)𝐵) finSupp (0g‘𝑀))
Theorem	scmsuppfi 43927*	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 ∈ (𝑅 ↑𝑚 𝑉) ∧ (𝐴 supp (0g‘𝑆)) ∈ Fin) → ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) supp (0g‘𝑀)) ∈ Fin)
Theorem	scmfsupp 43928*	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 ∈ (𝑅 ↑𝑚 𝑉) ∧ 𝐴 finSupp (0g‘𝑆)) → (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) finSupp (0g‘𝑀))
Theorem	suppmptcfin 43929*	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 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑋, 1 , 0 ))    ⇒   ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹 supp 0 ) ∈ Fin)
Theorem	mptcfsupp 43930*	A mapping with value 0 except of one is finitely supported. (Contributed by AV, 9-Jun-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑅 = (Scalar‘𝑀)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑋, 1 , 0 ))    ⇒   ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝐹 finSupp 0 )
Theorem	fsuppmptdmf 43931*	A mapping with a finite domain is finitely supported. (Contributed by AV, 4-Sep-2019.)
⊢ Ⅎ𝑥𝜑    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝑌)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑊)    ⇒   ⊢ (𝜑 → 𝐹 finSupp 𝑍)
20.38.19.9  Left modules (extension)
Theorem	lmodvsmdi 43932	Multiple distributive law for scalar product (left-distributivity). (Contributed by AV, 5-Sep-2019.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ ↑ = (.g‘𝑊)    &   ⊢ 𝐸 = (.g‘𝐹)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑉)) → (𝑅 · (𝑁 ↑ 𝑋)) = ((𝑁𝐸𝑅) · 𝑋))
Theorem	gsumlsscl 43933*	Closure of a group sum in a linear subspace: A (finitely supported) sum of scalar multiplications of vectors of a subset of a linear subspace is also contained in the linear subspace. (Contributed by AV, 20-Apr-2019.) (Revised by AV, 28-Jul-2019.)
⊢ 𝑆 = (LSubSp‘𝑀)    &   ⊢ 𝑅 = (Scalar‘𝑀)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ ((𝑀 ∈ LMod ∧ 𝑍 ∈ 𝑆 ∧ 𝑉 ⊆ 𝑍) → ((𝐹 ∈ (𝐵 ↑𝑚 𝑉) ∧ 𝐹 finSupp (0g‘𝑅)) → (𝑀 Σg (𝑣 ∈ 𝑉 ↦ ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣))) ∈ 𝑍))
20.38.19.10  Associative algebras (extension)
Theorem	ascl1 43934	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)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑊 ∈ Ring)    ⇒   ⊢ (𝜑 → (𝐴‘(1r‘𝐹)) = (1r‘𝑊))
Theorem	assaascl0 43935	The scalar 0 embedded into an associative algebra corresponds to the 0 of the associative algebra. (Contributed by AV, 31-Jul-2019.)
⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ AssAlg)    ⇒   ⊢ (𝜑 → (𝐴‘(0g‘𝐹)) = (0g‘𝑊))
Theorem	assaascl1 43936	The scalar 1 embedded into an associative algebra corresponds to the 1 of the an associative algebra. (Contributed by AV, 31-Jul-2019.)
⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ AssAlg)    ⇒   ⊢ (𝜑 → (𝐴‘(1r‘𝐹)) = (1r‘𝑊))
20.38.19.11  Univariate polynomials (extension)
Theorem	ply1vr1smo 43937	The variable in a polynomial expressed as scaled monomial. (Contributed by AV, 12-Aug-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ 𝐺 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝐺)    &   ⊢ 𝑋 = (var1‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → ( 1 · (1 ↑ 𝑋)) = 𝑋)
Theorem	ply1ass23l 43938	Associative identity with scalar and ring multiplication for the polynomial ring. (Contributed by AV, 14-Aug-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ × = (.r‘𝑃)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑃)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)))
Theorem	ply1sclrmsm 43939	The ring multiplication of a polynomial with a scalar polynomial is equal to the scalar multiplication of the polynomial with the corresponding scalar. (Contributed by AV, 14-Aug-2019.)
⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐸 = (Base‘𝑃)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ × = (.r‘𝑃)    &   ⊢ 𝑁 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝑁)    &   ⊢ 𝐴 = (algSc‘𝑃)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐾 ∧ 𝑍 ∈ 𝐸) → ((𝐴‘𝐹) × 𝑍) = (𝐹 · 𝑍))
Theorem	coe1id 43940*	Coefficient vector of the unit polynomial. (Contributed by AV, 9-Aug-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐼 = (1r‘𝑃)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (coe1‘𝐼) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, 1 , 0 )))
Theorem	coe1sclmulval 43941	The value of the coefficient vector of a polynomial multiplied on the left by a scalar. (Contributed by AV, 14-Aug-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ 𝑆 = ( ·𝑠 ‘𝑃)    &   ⊢ ∙ = (.r‘𝑃)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ (𝑌 ∈ 𝐾 ∧ 𝑍 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) → ((coe1‘(𝑌𝑆𝑍))‘𝑁) = (𝑌 · ((coe1‘𝑍)‘𝑁)))
Theorem	ply1mulgsumlem1 43942*	Lemma 1 for ply1mulgsum 43946. (Contributed by AV, 19-Oct-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐴 = (coe1‘𝐾)    &   ⊢ 𝐶 = (coe1‘𝐿)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ × = (.r‘𝑃)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ ∗ = (.r‘𝑅)    &   ⊢ 𝑀 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝑀)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅))))
Theorem	ply1mulgsumlem2 43943*	Lemma 2 for ply1mulgsum 43946. (Contributed by AV, 19-Oct-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐴 = (coe1‘𝐾)    &   ⊢ 𝐶 = (coe1‘𝐿)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ × = (.r‘𝑃)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ ∗ = (.r‘𝑅)    &   ⊢ 𝑀 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝑀)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙))))) = (0g‘𝑅)))
Theorem	ply1mulgsumlem3 43944*	Lemma 3 for ply1mulgsum 43946. (Contributed by AV, 20-Oct-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐴 = (coe1‘𝐾)    &   ⊢ 𝐶 = (coe1‘𝐿)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ × = (.r‘𝑃)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ ∗ = (.r‘𝑅)    &   ⊢ 𝑀 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝑀)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ (𝑅 Σg (𝑙 ∈ (0...𝑘) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙)))))) finSupp (0g‘𝑅))
Theorem	ply1mulgsumlem4 43945*	Lemma 4 for ply1mulgsum 43946. (Contributed by AV, 19-Oct-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐴 = (coe1‘𝐾)    &   ⊢ 𝐶 = (coe1‘𝐿)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ × = (.r‘𝑃)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ ∗ = (.r‘𝑅)    &   ⊢ 𝑀 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝑀)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝑅 Σg (𝑙 ∈ (0...𝑘) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙))))) · (𝑘 ↑ 𝑋))) finSupp (0g‘𝑃))
Theorem	ply1mulgsum 43946*	The product of two polynomials expressed as group sum of scaled monomials. (Contributed by AV, 20-Oct-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐴 = (coe1‘𝐾)    &   ⊢ 𝐶 = (coe1‘𝐿)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ × = (.r‘𝑃)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ ∗ = (.r‘𝑅)    &   ⊢ 𝑀 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝑀)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → (𝐾 × 𝐿) = (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑅 Σg (𝑙 ∈ (0...𝑘) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙))))) · (𝑘 ↑ 𝑋)))))
Theorem	evl1at0 43947	Polynomial evaluation for the 0 scalar. (Contributed by AV, 10-Aug-2019.)
⊢ 𝑂 = (eval1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑍 = (0g‘𝑃)    ⇒   ⊢ (𝑅 ∈ CRing → ((𝑂‘𝑍)‘ 0 ) = 0 )
Theorem	evl1at1 43948	Polynomial evaluation for the 1 scalar. (Contributed by AV, 10-Aug-2019.)
⊢ 𝑂 = (eval1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝐼 = (1r‘𝑃)    ⇒   ⊢ (𝑅 ∈ CRing → ((𝑂‘𝐼)‘ 1 ) = 1 )
20.38.19.12  Univariate polynomials (examples)
Theorem	linply1 43949	A term of the form 𝑥 − 𝐶 is a (univariate) polynomial, also called "linear polynomial". (Part of ply1remlem 24443). (Contributed by AV, 3-Jul-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ − = (-g‘𝑃)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ 𝐺 = (𝑋 − (𝐴‘𝐶))    &   ⊢ (𝜑 → 𝐶 ∈ 𝐾)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    ⇒   ⊢ (𝜑 → 𝐺 ∈ 𝐵)
Theorem	lineval 43950	A term of the form 𝑥 − 𝐶 evaluated for 𝑥 = 𝑉 results in 𝑉 − 𝐶 (part of ply1remlem 24443). (Contributed by AV, 3-Jul-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ − = (-g‘𝑃)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ 𝐺 = (𝑋 − (𝐴‘𝐶))    &   ⊢ (𝜑 → 𝐶 ∈ 𝐾)    &   ⊢ 𝑂 = (eval1‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑉 ∈ 𝐾)    ⇒   ⊢ (𝜑 → ((𝑂‘𝐺)‘𝑉) = (𝑉(-g‘𝑅)𝐶))
Theorem	zringsubgval 43951	Subtraction in the ring of integers. (Contributed by AV, 3-Aug-2019.)
