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Theorem List for Metamath Proof Explorer - 20301-20400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlmodsubvs 20301 Subtraction of a scalar product in terms of addition. (Contributed by NM, 9-Apr-2015.)
𝑉 = (Baseβ€˜π‘Š)    &    + = (+gβ€˜π‘Š)    &    βˆ’ = (-gβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    &   π‘ = (invgβ€˜πΉ)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ 𝐴 ∈ 𝐾)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    β‡’   (πœ‘ β†’ (𝑋 βˆ’ (𝐴 Β· π‘Œ)) = (𝑋 + ((π‘β€˜π΄) Β· π‘Œ)))
 
Theoremlmodsubdi 20302 Scalar multiplication distributive law for subtraction. (hvsubdistr1 29777 analogue, with longer proof since our scalar multiplication is not commutative.) (Contributed by NM, 2-Jul-2014.)
𝑉 = (Baseβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    &    βˆ’ = (-gβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ 𝐴 ∈ 𝐾)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    β‡’   (πœ‘ β†’ (𝐴 Β· (𝑋 βˆ’ π‘Œ)) = ((𝐴 Β· 𝑋) βˆ’ (𝐴 Β· π‘Œ)))
 
Theoremlmodsubdir 20303 Scalar multiplication distributive law for subtraction. (hvsubdistr2 29778 analog.) (Contributed by NM, 2-Jul-2014.)
𝑉 = (Baseβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    &    βˆ’ = (-gβ€˜π‘Š)    &   π‘† = (-gβ€˜πΉ)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ 𝐴 ∈ 𝐾)    &   (πœ‘ β†’ 𝐡 ∈ 𝐾)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    β‡’   (πœ‘ β†’ ((𝐴𝑆𝐡) Β· 𝑋) = ((𝐴 Β· 𝑋) βˆ’ (𝐡 Β· 𝑋)))
 
Theoremlmodsubeq0 20304 If the difference between two vectors is zero, they are equal. (hvsubeq0 29796 analog.) (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &    βˆ’ = (-gβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ ((𝐴 βˆ’ 𝐡) = 0 ↔ 𝐴 = 𝐡))
 
Theoremlmodsubid 20305 Subtraction of a vector from itself. (hvsubid 29754 analog.) (Contributed by NM, 16-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &    βˆ’ = (-gβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ 𝐴 ∈ 𝑉) β†’ (𝐴 βˆ’ 𝐴) = 0 )
 
Theoremlmodvsghm 20306* Scalar multiplication of the vector space by a fixed scalar is an endomorphism of the additive group of vectors. (Contributed by Mario Carneiro, 5-May-2015.)
𝑉 = (Baseβ€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    β‡’   ((π‘Š ∈ LMod ∧ 𝑅 ∈ 𝐾) β†’ (π‘₯ ∈ 𝑉 ↦ (𝑅 Β· π‘₯)) ∈ (π‘Š GrpHom π‘Š))
 
Theoremlmodprop2d 20307* If two structures have the same components (properties), one is a left module iff the other one is. This version of lmodpropd 20308 also breaks up the components of the scalar ring. (Contributed by Mario Carneiro, 27-Jun-2015.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜πΎ))    &   (πœ‘ β†’ 𝐡 = (Baseβ€˜πΏ))    &   πΉ = (Scalarβ€˜πΎ)    &   πΊ = (Scalarβ€˜πΏ)    &   (πœ‘ β†’ 𝑃 = (Baseβ€˜πΉ))    &   (πœ‘ β†’ 𝑃 = (Baseβ€˜πΊ))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(+gβ€˜πΎ)𝑦) = (π‘₯(+gβ€˜πΏ)𝑦))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) β†’ (π‘₯(+gβ€˜πΉ)𝑦) = (π‘₯(+gβ€˜πΊ)𝑦))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) β†’ (π‘₯(.rβ€˜πΉ)𝑦) = (π‘₯(.rβ€˜πΊ)𝑦))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑃 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯( ·𝑠 β€˜πΎ)𝑦) = (π‘₯( ·𝑠 β€˜πΏ)𝑦))    β‡’   (πœ‘ β†’ (𝐾 ∈ LMod ↔ 𝐿 ∈ LMod))
 
Theoremlmodpropd 20308* If two structures have the same components (properties), one is a left module iff the other one is. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 27-Jun-2015.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜πΎ))    &   (πœ‘ β†’ 𝐡 = (Baseβ€˜πΏ))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(+gβ€˜πΎ)𝑦) = (π‘₯(+gβ€˜πΏ)𝑦))    &   (πœ‘ β†’ 𝐹 = (Scalarβ€˜πΎ))    &   (πœ‘ β†’ 𝐹 = (Scalarβ€˜πΏ))    &   π‘ƒ = (Baseβ€˜πΉ)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑃 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯( ·𝑠 β€˜πΎ)𝑦) = (π‘₯( ·𝑠 β€˜πΏ)𝑦))    β‡’   (πœ‘ β†’ (𝐾 ∈ LMod ↔ 𝐿 ∈ LMod))
 
Theoremgsumvsmul 20309* Pull a scalar multiplication out of a sum of vectors. This theorem properly generalizes gsummulc2 19954, since every ring is a left module over itself. (Contributed by Stefan O'Rear, 6-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.) (Revised by AV, 10-Jul-2019.)
𝐡 = (Baseβ€˜π‘…)    &   π‘† = (Scalarβ€˜π‘…)    &   πΎ = (Baseβ€˜π‘†)    &    0 = (0gβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &    Β· = ( ·𝑠 β€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ LMod)    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝑋 ∈ 𝐾)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ (π‘˜ ∈ 𝐴 ↦ π‘Œ) finSupp 0 )    β‡’   (πœ‘ β†’ (𝑅 Ξ£g (π‘˜ ∈ 𝐴 ↦ (𝑋 Β· π‘Œ))) = (𝑋 Β· (𝑅 Ξ£g (π‘˜ ∈ 𝐴 ↦ π‘Œ))))
 
