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Theorem List for Metamath Proof Explorer - 19301-19400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlmodvnegid 19301 Addition of a vector with its negative. (Contributed by NM, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    0 = (0g𝑊)    &   𝑁 = (invg𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑉) → (𝑋 + (𝑁𝑋)) = 0 )

Theoremlmodvneg1 19302 Minus 1 times a vector is the negative of the vector. Equation 2 of [Kreyszig] p. 51. (Contributed by NM, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (invg𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &    1 = (1r𝐹)    &   𝑀 = (invg𝐹)       ((𝑊 ∈ LMod ∧ 𝑋𝑉) → ((𝑀1 ) · 𝑋) = (𝑁𝑋))

Theoremlmodvsneg 19303 Multiplication of a vector by a negated scalar. (Contributed by Stefan O'Rear, 28-Feb-2015.)
𝐵 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝑁 = (invg𝑊)    &   𝐾 = (Base‘𝐹)    &   𝑀 = (invg𝐹)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝐵)    &   (𝜑𝑅𝐾)       (𝜑 → (𝑁‘(𝑅 · 𝑋)) = ((𝑀𝑅) · 𝑋))

Theoremlmodvsubcl 19304 Closure of vector subtraction. (hvsubcl 28450 analog.) (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &    = (-g𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑉𝑌𝑉) → (𝑋 𝑌) ∈ 𝑉)

Theoremlmodcom 19305 Left module vector sum is commutative. (Contributed by Gérard Lang, 25-Jun-2014.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑉𝑌𝑉) → (𝑋 + 𝑌) = (𝑌 + 𝑋))

Theoremlmodabl 19306 A left module is an abelian group (of vectors, under addition). (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 25-Jun-2014.)
(𝑊 ∈ LMod → 𝑊 ∈ Abel)

Theoremlmodcmn 19307 A left module is a commutative monoid under addition. (Contributed by NM, 7-Jan-2015.)
(𝑊 ∈ LMod → 𝑊 ∈ CMnd)

Theoremlmodnegadd 19308 Distribute negation through addition of scalar products. (Contributed by NM, 9-Apr-2015.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    · = ( ·𝑠𝑊)    &   𝑁 = (invg𝑊)    &   𝑅 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑅)    &   𝐼 = (invg𝑅)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐴𝐾)    &   (𝜑𝐵𝐾)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → (𝑁‘((𝐴 · 𝑋) + (𝐵 · 𝑌))) = (((𝐼𝐴) · 𝑋) + ((𝐼𝐵) · 𝑌)))

Theoremlmod4 19309 Commutative/associative law for left module vector sum. (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)       ((𝑊 ∈ LMod ∧ (𝑋𝑉𝑌𝑉) ∧ (𝑍𝑉𝑈𝑉)) → ((𝑋 + 𝑌) + (𝑍 + 𝑈)) = ((𝑋 + 𝑍) + (𝑌 + 𝑈)))

Theoremlmodvsubadd 19310 Relationship between vector subtraction and addition. (hvsubadd 28510 analog.) (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    = (-g𝑊)       ((𝑊 ∈ LMod ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐴 𝐵) = 𝐶 ↔ (𝐵 + 𝐶) = 𝐴))

Theoremlmodvaddsub4 19311 Vector addition/subtraction law. (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    = (-g𝑊)       ((𝑊 ∈ LMod ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → ((𝐴 + 𝐵) = (𝐶 + 𝐷) ↔ (𝐴 𝐶) = (𝐷 𝐵)))

Theoremlmodvpncan 19312 Addition/subtraction cancellation law for vectors. (hvpncan 28472 analog.) (Contributed by NM, 16-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    = (-g𝑊)       ((𝑊 ∈ LMod ∧ 𝐴𝑉𝐵𝑉) → ((𝐴 + 𝐵) 𝐵) = 𝐴)

