P. 211 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 21001-21100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	lspprel 21001*	Member of the span of a pair of vectors. (Contributed by NM, 10-Apr-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑍 ∈ (𝑁‘{𝑋, 𝑌}) ↔ ∃𝑘 ∈ 𝐾 ∃𝑙 ∈ 𝐾 𝑍 = ((𝑘 · 𝑋) + (𝑙 · 𝑌))))
Theorem	lspprabs 21002	Absorption of vector sum into span of pair. (Contributed by NM, 27-Apr-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑁‘{𝑋, (𝑋 + 𝑌)}) = (𝑁‘{𝑋, 𝑌}))
Theorem	lspvadd 21003	The span of a vector sum is included in the span of its arguments. (Contributed by NM, 22-Feb-2014.) (Proof shortened by Mario Carneiro, 21-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑁‘{(𝑋 + 𝑌)}) ⊆ (𝑁‘{𝑋, 𝑌}))
Theorem	lspsntri 21004	Triangle-type inequality for span of a singleton. (Contributed by NM, 24-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑁‘{(𝑋 + 𝑌)}) ⊆ ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})))
Theorem	lspsntrim 21005	Triangle-type inequality for span of a singleton of vector difference. (Contributed by NM, 25-Apr-2014.) (Revised by Mario Carneiro, 21-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ − = (-g‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑁‘{(𝑋 − 𝑌)}) ⊆ ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})))
Theorem	lbspropd 21006*	If two structures have the same components (properties), they have the same set of bases. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 24-Apr-2024.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐿))    &   ⊢ (𝜑 → 𝐵 ⊆ 𝑊)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝑊)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦))    &   ⊢ 𝐹 = (Scalar‘𝐾)    &   ⊢ 𝐺 = (Scalar‘𝐿)    &   ⊢ (𝜑 → 𝑃 = (Base‘𝐹))    &   ⊢ (𝜑 → 𝑃 = (Base‘𝐺))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → (𝑥(+g‘𝐹)𝑦) = (𝑥(+g‘𝐺)𝑦))    &   ⊢ (𝜑 → 𝐾 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐿 ∈ 𝑌)    ⇒   ⊢ (𝜑 → (LBasis‘𝐾) = (LBasis‘𝐿))
Theorem	pj1lmhm 21007	The left projection function is a linear operator. (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
⊢ 𝐿 = (LSubSp‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑃 = (proj1‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑇 ∈ 𝐿)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐿)    &   ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 })    ⇒   ⊢ (𝜑 → (𝑇𝑃𝑈) ∈ ((𝑊 ↾s (𝑇 ⊕ 𝑈)) LMHom 𝑊))
Theorem	pj1lmhm2 21008	The left projection function is a linear operator. (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
⊢ 𝐿 = (LSubSp‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑃 = (proj1‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑇 ∈ 𝐿)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐿)    &   ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 })    ⇒   ⊢ (𝜑 → (𝑇𝑃𝑈) ∈ ((𝑊 ↾s (𝑇 ⊕ 𝑈)) LMHom (𝑊 ↾s 𝑇)))
10.6  Vector spaces
10.6.1  Definition and basic properties
Syntax	clvec 21009	Extend class notation with class of all left vector spaces.
class LVec
Definition	df-lvec 21010	Define the class of all left vector spaces. A left vector space over a division ring is an Abelian group (vectors) together with a division ring (scalars) and a left scalar product connecting them. Some authors call this a "left module over a division ring", reserving "vector space" for those where the division ring is commutative, i.e., is a field. (Contributed by NM, 11-Nov-2013.)
⊢ LVec = {𝑓 ∈ LMod ∣ (Scalar‘𝑓) ∈ DivRing}
Theorem	islvec 21011	The predicate "is a left vector space". (Contributed by NM, 11-Nov-2013.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ (𝑊 ∈ LVec ↔ (𝑊 ∈ LMod ∧ 𝐹 ∈ DivRing))
Theorem	lvecdrng 21012	The set of scalars of a left vector space is a division ring. (Contributed by NM, 17-Apr-2014.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ (𝑊 ∈ LVec → 𝐹 ∈ DivRing)
Theorem	lveclmod 21013	A left vector space is a left module. (Contributed by NM, 9-Dec-2013.)
⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
Theorem	lveclmodd 21014	A vector space is a left module. (Contributed by SN, 16-May-2024.)
⊢ (𝜑 → 𝑊 ∈ LVec)    ⇒   ⊢ (𝜑 → 𝑊 ∈ LMod)
Theorem	lvecgrpd 21015	A vector space is a group. (Contributed by SN, 16-May-2024.)
⊢ (𝜑 → 𝑊 ∈ LVec)    ⇒   ⊢ (𝜑 → 𝑊 ∈ Grp)
Theorem	lsslvec 21016	A vector subspace is a vector space. (Contributed by NM, 14-Mar-2015.)