Theoremmptscmfsupp0 20310* A mapping to a scalar product is finitely supported if the mapping to the scalar is finitely supported. (Contributed by AV, 5-Oct-2019.)
(πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ 𝑄 ∈ LMod)    &   (πœ‘ β†’ 𝑅 = (Scalarβ€˜π‘„))    &   πΎ = (Baseβ€˜π‘„)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐷) β†’ 𝑆 ∈ 𝐡)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐷) β†’ π‘Š ∈ 𝐾)    &    0 = (0gβ€˜π‘„)    &   π‘ = (0gβ€˜π‘…)    &    βˆ— = ( ·𝑠 β€˜π‘„)    &   (πœ‘ β†’ (π‘˜ ∈ 𝐷 ↦ 𝑆) finSupp 𝑍)    β‡’   (πœ‘ β†’ (π‘˜ ∈ 𝐷 ↦ (𝑆 βˆ— π‘Š)) finSupp 0 )
 
Theoremmptscmfsuppd 20311* A function mapping to a scalar product in which one factor is finitely supported is finitely supported. Formerly part of proof for ply1coe 21589. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by AV, 8-Aug-2019.) (Proof shortened by AV, 18-Oct-2019.)
𝐡 = (Baseβ€˜π‘ƒ)    &   π‘† = (Scalarβ€˜π‘ƒ)    &    Β· = ( ·𝑠 β€˜π‘ƒ)    &   (πœ‘ β†’ 𝑃 ∈ LMod)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑋) β†’ 𝑍 ∈ 𝐡)    &   (πœ‘ β†’ 𝐴:π‘‹βŸΆπ‘Œ)    &   (πœ‘ β†’ 𝐴 finSupp (0gβ€˜π‘†))    β‡’   (πœ‘ β†’ (π‘˜ ∈ 𝑋 ↦ ((π΄β€˜π‘˜) Β· 𝑍)) finSupp (0gβ€˜π‘ƒ))
 
Theoremrmodislmodlem 20312* Lemma for rmodislmod 20313. This is the part of the proof of rmodislmod 20313 which requires the scalar ring to be commutative. (Contributed by AV, 3-Dec-2021.)
𝑉 = (Baseβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &    Β· = ( ·𝑠 β€˜π‘…)    &   πΉ = (Scalarβ€˜π‘…)    &   πΎ = (Baseβ€˜πΉ)    &    ⨣ = (+gβ€˜πΉ)    &    Γ— = (.rβ€˜πΉ)    &    1 = (1rβ€˜πΉ)    &   (𝑅 ∈ Grp ∧ 𝐹 ∈ Ring ∧ βˆ€π‘ž ∈ 𝐾 βˆ€π‘Ÿ ∈ 𝐾 βˆ€π‘₯ ∈ 𝑉 βˆ€π‘€ ∈ 𝑉 (((𝑀 Β· π‘Ÿ) ∈ 𝑉 ∧ ((𝑀 + π‘₯) Β· π‘Ÿ) = ((𝑀 Β· π‘Ÿ) + (π‘₯ Β· π‘Ÿ)) ∧ (𝑀 Β· (π‘ž ⨣ π‘Ÿ)) = ((𝑀 Β· π‘ž) + (𝑀 Β· π‘Ÿ))) ∧ ((𝑀 Β· (π‘ž Γ— π‘Ÿ)) = ((𝑀 Β· π‘ž) Β· π‘Ÿ) ∧ (𝑀 Β· 1 ) = 𝑀)))    &    βˆ— = (𝑠 ∈ 𝐾, 𝑣 ∈ 𝑉 ↦ (𝑣 Β· 𝑠))    &   πΏ = (𝑅 sSet ⟨( ·𝑠 β€˜ndx), βˆ— ⟩)    β‡’   ((𝐹 ∈ CRing ∧ (π‘Ž ∈ 𝐾 ∧ 𝑏 ∈ 𝐾 ∧ 𝑐 ∈ 𝑉)) β†’ ((π‘Ž Γ— 𝑏) βˆ— 𝑐) = (π‘Ž βˆ— (𝑏 βˆ— 𝑐)))
 
Theoremrmodislmod 20313* The right module 𝑅 induces a left module 𝐿 by replacing the scalar multiplication with a reversed multiplication if the scalar ring is commutative. The hypothesis "rmodislmod.r" is a definition of a right module analogous to Definition df-lmod 20247 of a left module, see also islmod 20249. (Contributed by AV, 3-Dec-2021.) (Proof shortened by AV, 18-Oct-2024.)
𝑉 = (Baseβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &    Β· = ( ·𝑠 β€˜π‘…)    &   πΉ = (Scalarβ€˜π‘…)    &   πΎ = (Baseβ€˜πΉ)    &    ⨣ = (+gβ€˜πΉ)    &    Γ— = (.rβ€˜πΉ)    &    1 = (1rβ€˜πΉ)    &   (𝑅 ∈ Grp ∧ 𝐹 ∈ Ring ∧ βˆ€π‘ž ∈ 𝐾 βˆ€π‘Ÿ ∈ 𝐾 βˆ€π‘₯ ∈ 𝑉 βˆ€π‘€ ∈ 𝑉 (((𝑀 Β· π‘Ÿ) ∈ 𝑉 ∧ ((𝑀 + π‘₯) Β· π‘Ÿ) = ((𝑀 Β· π‘Ÿ) + (π‘₯ Β· π‘Ÿ)) ∧ (𝑀 Β· (π‘ž ⨣ π‘Ÿ)) = ((𝑀 Β· π‘ž) + (𝑀 Β· π‘Ÿ))) ∧ ((𝑀 Β· (π‘ž Γ— π‘Ÿ)) = ((𝑀 Β· π‘ž) Β· π‘Ÿ) ∧ (𝑀 Β· 1 ) = 𝑀)))    &    βˆ— = (𝑠 ∈ 𝐾, 𝑣 ∈ 𝑉 ↦ (𝑣 Β· 𝑠))    &   πΏ = (𝑅 sSet ⟨( ·𝑠 β€˜ndx), βˆ— ⟩)    β‡’   (𝐹 ∈ CRing β†’ 𝐿 ∈ LMod)
 