Theoremlmodvnpcan 19313 Cancellation law for vector subtraction (npcan 10634 analog). (Contributed by NM, 19-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    = (-g𝑊)       ((𝑊 ∈ LMod ∧ 𝐴𝑉𝐵𝑉) → ((𝐴 𝐵) + 𝐵) = 𝐴)

Theoremlmodvsubval2 19314 Value of vector subtraction in terms of addition. (hvsubval 28449 analog.) (Contributed by NM, 31-Mar-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    = (-g𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝑁 = (invg𝐹)    &    1 = (1r𝐹)       ((𝑊 ∈ LMod ∧ 𝐴𝑉𝐵𝑉) → (𝐴 𝐵) = (𝐴 + ((𝑁1 ) · 𝐵)))

Theoremlmodsubvs 19315 Subtraction of a scalar product in terms of addition. (Contributed by NM, 9-Apr-2015.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    = (-g𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &   𝑁 = (invg𝐹)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐴𝐾)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → (𝑋 (𝐴 · 𝑌)) = (𝑋 + ((𝑁𝐴) · 𝑌)))

Theoremlmodsubdi 19316 Scalar multiplication distributive law for subtraction. (hvsubdistr1 28482 analogue, with longer proof since our scalar multiplication is not commutative.) (Contributed by NM, 2-Jul-2014.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    = (-g𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐴𝐾)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → (𝐴 · (𝑋 𝑌)) = ((𝐴 · 𝑋) (𝐴 · 𝑌)))

Theoremlmodsubdir 19317 Scalar multiplication distributive law for subtraction. (hvsubdistr2 28483 analog.) (Contributed by NM, 2-Jul-2014.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    = (-g𝑊)    &   𝑆 = (-g𝐹)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐴𝐾)    &   (𝜑𝐵𝐾)    &   (𝜑𝑋𝑉)       (𝜑 → ((𝐴𝑆𝐵) · 𝑋) = ((𝐴 · 𝑋) (𝐵 · 𝑋)))

Theoremlmodsubeq0 19318 If the difference between two vectors is zero, they are equal. (hvsubeq0 28501 analog.) (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)    &    = (-g𝑊)       ((𝑊 ∈ LMod ∧ 𝐴𝑉𝐵𝑉) → ((𝐴 𝐵) = 0𝐴 = 𝐵))

Theoremlmodsubid 19319 Subtraction of a vector from itself. (hvsubid 28459 analog.) (Contributed by NM, 16-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)    &    = (-g𝑊)       ((𝑊 ∈ LMod ∧ 𝐴𝑉) → (𝐴 𝐴) = 0 )

Theoremlmodvsghm 19320* 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 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ LMod ∧ 𝑅𝐾) → (𝑥𝑉 ↦ (𝑅 · 𝑥)) ∈ (𝑊 GrpHom 𝑊))

Theoremlmodprop2d 19321* If two structures have the same components (properties), one is a left module iff the other one is. This version of lmodpropd 19322 also breaks up the components of the scalar ring. (Contributed by Mario Carneiro, 27-Jun-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   𝐹 = (Scalar‘𝐾)    &   𝐺 = (Scalar‘𝐿)    &   (𝜑𝑃 = (Base‘𝐹))    &   (𝜑𝑃 = (Base‘𝐺))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))    &   ((𝜑 ∧ (𝑥𝑃𝑦𝑃)) → (𝑥(+g𝐹)𝑦) = (𝑥(+g𝐺)𝑦))    &   ((𝜑 ∧ (𝑥𝑃𝑦𝑃)) → (𝑥(.r𝐹)𝑦) = (𝑥(.r𝐺)𝑦))    &   ((𝜑 ∧ (𝑥𝑃𝑦𝐵)) → (𝑥( ·𝑠𝐾)𝑦) = (𝑥( ·𝑠𝐿)𝑦))       (𝜑 → (𝐾 ∈ LMod ↔ 𝐿 ∈ LMod))