⊢ 𝑋 = (𝑊 ↾s 𝑈)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LVec ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ LVec)
Theorem	lmhmlvec 21017	The property for modules to be vector spaces is invariant under module isomorphism. (Contributed by Steven Nguyen, 15-Aug-2023.)
⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝑆 ∈ LVec ↔ 𝑇 ∈ LVec))
Theorem	lvecvs0or 21018	If a scalar product is zero, one of its factors must be zero. (hvmul0or 30954 analog.) (Contributed by NM, 2-Jul-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ 𝑂 = (0g‘𝐹)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((𝐴 · 𝑋) = 0 ↔ (𝐴 = 𝑂 ∨ 𝑋 = 0 )))
Theorem	lvecvsn0 21019	A scalar product is nonzero iff both of its factors are nonzero. (Contributed by NM, 3-Jan-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ 𝑂 = (0g‘𝐹)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((𝐴 · 𝑋) ≠ 0 ↔ (𝐴 ≠ 𝑂 ∧ 𝑋 ≠ 0 )))
Theorem	lssvs0or 21020	If a scalar product belongs to a subspace, either the scalar component is zero or the vector component also belongs to the subspace. (Contributed by NM, 5-Apr-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ 0 = (0g‘𝐹)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐾)    ⇒   ⊢ (𝜑 → ((𝐴 · 𝑋) ∈ 𝑈 ↔ (𝐴 = 0 ∨ 𝑋 ∈ 𝑈)))
Theorem	lvecvscan 21021	Cancellation law for scalar multiplication. (hvmulcan 31001 analog.) (Contributed by NM, 2-Jul-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ 0 = (0g‘𝐹)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ≠ 0 )    ⇒   ⊢ (𝜑 → ((𝐴 · 𝑋) = (𝐴 · 𝑌) ↔ 𝑋 = 𝑌))
Theorem	lvecvscan2 21022	Cancellation law for scalar multiplication. (hvmulcan2 31002 analog.) (Contributed by NM, 2-Jul-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐾)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ≠ 0 )    ⇒   ⊢ (𝜑 → ((𝐴 · 𝑋) = (𝐵 · 𝑋) ↔ 𝐴 = 𝐵))
Theorem	lvecinv 21023	Invert coefficient of scalar product. (Contributed by NM, 11-Apr-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ 0 = (0g‘𝐹)    &   ⊢ 𝐼 = (invr‘𝐹)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝐴 ∈ (𝐾 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑋 = (𝐴 · 𝑌) ↔ 𝑌 = ((𝐼‘𝐴) · 𝑋)))
Theorem	lspsnvs 21024	A nonzero scalar product does not change the span of a singleton. (spansncol 31497 analog.) (Contributed by NM, 23-Apr-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ 0 = (0g‘𝐹)    &   ⊢ 𝑁 = (LSpan‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LVec ∧ (𝑅 ∈ 𝐾 ∧ 𝑅 ≠ 0 ) ∧ 𝑋 ∈ 𝑉) → (𝑁‘{(𝑅 · 𝑋)}) = (𝑁‘{𝑋}))
Theorem	lspsneleq 21025	Membership relation that implies equality of spans. (spansneleq 31499 analog.) (Contributed by NM, 4-Jul-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑋}))    &   ⊢ (𝜑 → 𝑌 ≠ 0 )    ⇒   ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑋}))
Theorem	lspsncmp 21026	Comparable spans of nonzero singletons are equal. (Contributed by NM, 27-Apr-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌}) ↔ (𝑁‘{𝑋}) = (𝑁‘{𝑌})))
Theorem	lspsnne1 21027	Two ways to express that vectors have different spans. (Contributed by NM, 28-May-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    ⇒   ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌}))
Theorem	lspsnne2 21028	Two ways to express that vectors have different spans. (Contributed by NM, 20-May-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌}))    ⇒   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
Theorem	lspsnnecom 21029	Swap two vectors with different spans. (Contributed by NM, 20-May-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌}))    ⇒   ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑁‘{𝑋}))
Theorem	lspabs2 21030	Absorption law for span of vector sum. (Contributed by NM, 30-Apr-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{(𝑋 + 𝑌)}))    ⇒   ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))
Theorem	lspabs3 21031	Absorption law for span of vector sum. (Contributed by NM, 30-Apr-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑋 + 𝑌) ≠ 0 )    &   ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))    ⇒   ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{(𝑋 + 𝑌)}))
