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	pwsdiaglmhm 21001*	Diagonal homomorphism into a structure power. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ 𝑌 = (𝑅 ↑s 𝐼)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐼 × {𝑥}))    ⇒   ⊢ ((𝑅 ∈ LMod ∧ 𝐼 ∈ 𝑊) → 𝐹 ∈ (𝑅 LMHom 𝑌))
Theorem	pwssplit0 21002*	Splitting for structure powers, part 0: restriction is a function. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ 𝑌 = (𝑊 ↑s 𝑈)    &   ⊢ 𝑍 = (𝑊 ↑s 𝑉)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ 𝐶 = (Base‘𝑍)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝑥 ↾ 𝑉))    ⇒   ⊢ ((𝑊 ∈ 𝑇 ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → 𝐹:𝐵⟶𝐶)
Theorem	pwssplit1 21003*	Splitting for structure powers, part 1: restriction is an onto function. The only actual monoid law we need here is that the base set is nonempty. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ 𝑌 = (𝑊 ↑s 𝑈)    &   ⊢ 𝑍 = (𝑊 ↑s 𝑉)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ 𝐶 = (Base‘𝑍)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝑥 ↾ 𝑉))    ⇒   ⊢ ((𝑊 ∈ Mnd ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → 𝐹:𝐵–onto→𝐶)
Theorem	pwssplit2 21004*	Splitting for structure powers, part 2: restriction is a group homomorphism. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ 𝑌 = (𝑊 ↑s 𝑈)    &   ⊢ 𝑍 = (𝑊 ↑s 𝑉)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ 𝐶 = (Base‘𝑍)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝑥 ↾ 𝑉))    ⇒   ⊢ ((𝑊 ∈ Grp ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → 𝐹 ∈ (𝑌 GrpHom 𝑍))
Theorem	pwssplit3 21005*	Splitting for structure powers, part 3: restriction is a module homomorphism. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ 𝑌 = (𝑊 ↑s 𝑈)    &   ⊢ 𝑍 = (𝑊 ↑s 𝑉)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ 𝐶 = (Base‘𝑍)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝑥 ↾ 𝑉))    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → 𝐹 ∈ (𝑌 LMHom 𝑍))
Theorem	islmim 21006	An isomorphism of left modules is a bijective homomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    ⇒   ⊢ (𝐹 ∈ (𝑅 LMIso 𝑆) ↔ (𝐹 ∈ (𝑅 LMHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶))
Theorem	lmimf1o 21007	An isomorphism of left modules is a bijection. (Contributed by Stefan O'Rear, 21-Jan-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    ⇒   ⊢ (𝐹 ∈ (𝑅 LMIso 𝑆) → 𝐹:𝐵–1-1-onto→𝐶)
Theorem	lmimlmhm 21008	An isomorphism of modules is a homomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.)
⊢ (𝐹 ∈ (𝑅 LMIso 𝑆) → 𝐹 ∈ (𝑅 LMHom 𝑆))
Theorem	lmimgim 21009	An isomorphism of modules is an isomorphism of groups. (Contributed by Stefan O'Rear, 21-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
⊢ (𝐹 ∈ (𝑅 LMIso 𝑆) → 𝐹 ∈ (𝑅 GrpIso 𝑆))
Theorem	islmim2 21010	An isomorphism of left modules is a homomorphism whose converse is a homomorphism. (Contributed by Mario Carneiro, 6-May-2015.)
⊢ (𝐹 ∈ (𝑅 LMIso 𝑆) ↔ (𝐹 ∈ (𝑅 LMHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 LMHom 𝑅)))
Theorem	lmimcnv 21011	The converse of a bijective module homomorphism is a bijective module homomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
⊢ (𝐹 ∈ (𝑆 LMIso 𝑇) → ◡𝐹 ∈ (𝑇 LMIso 𝑆))
Theorem	brlmic 21012	The relation "is isomorphic to" for modules. (Contributed by Stefan O'Rear, 25-Jan-2015.)
⊢ (𝑅 ≃𝑚 𝑆 ↔ (𝑅 LMIso 𝑆) ≠ ∅)
Theorem	brlmici 21013	Prove isomorphic by an explicit isomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.)
⊢ (𝐹 ∈ (𝑅 LMIso 𝑆) → 𝑅 ≃𝑚 𝑆)
Theorem	lmiclcl 21014	Isomorphism implies the left side is a module. (Contributed by Stefan O'Rear, 25-Jan-2015.)
⊢ (𝑅 ≃𝑚 𝑆 → 𝑅 ∈ LMod)
Theorem	lmicrcl 21015	Isomorphism implies the right side is a module. (Contributed by Mario Carneiro, 6-May-2015.)