TheoremrmodislmodOLD 20314* Obsolete version of rmodislmod 20313 as of 18-Oct-2024. The right module 𝑅 induces a left module 𝐿 by replacing the scalar multiplication with a reversed multiplication if the scalar ring is commutative. The hypothesis "rmodislmod.r" is a definition of a right module analogous to Definition df-lmod 20247 of a left module, see also islmod 20249. (Contributed by AV, 3-Dec-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑉 = (Baseβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &    Β· = ( ·𝑠 β€˜π‘…)    &   πΉ = (Scalarβ€˜π‘…)    &   πΎ = (Baseβ€˜πΉ)    &    ⨣ = (+gβ€˜πΉ)    &    Γ— = (.rβ€˜πΉ)    &    1 = (1rβ€˜πΉ)    &   (𝑅 ∈ Grp ∧ 𝐹 ∈ Ring ∧ βˆ€π‘ž ∈ 𝐾 βˆ€π‘Ÿ ∈ 𝐾 βˆ€π‘₯ ∈ 𝑉 βˆ€π‘€ ∈ 𝑉 (((𝑀 Β· π‘Ÿ) ∈ 𝑉 ∧ ((𝑀 + π‘₯) Β· π‘Ÿ) = ((𝑀 Β· π‘Ÿ) + (π‘₯ Β· π‘Ÿ)) ∧ (𝑀 Β· (π‘ž ⨣ π‘Ÿ)) = ((𝑀 Β· π‘ž) + (𝑀 Β· π‘Ÿ))) ∧ ((𝑀 Β· (π‘ž Γ— π‘Ÿ)) = ((𝑀 Β· π‘ž) Β· π‘Ÿ) ∧ (𝑀 Β· 1 ) = 𝑀)))    &    βˆ— = (𝑠 ∈ 𝐾, 𝑣 ∈ 𝑉 ↦ (𝑣 Β· 𝑠))    &   πΏ = (𝑅 sSet ⟨( ·𝑠 β€˜ndx), βˆ— ⟩)    β‡’   (𝐹 ∈ CRing β†’ 𝐿 ∈ LMod)
 
10.5.2  Subspaces and spans in a left module
 
Syntaxclss 20315 Extend class notation with linear subspaces of a left module or left vector space.
class LSubSp
 
Definitiondf-lss 20316* Define the set of linear subspaces of a left module or left vector space. (Contributed by NM, 8-Dec-2013.)
LSubSp = (𝑀 ∈ V ↦ {𝑠 ∈ (𝒫 (Baseβ€˜π‘€) βˆ– {βˆ…}) ∣ βˆ€π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€))βˆ€π‘Ž ∈ 𝑠 βˆ€π‘ ∈ 𝑠 ((π‘₯( ·𝑠 β€˜π‘€)π‘Ž)(+gβ€˜π‘€)𝑏) ∈ 𝑠})
 
Theoremlssset 20317* The set of all (not necessarily closed) linear subspaces of a left module or left vector space. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 15-Jul-2014.)
𝐹 = (Scalarβ€˜π‘Š)    &   π΅ = (Baseβ€˜πΉ)    &   π‘‰ = (Baseβ€˜π‘Š)    &    + = (+gβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    β‡’   (π‘Š ∈ 𝑋 β†’ 𝑆 = {𝑠 ∈ (𝒫 𝑉 βˆ– {βˆ…}) ∣ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘Ž ∈ 𝑠 βˆ€π‘ ∈ 𝑠 ((π‘₯ Β· π‘Ž) + 𝑏) ∈ 𝑠})
 
Theoremislss 20318* The predicate "is a subspace" (of a left module or left vector space). (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)
𝐹 = (Scalarβ€˜π‘Š)    &   π΅ = (Baseβ€˜πΉ)    &   π‘‰ = (Baseβ€˜π‘Š)    &    + = (+gβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    β‡’   (π‘ˆ ∈ 𝑆 ↔ (π‘ˆ βŠ† 𝑉 ∧ π‘ˆ β‰  βˆ… ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘Ž ∈ π‘ˆ βˆ€π‘ ∈ π‘ˆ ((π‘₯ Β· π‘Ž) + 𝑏) ∈ π‘ˆ))
 
Theoremislssd 20319* Properties that determine a subspace of a left module or left vector space. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)
(πœ‘ β†’ 𝐹 = (Scalarβ€˜π‘Š))    &   (πœ‘ β†’ 𝐡 = (Baseβ€˜πΉ))    &   (πœ‘ β†’ 𝑉 = (Baseβ€˜π‘Š))    &   (πœ‘ β†’ + = (+gβ€˜π‘Š))    &   (πœ‘ β†’ Β· = ( ·𝑠 β€˜π‘Š))    &   (πœ‘ β†’ 𝑆 = (LSubSpβ€˜π‘Š))    &   (πœ‘ β†’ π‘ˆ βŠ† 𝑉)    &   (πœ‘ β†’ π‘ˆ β‰  βˆ…)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ π‘Ž ∈ π‘ˆ ∧ 𝑏 ∈ π‘ˆ)) β†’ ((π‘₯ Β· π‘Ž) + 𝑏) ∈ π‘ˆ)    β‡’   (πœ‘ β†’ π‘ˆ ∈ 𝑆)
 