Theorem	lspsneq 21032*	Equal spans of singletons must have proportional vectors. See lspsnss2 20911 for comparable span version. TODO: can proof be shortened? (Contributed by NM, 21-Mar-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑆)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((𝑁‘{𝑋}) = (𝑁‘{𝑌}) ↔ ∃𝑘 ∈ (𝐾 ∖ { 0 })𝑋 = (𝑘 · 𝑌)))
Theorem	lspsneu 21033*	Nonzero vectors with equal singleton spans have a unique proportionality constant. (Contributed by NM, 31-May-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝑂 = (0g‘𝑆)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    ⇒   ⊢ (𝜑 → ((𝑁‘{𝑋}) = (𝑁‘{𝑌}) ↔ ∃!𝑘 ∈ (𝐾 ∖ {𝑂})𝑋 = (𝑘 · 𝑌)))
Theorem	ellspsn4 21034	A member of the span of the singleton of a vector is a member of a subspace containing the vector. (elspansn4 31502 analog.) (Contributed by NM, 4-Jul-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑋}))    &   ⊢ (𝜑 → 𝑌 ≠ 0 )    ⇒   ⊢ (𝜑 → (𝑋 ∈ 𝑈 ↔ 𝑌 ∈ 𝑈))
Theorem	lspdisj 21035	The span of a vector not in a subspace is disjoint with the subspace. (Contributed by NM, 6-Apr-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈)    ⇒   ⊢ (𝜑 → ((𝑁‘{𝑋}) ∩ 𝑈) = { 0 })
Theorem	lspdisjb 21036	A nonzero vector is not in a subspace iff its span is disjoint with the subspace. (Contributed by NM, 23-Apr-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    ⇒   ⊢ (𝜑 → (¬ 𝑋 ∈ 𝑈 ↔ ((𝑁‘{𝑋}) ∩ 𝑈) = { 0 }))
Theorem	lspdisj2 21037	Unequal spans are disjoint (share only the zero vector). (Contributed by NM, 22-Mar-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    ⇒   ⊢ (𝜑 → ((𝑁‘{𝑋}) ∩ (𝑁‘{𝑌})) = { 0 })
Theorem	lspfixed 21038*	Show membership in the span of the sum of two vectors, one of which (𝑌) is fixed in advance. (Contributed by NM, 27-May-2015.) (Revised by AV, 12-Jul-2022.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌}))    &   ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑍}))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    ⇒   ⊢ (𝜑 → ∃𝑧 ∈ ((𝑁‘{𝑍}) ∖ { 0 })𝑋 ∈ (𝑁‘{(𝑌 + 𝑧)}))
Theorem	lspexch 21039	Exchange property for span of a pair. TODO: see if a version with Y,Z and X,Z reversed will shorten proofs (analogous to lspexchn1 21040 versus lspexchn2 21041); look for lspexch 21039 and prcom 4696 in same proof. TODO: would a hypothesis of ¬ 𝑋 ∈ (𝑁‘{𝑍}) instead of (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}) be better overall? This would be shorter and also satisfy the 𝑋 ≠ 0 condition. Here and also lspindp* and all proofs affected by them (all in NM's mathbox); there are 58 hypotheses with the ≠ pattern as of 24-May-2015. (Contributed by NM, 11-Apr-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    ⇒   ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑋, 𝑍}))
Theorem	lspexchn1 21040	Exchange property for span of a pair with negated membership. TODO: look at uses of lspexch 21039 to see if this will shorten proofs. (Contributed by NM, 20-May-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑁‘{𝑍}))    &   ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    ⇒   ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑁‘{𝑋, 𝑍}))
Theorem	lspexchn2 21041	Exchange property for span of a pair with negated membership. TODO: look at uses of lspexch 21039 to see if this will shorten proofs. (Contributed by NM, 24-May-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑁‘{𝑍}))    &   ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑍, 𝑌}))    ⇒   ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑋}))
Theorem	lspindpi 21042	Partial independence property. (Contributed by NM, 23-Apr-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    ⇒   ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})))
Theorem	lspindp1 21043	Alternate way to say 3 vectors are mutually independent (swap 1st and 2nd). (Contributed by NM, 11-Apr-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑁‘{𝑋, 𝑌}))    ⇒   ⊢ (𝜑 → ((𝑁‘{𝑍}) ≠ (𝑁‘{𝑌}) ∧ ¬ 𝑋 ∈ (𝑁‘{𝑍, 𝑌})))
Theorem	lspindp2l 21044	Alternate way to say 3 vectors are mutually independent (rotate left). (Contributed by NM, 10-May-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑁‘{𝑋, 𝑌}))    ⇒   ⊢ (𝜑 → ((𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}) ∧ ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})))
Theorem	lspindp2 21045	Alternate way to say 3 vectors are mutually independent (rotate right). (Contributed by NM, 12-Apr-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑁‘{𝑋, 𝑌}))    ⇒   ⊢ (𝜑 → ((𝑁‘{𝑍}) ≠ (𝑁‘{𝑋}) ∧ ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑋})))
