P. 218 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 21701-21800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	uvcvvcl 21701	A coordinate of a unit vector is either 0 or 1. (Contributed by Stefan O'Rear, 3-Feb-2015.)
⊢ 𝑈 = (𝑅 unitVec 𝐼)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) ∧ 𝐾 ∈ 𝐼) → ((𝑈‘𝐽)‘𝐾) ∈ { 0 , 1 })
Theorem	uvcvvcl2 21702	A unit vector coordinate is a ring element. (Contributed by Stefan O'Rear, 3-Feb-2015.)
⊢ 𝑈 = (𝑅 unitVec 𝐼)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐼)    &   ⊢ (𝜑 → 𝐾 ∈ 𝐼)    ⇒   ⊢ (𝜑 → ((𝑈‘𝐽)‘𝐾) ∈ 𝐵)
Theorem	uvcvv1 21703	The unit vector is one at its designated coordinate. (Contributed by Stefan O'Rear, 3-Feb-2015.)
⊢ 𝑈 = (𝑅 unitVec 𝐼)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐼)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝜑 → ((𝑈‘𝐽)‘𝐽) = 1 )
Theorem	uvcvv0 21704	The unit vector is zero at its designated coordinate. (Contributed by Stefan O'Rear, 3-Feb-2015.)
⊢ 𝑈 = (𝑅 unitVec 𝐼)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐼)    &   ⊢ (𝜑 → 𝐾 ∈ 𝐼)    &   ⊢ (𝜑 → 𝐽 ≠ 𝐾)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝜑 → ((𝑈‘𝐽)‘𝐾) = 0 )
Theorem	uvcff 21705	Domain and codomain of the unit vector generator; ring condition required to be sure 1 and 0 are actually in the ring. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.)
⊢ 𝑈 = (𝑅 unitVec 𝐼)    &   ⊢ 𝑌 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝑌)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝑈:𝐼⟶𝐵)
Theorem	uvcf1 21706	In a nonzero ring, each unit vector is different. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.)
⊢ 𝑈 = (𝑅 unitVec 𝐼)    &   ⊢ 𝑌 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝑌)    ⇒   ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊) → 𝑈:𝐼–1-1→𝐵)
Theorem	uvcresum 21707	Any element of a free module can be expressed as a finite linear combination of unit vectors. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Proof shortened by Mario Carneiro, 5-Jul-2015.)
⊢ 𝑈 = (𝑅 unitVec 𝐼)    &   ⊢ 𝑌 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ · = ( ·𝑠 ‘𝑌)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵) → 𝑋 = (𝑌 Σg (𝑋 ∘f · 𝑈)))
Theorem	frlmssuvc1 21708*	A scalar multiple of a unit vector included in a support-restriction subspace is included in the subspace. (Contributed by Stefan O'Rear, 5-Feb-2015.) (Revised by AV, 24-Jun-2019.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝑈 = (𝑅 unitVec 𝐼)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝐹)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐶 = {𝑥 ∈ 𝐵 ∣ (𝑥 supp 0 ) ⊆ 𝐽}    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐽 ⊆ 𝐼)    &   ⊢ (𝜑 → 𝐿 ∈ 𝐽)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐾)    ⇒   ⊢ (𝜑 → (𝑋 · (𝑈‘𝐿)) ∈ 𝐶)
Theorem	frlmssuvc2 21709*	A nonzero scalar multiple of a unit vector not included in a support-restriction subspace is not included in the subspace. (Contributed by Stefan O'Rear, 5-Feb-2015.) (Revised by AV, 24-Jun-2019.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝑈 = (𝑅 unitVec 𝐼)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝐹)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐶 = {𝑥 ∈ 𝐵 ∣ (𝑥 supp 0 ) ⊆ 𝐽}    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐽 ⊆ 𝐼)    &   ⊢ (𝜑 → 𝐿 ∈ (𝐼 ∖ 𝐽))    &   ⊢ (𝜑 → 𝑋 ∈ (𝐾 ∖ { 0 }))    ⇒   ⊢ (𝜑 → ¬ (𝑋 · (𝑈‘𝐿)) ∈ 𝐶)
Theorem	frlmsslsp 21710*	A subset of a free module obtained by restricting the support set is spanned by the relevant unit vectors. (Contributed by Stefan O'Rear, 6-Feb-2015.) (Revised by AV, 24-Jun-2019.)
