P. 212 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 21101-21200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	lsmelval2 21101*	Subspace sum membership in terms of a sum of 1-dim subspaces (atoms), which can be useful for treating subspaces as projective lattice elements. (Contributed by NM, 9-Aug-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝑋 ∈ (𝑇 ⊕ 𝑈) ↔ (𝑋 ∈ 𝑉 ∧ ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑈 (𝑁‘{𝑋}) ⊆ ((𝑁‘{𝑦}) ⊕ (𝑁‘{𝑧})))))
Theorem	lsmsp 21102	Subspace sum in terms of span. (Contributed by NM, 6-Feb-2014.) (Proof shortened by Mario Carneiro, 21-Jun-2014.)
⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → (𝑇 ⊕ 𝑈) = (𝑁‘(𝑇 ∪ 𝑈)))
Theorem	lsmsp2 21103	Subspace sum of spans of subsets is the span of their union. (spanuni 31572 analog.) (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ⊆ 𝑉 ∧ 𝑈 ⊆ 𝑉) → ((𝑁‘𝑇) ⊕ (𝑁‘𝑈)) = (𝑁‘(𝑇 ∪ 𝑈)))
Theorem	lsmssspx 21104	Subspace sum (in its extended domain) is a subset of the span of the union of its arguments. (Contributed by NM, 6-Aug-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ (𝜑 → 𝑇 ⊆ 𝑉)    &   ⊢ (𝜑 → 𝑈 ⊆ 𝑉)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    ⇒   ⊢ (𝜑 → (𝑇 ⊕ 𝑈) ⊆ (𝑁‘(𝑇 ∪ 𝑈)))
Theorem	lsmpr 21105	The span of a pair of vectors equals the sum of the spans of their singletons. (Contributed by NM, 13-Jan-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})))
Theorem	lsppreli 21106	A vector expressed as a sum belongs to the span of its components. (Contributed by NM, 9-Apr-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐾)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((𝐴 · 𝑋) + (𝐵 · 𝑌)) ∈ (𝑁‘{𝑋, 𝑌}))
Theorem	lsmelpr 21107	Two ways to say that a vector belongs to the span of a pair of vectors. (Contributed by NM, 14-Jan-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑋 ∈ (𝑁‘{𝑌, 𝑍}) ↔ (𝑁‘{𝑋}) ⊆ ((𝑁‘{𝑌}) ⊕ (𝑁‘{𝑍}))))
Theorem	lsppr0 21108	The span of a vector paired with zero equals the span of the singleton of the vector. (Contributed by NM, 29-Aug-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑁‘{𝑋, 0 }) = (𝑁‘{𝑋}))
Theorem	lsppr 21109*	Span of a pair of vectors. (Contributed by NM, 22-Aug-2014.)
⊢ 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 → (𝑁‘{𝑋, 𝑌}) = {𝑣 ∣ ∃𝑘 ∈ 𝐾 ∃𝑙 ∈ 𝐾 𝑣 = ((𝑘 · 𝑋) + (𝑙 · 𝑌))})
Theorem	lspprel 21110*	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 21111	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 21112	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 21113	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 21114	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 21115*	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 21116	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 21117	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 21118	Extend class notation with class of all left vector spaces.
class LVec
Definition	df-lvec 21119	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 21120	The predicate "is a left vector space". (Contributed by NM, 11-Nov-2013.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ (𝑊 ∈ LVec ↔ (𝑊 ∈ LMod ∧ 𝐹 ∈ DivRing))
Theorem	lvecdrng 21121	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 21122	A left vector space is a left module. (Contributed by NM, 9-Dec-2013.)
⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
Theorem	lveclmodd 21123	A vector space is a left module. (Contributed by SN, 16-May-2024.)
⊢ (𝜑 → 𝑊 ∈ LVec)    ⇒   ⊢ (𝜑 → 𝑊 ∈ LMod)
Theorem	lvecgrpd 21124	A vector space is a group. (Contributed by SN, 16-May-2024.)