Theorem	lspindp3 21046	Independence of 2 vectors is preserved by vector sum. (Contributed by NM, 26-Apr-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    ⇒   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{(𝑋 + 𝑌)}))
Theorem	lspindp4 21047	(Partial) independence of 3 vectors is preserved by vector sum. (Contributed by NM, 26-Apr-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑁‘{𝑋, 𝑌}))    ⇒   ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑁‘{𝑋, (𝑋 + 𝑌)}))
Theorem	lvecindp 21048	Compute the 𝑋 coefficient in a sum with an independent vector 𝑋 (first conjunct), which can then be removed to continue with the remaining vectors summed in expressions 𝑌 and 𝑍 (second conjunct). Typically, 𝑈 is the span of the remaining vectors. (Contributed by NM, 5-Apr-2015.) (Revised by Mario Carneiro, 21-Apr-2016.) (Proof shortened by AV, 19-Jul-2022.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐾)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐾)    &   ⊢ (𝜑 → ((𝐴 · 𝑋) + 𝑌) = ((𝐵 · 𝑋) + 𝑍))    ⇒   ⊢ (𝜑 → (𝐴 = 𝐵 ∧ 𝑌 = 𝑍))
Theorem	lvecindp2 21049	Sums of independent vectors must have equal coefficients. (Contributed by NM, 22-Mar-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐾)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐾)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐾)    &   ⊢ (𝜑 → 𝐷 ∈ 𝐾)    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   ⊢ (𝜑 → ((𝐴 · 𝑋) + (𝐵 · 𝑌)) = ((𝐶 · 𝑋) + (𝐷 · 𝑌)))    ⇒   ⊢ (𝜑 → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
Theorem	lspsnsubn0 21050	Unequal singleton spans imply nonzero vector subtraction. (Contributed by NM, 19-Mar-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ − = (-g‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    ⇒   ⊢ (𝜑 → (𝑋 − 𝑌) ≠ 0 )
Theorem	lsmcv 21051	Subspace sum has the covering property (using spans of singletons to represent atoms). Similar to Exercise 5 of [Kalmbach] p. 153. (spansncvi 31581 analog.) TODO: ugly proof; can it be shortened? (Contributed by NM, 2-Oct-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ ((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) → 𝑈 = (𝑇 ⊕ (𝑁‘{𝑋})))
Theorem	lspsolvlem 21052*	Lemma for lspsolv 21053. (Contributed by Mario Carneiro, 25-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ + = (+g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝑄 = {𝑧 ∈ 𝑉 ∣ ∃𝑟 ∈ 𝐵 (𝑧 + (𝑟 · 𝑌)) ∈ (𝑁‘𝐴)}    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑁‘(𝐴 ∪ {𝑌})))    ⇒   ⊢ (𝜑 → ∃𝑟 ∈ 𝐵 (𝑋 + (𝑟 · 𝑌)) ∈ (𝑁‘𝐴))
Theorem	lspsolv 21053	If 𝑋 is in the span of 𝐴 ∪ {𝑌} but not 𝐴, then 𝑌 is in the span of 𝐴 ∪ {𝑋}. (Contributed by Mario Carneiro, 25-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) → 𝑌 ∈ (𝑁‘(𝐴 ∪ {𝑋})))
Theorem	lssacsex 21054*	In a vector space, subspaces form an algebraic closure system whose closure operator has the exchange property. Strengthening of lssacs 20873 by lspsolv 21053. (Contributed by David Moews, 1-May-2017.)
⊢ 𝐴 = (LSubSp‘𝑊)    &   ⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ 𝑋 = (Base‘𝑊)    ⇒   ⊢ (𝑊 ∈ LVec → (𝐴 ∈ (ACS‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧}))))
Theorem	lspsnat 21055	There is no subspace strictly between the zero subspace and the span of a vector (i.e. a 1-dimensional subspace is an atom). (h1datomi 31510 analog.) (Contributed by NM, 20-Apr-2014.) (Proof shortened by Mario Carneiro, 22-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    ⇒   ⊢ (((𝑊 ∈ LVec ∧ 𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑉) ∧ 𝑈 ⊆ (𝑁‘{𝑋})) → (𝑈 = (𝑁‘{𝑋}) ∨ 𝑈 = { 0 }))
Theorem	lspsncv0 21056*	The span of a singleton covers the zero subspace, using Definition 3.2.18 of [PtakPulmannova] p. 68 for "covers".) (Contributed by NM, 12-Aug-2014.) (Revised by AV, 13-Jul-2022.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ¬ ∃𝑦 ∈ 𝑆 ({ 0 } ⊊ 𝑦 ∧ 𝑦 ⊊ (𝑁‘{𝑋})))