⊢ 𝑌 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝑈 = (𝑅 unitVec 𝐼)    &   ⊢ 𝐾 = (LSpan‘𝑌)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐶 = {𝑥 ∈ 𝐵 ∣ (𝑥 supp 0 ) ⊆ 𝐽}    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (𝐾‘(𝑈 “ 𝐽)) = 𝐶)
Theorem	frlmlbs 21711	The unit vectors comprise a basis for a free module. (Contributed by Stefan O'Rear, 6-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝑈 = (𝑅 unitVec 𝐼)    &   ⊢ 𝐽 = (LBasis‘𝐹)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → ran 𝑈 ∈ 𝐽)
Theorem	frlmup1 21712*	Any assignment of unit vectors to target vectors can be extended (uniquely) to a homomorphism from a free module to an arbitrary other module on the same base ring. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ 𝐶 = (Base‘𝑇)    &   ⊢ · = ( ·𝑠 ‘𝑇)    &   ⊢ 𝐸 = (𝑥 ∈ 𝐵 ↦ (𝑇 Σg (𝑥 ∘f · 𝐴)))    &   ⊢ (𝜑 → 𝑇 ∈ LMod)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑅 = (Scalar‘𝑇))    &   ⊢ (𝜑 → 𝐴:𝐼⟶𝐶)    ⇒   ⊢ (𝜑 → 𝐸 ∈ (𝐹 LMHom 𝑇))
Theorem	frlmup2 21713*	The evaluation map has the intended behavior on the unit vectors. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ 𝐶 = (Base‘𝑇)    &   ⊢ · = ( ·𝑠 ‘𝑇)    &   ⊢ 𝐸 = (𝑥 ∈ 𝐵 ↦ (𝑇 Σg (𝑥 ∘f · 𝐴)))    &   ⊢ (𝜑 → 𝑇 ∈ LMod)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑅 = (Scalar‘𝑇))    &   ⊢ (𝜑 → 𝐴:𝐼⟶𝐶)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐼)    &   ⊢ 𝑈 = (𝑅 unitVec 𝐼)    ⇒   ⊢ (𝜑 → (𝐸‘(𝑈‘𝑌)) = (𝐴‘𝑌))
Theorem	frlmup3 21714*	The range of such an evaluation map is the finite linear combinations of the target vectors and also the span of the target vectors. (Contributed by Stefan O'Rear, 6-Feb-2015.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ 𝐶 = (Base‘𝑇)    &   ⊢ · = ( ·𝑠 ‘𝑇)    &   ⊢ 𝐸 = (𝑥 ∈ 𝐵 ↦ (𝑇 Σg (𝑥 ∘f · 𝐴)))    &   ⊢ (𝜑 → 𝑇 ∈ LMod)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑅 = (Scalar‘𝑇))    &   ⊢ (𝜑 → 𝐴:𝐼⟶𝐶)    &   ⊢ 𝐾 = (LSpan‘𝑇)    ⇒   ⊢ (𝜑 → ran 𝐸 = (𝐾‘ran 𝐴))
Theorem	frlmup4 21715*	Universal property of the free module by existential uniqueness. (Contributed by Stefan O'Rear, 7-Mar-2015.)
⊢ 𝑅 = (Scalar‘𝑇)    &   ⊢ 𝐹 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝑈 = (𝑅 unitVec 𝐼)    &   ⊢ 𝐶 = (Base‘𝑇)    ⇒   ⊢ ((𝑇 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐴:𝐼⟶𝐶) → ∃!𝑚 ∈ (𝐹 LMHom 𝑇)(𝑚 ∘ 𝑈) = 𝐴)
Theorem	ellspd 21716*	The elements of the span of an indexed collection of basic vectors are those vectors which can be written as finite linear combinations of basic vectors. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Revised by AV, 24-Jun-2019.) (Revised by AV, 11-Apr-2024.)
⊢ 𝑁 = (LSpan‘𝑀)    &   ⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝑆 = (Scalar‘𝑀)    &   ⊢ 0 = (0g‘𝑆)    &   ⊢ · = ( ·𝑠 ‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝐼⟶𝐵)    &   ⊢ (𝜑 → 𝑀 ∈ LMod)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(𝐹 “ 𝐼)) ↔ ∃𝑓 ∈ (𝐾 ↑m 𝐼)(𝑓 finSupp 0 ∧ 𝑋 = (𝑀 Σg (𝑓 ∘f · 𝐹)))))
Theorem	elfilspd 21717*	Simplified version of ellspd 21716 when the spanning set is finite: all linear combinations are then acceptable. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.)
⊢ 𝑁 = (LSpan‘𝑀)    &   ⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝑆 = (Scalar‘𝑀)    &   ⊢ 0 = (0g‘𝑆)    &   ⊢ · = ( ·𝑠 ‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝐼⟶𝐵)    &   ⊢ (𝜑 → 𝑀 ∈ LMod)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    ⇒   ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(𝐹 “ 𝐼)) ↔ ∃𝑓 ∈ (𝐾 ↑m 𝐼)𝑋 = (𝑀 Σg (𝑓 ∘f · 𝐹))))
11.1.4  Independent sets and families According to the definition in [Lang] p. 129: "A subset S of a module M is said to be linearly independent (over A) if whenever we have a linear combination ∑x∈Saxx which is equal to 0, then ax = 0 for all x ∈ S", and according to the Definition in [Lang] p. 130: "a familiy {xi}i∈I of elements of M is said to be linearly independent (over A) if whenever we have a linear combination ∑i∈Iaixi = 0, then ai = 0 for all i ∈ I." These definitions correspond to Definitions df-linds 21721 and df-lindf 21720 respectively, where it is claimed that a nonzero summand can be extracted (∑i∈{I\{j}}aixi = -ajxj) and be represented as a linear combination of the remaining elements of the family.