⊢ (𝑅 ≃𝑚 𝑆 → 𝑆 ∈ LMod)
Theorem	lmicsym 21016	Module isomorphism is symmetric. (Contributed by Stefan O'Rear, 26-Feb-2015.)
⊢ (𝑅 ≃𝑚 𝑆 → 𝑆 ≃𝑚 𝑅)
Theorem	lmhmpropd 21017*	Module homomorphism depends only on the module attributes of structures. (Contributed by Mario Carneiro, 8-Oct-2015.)
⊢ (𝜑 → 𝐵 = (Base‘𝐽))    &   ⊢ (𝜑 → 𝐶 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐿))    &   ⊢ (𝜑 → 𝐶 = (Base‘𝑀))    &   ⊢ (𝜑 → 𝐹 = (Scalar‘𝐽))    &   ⊢ (𝜑 → 𝐺 = (Scalar‘𝐾))    &   ⊢ (𝜑 → 𝐹 = (Scalar‘𝐿))    &   ⊢ (𝜑 → 𝐺 = (Scalar‘𝑀))    &   ⊢ 𝑃 = (Base‘𝐹)    &   ⊢ 𝑄 = (Base‘𝐺)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐽)𝑦) = (𝑥(+g‘𝐿)𝑦))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝑀)𝑦))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐽)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑄 ∧ 𝑦 ∈ 𝐶)) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝑀)𝑦))    ⇒   ⊢ (𝜑 → (𝐽 LMHom 𝐾) = (𝐿 LMHom 𝑀))
10.5.4  Subspace sum; bases for a left module
Syntax	clbs 21018	Extend class notation with the set of bases for a vector space.
class LBasis
Definition	df-lbs 21019*	Define the set of bases to a left module or left vector space. (Contributed by Mario Carneiro, 24-Jun-2014.)
⊢ LBasis = (𝑤 ∈ V ↦ {𝑏 ∈ 𝒫 (Base‘𝑤) ∣ [(LSpan‘𝑤) / 𝑛][(Scalar‘𝑤) / 𝑠]((𝑛‘𝑏) = (Base‘𝑤) ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ ((Base‘𝑠) ∖ {(0g‘𝑠)}) ¬ (𝑦( ·𝑠 ‘𝑤)𝑥) ∈ (𝑛‘(𝑏 ∖ {𝑥})))})
Theorem	islbs 21020*	The predicate "𝐵 is a basis for the left module or vector space 𝑊". A subset of the base set is a basis if zero is not in the set, it spans the set, and no nonzero multiple of an element of the basis is in the span of the rest of the family. (Contributed by Mario Carneiro, 24-Jun-2014.) (Revised by Mario Carneiro, 14-Jan-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ 𝐽 = (LBasis‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ 0 = (0g‘𝐹)    ⇒   ⊢ (𝑊 ∈ 𝑋 → (𝐵 ∈ 𝐽 ↔ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})))))
Theorem	lbsss 21021	A basis is a set of vectors. (Contributed by Mario Carneiro, 24-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐽 = (LBasis‘𝑊)    ⇒   ⊢ (𝐵 ∈ 𝐽 → 𝐵 ⊆ 𝑉)
Theorem	lbsel 21022	An element of a basis is a vector. (Contributed by Mario Carneiro, 24-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐽 = (LBasis‘𝑊)    ⇒   ⊢ ((𝐵 ∈ 𝐽 ∧ 𝐸 ∈ 𝐵) → 𝐸 ∈ 𝑉)
Theorem	lbssp 21023	The span of a basis is the whole space. (Contributed by Mario Carneiro, 24-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐽 = (LBasis‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    ⇒   ⊢ (𝐵 ∈ 𝐽 → (𝑁‘𝐵) = 𝑉)
Theorem	lbsind 21024	A basis is linearly independent; that is, every element has a span which trivially intersects the span of the remainder of the basis. (Contributed by Mario Carneiro, 12-Jan-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐽 = (LBasis‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ 0 = (0g‘𝐹)    ⇒   ⊢ (((𝐵 ∈ 𝐽 ∧ 𝐸 ∈ 𝐵) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸})))
Theorem	lbsind2 21025	A basis is linearly independent; that is, every element is not in the span of the remainder of the basis. (Contributed by Mario Carneiro, 25-Jun-2014.) (Revised by Mario Carneiro, 12-Jan-2015.)