Theoremlssss 20320 A subspace is a set of vectors. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    β‡’   (π‘ˆ ∈ 𝑆 β†’ π‘ˆ βŠ† 𝑉)
 
Theoremlssel 20321 A subspace member is a vector. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 8-Jan-2015.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    β‡’   ((π‘ˆ ∈ 𝑆 ∧ 𝑋 ∈ π‘ˆ) β†’ 𝑋 ∈ 𝑉)
 
Theoremlss1 20322 The set of vectors in a left module is a subspace. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    β‡’   (π‘Š ∈ LMod β†’ 𝑉 ∈ 𝑆)
 
Theoremlssuni 20323 The union of all subspaces is the vector space. (Contributed by NM, 13-Mar-2015.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    β‡’   (πœ‘ β†’ βˆͺ 𝑆 = 𝑉)
 
Theoremlssn0 20324 A subspace is not empty. (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 8-Jan-2015.)
𝑆 = (LSubSpβ€˜π‘Š)    β‡’   (π‘ˆ ∈ 𝑆 β†’ π‘ˆ β‰  βˆ…)
 
Theorem00lss 20325 The empty structure has no subspaces (for use with fvco4i 6938). (Contributed by Stefan O'Rear, 31-Mar-2015.)
βˆ… = (LSubSpβ€˜βˆ…)
 
Theoremlsscl 20326 Closure property of a subspace. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)
𝐹 = (Scalarβ€˜π‘Š)    &   π΅ = (Baseβ€˜πΉ)    &    + = (+gβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    β‡’   ((π‘ˆ ∈ 𝑆 ∧ (𝑍 ∈ 𝐡 ∧ 𝑋 ∈ π‘ˆ ∧ π‘Œ ∈ π‘ˆ)) β†’ ((𝑍 Β· 𝑋) + π‘Œ) ∈ π‘ˆ)
 
Theoremlssvsubcl 20327 Closure of vector subtraction in a subspace. (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
βˆ’ = (-gβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    β‡’   (((π‘Š ∈ LMod ∧ π‘ˆ ∈ 𝑆) ∧ (𝑋 ∈ π‘ˆ ∧ π‘Œ ∈ π‘ˆ)) β†’ (𝑋 βˆ’ π‘Œ) ∈ π‘ˆ)
 
Theoremlssvancl1 20328 Non-closure: if one vector belongs to a subspace but another does not, their sum does not belong. Useful for obtaining a new vector not in a subspace. TODO: notice similarity to lspindp3 20520. Can it be used along with lspsnne1 20501, lspsnne2 20502 to shorten this proof? (Contributed by NM, 14-May-2015.)
𝑉 = (Baseβ€˜π‘Š)    &    + = (+gβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ π‘ˆ ∈ 𝑆)    &   (πœ‘ β†’ 𝑋 ∈ π‘ˆ)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    &   (πœ‘ β†’ Β¬ π‘Œ ∈ π‘ˆ)    β‡’   (πœ‘ β†’ Β¬ (𝑋 + π‘Œ) ∈ π‘ˆ)
 
Theoremlssvancl2 20329 Non-closure: if one vector belongs to a subspace but another does not, their sum does not belong. Useful for obtaining a new vector not in a subspace. (Contributed by NM, 20-May-2015.)
𝑉 = (Baseβ€˜π‘Š)    &    + = (+gβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ π‘ˆ ∈ 𝑆)    &   (πœ‘ β†’ 𝑋 ∈ π‘ˆ)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    &   (πœ‘ β†’ Β¬ π‘Œ ∈ π‘ˆ)    β‡’   (πœ‘ β†’ Β¬ (π‘Œ + 𝑋) ∈ π‘ˆ)
 
Theoremlss0cl 20330 The zero vector belongs to every subspace. (Contributed by NM, 12-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
0 = (0gβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ π‘ˆ ∈ 𝑆) β†’ 0 ∈ π‘ˆ)
 
Theoremlsssn0 20331 The singleton of the zero vector is a subspace. (Contributed by NM, 13-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
0 = (0gβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    β‡’   (π‘Š ∈ LMod β†’ { 0 } ∈ 𝑆)
 
Theoremlss0ss 20332 The zero subspace is included in every subspace. (sh0le 30168 analog.) (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
0 = (0gβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑆) β†’ { 0 } βŠ† 𝑋)
 
Theoremlssle0 20333 No subspace is smaller than the zero subspace. (shle0 30170 analog.) (Contributed by NM, 20-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
0 = (0gβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑆) β†’ (𝑋 βŠ† { 0 } ↔ 𝑋 = { 0 }))
 
Theoremlssne0 20334* A nonzero subspace has a nonzero vector. (shne0i 30176 analog.) (Contributed by NM, 20-Apr-2014.) (Proof shortened by Mario Carneiro, 8-Jan-2015.)
0 = (0gβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    β‡’   (𝑋 ∈ 𝑆 β†’ (𝑋 β‰  { 0 } ↔ βˆƒπ‘¦ ∈ 𝑋 𝑦 β‰  0 ))
 
Theoremlssvneln0 20335 A vector 𝑋 which doesn't belong to a subspace π‘ˆ is nonzero. (Contributed by NM, 14-May-2015.) (Revised by AV, 19-Jul-2022.)
0 = (0gβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ π‘ˆ ∈ 𝑆)    &   (πœ‘ β†’ Β¬ 𝑋 ∈ π‘ˆ)    β‡’   (πœ‘ β†’ 𝑋 β‰  0 )
 
Theoremlssneln0 20336 A vector 𝑋 which doesn't belong to a subspace π‘ˆ is nonzero. (Contributed by NM, 14-May-2015.) (Revised by AV, 17-Jul-2022.) (Proof shortened by AV, 19-Jul-2022.)
0 = (0gβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ π‘ˆ ∈ 𝑆)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ Β¬ 𝑋 ∈ π‘ˆ)    β‡’   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))
 