Syntax	clindf 21718	The class relationship of independent families in a module.
class LIndF
Syntax	clinds 21719	The class generator of independent sets in a module.
class LIndS
Definition	df-lindf 21720*	An independent family is a family of vectors, no nonzero multiple of which can be expressed as a linear combination of other elements of the family. This is almost, but not quite, the same as a function into an independent set. This is a defined concept because it matters in many cases whether independence is taken at a set or family level. For instance, a number is transcedental iff its nonzero powers are linearly independent. Is 1 transcedental? It has only one nonzero power. We can almost define family independence as a family of unequal elements with independent range, as islindf3 21740, but taking that as primitive would lead to unpleasant corner case behavior with the zero ring. This is equivalent to the common definition of having no nontrivial representations of zero (islindf4 21752) and only one representation for each element of the range (islindf5 21753). (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ LIndF = {⟨𝑓, 𝑤⟩ ∣ (𝑓:dom 𝑓⟶(Base‘𝑤) ∧ [(Scalar‘𝑤) / 𝑠]∀𝑥 ∈ dom 𝑓∀𝑘 ∈ ((Base‘𝑠) ∖ {(0g‘𝑠)}) ¬ (𝑘( ·𝑠 ‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))))}
Definition	df-linds 21721*	An independent set is a set which is independent as a family. See also islinds3 21748 and islinds4 21749. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ LIndS = (𝑤 ∈ V ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ( I ↾ 𝑠) LIndF 𝑤})
Theorem	rellindf 21722	The independent-family predicate is a proper relation and can be used with brrelex1i 5728. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ Rel LIndF
Theorem	islinds 21723	Property of an independent set of vectors in terms of an independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ (𝑊 ∈ 𝑉 → (𝑋 ∈ (LIndS‘𝑊) ↔ (𝑋 ⊆ 𝐵 ∧ ( I ↾ 𝑋) LIndF 𝑊)))
Theorem	linds1 21724	An independent set of vectors is a set of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ (𝑋 ∈ (LIndS‘𝑊) → 𝑋 ⊆ 𝐵)
Theorem	linds2 21725	An independent set of vectors is independent as a family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ (𝑋 ∈ (LIndS‘𝑊) → ( I ↾ 𝑋) LIndF 𝑊)
Theorem	islindf 21726*	Property of an independent family of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐾 = (LSpan‘𝑊)    &   ⊢ 𝑆 = (Scalar‘𝑊)    &   ⊢ 𝑁 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑆)    ⇒   ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐹 ∈ 𝑋) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹⟶𝐵 ∧ ∀𝑥 ∈ dom 𝐹∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹‘𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))))
Theorem	islinds2 21727*	Expanded property of an independent set of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐾 = (LSpan‘𝑊)    &   ⊢ 𝑆 = (Scalar‘𝑊)    &   ⊢ 𝑁 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑆)    ⇒   ⊢ (𝑊 ∈ 𝑌 → (𝐹 ∈ (LIndS‘𝑊) ↔ (𝐹 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐹 ∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · 𝑥) ∈ (𝐾‘(𝐹 ∖ {𝑥})))))
Theorem	islindf2 21728*	Property of an independent family of vectors with prior constrained domain and codomain. (Contributed by Stefan O'Rear, 26-Feb-2015.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐾 = (LSpan‘𝑊)    &   ⊢ 𝑆 = (Scalar‘𝑊)    &   ⊢ 𝑁 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑆)    ⇒   ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) → (𝐹 LIndF 𝑊 ↔ ∀𝑥 ∈ 𝐼 ∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹‘𝑥)) ∈ (𝐾‘(𝐹 “ (𝐼 ∖ {𝑥})))))
Theorem	lindff 21729	Functional property of a linearly independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ ((𝐹 LIndF 𝑊 ∧ 𝑊 ∈ 𝑌) → 𝐹:dom 𝐹⟶𝐵)