⊢ 𝐽 = (LBasis‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 1 = (1r‘𝐹)    &   ⊢ 0 = (0g‘𝐹)    ⇒   ⊢ (((𝑊 ∈ LMod ∧ 1 ≠ 0 ) ∧ 𝐵 ∈ 𝐽 ∧ 𝐸 ∈ 𝐵) → ¬ 𝐸 ∈ (𝑁‘(𝐵 ∖ {𝐸})))
Theorem	lbspss 21026	No proper subset of a basis spans the space. (Contributed by Mario Carneiro, 25-Jun-2014.)
⊢ 𝐽 = (LBasis‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 1 = (1r‘𝐹)    &   ⊢ 0 = (0g‘𝐹)    &   ⊢ 𝑉 = (Base‘𝑊)    ⇒   ⊢ (((𝑊 ∈ LMod ∧ 1 ≠ 0 ) ∧ 𝐵 ∈ 𝐽 ∧ 𝐶 ⊊ 𝐵) → (𝑁‘𝐶) ≠ 𝑉)
Theorem	lsmcl 21027	The sum of two subspaces is a subspace. (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → (𝑇 ⊕ 𝑈) ∈ 𝑆)
Theorem	lsmspsn 21028*	Member of subspace sum of spans of singletons. (Contributed by NM, 8-Apr-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑈 ∈ ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ↔ ∃𝑗 ∈ 𝐾 ∃𝑘 ∈ 𝐾 𝑈 = ((𝑗 · 𝑋) + (𝑘 · 𝑌))))
Theorem	lsmelval2 21029*	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 21030	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 21031	Subspace sum of spans of subsets is the span of their union. (spanuni 31535 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 21032	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 21033	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 21034	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 21035	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 21036	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 21037*	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 21038*	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 21039	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 21040	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 21041	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 21042	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 21043*	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 21044	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 21045	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 21046	Extend class notation with class of all left vector spaces.
class LVec
Definition	df-lvec 21047	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 21048	The predicate "is a left vector space". (Contributed by NM, 11-Nov-2013.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ (𝑊 ∈ LVec ↔ (𝑊 ∈ LMod ∧ 𝐹 ∈ DivRing))
Theorem	lvecdrng 21049	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 21050	A left vector space is a left module. (Contributed by NM, 9-Dec-2013.)
⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
Theorem	lveclmodd 21051	A vector space is a left module. (Contributed by SN, 16-May-2024.)
⊢ (𝜑 → 𝑊 ∈ LVec)    ⇒   ⊢ (𝜑 → 𝑊 ∈ LMod)
Theorem	lvecgrpd 21052	A vector space is a group. (Contributed by SN, 16-May-2024.)
⊢ (𝜑 → 𝑊 ∈ LVec)    ⇒   ⊢ (𝜑 → 𝑊 ∈ Grp)
Theorem	lsslvec 21053	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 21054	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 21055	If a scalar product is zero, one of its factors must be zero. (hvmul0or 31016 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 21056	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 21057	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 21058	Cancellation law for scalar multiplication. (hvmulcan 31063 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 21059	Cancellation law for scalar multiplication. (hvmulcan2 31064 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 21060	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 21061	A nonzero scalar product does not change the span of a singleton. (spansncol 31559 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 21062	Membership relation that implies equality of spans. (spansneleq 31561 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 21063	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 21064	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 21065	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 21066	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 21067	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 21068	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 21069*	Equal spans of singletons must have proportional vectors. See lspsnss2 20948 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 21070*	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 21071	A member of the span of the singleton of a vector is a member of a subspace containing the vector. (elspansn4 31564 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 21072	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 21073	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 21074	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 21075*	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 21076	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 21077 versus lspexchn2 21078); look for lspexch 21076 and prcom 4686 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 21077	Exchange property for span of a pair with negated membership. TODO: look at uses of lspexch 21076 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 21078	Exchange property for span of a pair with negated membership. TODO: look at uses of lspexch 21076 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 21079	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 21080	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 21081	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 21082	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 21083	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 21084	(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 21085	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 21086	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 21087	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 21088	Subspace sum has the covering property (using spans of singletons to represent atoms). Similar to Exercise 5 of [Kalmbach] p. 153. (spansncvi 31643 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 21089*	Lemma for lspsolv 21090. (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 21090	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 21091*	In a vector space, subspaces form an algebraic closure system whose closure operator has the exchange property. Strengthening of lssacs 20910 by lspsolv 21090. (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 21092	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 31572 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 21093*	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 21094	Lemma for lspprat 21100. 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 21090 (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 21095	Lemma for lspprat 21100. 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 21096	Lemma for lspprat 21100. In the first case of lsppratlem1 21094, 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 21097	Lemma for lspprat 21100. In the second case of lsppratlem1 21094, 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 21098	Lemma for lspprat 21100. 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 21099	Lemma for lspprat 21100. 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 21100*	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}))