Theoremlmodpropd 19322* 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.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))    &   (𝜑𝐹 = (Scalar‘𝐾))    &   (𝜑𝐹 = (Scalar‘𝐿))    &   𝑃 = (Base‘𝐹)    &   ((𝜑 ∧ (𝑥𝑃𝑦𝐵)) → (𝑥( ·𝑠𝐾)𝑦) = (𝑥( ·𝑠𝐿)𝑦))       (𝜑 → (𝐾 ∈ LMod ↔ 𝐿 ∈ LMod))

Theoremgsumvsmul 19323* Pull a scalar multiplication out of a sum of vectors. This theorem properly generalizes gsummulc2 18998, 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.)
𝐵 = (Base‘𝑅)    &   𝑆 = (Scalar‘𝑅)    &   𝐾 = (Base‘𝑆)    &    0 = (0g𝑅)    &    + = (+g𝑅)    &    · = ( ·𝑠𝑅)    &   (𝜑𝑅 ∈ LMod)    &   (𝜑𝐴𝑉)    &   (𝜑𝑋𝐾)    &   ((𝜑𝑘𝐴) → 𝑌𝐵)    &   (𝜑 → (𝑘𝐴𝑌) finSupp 0 )       (𝜑 → (𝑅 Σg (𝑘𝐴 ↦ (𝑋 · 𝑌))) = (𝑋 · (𝑅 Σg (𝑘𝐴𝑌))))

Theoremmptscmfsupp0 19324* A mapping to a scalar product is finitely supported if the mapping to the scalar is finitely supported. (Contributed by AV, 5-Oct-2019.)
(𝜑𝐷𝑉)    &   (𝜑𝑄 ∈ LMod)    &   (𝜑𝑅 = (Scalar‘𝑄))    &   𝐾 = (Base‘𝑄)    &   ((𝜑𝑘𝐷) → 𝑆𝐵)    &   ((𝜑𝑘𝐷) → 𝑊𝐾)    &    0 = (0g𝑄)    &   𝑍 = (0g𝑅)    &    = ( ·𝑠𝑄)    &   (𝜑 → (𝑘𝐷𝑆) finSupp 𝑍)       (𝜑 → (𝑘𝐷 ↦ (𝑆 𝑊)) finSupp 0 )

Theoremmptscmfsuppd 19325* A function mapping to a scalar product in which one factor is finitely supported is finitely supported. Formerly part of proof for ply1coe 20066. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by AV, 8-Aug-2019.) (Proof shortened by AV, 18-Oct-2019.)
𝐵 = (Base‘𝑃)    &   𝑆 = (Scalar‘𝑃)    &    · = ( ·𝑠𝑃)    &   (𝜑𝑃 ∈ LMod)    &   (𝜑𝑋𝑉)    &   ((𝜑𝑘𝑋) → 𝑍𝐵)    &   (𝜑𝐴:𝑋𝑌)    &   (𝜑𝐴 finSupp (0g𝑆))       (𝜑 → (𝑘𝑋 ↦ ((𝐴𝑘) · 𝑍)) finSupp (0g𝑃))