Theorem	lindfind 21730	A linearly independent family is independent: no nonzero element multiple can be expressed as a linear combination of the others. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ 𝐿 = (Scalar‘𝑊)    &   ⊢ 0 = (0g‘𝐿)    &   ⊢ 𝐾 = (Base‘𝐿)    ⇒   ⊢ (((𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → ¬ (𝐴 · (𝐹‘𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸}))))
Theorem	lindsind 21731	A linearly independent set is independent: no nonzero element multiple can be expressed as a linear combination of the others. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ 𝐿 = (Scalar‘𝑊)    &   ⊢ 0 = (0g‘𝐿)    &   ⊢ 𝐾 = (Base‘𝐿)    ⇒   ⊢ (((𝐹 ∈ (LIndS‘𝑊) ∧ 𝐸 ∈ 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐹 ∖ {𝐸})))
Theorem	lindfind2 21732	In a linearly independent family in a module over a nonzero ring, no element is contained in the span of any non-containing set. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ 𝐾 = (LSpan‘𝑊)    &   ⊢ 𝐿 = (Scalar‘𝑊)    ⇒   ⊢ (((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) ∧ 𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) → ¬ (𝐹‘𝐸) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝐸}))))
Theorem	lindsind2 21733	In a linearly independent set in a module over a nonzero ring, no element is contained in the span of any non-containing set. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ 𝐾 = (LSpan‘𝑊)    &   ⊢ 𝐿 = (Scalar‘𝑊)    ⇒   ⊢ (((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝐸 ∈ 𝐹) → ¬ 𝐸 ∈ (𝐾‘(𝐹 ∖ {𝐸})))
Theorem	lindff1 21734	A linearly independent family over a nonzero ring has no repeated elements. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐿 = (Scalar‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing ∧ 𝐹 LIndF 𝑊) → 𝐹:dom 𝐹–1-1→𝐵)
Theorem	lindfrn 21735	The range of an independent family is an independent set. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → ran 𝐹 ∈ (LIndS‘𝑊))
Theorem	f1lindf 21736	Rearranging and deleting elements from an independent family gives an independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (𝐹 ∘ 𝐺) LIndF 𝑊)
Theorem	lindfres 21737	Any restriction of an independent family is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → (𝐹 ↾ 𝑋) LIndF 𝑊)
Theorem	lindsss 21738	Any subset of an independent set is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ ((𝑊 ∈ LMod ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝐺 ⊆ 𝐹) → 𝐺 ∈ (LIndS‘𝑊))
Theorem	f1linds 21739	A family constructed from non-repeated elements of an independent set is independent. (Contributed by Stefan O'Rear, 26-Feb-2015.)
⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ (LIndS‘𝑊) ∧ 𝐹:𝐷–1-1→𝑆) → 𝐹 LIndF 𝑊)
Theorem	islindf3 21740	In a nonzero ring, independent families can be equivalently characterized as renamings of independent sets. (Contributed by Stefan O'Rear, 26-Feb-2015.)
⊢ 𝐿 = (Scalar‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹–1-1→V ∧ ran 𝐹 ∈ (LIndS‘𝑊))))
Theorem	lindfmm 21741	Linear independence of a family is unchanged by injective linear functions. (Contributed by Stefan O'Rear, 26-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 𝐶 = (Base‘𝑇)    ⇒   ⊢ ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶 ∧ 𝐹:𝐼⟶𝐵) → (𝐹 LIndF 𝑆 ↔ (𝐺 ∘ 𝐹) LIndF 𝑇))
Theorem	lindsmm 21742	Linear independence of a set is unchanged by injective linear functions. (Contributed by Stefan O'Rear, 26-Feb-2015.)
⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 𝐶 = (Base‘𝑇)    ⇒   ⊢ ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶 ∧ 𝐹 ⊆ 𝐵) → (𝐹 ∈ (LIndS‘𝑆) ↔ (𝐺 “ 𝐹) ∈ (LIndS‘𝑇)))
Theorem	lindsmm2 21743	The monomorphic image of an independent set is independent. (Contributed by Stefan O'Rear, 26-Feb-2015.)
⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 𝐶 = (Base‘𝑇)    ⇒   ⊢ ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶 ∧ 𝐹 ∈ (LIndS‘𝑆)) → (𝐺 “ 𝐹) ∈ (LIndS‘𝑇))
Theorem	lsslindf 21744	Linear independence is unchanged by working in a subspace. (Contributed by Stefan O'Rear, 24-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
⊢ 𝑈 = (LSubSp‘𝑊)    &   ⊢ 𝑋 = (𝑊 ↾s 𝑆)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) → (𝐹 LIndF 𝑋 ↔ 𝐹 LIndF 𝑊))
Theorem	lsslinds 21745	Linear independence is unchanged by working in a subspace. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ 𝑈 = (LSubSp‘𝑊)    &   ⊢ 𝑋 = (𝑊 ↾s 𝑆)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ 𝐹 ⊆ 𝑆) → (𝐹 ∈ (LIndS‘𝑋) ↔ 𝐹 ∈ (LIndS‘𝑊)))
Theorem	islbs4 21746	A basis is an independent spanning set. This could have been used as alternative definition of a basis: LBasis = (𝑤 ∈ V ↦ {𝑏 ∈ 𝒫 (Base‘𝑤) ∣ (((LSpan‘𝑤) ‘𝑏) = (Base‘𝑤) ∧ 𝑏 ∈ (LIndS‘𝑤))}). (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐽 = (LBasis‘𝑊)    &   ⊢ 𝐾 = (LSpan‘𝑊)    ⇒   ⊢ (𝑋 ∈ 𝐽 ↔ (𝑋 ∈ (LIndS‘𝑊) ∧ (𝐾‘𝑋) = 𝐵))
Theorem	lbslinds 21747	A basis is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ 𝐽 = (LBasis‘𝑊)    ⇒   ⊢ 𝐽 ⊆ (LIndS‘𝑊)
Theorem	islinds3 21748	A subset is linearly independent iff it is a basis of its span. (Contributed by Stefan O'Rear, 25-Feb-2015.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐾 = (LSpan‘𝑊)    &   ⊢ 𝑋 = (𝑊 ↾s (𝐾‘𝑌))    &   ⊢ 𝐽 = (LBasis‘𝑋)    ⇒   ⊢ (𝑊 ∈ LMod → (𝑌 ∈ (LIndS‘𝑊) ↔ 𝑌 ∈ 𝐽))
Theorem	islinds4 21749*	A set is independent in a vector space iff it is a subset of some basis. This is an axiom of choice equivalent. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ 𝐽 = (LBasis‘𝑊)    ⇒   ⊢ (𝑊 ∈ LVec → (𝑌 ∈ (LIndS‘𝑊) ↔ ∃𝑏 ∈ 𝐽 𝑌 ⊆ 𝑏))
11.1.5  Characterization of free modules
Theorem	lmimlbs 21750	The isomorphic image of a basis is a basis. (Contributed by Stefan O'Rear, 26-Feb-2015.)
⊢ 𝐽 = (LBasis‘𝑆)    &   ⊢ 𝐾 = (LBasis‘𝑇)    ⇒   ⊢ ((𝐹 ∈ (𝑆 LMIso 𝑇) ∧ 𝐵 ∈ 𝐽) → (𝐹 “ 𝐵) ∈ 𝐾)
Theorem	lmiclbs 21751	Having a basis is an isomorphism invariant. (Contributed by Stefan O'Rear, 26-Feb-2015.)
⊢ 𝐽 = (LBasis‘𝑆)    &   ⊢ 𝐾 = (LBasis‘𝑇)    ⇒   ⊢ (𝑆 ≃𝑚 𝑇 → (𝐽 ≠ ∅ → 𝐾 ≠ ∅))
Theorem	islindf4 21752*	A family is independent iff it has no nontrivial representations of zero. (Contributed by Stefan O'Rear, 28-Feb-2015.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝑅 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑌 = (0g‘𝑅)    &   ⊢ 𝐿 = (Base‘(𝑅 freeLMod 𝐼))    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) → (𝐹 LIndF 𝑊 ↔ ∀𝑥 ∈ 𝐿 ((𝑊 Σg (𝑥 ∘f · 𝐹)) = 0 → 𝑥 = (𝐼 × {𝑌}))))
Theorem	islindf5 21753*	A family is independent iff the linear combinations homomorphism is injective. (Contributed by Stefan O'Rear, 28-Feb-2015.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ 𝐶 = (Base‘𝑇)    &   ⊢ · = ( ·𝑠 ‘𝑇)    &   ⊢ 𝐸 = (𝑥 ∈ 𝐵 ↦ (𝑇 Σg (𝑥 ∘f · 𝐴)))    &   ⊢ (𝜑 → 𝑇 ∈ LMod)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑅 = (Scalar‘𝑇))    &   ⊢ (𝜑 → 𝐴:𝐼⟶𝐶)    ⇒   ⊢ (𝜑 → (𝐴 LIndF 𝑇 ↔ 𝐸:𝐵–1-1→𝐶))
Theorem	indlcim 21754*	An independent, spanning family extends to an isomorphism from a free module. (Contributed by Stefan O'Rear, 26-Feb-2015.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ 𝐶 = (Base‘𝑇)    &   ⊢ · = ( ·𝑠 ‘𝑇)    &   ⊢ 𝑁 = (LSpan‘𝑇)    &   ⊢ 𝐸 = (𝑥 ∈ 𝐵 ↦ (𝑇 Σg (𝑥 ∘f · 𝐴)))    &   ⊢ (𝜑 → 𝑇 ∈ LMod)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑅 = (Scalar‘𝑇))    &   ⊢ (𝜑 → 𝐴:𝐼–onto→𝐽)    &   ⊢ (𝜑 → 𝐴 LIndF 𝑇)    &   ⊢ (𝜑 → (𝑁‘𝐽) = 𝐶)    ⇒   ⊢ (𝜑 → 𝐸 ∈ (𝐹 LMIso 𝑇))
Theorem	lbslcic 21755	A module with a basis is isomorphic to a free module with the same cardinality. (Contributed by Stefan O'Rear, 26-Feb-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐽 = (LBasis‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) → 𝑊 ≃𝑚 (𝐹 freeLMod 𝐼))
Theorem	lmisfree 21756*	A module has a basis iff it is isomorphic to a free module. In settings where isomorphic objects are not distinguished, it is common to define "free module" as any module with a basis; thus for instance lbsex 21035 might be described as "every vector space is free". (Contributed by Stefan O'Rear, 26-Feb-2015.)