Theoremlssssr 20337* Conclude subspace ordering from nonzero vector membership. (ssrdv 3949 analog.) (Contributed by NM, 17-Aug-2014.) (Revised by AV, 13-Jul-2022.)
0 = (0gβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ 𝑇 βŠ† 𝑉)    &   (πœ‘ β†’ π‘ˆ ∈ 𝑆)    &   ((πœ‘ ∧ π‘₯ ∈ (𝑉 βˆ– { 0 })) β†’ (π‘₯ ∈ 𝑇 β†’ π‘₯ ∈ π‘ˆ))    β‡’   (πœ‘ β†’ 𝑇 βŠ† π‘ˆ)
 
Theoremlssvacl 20338 Closure of vector addition in a subspace. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
+ = (+gβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    β‡’   (((π‘Š ∈ LMod ∧ π‘ˆ ∈ 𝑆) ∧ (𝑋 ∈ π‘ˆ ∧ π‘Œ ∈ π‘ˆ)) β†’ (𝑋 + π‘Œ) ∈ π‘ˆ)
 
Theoremlssvscl 20339 Closure of scalar product in a subspace. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝐹 = (Scalarβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   π΅ = (Baseβ€˜πΉ)    &   π‘† = (LSubSpβ€˜π‘Š)    β‡’   (((π‘Š ∈ LMod ∧ π‘ˆ ∈ 𝑆) ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ π‘ˆ)) β†’ (𝑋 Β· π‘Œ) ∈ π‘ˆ)
 
Theoremlssvnegcl 20340 Closure of negative vectors in a subspace. (Contributed by Stefan O'Rear, 11-Dec-2014.)
𝑆 = (LSubSpβ€˜π‘Š)    &   π‘ = (invgβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ π‘ˆ ∈ 𝑆 ∧ 𝑋 ∈ π‘ˆ) β†’ (π‘β€˜π‘‹) ∈ π‘ˆ)
 
Theoremlsssubg 20341 All subspaces are subgroups. (Contributed by Stefan O'Rear, 11-Dec-2014.)
𝑆 = (LSubSpβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ π‘ˆ ∈ 𝑆) β†’ π‘ˆ ∈ (SubGrpβ€˜π‘Š))
 
Theoremlsssssubg 20342 All subspaces are subgroups. (Contributed by Mario Carneiro, 19-Apr-2016.)
𝑆 = (LSubSpβ€˜π‘Š)    β‡’   (π‘Š ∈ LMod β†’ 𝑆 βŠ† (SubGrpβ€˜π‘Š))
 
Theoremislss3 20343 A linear subspace of a module is a subset which is a module in its own right. (Contributed by Stefan O'Rear, 6-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
𝑋 = (π‘Š β†Ύs π‘ˆ)    &   π‘‰ = (Baseβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    β‡’   (π‘Š ∈ LMod β†’ (π‘ˆ ∈ 𝑆 ↔ (π‘ˆ βŠ† 𝑉 ∧ 𝑋 ∈ LMod)))
 
Theoremlsslmod 20344 A submodule is a module. (Contributed by Stefan O'Rear, 12-Dec-2014.)
𝑋 = (π‘Š β†Ύs π‘ˆ)    &   π‘† = (LSubSpβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ π‘ˆ ∈ 𝑆) β†’ 𝑋 ∈ LMod)
 
Theoremlsslss 20345 The subspaces of a subspace are the smaller subspaces. (Contributed by Stefan O'Rear, 12-Dec-2014.)
𝑋 = (π‘Š β†Ύs π‘ˆ)    &   π‘† = (LSubSpβ€˜π‘Š)    &   π‘‡ = (LSubSpβ€˜π‘‹)    β‡’   ((π‘Š ∈ LMod ∧ π‘ˆ ∈ 𝑆) β†’ (𝑉 ∈ 𝑇 ↔ (𝑉 ∈ 𝑆 ∧ 𝑉 βŠ† π‘ˆ)))
 
Theoremislss4 20346* A linear subspace is a subgroup which respects scalar multiplication. (Contributed by Stefan O'Rear, 11-Dec-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
𝐹 = (Scalarβ€˜π‘Š)    &   π΅ = (Baseβ€˜πΉ)    &   π‘‰ = (Baseβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    β‡’   (π‘Š ∈ LMod β†’ (π‘ˆ ∈ 𝑆 ↔ (π‘ˆ ∈ (SubGrpβ€˜π‘Š) ∧ βˆ€π‘Ž ∈ 𝐡 βˆ€π‘ ∈ π‘ˆ (π‘Ž Β· 𝑏) ∈ π‘ˆ)))
 
Theoremlss1d 20347* One-dimensional subspace (or zero-dimensional if 𝑋 is the zero vector). (Contributed by NM, 14-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Baseβ€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    &   π‘† = (LSubSpβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ {𝑣 ∣ βˆƒπ‘˜ ∈ 𝐾 𝑣 = (π‘˜ Β· 𝑋)} ∈ 𝑆)
 
Theoremlssintcl 20348 The intersection of a nonempty set of subspaces is a subspace. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑆 = (LSubSpβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ 𝐴 βŠ† 𝑆 ∧ 𝐴 β‰  βˆ…) β†’ ∩ 𝐴 ∈ 𝑆)
 
Theoremlssincl 20349 The intersection of two subspaces is a subspace. (Contributed by NM, 7-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑆 = (LSubSpβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ π‘ˆ ∈ 𝑆) β†’ (𝑇 ∩ π‘ˆ) ∈ 𝑆)
 