Theorem	lsppratlem1 21057	Lemma for lspprat 21063. Let 𝑥 ∈ (𝑈 ∖ {0}) (if there is no such 𝑥 then 𝑈 is the zero subspace), and let 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥})) (assuming the conclusion is false). The goal is to write 𝑋, 𝑌 in terms of 𝑥, 𝑦, which would normally be done by solving the system of linear equations. The span equivalent of this process is lspsolv 21053 (hence the name), which we use extensively below. In this lemma, we show that since 𝑥 ∈ (𝑁‘{𝑋, 𝑌}), either 𝑥 ∈ (𝑁‘{𝑌}) or 𝑋 ∈ (𝑁‘{𝑥, 𝑌}). (Contributed by NM, 29-Aug-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑈 ⊊ (𝑁‘{𝑋, 𝑌}))    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ (𝜑 → 𝑥 ∈ (𝑈 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥})))    ⇒   ⊢ (𝜑 → (𝑥 ∈ (𝑁‘{𝑌}) ∨ 𝑋 ∈ (𝑁‘{𝑥, 𝑌})))
Theorem	lsppratlem2 21058	Lemma for lspprat 21063. Show that if 𝑋 and 𝑌 are both in (𝑁‘{𝑥, 𝑦}) (which will be our goal for each of the two cases above), then (𝑁‘{𝑋, 𝑌}) ⊆ 𝑈, contradicting the hypothesis for 𝑈. (Contributed by NM, 29-Aug-2014.) (Revised by Mario Carneiro, 5-Sep-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑈 ⊊ (𝑁‘{𝑋, 𝑌}))    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ (𝜑 → 𝑥 ∈ (𝑈 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥})))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑥, 𝑦}))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑥, 𝑦}))    ⇒   ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ 𝑈)
Theorem	lsppratlem3 21059	Lemma for lspprat 21063. In the first case of lsppratlem1 21057, since 𝑥 ∉ (𝑁‘∅), also 𝑌 ∈ (𝑁‘{𝑥}), and since 𝑦 ∈ (𝑁‘{𝑋, 𝑌}) ⊆ (𝑁‘{𝑋, 𝑥}) and 𝑦 ∉ (𝑁‘{𝑥}), we have 𝑋 ∈ (𝑁‘{𝑥, 𝑦}) as desired. (Contributed by NM, 29-Aug-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑈 ⊊ (𝑁‘{𝑋, 𝑌}))    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ (𝜑 → 𝑥 ∈ (𝑈 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥})))    &   ⊢ (𝜑 → 𝑥 ∈ (𝑁‘{𝑌}))    ⇒   ⊢ (𝜑 → (𝑋 ∈ (𝑁‘{𝑥, 𝑦}) ∧ 𝑌 ∈ (𝑁‘{𝑥, 𝑦})))
Theorem	lsppratlem4 21060	Lemma for lspprat 21063. In the second case of lsppratlem1 21057, 𝑦 ∈ (𝑁‘{𝑋, 𝑌}) ⊆ (𝑁‘{𝑥, 𝑌}) and 𝑦 ∉ (𝑁‘{𝑥}) implies 𝑌 ∈ (𝑁‘{𝑥, 𝑦}) and thus 𝑋 ∈ (𝑁‘{𝑥, 𝑌}) ⊆ (𝑁‘{𝑥, 𝑦}) as well. (Contributed by NM, 29-Aug-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑈 ⊊ (𝑁‘{𝑋, 𝑌}))    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ (𝜑 → 𝑥 ∈ (𝑈 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥})))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑥, 𝑌}))    ⇒   ⊢ (𝜑 → (𝑋 ∈ (𝑁‘{𝑥, 𝑦}) ∧ 𝑌 ∈ (𝑁‘{𝑥, 𝑦})))
Theorem	lsppratlem5 21061	Lemma for lspprat 21063. Combine the two cases and show a contradiction to 𝑈 ⊊ (𝑁‘{𝑋, 𝑌}) under the assumptions on 𝑥 and 𝑦. (Contributed by NM, 29-Aug-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑈 ⊊ (𝑁‘{𝑋, 𝑌}))    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ (𝜑 → 𝑥 ∈ (𝑈 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥})))    ⇒   ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ 𝑈)
Theorem	lsppratlem6 21062	Lemma for lspprat 21063. Negating the assumption on 𝑦, we arrive close to the desired conclusion. (Contributed by NM, 29-Aug-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑈 ⊊ (𝑁‘{𝑋, 𝑌}))    &   ⊢ 0 = (0g‘𝑊)    ⇒   ⊢ (𝜑 → (𝑥 ∈ (𝑈 ∖ { 0 }) → 𝑈 = (𝑁‘{𝑥})))
Theorem	lspprat 21063*	A proper subspace of the span of a pair of vectors is the span of a singleton (an atom) or the zero subspace (if 𝑧 is zero). Proof suggested by Mario Carneiro, 28-Aug-2014. (Contributed by NM, 29-Aug-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑈 ⊊ (𝑁‘{𝑋, 𝑌}))    ⇒   ⊢ (𝜑 → ∃𝑧 ∈ 𝑉 𝑈 = (𝑁‘{𝑧}))
Theorem	islbs2 21064*	An equivalent formulation of the basis predicate in a vector space: a subset is a basis iff no element is in the span of the rest of the set. (Contributed by Mario Carneiro, 14-Jan-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐽 = (LBasis‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    ⇒   ⊢ (𝑊 ∈ LVec → (𝐵 ∈ 𝐽 ↔ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))))
Theorem	islbs3 21065*	An equivalent formulation of the basis predicate: a subset is a basis iff it is a minimal spanning set. (Contributed by Mario Carneiro, 25-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐽 = (LBasis‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    ⇒   ⊢ (𝑊 ∈ LVec → (𝐵 ∈ 𝐽 ↔ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑠(𝑠 ⊊ 𝐵 → (𝑁‘𝑠) ⊊ 𝑉))))
Theorem	lbsacsbs 21066	Being a basis in a vector space is equivalent to being a basis in the associated algebraic closure system. Equivalent to islbs2 21064. (Contributed by David Moews, 1-May-2017.)