Theoremrmodislmodlem 19326* Lemma for rmodislmod 19327. This is the part of the proof of rmodislmod 19327 which requires the scalar ring to be commutative. (Contributed by AV, 3-Dec-2021.)
𝑉 = (Base‘𝑅)    &    + = (+g𝑅)    &    · = ( ·𝑠𝑅)    &   𝐹 = (Scalar‘𝑅)    &   𝐾 = (Base‘𝐹)    &    = (+g𝐹)    &    × = (.r𝐹)    &    1 = (1r𝐹)    &   (𝑅 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑞𝐾𝑟𝐾𝑥𝑉𝑤𝑉 (((𝑤 · 𝑟) ∈ 𝑉 ∧ ((𝑤 + 𝑥) · 𝑟) = ((𝑤 · 𝑟) + (𝑥 · 𝑟)) ∧ (𝑤 · (𝑞 𝑟)) = ((𝑤 · 𝑞) + (𝑤 · 𝑟))) ∧ ((𝑤 · (𝑞 × 𝑟)) = ((𝑤 · 𝑞) · 𝑟) ∧ (𝑤 · 1 ) = 𝑤)))    &    = (𝑠𝐾, 𝑣𝑉 ↦ (𝑣 · 𝑠))    &   𝐿 = (𝑅 sSet ⟨( ·𝑠 ‘ndx), ⟩)       ((𝐹 ∈ CRing ∧ (𝑎𝐾𝑏𝐾𝑐𝑉)) → ((𝑎 × 𝑏) 𝑐) = (𝑎 (𝑏 𝑐)))

Theoremrmodislmod 19327* 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 the definition df-lmod 19261 of a left module, see also islmod 19263. (Contributed by AV, 3-Dec-2021.)
𝑉 = (Base‘𝑅)    &    + = (+g𝑅)    &    · = ( ·𝑠𝑅)    &   𝐹 = (Scalar‘𝑅)    &   𝐾 = (Base‘𝐹)    &    = (+g𝐹)    &    × = (.r𝐹)    &    1 = (1r𝐹)    &   (𝑅 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑞𝐾𝑟𝐾𝑥𝑉𝑤𝑉 (((𝑤 · 𝑟) ∈ 𝑉 ∧ ((𝑤 + 𝑥) · 𝑟) = ((𝑤 · 𝑟) + (𝑥 · 𝑟)) ∧ (𝑤 · (𝑞 𝑟)) = ((𝑤 · 𝑞) + (𝑤 · 𝑟))) ∧ ((𝑤 · (𝑞 × 𝑟)) = ((𝑤 · 𝑞) · 𝑟) ∧ (𝑤 · 1 ) = 𝑤)))    &    = (𝑠𝐾, 𝑣𝑉 ↦ (𝑣 · 𝑠))    &   𝐿 = (𝑅 sSet ⟨( ·𝑠 ‘ndx), ⟩)       (𝐹 ∈ CRing → 𝐿 ∈ LMod)

10.6.2  Subspaces and spans in a left module

Syntaxclss 19328 Extend class notation with linear subspaces of a left module or left vector space.
class LSubSp

Definitiondf-lss 19329* Define the set of linear subspaces of a left module or left vector space. (Contributed by NM, 8-Dec-2013.)
LSubSp = (𝑤 ∈ V ↦ {𝑠 ∈ (𝒫 (Base‘𝑤) ∖ {∅}) ∣ ∀𝑥 ∈ (Base‘(Scalar‘𝑤))∀𝑎𝑠𝑏𝑠 ((𝑥( ·𝑠𝑤)𝑎)(+g𝑤)𝑏) ∈ 𝑠})

Theoremlssset 19330* 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 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝐹)    &   𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    · = ( ·𝑠𝑊)    &   𝑆 = (LSubSp‘𝑊)       (𝑊𝑋𝑆 = {𝑠 ∈ (𝒫 𝑉 ∖ {∅}) ∣ ∀𝑥𝐵𝑎𝑠𝑏𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠})

Theoremislss 19331* 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 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝐹)    &   𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    · = ( ·𝑠𝑊)    &   𝑆 = (LSubSp‘𝑊)       (𝑈𝑆 ↔ (𝑈𝑉𝑈 ≠ ∅ ∧ ∀𝑥𝐵𝑎𝑈𝑏𝑈 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑈))