⊢ 𝐽 = (LBasis‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ (𝑊 ∈ LMod → (𝐽 ≠ ∅ ↔ ∃𝑘 𝑊 ≃𝑚 (𝐹 freeLMod 𝑘)))
Theorem	lvecisfrlm 21757*	Every vector space is isomorphic to a free module. (Contributed by AV, 7-Mar-2019.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ (𝑊 ∈ LVec → ∃𝑘 𝑊 ≃𝑚 (𝐹 freeLMod 𝑘))
Theorem	lmimco 21758	The composition of two isomorphisms of modules is an isomorphism of modules. (Contributed by AV, 10-Mar-2019.)
⊢ ((𝐹 ∈ (𝑆 LMIso 𝑇) ∧ 𝐺 ∈ (𝑅 LMIso 𝑆)) → (𝐹 ∘ 𝐺) ∈ (𝑅 LMIso 𝑇))
Theorem	lmictra 21759	Module isomorphism is transitive. (Contributed by AV, 10-Mar-2019.)
⊢ ((𝑅 ≃𝑚 𝑆 ∧ 𝑆 ≃𝑚 𝑇) → 𝑅 ≃𝑚 𝑇)
Theorem	uvcf1o 21760	In a nonzero ring, the mapping of the index set of a free module onto the unit vectors of the free module is a 1-1 onto function. (Contributed by AV, 10-Mar-2019.)
⊢ 𝑈 = (𝑅 unitVec 𝐼)    ⇒   ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊) → 𝑈:𝐼–1-1-onto→ran 𝑈)
Theorem	uvcendim 21761	In a nonzero ring, the number of unit vectors of a free module corresponds to the dimension of the free module. (Contributed by AV, 10-Mar-2019.)
⊢ 𝑈 = (𝑅 unitVec 𝐼)    ⇒   ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊) → 𝐼 ≈ ran 𝑈)
Theorem	frlmisfrlm 21762	A free module is isomorphic to a free module over the same (nonzero) ring, with the same cardinality. (Contributed by AV, 10-Mar-2019.)
⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑌 ∧ 𝐼 ≈ 𝐽) → (𝑅 freeLMod 𝐼) ≃𝑚 (𝑅 freeLMod 𝐽))
Theorem	frlmiscvec 21763	Every free module is isomorphic to the free module of "column vectors" of the same dimension over the same (nonzero) ring. (Contributed by AV, 10-Mar-2019.)
⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑌) → (𝑅 freeLMod 𝐼) ≃𝑚 (𝑅 freeLMod (𝐼 × {∅})))
11.2  Associative algebras
11.2.1  Definition and basic properties
Syntax	casa 21764	Associative algebra.
class AssAlg
Syntax	casp 21765	Algebraic span function.
class AlgSpan
Syntax	cascl 21766	Class of algebra scalar injection function.
class algSc
Definition	df-assa 21767*	Definition of an associative algebra. An associative algebra is a set equipped with a left-module structure on a ring, coupled with a multiplicative internal operation on the vectors of the module that is associative and distributive for the additive structure of the left-module (so giving the vectors a ring structure) and that is also bilinear under the scalar product. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by SN, 2-Mar-2025.)
⊢ AssAlg = {𝑤 ∈ (LMod ∩ Ring) ∣ [(Scalar‘𝑤) / 𝑓]∀𝑟 ∈ (Base‘𝑓)∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑤)[( ·𝑠 ‘𝑤) / 𝑠][(.r‘𝑤) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦)))}
Definition	df-asp 21768*	Define the algebraic span of a set of vectors in an algebra. (Contributed by Mario Carneiro, 7-Jan-2015.)
⊢ AlgSpan = (𝑤 ∈ AssAlg ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ ∩ {𝑡 ∈ ((SubRing‘𝑤) ∩ (LSubSp‘𝑤)) ∣ 𝑠 ⊆ 𝑡}))
Definition	df-ascl 21769*	Every unital algebra contains a canonical homomorphic image of its ring of scalars as scalar multiples of the unity element. This names the homomorphism. (Contributed by Mario Carneiro, 8-Mar-2015.)