⊢ − = (-g‘ℤring)    ⇒   ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (𝑋 − 𝑌) = (𝑋 − 𝑌))
Theorem	linevalexample 43952	The polynomial 𝑥 − 3 over ℤ evaluated for 𝑥 = 5 results in 2. (Contributed by AV, 3-Jul-2019.)
⊢ 𝑃 = (Poly1‘ℤring)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝑋 = (var1‘ℤring)    &   ⊢ − = (-g‘𝑃)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ 𝐺 = (𝑋 − (𝐴‘3))    &   ⊢ 𝑂 = (eval1‘ℤring)    ⇒   ⊢ ((𝑂‘(𝑋 − (𝐴‘3)))‘5) = 2
20.38.20  Linear algebra (extension)
20.38.20.1  The subalgebras of diagonal and scalar matrices (extension) In the following, alternative definitions for diagonal and scalar matrices are provided. These definitions define diagonal and scalar matrices as extensible structures, whereas the definitions df-dmat 20787 and df-scmat 20788 define diagonal and scalar matrices as sets.
Syntax	cdmatalt 43953	Alternative notation for the algebra of diagonal matrices.
class DMatALT
Syntax	cscmatalt 43954	Alternative notation for the algebra of scalar matrices.
class ScMatALT
Definition	df-dmatalt 43955*	Define the set of n x n diagonal (square) matrices over a set (usually a ring) r, see definition in [Roman] p. 4 or Definition 3.12 in [Hefferon] p. 240. (Contributed by AV, 8-Dec-2019.)
⊢ DMatALT = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ⦋(𝑛 Mat 𝑟) / 𝑎⦌(𝑎 ↾s {𝑚 ∈ (Base‘𝑎) ∣ ∀𝑖 ∈ 𝑛 ∀𝑗 ∈ 𝑛 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = (0g‘𝑟))}))
Definition	df-scmatalt 43956*	Define the algebra of n x n scalar matrices over a set (usually a ring) r, see definition in [Connell] p. 57: "A scalar matrix is a diagonal matrix for which all the diagonal terms are equal, i.e., a matrix of the form cIn";. (Contributed by AV, 8-Dec-2019.)
⊢ ScMatALT = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ⦋(𝑛 Mat 𝑟) / 𝑎⦌(𝑎 ↾s {𝑚 ∈ (Base‘𝑎) ∣ ∃𝑐 ∈ (Base‘𝑟)∀𝑖 ∈ 𝑛 ∀𝑗 ∈ 𝑛 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑟))}))
Theorem	dmatALTval 43957*	The algebra of 𝑁 x 𝑁 diagonal matrices over a ring 𝑅. (Contributed by AV, 8-Dec-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐷 = (𝑁 DMatALT 𝑅)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → 𝐷 = (𝐴 ↾s {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )}))
Theorem	dmatALTbas 43958*	The base set of the algebra of 𝑁 x 𝑁 diagonal matrices over a ring 𝑅, i.e. the set of all 𝑁 x 𝑁 diagonal matrices over the ring 𝑅. (Contributed by AV, 8-Dec-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐷 = (𝑁 DMatALT 𝑅)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (Base‘𝐷) = {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )})
Theorem	dmatALTbasel 43959*	An element of the base set of the algebra of 𝑁 x 𝑁 diagonal matrices over a ring 𝑅, i.e. an 𝑁 x 𝑁 diagonal matrix over the ring 𝑅. (Contributed by AV, 8-Dec-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐷 = (𝑁 DMatALT 𝑅)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (𝑀 ∈ (Base‘𝐷) ↔ (𝑀 ∈ 𝐵 ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 ))))
Theorem	dmatbas 43960	The set of all 𝑁 x 𝑁 diagonal matrices over (the ring) 𝑅 is the base set of the algebra of 𝑁 x 𝑁 diagonal matrices over (the ring) 𝑅. (Contributed by AV, 8-Dec-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐷 = (𝑁 DMat 𝑅)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐷 = (Base‘(𝑁 DMatALT 𝑅)))