Theoremislssd 19332* 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.)
(𝜑𝐹 = (Scalar‘𝑊))    &   (𝜑𝐵 = (Base‘𝐹))    &   (𝜑𝑉 = (Base‘𝑊))    &   (𝜑+ = (+g𝑊))    &   (𝜑· = ( ·𝑠𝑊))    &   (𝜑𝑆 = (LSubSp‘𝑊))    &   (𝜑𝑈𝑉)    &   (𝜑𝑈 ≠ ∅)    &   ((𝜑 ∧ (𝑥𝐵𝑎𝑈𝑏𝑈)) → ((𝑥 · 𝑎) + 𝑏) ∈ 𝑈)       (𝜑𝑈𝑆)

Theoremlssss 19333 A subspace is a set of vectors. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)       (𝑈𝑆𝑈𝑉)

Theoremlssel 19334 A subspace member is a vector. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 8-Jan-2015.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)       ((𝑈𝑆𝑋𝑈) → 𝑋𝑉)

Theoremlss1 19335 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 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)       (𝑊 ∈ LMod → 𝑉𝑆)

Theoremlssuni 19336 The union of all subspaces is the vector space. (Contributed by NM, 13-Mar-2015.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   (𝜑𝑊 ∈ LMod)       (𝜑 𝑆 = 𝑉)

Theoremlssn0 19337 A subspace is not empty. (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 8-Jan-2015.)
𝑆 = (LSubSp‘𝑊)       (𝑈𝑆𝑈 ≠ ∅)

Theorem00lss 19338 The empty structure has no subspaces (for use with fvco4i 6538). (Contributed by Stefan O'Rear, 31-Mar-2015.)
∅ = (LSubSp‘∅)

Theoremlsscl 19339 Closure property of a subspace. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝐹)    &    + = (+g𝑊)    &    · = ( ·𝑠𝑊)    &   𝑆 = (LSubSp‘𝑊)       ((𝑈𝑆 ∧ (𝑍𝐵𝑋𝑈𝑌𝑈)) → ((𝑍 · 𝑋) + 𝑌) ∈ 𝑈)

Theoremlssvsubcl 19340 Closure of vector subtraction in a subspace. (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
= (-g𝑊)    &   𝑆 = (LSubSp‘𝑊)       (((𝑊 ∈ LMod ∧ 𝑈𝑆) ∧ (𝑋𝑈𝑌𝑈)) → (𝑋 𝑌) ∈ 𝑈)

Theoremlssvancl1 19341 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 19536. Can it be used along with lspsnne1 19516, lspsnne2 19517 to shorten this proof? (Contributed by NM, 14-May-2015.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝑆)    &   (𝜑𝑋𝑈)    &   (𝜑𝑌𝑉)    &   (𝜑 → ¬ 𝑌𝑈)       (𝜑 → ¬ (𝑋 + 𝑌) ∈ 𝑈)

Theoremlssvancl2 19342 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 = (Base‘𝑊)    &    + = (+g𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝑆)    &   (𝜑𝑋𝑈)    &   (𝜑𝑌𝑉)    &   (𝜑 → ¬ 𝑌𝑈)       (𝜑 → ¬ (𝑌 + 𝑋) ∈ 𝑈)

Theoremlss0cl 19343 The zero vector belongs to every subspace. (Contributed by NM, 12-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
0 = (0g𝑊)    &   𝑆 = (LSubSp‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑈𝑆) → 0𝑈)

Theoremlsssn0 19344 The singleton of the zero vector is a subspace. (Contributed by NM, 13-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
0 = (0g𝑊)    &   𝑆 = (LSubSp‘𝑊)       (𝑊 ∈ LMod → { 0 } ∈ 𝑆)

Theoremlss0ss 19345 The zero subspace is included in every subspace. (sh0le 28875 analog.) (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
0 = (0g𝑊)    &   𝑆 = (LSubSp‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑆) → { 0 } ⊆ 𝑋)

Theoremlssle0 19346 No subspace is smaller than the zero subspace. (shle0 28877 analog.) (Contributed by NM, 20-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
0 = (0g𝑊)    &   𝑆 = (LSubSp‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑆) → (𝑋 ⊆ { 0 } ↔ 𝑋 = { 0 }))