⊢ 𝐴 = (LSubSp‘𝑊)    &   ⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ 𝑋 = (Base‘𝑊)    &   ⊢ 𝐼 = (mrInd‘𝐴)    &   ⊢ 𝐽 = (LBasis‘𝑊)    ⇒   ⊢ (𝑊 ∈ LVec → (𝑆 ∈ 𝐽 ↔ (𝑆 ∈ 𝐼 ∧ (𝑁‘𝑆) = 𝑋)))
Theorem	lvecdim 21067	The dimension theorem for vector spaces: any two bases of the same vector space are equinumerous. Proven by using lssacsex 21054 and lbsacsbs 21066 to show that being a basis for a vector space is equivalent to being a basis for the associated algebraic closure system, and then using acsexdimd 18518. (Contributed by David Moews, 1-May-2017.)
⊢ 𝐽 = (LBasis‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑇 ∈ 𝐽) → 𝑆 ≈ 𝑇)
Theorem	lbsextlem1 21068*	Lemma for lbsext 21073. The set 𝑆 is the set of all linearly independent sets containing 𝐶; we show here that it is nonempty. (Contributed by Mario Carneiro, 25-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐽 = (LBasis‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝐶 ⊆ 𝑉)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥})))    &   ⊢ 𝑆 = {𝑧 ∈ 𝒫 𝑉 ∣ (𝐶 ⊆ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})))}    ⇒   ⊢ (𝜑 → 𝑆 ≠ ∅)
Theorem	lbsextlem2 21069*	Lemma for lbsext 21073. Since 𝐴 is a chain (actually, we only need it to be closed under binary union), the union 𝑇 of the spans of each individual element of 𝐴 is a subspace, and it contains all of ∪ 𝐴 (except for our target vector 𝑥- we are trying to make 𝑥 a linear combination of all the other vectors in some set from 𝐴). (Contributed by Mario Carneiro, 25-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐽 = (LBasis‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝐶 ⊆ 𝑉)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥})))    &   ⊢ 𝑆 = {𝑧 ∈ 𝒫 𝑉 ∣ (𝐶 ⊆ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})))}    &   ⊢ 𝑃 = (LSubSp‘𝑊)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝑆)    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    &   ⊢ (𝜑 → [⊊] Or 𝐴)    &   ⊢ 𝑇 = ∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥}))    ⇒   ⊢ (𝜑 → (𝑇 ∈ 𝑃 ∧ (∪ 𝐴 ∖ {𝑥}) ⊆ 𝑇))
Theorem	lbsextlem3 21070*	Lemma for lbsext 21073. A chain in 𝑆 has an upper bound in 𝑆. (Contributed by Mario Carneiro, 25-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐽 = (LBasis‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝐶 ⊆ 𝑉)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥})))    &   ⊢ 𝑆 = {𝑧 ∈ 𝒫 𝑉 ∣ (𝐶 ⊆ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})))}    &   ⊢ 𝑃 = (LSubSp‘𝑊)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝑆)    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    &   ⊢ (𝜑 → [⊊] Or 𝐴)    &   ⊢ 𝑇 = ∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥}))    ⇒   ⊢ (𝜑 → ∪ 𝐴 ∈ 𝑆)
Theorem	lbsextlem4 21071*	Lemma for lbsext 21073. lbsextlem3 21070 satisfies the conditions for the application of Zorn's lemma zorn 10460 (thus invoking AC), and so there is a maximal linearly independent set extending 𝐶. Here we prove that such a set is a basis. (Contributed by Mario Carneiro, 25-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐽 = (LBasis‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝐶 ⊆ 𝑉)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥})))    &   ⊢ 𝑆 = {𝑧 ∈ 𝒫 𝑉 ∣ (𝐶 ⊆ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})))}    &   ⊢ (𝜑 → 𝒫 𝑉 ∈ dom card)    ⇒   ⊢ (𝜑 → ∃𝑠 ∈ 𝐽 𝐶 ⊆ 𝑠)
Theorem	lbsextg 21072*	For any linearly independent subset 𝐶 of 𝑉, there is a basis containing the vectors in 𝐶. (Contributed by Mario Carneiro, 17-May-2015.)