⊢ algSc = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑤)) ↦ (𝑥( ·𝑠 ‘𝑤)(1r‘𝑤))))
Theorem	isassa 21770*	The properties of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by SN, 2-Mar-2025.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ × = (.r‘𝑊)    ⇒   ⊢ (𝑊 ∈ AssAlg ↔ ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring) ∧ ∀𝑟 ∈ 𝐵 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))))
Theorem	assalem 21771	The properties of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ × = (.r‘𝑊)    ⇒   ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)) ∧ (𝑋 × (𝐴 · 𝑌)) = (𝐴 · (𝑋 × 𝑌))))
Theorem	assaass 21772	Left-associative property of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ × = (.r‘𝑊)    ⇒   ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → ((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)))
Theorem	assaassr 21773	Right-associative property of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ × = (.r‘𝑊)    ⇒   ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝑋 × (𝐴 · 𝑌)) = (𝐴 · (𝑋 × 𝑌)))
Theorem	assalmod 21774	An associative algebra is a left module. (Contributed by Mario Carneiro, 5-Dec-2014.)
⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ LMod)
Theorem	assaring 21775	An associative algebra is a ring. (Contributed by Mario Carneiro, 5-Dec-2014.)
⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ Ring)
Theorem	assasca 21776	The scalars of an associative algebra form a ring. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by SN, 2-Mar-2025.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ (𝑊 ∈ AssAlg → 𝐹 ∈ Ring)
Theorem	assa2ass 21777	Left- and right-associative property of an associative algebra. Notice that the scalars are commuted! (Contributed by AV, 14-Aug-2019.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ ∗ = (.r‘𝐹)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ × = (.r‘𝑊)    ⇒   ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → ((𝐴 · 𝑋) × (𝐶 · 𝑌)) = ((𝐶 ∗ 𝐴) · (𝑋 × 𝑌)))
Theorem	isassad 21778*	Sufficient condition for being an associative algebra. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by SN, 2-Mar-2025.)
⊢ (𝜑 → 𝑉 = (Base‘𝑊))    &   ⊢ (𝜑 → 𝐹 = (Scalar‘𝑊))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐹))    &   ⊢ (𝜑 → · = ( ·𝑠 ‘𝑊))    &   ⊢ (𝜑 → × = (.r‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑊 ∈ Ring)    &   ⊢ ((𝜑 ∧ (𝑟 ∈ 𝐵 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → ((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)))    &   ⊢ ((𝜑 ∧ (𝑟 ∈ 𝐵 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))    ⇒   ⊢ (𝜑 → 𝑊 ∈ AssAlg)
Theorem	issubassa3 21779	A subring that is also a subspace is a subalgebra. The key theorem is islss3 20825. (Contributed by Mario Carneiro, 7-Jan-2015.)
⊢ 𝑆 = (𝑊 ↾s 𝐴)    &   ⊢ 𝐿 = (LSubSp‘𝑊)    ⇒   ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)) → 𝑆 ∈ AssAlg)
Theorem	issubassa 21780	The subalgebras of an associative algebra are exactly the subrings (under the ring multiplication) that are simultaneously subspaces (under the scalar multiplication from the vector space). (Contributed by Mario Carneiro, 7-Jan-2015.)
⊢ 𝑆 = (𝑊 ↾s 𝐴)    &   ⊢ 𝐿 = (LSubSp‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 1 = (1r‘𝑊)    ⇒   ⊢ ((𝑊 ∈ AssAlg ∧ 1 ∈ 𝐴 ∧ 𝐴 ⊆ 𝑉) → (𝑆 ∈ AssAlg ↔ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)))
Theorem	sraassab 21781	A subring algebra is an associative algebra if and only if the subring is included in the ring's center. (Contributed by SN, 21-Mar-2025.)
⊢ 𝐴 = ((subringAlg ‘𝑊)‘𝑆)    &   ⊢ 𝑍 = (Cntr‘(mulGrp‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ Ring)    &   ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑊))    ⇒   ⊢ (𝜑 → (𝐴 ∈ AssAlg ↔ 𝑆 ⊆ 𝑍))
Theorem	sraassa 21782	The subring algebra over a commutative ring is an associative algebra. (Contributed by Mario Carneiro, 6-Oct-2015.) (Proof shortened by SN, 21-Mar-2025.)
⊢ 𝐴 = ((subringAlg ‘𝑊)‘𝑆)    ⇒   ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ AssAlg)
Theorem	sraassaOLD 21783	Obsolete version of sraassa 21782 as of 21-Mar-2025. (Contributed by Mario Carneiro, 6-Oct-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐴 = ((subringAlg ‘𝑊)‘𝑆)    ⇒   ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ AssAlg)
Theorem	rlmassa 21784	The ring module over a commutative ring is an associative algebra. (Contributed by Mario Carneiro, 6-Oct-2015.)