Theoremlssne0 19347* A nonzero subspace has a nonzero vector. (shne0i 28883 analog.) (Contributed by NM, 20-Apr-2014.) (Proof shortened by Mario Carneiro, 8-Jan-2015.)
0 = (0g𝑊)    &   𝑆 = (LSubSp‘𝑊)       (𝑋𝑆 → (𝑋 ≠ { 0 } ↔ ∃𝑦𝑋 𝑦0 ))

Theoremlssvneln0 19348 A vector 𝑋 which doesn't belong to a subspace 𝑈 is nonzero. (Contributed by NM, 14-May-2015.) (Revised by AV, 19-Jul-2022.)
0 = (0g𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝑆)    &   (𝜑 → ¬ 𝑋𝑈)       (𝜑𝑋0 )

Theoremlssneln0 19349 A vector 𝑋 which doesn't belong to a subspace 𝑈 is nonzero. (Contributed by NM, 14-May-2015.) (Revised by AV, 17-Jul-2022.) (Proof shortened by AV, 19-Jul-2022.)
0 = (0g𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝑆)    &   (𝜑𝑋𝑉)    &   (𝜑 → ¬ 𝑋𝑈)       (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))

Theoremlssneln0OLD 19350 Obsolete version of lssneln0 19349 as of 17-Jul-2022. (Contributed by NM, 14-May-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝑆)    &   (𝜑𝑋𝑉)    &   (𝜑 → ¬ 𝑋𝑈)       (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))

Theoremlssssr 19351* Conclude subspace ordering from nonzero vector membership. (ssrdv 3827 analog.) (Contributed by NM, 17-Aug-2014.) (Revised by AV, 13-Jul-2022.)
0 = (0g𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑇𝑉)    &   (𝜑𝑈𝑆)    &   ((𝜑𝑥 ∈ (𝑉 ∖ { 0 })) → (𝑥𝑇𝑥𝑈))       (𝜑𝑇𝑈)

TheoremlssssrOLD 19352* Obsolete version of lssssr 19351 as of 13-Jul-2022. (Contributed by NM, 17-Aug-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑇𝑉)    &   (𝜑𝑈𝑆)    &   ((𝜑𝑥 ∈ (𝑉 ∖ { 0 })) → (𝑥𝑇𝑥𝑈))       (𝜑𝑇𝑈)

Theoremlssvacl 19353 Closure of vector addition in a subspace. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
+ = (+g𝑊)    &   𝑆 = (LSubSp‘𝑊)       (((𝑊 ∈ LMod ∧ 𝑈𝑆) ∧ (𝑋𝑈𝑌𝑈)) → (𝑋 + 𝑌) ∈ 𝑈)

Theoremlssvscl 19354 Closure of scalar product in a subspace. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐵 = (Base‘𝐹)    &   𝑆 = (LSubSp‘𝑊)       (((𝑊 ∈ LMod ∧ 𝑈𝑆) ∧ (𝑋𝐵𝑌𝑈)) → (𝑋 · 𝑌) ∈ 𝑈)

Theoremlssvnegcl 19355 Closure of negative vectors in a subspace. (Contributed by Stefan O'Rear, 11-Dec-2014.)
𝑆 = (LSubSp‘𝑊)    &   𝑁 = (invg𝑊)       ((𝑊 ∈ LMod ∧ 𝑈𝑆𝑋𝑈) → (𝑁𝑋) ∈ 𝑈)

Theoremlsssubg 19356 All subspaces are subgroups. (Contributed by Stefan O'Rear, 11-Dec-2014.)
𝑆 = (LSubSp‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑈𝑆) → 𝑈 ∈ (SubGrp‘𝑊))

Theoremlsssssubg 19357 All subspaces are subgroups. (Contributed by Mario Carneiro, 19-Apr-2016.)
𝑆 = (LSubSp‘𝑊)       (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊))