20.38.20.2  Linear combinations According to Wikipedia ("Linear combination", 29-Mar-2019, https://en.wikipedia.org/wiki/Linear_combination) "In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g., a linear combination of x and y would be any expression of the form ax + by, where a and b are constants). The concept of linear combinations is central to linear algebra and related fields of mathematics." In linear algebra, these "terms" are "vectors" (elements from vector spaces or left modules), and the constants are elements of the underlying field resp. ring. This corresponds to the definition in [Lang] p. 129: "Let M be a module over a ring A and let S be a subset of M. By a linear combination of elements of S (with coefficients in A) one means a sum ∑x ∈S axx where {ax} is a set of elements of A, ...". In the definition in [Lang] p. 129, it is additionally claimed that "..., almost all of which [elements of A] are equal to 0.". This is not necessarily required in the following definition df-linc 43963, 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 43964, so that we can show that such sets are submodules of the corresponding modules, see lincolss 43991. Remark:According to Wikipedia ("Linear span", 28-Apr-2019, https://en.wikipedia.org/wiki/Linear_span 43991) "In linear algebra, the linear span (also called the linear hull or just span) of a set of vectors in a vector space [or module] is the intersection of all linear subspaces which each contain every vector in that set.", and "Alternately, the span of [a set] S may be defined as the set of all finite linear combinations of elements (vectors) of S". Whereas spans are defined according to the first approach in df-lsp 19438, the set of all linear combinations as defined by df-lco 43964 follows the alternative approach. That both definitions are equivalent is shown by lspeqlco 43996.
Syntax	clinc 43961	Extend class notation with the operation constructing a linear combination (of vectors from a left module).
class linC
Syntax	clinco 43962	Extend class notation with the operation constructing a set of linear combinations (of vectors from a left module) with finite support.
class LinCo
Definition	df-linc 43963*	Define the operation constructing a linear combination. Although this definition is taylored for linear combinations of vectors from left modules, it can be used for any structure having a Base, Scalar s and a scalar multiplication ·𝑠. (Contributed by AV, 29-Mar-2019.)
⊢ linC = (𝑚 ∈ V ↦ (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ (𝑚 Σg (𝑥 ∈ 𝑣 ↦ ((𝑠‘𝑥)( ·𝑠 ‘𝑚)𝑥)))))
Definition	df-lco 43964*	Define the operation constructing the set of all linear combinations for a set of vectors. (Contributed by AV, 31-Mar-2019.) (Revised by AV, 28-Jul-2019.)
⊢ LinCo = (𝑚 ∈ V, 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ {𝑐 ∈ (Base‘𝑚) ∣ ∃𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣)(𝑠 finSupp (0g‘(Scalar‘𝑚)) ∧ 𝑐 = (𝑠( linC ‘𝑚)𝑣))})
Theorem	lincop 43965*	A linear combination as operation. (Contributed by AV, 30-Mar-2019.)
⊢ (𝑀 ∈ 𝑋 → ( linC ‘𝑀) = (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥 ∈ 𝑣 ↦ ((𝑠‘𝑥)( ·𝑠 ‘𝑀)𝑥)))))
Theorem	lincval 43966*	The value of a linear combination. (Contributed by AV, 30-Mar-2019.)
⊢ ((𝑀 ∈ 𝑋 ∧ 𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑆( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑥 ∈ 𝑉 ↦ ((𝑆‘𝑥)( ·𝑠 ‘𝑀)𝑥))))
Theorem	dflinc2 43967*	Alternative definition of linear combinations using the function operation. (Contributed by AV, 1-Apr-2019.)