Theoremlssmre 20350 The subspaces of a module comprise a Moore system on the vectors of the module. (Contributed by Stefan O'Rear, 31-Jan-2015.)
𝐡 = (Baseβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    β‡’   (π‘Š ∈ LMod β†’ 𝑆 ∈ (Mooreβ€˜π΅))
 
Theoremlssacs 20351 Submodules are an algebraic closure system. (Contributed by Stefan O'Rear, 4-Apr-2015.)
𝐡 = (Baseβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    β‡’   (π‘Š ∈ LMod β†’ 𝑆 ∈ (ACSβ€˜π΅))
 
Theoremprdsvscacl 20352* Pointwise scalar multiplication is closed in products of modules. (Contributed by Stefan O'Rear, 10-Jan-2015.)
π‘Œ = (𝑆Xs𝑅)    &   π΅ = (Baseβ€˜π‘Œ)    &    Β· = ( ·𝑠 β€˜π‘Œ)    &   πΎ = (Baseβ€˜π‘†)    &   (πœ‘ β†’ 𝑆 ∈ Ring)    &   (πœ‘ β†’ 𝐼 ∈ π‘Š)    &   (πœ‘ β†’ 𝑅:𝐼⟢LMod)    &   (πœ‘ β†’ 𝐹 ∈ 𝐾)    &   (πœ‘ β†’ 𝐺 ∈ 𝐡)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐼) β†’ (Scalarβ€˜(π‘…β€˜π‘₯)) = 𝑆)    β‡’   (πœ‘ β†’ (𝐹 Β· 𝐺) ∈ 𝐡)
 
Theoremprdslmodd 20353* The product of a family of left modules is a left module. (Contributed by Stefan O'Rear, 10-Jan-2015.)
π‘Œ = (𝑆Xs𝑅)    &   (πœ‘ β†’ 𝑆 ∈ Ring)    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝑅:𝐼⟢LMod)    &   ((πœ‘ ∧ 𝑦 ∈ 𝐼) β†’ (Scalarβ€˜(π‘…β€˜π‘¦)) = 𝑆)    β‡’   (πœ‘ β†’ π‘Œ ∈ LMod)
 
Theorempwslmod 20354 A structure power of a left module is a left module. (Contributed by Mario Carneiro, 11-Jan-2015.)
π‘Œ = (𝑅 ↑s 𝐼)    β‡’   ((𝑅 ∈ LMod ∧ 𝐼 ∈ 𝑉) β†’ π‘Œ ∈ LMod)
 
Syntaxclspn 20355 Extend class notation with span of a set of vectors.
class LSpan
 
Definitiondf-lsp 20356* Define span of a set of vectors of a left module or left vector space. (Contributed by NM, 8-Dec-2013.)
LSpan = (𝑀 ∈ V ↦ (𝑠 ∈ 𝒫 (Baseβ€˜π‘€) ↦ ∩ {𝑑 ∈ (LSubSpβ€˜π‘€) ∣ 𝑠 βŠ† 𝑑}))
 
Theoremlspfval 20357* The span function for a left vector space (or a left module). (df-span 30037 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   (π‘Š ∈ 𝑋 β†’ 𝑁 = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑑 ∈ 𝑆 ∣ 𝑠 βŠ† 𝑑}))
 
Theoremlspf 20358 The span operator on a left module maps subsets to subsets. (Contributed by Stefan O'Rear, 12-Dec-2014.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   (π‘Š ∈ LMod β†’ 𝑁:𝒫 π‘‰βŸΆπ‘†)
 
Theoremlspval 20359* The span of a set of vectors (in a left module). (spanval 30061 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ π‘ˆ βŠ† 𝑉) β†’ (π‘β€˜π‘ˆ) = ∩ {𝑑 ∈ 𝑆 ∣ π‘ˆ βŠ† 𝑑})
 
Theoremlspcl 20360 The span of a set of vectors is a subspace. (spancl 30064 analog.) (Contributed by NM, 9-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ π‘ˆ βŠ† 𝑉) β†’ (π‘β€˜π‘ˆ) ∈ 𝑆)
 
Theoremlspsncl 20361 The span of a singleton is a subspace (frequently used special case of lspcl 20360). (Contributed by NM, 17-Jul-2014.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ (π‘β€˜{𝑋}) ∈ 𝑆)
 
Theoremlspprcl 20362 The span of a pair is a subspace (frequently used special case of lspcl 20360). (Contributed by NM, 11-Apr-2015.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    β‡’   (πœ‘ β†’ (π‘β€˜{𝑋, π‘Œ}) ∈ 𝑆)
 
Theoremlsptpcl 20363 The span of an unordered triple is a subspace (frequently used special case of lspcl 20360). (Contributed by NM, 22-May-2015.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    &   (πœ‘ β†’ 𝑍 ∈ 𝑉)    β‡’   (πœ‘ β†’ (π‘β€˜{𝑋, π‘Œ, 𝑍}) ∈ 𝑆)
 
Theoremlspsnsubg 20364 The span of a singleton is an additive subgroup (frequently used special case of lspcl 20360). (Contributed by Mario Carneiro, 21-Apr-2016.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ (π‘β€˜{𝑋}) ∈ (SubGrpβ€˜π‘Š))
 
Theorem00lsp 20365 fvco4i 6938 lemma for linear spans. (Contributed by Stefan O'Rear, 4-Apr-2015.)
βˆ… = (LSpanβ€˜βˆ…)
 
Theoremlspid 20366 The span of a subspace is itself. (spanid 30075 analog.) (Contributed by NM, 15-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑆 = (LSubSpβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ π‘ˆ ∈ 𝑆) β†’ (π‘β€˜π‘ˆ) = π‘ˆ)
 
Theoremlspssv 20367 A span is a set of vectors. (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ π‘ˆ βŠ† 𝑉) β†’ (π‘β€˜π‘ˆ) βŠ† 𝑉)
 