Theoremislss3 19358 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 = (𝑊s 𝑈)    &   𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)       (𝑊 ∈ LMod → (𝑈𝑆 ↔ (𝑈𝑉𝑋 ∈ LMod)))

Theoremlsslmod 19359 A submodule is a module. (Contributed by Stefan O'Rear, 12-Dec-2014.)
𝑋 = (𝑊s 𝑈)    &   𝑆 = (LSubSp‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑈𝑆) → 𝑋 ∈ LMod)

Theoremlsslss 19360 The subspaces of a subspace are the smaller subspaces. (Contributed by Stefan O'Rear, 12-Dec-2014.)
𝑋 = (𝑊s 𝑈)    &   𝑆 = (LSubSp‘𝑊)    &   𝑇 = (LSubSp‘𝑋)       ((𝑊 ∈ LMod ∧ 𝑈𝑆) → (𝑉𝑇 ↔ (𝑉𝑆𝑉𝑈)))

Theoremislss4 19361* 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 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝐹)    &   𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝑆 = (LSubSp‘𝑊)       (𝑊 ∈ LMod → (𝑈𝑆 ↔ (𝑈 ∈ (SubGrp‘𝑊) ∧ ∀𝑎𝐵𝑏𝑈 (𝑎 · 𝑏) ∈ 𝑈)))

Theoremlss1d 19362* 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 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)    &   𝑆 = (LSubSp‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑉) → {𝑣 ∣ ∃𝑘𝐾 𝑣 = (𝑘 · 𝑋)} ∈ 𝑆)

Theoremlssintcl 19363 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 = (LSubSp‘𝑊)       ((𝑊 ∈ LMod ∧ 𝐴𝑆𝐴 ≠ ∅) → 𝐴𝑆)

Theoremlssincl 19364 The intersection of two subspaces is a subspace. (Contributed by NM, 7-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑆 = (LSubSp‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑇𝑆𝑈𝑆) → (𝑇𝑈) ∈ 𝑆)

Theoremlssmre 19365 The subspaces of a module comprise a Moore system on the vectors of the module. (Contributed by Stefan O'Rear, 31-Jan-2015.)
𝐵 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)       (𝑊 ∈ LMod → 𝑆 ∈ (Moore‘𝐵))

Theoremlssacs 19366 Submodules are an algebraic closure system. (Contributed by Stefan O'Rear, 4-Apr-2015.)
𝐵 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)       (𝑊 ∈ LMod → 𝑆 ∈ (ACS‘𝐵))

Theoremprdsvscacl 19367* Pointwise scalar multiplication is closed in products of modules. (Contributed by Stefan O'Rear, 10-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   𝐵 = (Base‘𝑌)    &    · = ( ·𝑠𝑌)    &   𝐾 = (Base‘𝑆)    &   (𝜑𝑆 ∈ Ring)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅:𝐼⟶LMod)    &   (𝜑𝐹𝐾)    &   (𝜑𝐺𝐵)    &   ((𝜑𝑥𝐼) → (Scalar‘(𝑅𝑥)) = 𝑆)       (𝜑 → (𝐹 · 𝐺) ∈ 𝐵)

Theoremprdslmodd 19368* The product of a family of left modules is a left module. (Contributed by Stefan O'Rear, 10-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑𝑆 ∈ Ring)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅:𝐼⟶LMod)    &   ((𝜑𝑦𝐼) → (Scalar‘(𝑅𝑦)) = 𝑆)       (𝜑𝑌 ∈ LMod)

Theorempwslmod 19369 The product of a family of left modules is a left module. (Contributed by Mario Carneiro, 11-Jan-2015.)
𝑌 = (𝑅s 𝐼)       ((𝑅 ∈ LMod ∧ 𝐼𝑉) → 𝑌 ∈ LMod)

Syntaxclspn 19370 Extend class notation with span of a set of vectors.
class LSpan