⊢ (𝜑 → 𝑊 ∈ LVec)    ⇒   ⊢ (𝜑 → 𝑊 ∈ Grp)
Theorem	lsslvec 21125	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 21126	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 21127	If a scalar product is zero, one of its factors must be zero. (hvmul0or 31053 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 21128	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 21129	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 21130	Cancellation law for scalar multiplication. (hvmulcan 31100 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 21131	Cancellation law for scalar multiplication. (hvmulcan2 31101 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 21132	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 21133	A nonzero scalar product does not change the span of a singleton. (spansncol 31596 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 21134	Membership relation that implies equality of spans. (spansneleq 31598 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 21135	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 21136	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 21137	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 21138	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 21139	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 21140	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 21141*	Equal spans of singletons must have proportional vectors. See lspsnss2 21020 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 21142*	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 21143	A member of the span of the singleton of a vector is a member of a subspace containing the vector. (elspansn4 31601 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 21144	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 21145	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 21146	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 21147*	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 21148	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 21149 versus lspexchn2 21150); look for lspexch 21148 and prcom 4736 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 21149	Exchange property for span of a pair with negated membership. TODO: look at uses of lspexch 21148 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 21150	Exchange property for span of a pair with negated membership. TODO: look at uses of lspexch 21148 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 21151	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 21152	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 21153	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 21154	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 21155	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 21156	(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 21157	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 21158	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 21159	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 21160	Subspace sum has the covering property (using spans of singletons to represent atoms). Similar to Exercise 5 of [Kalmbach] p. 153. (spansncvi 31680 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 21161*	Lemma for lspsolv 21162. (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 21162	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 21163*	In a vector space, subspaces form an algebraic closure system whose closure operator has the exchange property. Strengthening of lssacs 20982 by lspsolv 21162. (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 21164	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 31609 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 21165*	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 21166	Lemma for lspprat 21172. 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 21162 (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 21167	Lemma for lspprat 21172. 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 21168	Lemma for lspprat 21172. In the first case of lsppratlem1 21166, 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 21169	Lemma for lspprat 21172. In the second case of lsppratlem1 21166, 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 21170	Lemma for lspprat 21172. 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 21171	Lemma for lspprat 21172. 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 21172*	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 21173*	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 21174*	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 21175	Being a basis in a vector space is equivalent to being a basis in the associated algebraic closure system. Equivalent to islbs2 21173. (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 21176	The dimension theorem for vector spaces: any two bases of the same vector space are equinumerous. Proven by using lssacsex 21163 and lbsacsbs 21175 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 18616. (Contributed by David Moews, 1-May-2017.)
⊢ 𝐽 = (LBasis‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑇 ∈ 𝐽) → 𝑆 ≈ 𝑇)
Theorem	lbsextlem1 21177*	Lemma for lbsext 21182. 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 21178*	Lemma for lbsext 21182. 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 21179*	Lemma for lbsext 21182. 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 21180*	Lemma for lbsext 21182. lbsextlem3 21179 satisfies the conditions for the application of Zorn's lemma zorn 10544 (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 21181*	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 21182*	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 21183	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 21184	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 21185*	If two structures have the same components (properties), one is a left vector space iff the other one is. This version of lvecpropd 21186 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 21186*	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 21187	Extend class notation with the subring algebra generator.
class subringAlg
Syntax	crglmod 21188	Extend class notation with the left module induced by a ring over itself.
class ringLMod
Definition	df-sra 21189*	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 21190	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 21191	Lemma for srabase 21194 through sravsca 21202. (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 21192	Lemma for srabase 21194 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	sralemOLD 21193	Obsolete version of sralem 21192 as of 29-Oct-2024. Lemma for srabase 21194 and similar theorems. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊))    &   ⊢ 𝐸 = Slot 𝑁    &   ⊢ 𝑁 ∈ ℕ    &   ⊢ (𝑁 < 5 ∨ 8 < 𝑁)    ⇒   ⊢ (𝜑 → (𝐸‘𝑊) = (𝐸‘𝐴))
Theorem	srabase 21194	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	srabaseOLD 21195	Obsolete version of srabase 21194 as of 29-Oct-2024. 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.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊))    ⇒   ⊢ (𝜑 → (Base‘𝑊) = (Base‘𝐴))
Theorem	sraaddg 21196	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	sraaddgOLD 21197	Obsolete version of sraaddg 21196 as of 29-Oct-2024. 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.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊))    ⇒   ⊢ (𝜑 → (+g‘𝑊) = (+g‘𝐴))
Theorem	sramulr 21198	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	sramulrOLD 21199	Obsolete version of sramulr 21198 as of 29-Oct-2024. 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.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊))    ⇒   ⊢ (𝜑 → (.r‘𝑊) = (.r‘𝐴))
Theorem	srasca 21200	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))