⊢ 𝐽 = (LBasis‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    ⇒   ⊢ (((𝑊 ∈ LVec ∧ 𝒫 𝑉 ∈ dom card) ∧ 𝐶 ⊆ 𝑉 ∧ ∀𝑥 ∈ 𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))) → ∃𝑠 ∈ 𝐽 𝐶 ⊆ 𝑠)
Theorem	lbsext 21073*	For any linearly independent subset 𝐶 of 𝑉, there is a basis containing the vectors in 𝐶. (Contributed by Mario Carneiro, 25-Jun-2014.) (Revised by Mario Carneiro, 17-May-2015.)
⊢ 𝐽 = (LBasis‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LVec ∧ 𝐶 ⊆ 𝑉 ∧ ∀𝑥 ∈ 𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))) → ∃𝑠 ∈ 𝐽 𝐶 ⊆ 𝑠)
Theorem	lbsexg 21074	Every vector space has a basis. This theorem is an AC equivalent; this is the forward implication. (Contributed by Mario Carneiro, 17-May-2015.)
⊢ 𝐽 = (LBasis‘𝑊)    ⇒   ⊢ ((CHOICE ∧ 𝑊 ∈ LVec) → 𝐽 ≠ ∅)
Theorem	lbsex 21075	Every vector space has a basis. This theorem is an AC equivalent. (Contributed by Mario Carneiro, 25-Jun-2014.)
⊢ 𝐽 = (LBasis‘𝑊)    ⇒   ⊢ (𝑊 ∈ LVec → 𝐽 ≠ ∅)
Theorem	lvecprop2d 21076*	If two structures have the same components (properties), one is a left vector space iff the other one is. This version of lvecpropd 21077 also breaks up the components of the scalar ring. (Contributed by Mario Carneiro, 27-Jun-2015.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐿))    &   ⊢ 𝐹 = (Scalar‘𝐾)    &   ⊢ 𝐺 = (Scalar‘𝐿)    &   ⊢ (𝜑 → 𝑃 = (Base‘𝐹))    &   ⊢ (𝜑 → 𝑃 = (Base‘𝐺))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → (𝑥(+g‘𝐹)𝑦) = (𝑥(+g‘𝐺)𝑦))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → (𝑥(.r‘𝐹)𝑦) = (𝑥(.r‘𝐺)𝑦))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦))    ⇒   ⊢ (𝜑 → (𝐾 ∈ LVec ↔ 𝐿 ∈ LVec))
Theorem	lvecpropd 21077*	If two structures have the same components (properties), one is a left vector space iff the other one is. (Contributed by Mario Carneiro, 27-Jun-2015.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐿))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘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 ∈ LVec ↔ 𝐿 ∈ LVec))
10.7  Subring algebras and ideals
10.7.1  Subring algebras
Syntax	csra 21078	Extend class notation with the subring algebra generator.
class subringAlg
Syntax	crglmod 21079	Extend class notation with the left module induced by a ring over itself.
class ringLMod
Definition	df-sra 21080*	Any ring can be regarded as a left algebra over any of its subrings. The function subringAlg associates with any ring and any of its subrings the left algebra consisting in the ring itself regarded as a left algebra over the subring. It has an inner product which is simply the ring product. (Contributed by Mario Carneiro, 27-Nov-2014.) (Revised by Thierry Arnoux, 16-Jun-2019.)
⊢ subringAlg = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ (((𝑤 sSet ⟨(Scalar‘ndx), (𝑤 ↾s 𝑠)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑤)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑤)⟩)))
Definition	df-rgmod 21081	Any ring can be regarded as a left algebra over itself. The function ringLMod associates with any ring the left algebra consisting in the ring itself regarded as a left algebra over itself. It has an inner product which is simply the ring product. (Contributed by Stefan O'Rear, 6-Dec-2014.)
⊢ ringLMod = (𝑤 ∈ V ↦ ((subringAlg ‘𝑤)‘(Base‘𝑤)))
Theorem	sraval 21082	Lemma for srabase 21084 through sravsca 21088. (Contributed by Mario Carneiro, 27-Nov-2014.) (Revised by Thierry Arnoux, 16-Jun-2019.)