⊢ (𝑅 ∈ CRing → (ringLMod‘𝑅) ∈ AssAlg)
Theorem	assapropd 21785*	If two structures have the same components (properties), one is an associative algebra iff the other one is. (Contributed by Mario Carneiro, 8-Feb-2015.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐿))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))    &   ⊢ (𝜑 → 𝐹 = (Scalar‘𝐾))    &   ⊢ (𝜑 → 𝐹 = (Scalar‘𝐿))    &   ⊢ 𝑃 = (Base‘𝐹)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦))    ⇒   ⊢ (𝜑 → (𝐾 ∈ AssAlg ↔ 𝐿 ∈ AssAlg))
Theorem	aspval 21786*	Value of the algebraic closure operation inside an associative algebra. (Contributed by Mario Carneiro, 7-Jan-2015.)
⊢ 𝐴 = (AlgSpan‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐿 = (LSubSp‘𝑊)    ⇒   ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → (𝐴‘𝑆) = ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑆 ⊆ 𝑡})
Theorem	asplss 21787	The algebraic span of a set of vectors is a vector subspace. (Contributed by Mario Carneiro, 7-Jan-2015.)
⊢ 𝐴 = (AlgSpan‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐿 = (LSubSp‘𝑊)    ⇒   ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → (𝐴‘𝑆) ∈ 𝐿)
Theorem	aspid 21788	The algebraic span of a subalgebra is itself. (spanid 31131 analog.) (Contributed by Mario Carneiro, 7-Jan-2015.)
⊢ 𝐴 = (AlgSpan‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐿 = (LSubSp‘𝑊)    ⇒   ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊) ∧ 𝑆 ∈ 𝐿) → (𝐴‘𝑆) = 𝑆)
Theorem	aspsubrg 21789	The algebraic span of a set of vectors is a subring of the algebra. (Contributed by Mario Carneiro, 7-Jan-2015.)
⊢ 𝐴 = (AlgSpan‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    ⇒   ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → (𝐴‘𝑆) ∈ (SubRing‘𝑊))
Theorem	aspss 21790	Span preserves subset ordering. (spanss 31132 analog.) (Contributed by Mario Carneiro, 7-Jan-2015.)
⊢ 𝐴 = (AlgSpan‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    ⇒   ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉 ∧ 𝑇 ⊆ 𝑆) → (𝐴‘𝑇) ⊆ (𝐴‘𝑆))
Theorem	aspssid 21791	A set of vectors is a subset of its span. (spanss2 31129 analog.) (Contributed by Mario Carneiro, 7-Jan-2015.)
⊢ 𝐴 = (AlgSpan‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    ⇒   ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → 𝑆 ⊆ (𝐴‘𝑆))
Theorem	asclfval 21792*	Function value of the algebraic scalars function. (Contributed by Mario Carneiro, 8-Mar-2015.)
⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 1 = (1r‘𝑊)    ⇒   ⊢ 𝐴 = (𝑥 ∈ 𝐾 ↦ (𝑥 · 1 ))
Theorem	asclval 21793	Value of a mapped algebra scalar. (Contributed by Mario Carneiro, 8-Mar-2015.)
⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 1 = (1r‘𝑊)    ⇒   ⊢ (𝑋 ∈ 𝐾 → (𝐴‘𝑋) = (𝑋 · 1 ))
Theorem	asclfn 21794	Unconditional functionality of the algebra scalars function. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ 𝐴 Fn 𝐾
Theorem	asclf 21795	The algebra scalars function is a function into the base set. (Contributed by Mario Carneiro, 4-Jul-2015.)
⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ Ring)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ (𝜑 → 𝐴:𝐾⟶𝐵)
Theorem	asclghm 21796	The algebra scalars function is a group homomorphism. (Contributed by Mario Carneiro, 4-Jul-2015.)
⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ Ring)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    ⇒   ⊢ (𝜑 → 𝐴 ∈ (𝐹 GrpHom 𝑊))
Theorem	ascl0 21797	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)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑊 ∈ Ring)    ⇒   ⊢ (𝜑 → (𝐴‘(0g‘𝐹)) = (0g‘𝑊))
Theorem	ascl1 21798	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	asclmul1 21799	Left multiplication by a lifted scalar is the same as the scalar operation. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ × = (.r‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    ⇒   ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → ((𝐴‘𝑅) × 𝑋) = (𝑅 · 𝑋))
Theorem	asclmul2 21800	Right multiplication by a lifted scalar is the same as the scalar operation. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ × = (.r‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    ⇒   ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑋 × (𝐴‘𝑅)) = (𝑅 · 𝑋))