Theoremlspss 20368 Span preserves subset ordering. (spanss 30076 analog.) (Contributed by NM, 11-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ π‘ˆ βŠ† 𝑉 ∧ 𝑇 βŠ† π‘ˆ) β†’ (π‘β€˜π‘‡) βŠ† (π‘β€˜π‘ˆ))
 
Theoremlspssid 20369 A set of vectors is a subset of its span. (spanss2 30073 analog.) (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ π‘ˆ βŠ† 𝑉) β†’ π‘ˆ βŠ† (π‘β€˜π‘ˆ))
 
Theoremlspidm 20370 The span of a set of vectors is idempotent. (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ π‘ˆ βŠ† 𝑉) β†’ (π‘β€˜(π‘β€˜π‘ˆ)) = (π‘β€˜π‘ˆ))
 
Theoremlspun 20371 The span of union is the span of the union of spans. (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ 𝑇 βŠ† 𝑉 ∧ π‘ˆ βŠ† 𝑉) β†’ (π‘β€˜(𝑇 βˆͺ π‘ˆ)) = (π‘β€˜((π‘β€˜π‘‡) βˆͺ (π‘β€˜π‘ˆ))))
 
Theoremlspssp 20372 If a set of vectors is a subset of a subspace, then the span of those vectors is also contained in the subspace. (Contributed by Mario Carneiro, 4-Sep-2014.)
𝑆 = (LSubSpβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ π‘ˆ ∈ 𝑆 ∧ 𝑇 βŠ† π‘ˆ) β†’ (π‘β€˜π‘‡) βŠ† π‘ˆ)
 
Theoremmrclsp 20373 Moore closure generalizes module span. (Contributed by Stefan O'Rear, 31-Jan-2015.)
π‘ˆ = (LSubSpβ€˜π‘Š)    &   πΎ = (LSpanβ€˜π‘Š)    &   πΉ = (mrClsβ€˜π‘ˆ)    β‡’   (π‘Š ∈ LMod β†’ 𝐾 = 𝐹)
 
Theoremlspsnss 20374 The span of the singleton of a subspace member is included in the subspace. (spansnss 30299 analog.) (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 4-Sep-2014.)
𝑆 = (LSubSpβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ π‘ˆ ∈ 𝑆 ∧ 𝑋 ∈ π‘ˆ) β†’ (π‘β€˜{𝑋}) βŠ† π‘ˆ)
 
Theoremlspsnel3 20375 A member of the span of the singleton of a vector is a member of a subspace containing the vector. (elspansn3 30300 analog.) (Contributed by NM, 4-Jul-2014.)
𝑆 = (LSubSpβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ π‘ˆ ∈ 𝑆)    &   (πœ‘ β†’ 𝑋 ∈ π‘ˆ)    &   (πœ‘ β†’ π‘Œ ∈ (π‘β€˜{𝑋}))    β‡’   (πœ‘ β†’ π‘Œ ∈ π‘ˆ)
 
Theoremlspprss 20376 The span of a pair of vectors in a subspace belongs to the subspace. (Contributed by NM, 12-Jan-2015.)
𝑆 = (LSubSpβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ π‘ˆ ∈ 𝑆)    &   (πœ‘ β†’ 𝑋 ∈ π‘ˆ)    &   (πœ‘ β†’ π‘Œ ∈ π‘ˆ)    β‡’   (πœ‘ β†’ (π‘β€˜{𝑋, π‘Œ}) βŠ† π‘ˆ)
 
Theoremlspsnid 20377 A vector belongs to the span of its singleton. (spansnid 30291 analog.) (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ 𝑋 ∈ (π‘β€˜{𝑋}))
 
Theoremlspsnel6 20378 Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 8-Aug-2014.) (Revised by Mario Carneiro, 8-Jan-2015.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ π‘ˆ ∈ 𝑆)    β‡’   (πœ‘ β†’ (𝑋 ∈ π‘ˆ ↔ (𝑋 ∈ 𝑉 ∧ (π‘β€˜{𝑋}) βŠ† π‘ˆ)))
 
Theoremlspsnel5 20379 Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 8-Aug-2014.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ π‘ˆ ∈ 𝑆)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    β‡’   (πœ‘ β†’ (𝑋 ∈ π‘ˆ ↔ (π‘β€˜{𝑋}) βŠ† π‘ˆ))
 
Theoremlspsnel5a 20380 Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 20-Feb-2015.)
𝑆 = (LSubSpβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ π‘ˆ ∈ 𝑆)    &   (πœ‘ β†’ 𝑋 ∈ π‘ˆ)    β‡’   (πœ‘ β†’ (π‘β€˜{𝑋}) βŠ† π‘ˆ)
 
Theoremlspprid1 20381 A member of a pair of vectors belongs to their span. (Contributed by NM, 14-May-2015.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    β‡’   (πœ‘ β†’ 𝑋 ∈ (π‘β€˜{𝑋, π‘Œ}))
 
Theoremlspprid2 20382 A member of a pair of vectors belongs to their span. (Contributed by NM, 14-May-2015.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    β‡’   (πœ‘ β†’ π‘Œ ∈ (π‘β€˜{𝑋, π‘Œ}))
 
Theoremlspprvacl 20383 The sum of two vectors belongs to their span. (Contributed by NM, 20-May-2015.)
𝑉 = (Baseβ€˜π‘Š)    &    + = (+gβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    β‡’   (πœ‘ β†’ (𝑋 + π‘Œ) ∈ (π‘β€˜{𝑋, π‘Œ}))
 
Theoremlssats2 20384* A way to express atomisticity (a subspace is the union of its atoms). (Contributed by NM, 3-Feb-2015.)
𝑆 = (LSubSpβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ π‘ˆ ∈ 𝑆)    β‡’   (πœ‘ β†’ π‘ˆ = βˆͺ π‘₯ ∈ π‘ˆ (π‘β€˜{π‘₯}))
 