⊢ linC = (𝑚 ∈ V ↦ (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ (𝑚 Σg (𝑠 ∘𝑓 ( ·𝑠 ‘𝑚)( I ↾ 𝑣)))))
Theorem	lcoop 43968*	A linear combination as operation. (Contributed by AV, 5-Apr-2019.) (Revised by AV, 28-Jul-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑆 = (Scalar‘𝑀)    &   ⊢ 𝑅 = (Base‘𝑆)    ⇒   ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 LinCo 𝑉) = {𝑐 ∈ 𝐵 ∣ ∃𝑠 ∈ (𝑅 ↑𝑚 𝑉)(𝑠 finSupp (0g‘𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉))})
Theorem	lcoval 43969*	The value of a linear combination. (Contributed by AV, 5-Apr-2019.) (Revised by AV, 28-Jul-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑆 = (Scalar‘𝑀)    &   ⊢ 𝑅 = (Base‘𝑆)    ⇒   ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐶 ∈ (𝑀 LinCo 𝑉) ↔ (𝐶 ∈ 𝐵 ∧ ∃𝑠 ∈ (𝑅 ↑𝑚 𝑉)(𝑠 finSupp (0g‘𝑆) ∧ 𝐶 = (𝑠( linC ‘𝑀)𝑉)))))
Theorem	lincfsuppcl 43970	A linear combination of vectors (with finite support) is a vector. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 28-Jul-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑅 = (Scalar‘𝑀)    &   ⊢ 𝑆 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑀 ∈ LMod ∧ (𝑉 ∈ 𝑊 ∧ 𝑉 ⊆ 𝐵) ∧ (𝐹 ∈ (𝑆 ↑𝑚 𝑉) ∧ 𝐹 finSupp 0 )) → (𝐹( linC ‘𝑀)𝑉) ∈ 𝐵)
Theorem	linccl 43971	A linear combination of vectors is a vector. (Contributed by AV, 31-Mar-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑅 = (Base‘(Scalar‘𝑀))    ⇒   ⊢ ((𝑀 ∈ LMod ∧ (𝑉 ∈ Fin ∧ 𝑉 ⊆ 𝐵 ∧ 𝑆 ∈ (𝑅 ↑𝑚 𝑉))) → (𝑆( linC ‘𝑀)𝑉) ∈ 𝐵)
Theorem	lincval0 43972	The value of an empty linear combination. (Contributed by AV, 12-Apr-2019.)
⊢ (𝑀 ∈ 𝑋 → (∅( linC ‘𝑀)∅) = (0g‘𝑀))
Theorem	lincvalsng 43973	The linear combination over a singleton. (Contributed by AV, 25-May-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑆 = (Scalar‘𝑀)    &   ⊢ 𝑅 = (Base‘𝑆)    &   ⊢ · = ( ·𝑠 ‘𝑀)    ⇒   ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → ({⟨𝑉, 𝑌⟩} ( linC ‘𝑀){𝑉}) = (𝑌 · 𝑉))
Theorem	lincvalsn 43974	The linear combination over a singleton. (Contributed by AV, 12-Apr-2019.) (Proof shortened by AV, 25-May-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑆 = (Scalar‘𝑀)    &   ⊢ 𝑅 = (Base‘𝑆)    &   ⊢ · = ( ·𝑠 ‘𝑀)    &   ⊢ 𝐹 = {⟨𝑉, 𝑌⟩}    ⇒   ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝐵 ∧ 𝑌 ∈ 𝑅) → (𝐹( linC ‘𝑀){𝑉}) = (𝑌 · 𝑉))
Theorem	lincvalpr 43975	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 43976	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 43977	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 43978*	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 43979*	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 43980*	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 43981	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 43982*	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 43983	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 43984	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 43985	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 43986	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 43987*	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 43988	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 43989	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 43990	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 43991	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 43992*	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 43993	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 43994	Lemma for lspeqlco 43996. (Contributed by AV, 17-Apr-2019.)
⊢ 𝐵 = (Base‘𝑀)    ⇒   ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → ((LSpan‘𝑀)‘𝑉) ⊆ (𝑀 LinCo 𝑉))
Theorem	lcosslsp 43995	Lemma for lspeqlco 43996. (Contributed by AV, 20-Apr-2019.)
⊢ 𝐵 = (Base‘𝑀)    ⇒   ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 LinCo 𝑉) ⊆ ((LSpan‘𝑀)‘𝑉))
Theorem	lspeqlco 43996	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 19438) 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))
20.38.20.3  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 43999 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 43999): "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 20636) and (linearly) independent sets (df-linds 20637). 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 19547) "every element is not in the span of the remainder of the [set]". The equivalence of the definitions df-linds 20637 and df-lininds 43999 for (linear) independence for (left) modules is shown in lindslininds 44021. Remark 2: Subsets of the base set of a (left) module are linearly dependent if they are not linearly independent (see df-lindeps 44001) 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 44040. 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 44042) and not for (left) modules in general.
Syntax	clininds 43997	Extend class notation with the relation between a module and its linearly independent subsets.
class linIndS
Syntax	clindeps 43998	Extend class notation with the relation between a module and its linearly dependent subsets.
class linDepS
Definition	df-lininds 43999*	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 44000	The class defining the relation between a module and its linearly independent subsets is a relation. (Contributed by AV, 13-Apr-2019.)
⊢ Rel linIndS