⊢ ((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) → ((subringAlg ‘𝑊)‘𝑆) = (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩))
Theorem	sralem 21083	Lemma for srabase 21084 and similar theorems. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊))    &   ⊢ 𝐸 = Slot (𝐸‘ndx)    &   ⊢ (Scalar‘ndx) ≠ (𝐸‘ndx)    &   ⊢ ( ·𝑠 ‘ndx) ≠ (𝐸‘ndx)    &   ⊢ (·𝑖‘ndx) ≠ (𝐸‘ndx)    ⇒   ⊢ (𝜑 → (𝐸‘𝑊) = (𝐸‘𝐴))
Theorem	srabase 21084	Base set of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊))    ⇒   ⊢ (𝜑 → (Base‘𝑊) = (Base‘𝐴))
Theorem	sraaddg 21085	Additive operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊))    ⇒   ⊢ (𝜑 → (+g‘𝑊) = (+g‘𝐴))
Theorem	sramulr 21086	Multiplicative operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊))    ⇒   ⊢ (𝜑 → (.r‘𝑊) = (.r‘𝐴))
Theorem	srasca 21087	The set of scalars of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Proof shortened by AV, 12-Nov-2024.)
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊))    ⇒   ⊢ (𝜑 → (𝑊 ↾s 𝑆) = (Scalar‘𝐴))
Theorem	sravsca 21088	The scalar product operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Proof shortened by AV, 12-Nov-2024.)
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊))    ⇒   ⊢ (𝜑 → (.r‘𝑊) = ( ·𝑠 ‘𝐴))
Theorem	sraip 21089	The inner product operation of a subring algebra. (Contributed by Thierry Arnoux, 16-Jun-2019.)
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊))    ⇒   ⊢ (𝜑 → (.r‘𝑊) = (·𝑖‘𝐴))
Theorem	sratset 21090	Topology component of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊))    ⇒   ⊢ (𝜑 → (TopSet‘𝑊) = (TopSet‘𝐴))
Theorem	sratopn 21091	Topology component of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊))    ⇒   ⊢ (𝜑 → (TopOpen‘𝑊) = (TopOpen‘𝐴))
Theorem	srads 21092	Distance function of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊))    ⇒   ⊢ (𝜑 → (dist‘𝑊) = (dist‘𝐴))
Theorem	sraring 21093	Condition for a subring algebra to be a ring. (Contributed by Thierry Arnoux, 24-Jul-2023.)
⊢ 𝐴 = ((subringAlg ‘𝑅)‘𝑉)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑉 ⊆ 𝐵) → 𝐴 ∈ Ring)
Theorem	sralmod 21094	The subring algebra is a left module. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ 𝐴 = ((subringAlg ‘𝑊)‘𝑆)    ⇒   ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝐴 ∈ LMod)
Theorem	sralmod0 21095	The subring module inherits a zero from its ring. (Contributed by Stefan O'Rear, 27-Dec-2014.)
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   ⊢ (𝜑 → 0 = (0g‘𝑊))    &   ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊))    ⇒   ⊢ (𝜑 → 0 = (0g‘𝐴))
Theorem	issubrgd 21096*	Prove a subring by closure (definition version). (Contributed by Stefan O'Rear, 7-Dec-2014.)
⊢ (𝜑 → 𝑆 = (𝐼 ↾s 𝐷))    &   ⊢ (𝜑 → 0 = (0g‘𝐼))    &   ⊢ (𝜑 → + = (+g‘𝐼))    &   ⊢ (𝜑 → 𝐷 ⊆ (Base‘𝐼))    &   ⊢ (𝜑 → 0 ∈ 𝐷)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → (𝑥 + 𝑦) ∈ 𝐷)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((invg‘𝐼)‘𝑥) ∈ 𝐷)    &   ⊢ (𝜑 → 1 = (1r‘𝐼))    &   ⊢ (𝜑 → · = (.r‘𝐼))    &   ⊢ (𝜑 → 1 ∈ 𝐷)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → (𝑥 · 𝑦) ∈ 𝐷)    &   ⊢ (𝜑 → 𝐼 ∈ Ring)    ⇒   ⊢ (𝜑 → 𝐷 ∈ (SubRing‘𝐼))
Theorem	rlmfn 21097	ringLMod is a function. (Contributed by Stefan O'Rear, 6-Dec-2014.)
⊢ ringLMod Fn V
Theorem	rlmval 21098	Value of the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
⊢ (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊))
Theorem	rlmval2 21099	Value of the ring module extended. (Contributed by AV, 2-Dec-2018.) (Revised by Thierry Arnoux, 16-Jun-2019.)
⊢ (𝑊 ∈ 𝑋 → (ringLMod‘𝑊) = (((𝑊 sSet ⟨(Scalar‘ndx), 𝑊⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩))
Theorem	rlmbas 21100	Base set of the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
⊢ (Base‘𝑅) = (Base‘(ringLMod‘𝑅))