Theoremlspsneli 20385 A scalar product with a vector belongs to the span of its singleton. (spansnmul 30292 analog.) (Contributed by NM, 2-Jul-2014.)
𝑉 = (Baseβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ 𝐴 ∈ 𝐾)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    β‡’   (πœ‘ β†’ (𝐴 Β· 𝑋) ∈ (π‘β€˜{𝑋}))
 
Theoremlspsn 20386* Span of the singleton of a vector. (Contributed by NM, 14-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
𝐹 = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    &   π‘‰ = (Baseβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ (π‘β€˜{𝑋}) = {𝑣 ∣ βˆƒπ‘˜ ∈ 𝐾 𝑣 = (π‘˜ Β· 𝑋)})
 
Theoremlspsnel 20387* Member of span of the singleton of a vector. (elspansn 30294 analog.) (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝐹 = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    &   π‘‰ = (Baseβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ (π‘ˆ ∈ (π‘β€˜{𝑋}) ↔ βˆƒπ‘˜ ∈ 𝐾 π‘ˆ = (π‘˜ Β· 𝑋)))
 
Theoremlspsnvsi 20388 Span of a scalar product of a singleton. (Contributed by NM, 23-Apr-2014.) (Proof shortened by Mario Carneiro, 4-Sep-2014.)
𝐹 = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    &   π‘‰ = (Baseβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) β†’ (π‘β€˜{(𝑅 Β· 𝑋)}) βŠ† (π‘β€˜{𝑋}))
 
Theoremlspsnss2 20389* Comparable spans of singletons must have proportional vectors. See lspsneq 20506 for equal span version. (Contributed by NM, 7-Jun-2015.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘† = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜π‘†)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    β‡’   (πœ‘ β†’ ((π‘β€˜{𝑋}) βŠ† (π‘β€˜{π‘Œ}) ↔ βˆƒπ‘˜ ∈ 𝐾 𝑋 = (π‘˜ Β· π‘Œ)))
 
Theoremlspsnneg 20390 Negation does not change the span of a singleton. (Contributed by NM, 24-Apr-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘€ = (invgβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ (π‘β€˜{(π‘€β€˜π‘‹)}) = (π‘β€˜{𝑋}))
 
Theoremlspsnsub 20391 Swapping subtraction order does not change the span of a singleton. (Contributed by NM, 4-Apr-2015.)
𝑉 = (Baseβ€˜π‘Š)    &    βˆ’ = (-gβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    β‡’   (πœ‘ β†’ (π‘β€˜{(𝑋 βˆ’ π‘Œ)}) = (π‘β€˜{(π‘Œ βˆ’ 𝑋)}))
 
Theoremlspsn0 20392 Span of the singleton of the zero vector. (spansn0 30269 analog.) (Contributed by NM, 15-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
0 = (0gβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   (π‘Š ∈ LMod β†’ (π‘β€˜{ 0 }) = { 0 })
 
Theoremlsp0 20393 Span of the empty set. (Contributed by Mario Carneiro, 5-Sep-2014.)
0 = (0gβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   (π‘Š ∈ LMod β†’ (π‘β€˜βˆ…) = { 0 })
 
Theoremlspuni0 20394 Union of the span of the empty set. (Contributed by NM, 14-Mar-2015.)
0 = (0gβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   (π‘Š ∈ LMod β†’ βˆͺ (π‘β€˜βˆ…) = 0 )
 
Theoremlspun0 20395 The span of a union with the zero subspace. (Contributed by NM, 22-May-2015.)
𝑉 = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ 𝑋 βŠ† 𝑉)    β‡’   (πœ‘ β†’ (π‘β€˜(𝑋 βˆͺ { 0 })) = (π‘β€˜π‘‹))
 
Theoremlspsneq0 20396 Span of the singleton is the zero subspace iff the vector is zero. (Contributed by NM, 27-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ ((π‘β€˜{𝑋}) = { 0 } ↔ 𝑋 = 0 ))
 
Theoremlspsneq0b 20397 Equal singleton spans imply both arguments are zero or both are nonzero. (Contributed by NM, 21-Mar-2015.)
𝑉 = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    &   (πœ‘ β†’ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ}))    β‡’   (πœ‘ β†’ (𝑋 = 0 ↔ π‘Œ = 0 ))
 
Theoremlmodindp1 20398 Two independent (non-colinear) vectors have nonzero sum. (Contributed by NM, 22-Apr-2015.)
𝑉 = (Baseβ€˜π‘Š)    &    + = (+gβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    β‡’   (πœ‘ β†’ (𝑋 + π‘Œ) β‰  0 )
 
Theoremlsslsp 20399 Spans in submodules correspond to spans in the containing module. (Contributed by Stefan O'Rear, 12-Dec-2014.) TODO: Shouldn't we swap π‘€β€˜πΊ and π‘β€˜πΊ since we are computing a property of π‘β€˜πΊ? (Like we say sin 0 = 0 and not 0 = sin 0.) - NM 15-Mar-2015.
𝑋 = (π‘Š β†Ύs π‘ˆ)    &   π‘€ = (LSpanβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘‹)    &   πΏ = (LSubSpβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ π‘ˆ ∈ 𝐿 ∧ 𝐺 βŠ† π‘ˆ) β†’ (π‘€β€˜πΊ) = (π‘β€˜πΊ))
 
Theoremlss0v 20400 The zero vector in a submodule equals the zero vector in the including module. (Contributed by NM, 15-Mar-2015.)
𝑋 = (π‘Š β†Ύs π‘ˆ)    &    0 = (0gβ€˜π‘Š)    &   π‘ = (0gβ€˜π‘‹)    &   πΏ = (LSubSpβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ π‘ˆ ∈ 𝐿) β†’ 𝑍 = 0 )
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 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330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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