Definitiondf-lsp 19371* Define span of a set of vectors of a left module or left vector space. (Contributed by NM, 8-Dec-2013.)
LSpan = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠𝑡}))

Theoremlspfval 19372* The span function for a left vector space (or a left module). (df-span 28744 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)       (𝑊𝑋𝑁 = (𝑠 ∈ 𝒫 𝑉 {𝑡𝑆𝑠𝑡}))

Theoremlspf 19373 The span operator on a left module maps subsets to subsets. (Contributed by Stefan O'Rear, 12-Dec-2014.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)       (𝑊 ∈ LMod → 𝑁:𝒫 𝑉𝑆)

Theoremlspval 19374* The span of a set of vectors (in a left module). (spanval 28768 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑈𝑉) → (𝑁𝑈) = {𝑡𝑆𝑈𝑡})

Theoremlspcl 19375 The span of a set of vectors is a subspace. (spancl 28771 analog.) (Contributed by NM, 9-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑈𝑉) → (𝑁𝑈) ∈ 𝑆)

Theoremlspsncl 19376 The span of a singleton is a subspace (frequently used special case of lspcl 19375). (Contributed by NM, 17-Jul-2014.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑉) → (𝑁‘{𝑋}) ∈ 𝑆)

Theoremlspprcl 19377 The span of a pair is a subspace (frequently used special case of lspcl 19375). (Contributed by NM, 11-Apr-2015.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ 𝑆)

Theoremlsptpcl 19378 The span of an unordered triple is a subspace (frequently used special case of lspcl 19375). (Contributed by NM, 22-May-2015.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑍𝑉)       (𝜑 → (𝑁‘{𝑋, 𝑌, 𝑍}) ∈ 𝑆)

Theoremlspsnsubg 19379 The span of a singleton is an additive subgroup (frequently used special case of lspcl 19375). (Contributed by Mario Carneiro, 21-Apr-2016.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑉) → (𝑁‘{𝑋}) ∈ (SubGrp‘𝑊))

Theorem00lsp 19380 fvco4i 6538 lemma for linear spans. (Contributed by Stefan O'Rear, 4-Apr-2015.)
∅ = (LSpan‘∅)

Theoremlspid 19381 The span of a subspace is itself. (spanid 28782 analog.) (Contributed by NM, 15-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑈𝑆) → (𝑁𝑈) = 𝑈)

Theoremlspssv 19382 A span is a set of vectors. (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑈𝑉) → (𝑁𝑈) ⊆ 𝑉)

Theoremlspss 19383 Span preserves subset ordering. (spanss 28783 analog.) (Contributed by NM, 11-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑈𝑉𝑇𝑈) → (𝑁𝑇) ⊆ (𝑁𝑈))

Theoremlspssid 19384 A set of vectors is a subset of its span. (spanss2 28780 analog.) (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑈𝑉) → 𝑈 ⊆ (𝑁𝑈))

Theoremlspidm 19385 The span of a set of vectors is idempotent. (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑈𝑉) → (𝑁‘(𝑁𝑈)) = (𝑁𝑈))

Theoremlspun 19386 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 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑇𝑉𝑈𝑉) → (𝑁‘(𝑇𝑈)) = (𝑁‘((𝑁𝑇) ∪ (𝑁𝑈))))

Theoremlspssp 19387 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 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑈𝑆𝑇𝑈) → (𝑁𝑇) ⊆ 𝑈)

Theoremmrclsp 19388 Moore closure generalizes module span. (Contributed by Stefan O'Rear, 31-Jan-2015.)
𝑈 = (LSubSp‘𝑊)    &   𝐾 = (LSpan‘𝑊)    &   𝐹 = (mrCls‘𝑈)       (𝑊 ∈ LMod → 𝐾 = 𝐹)

Theoremlspsnss 19389 The span of the singleton of a subspace member is included in the subspace. (spansnss 29006 analog.) (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 4-Sep-2014.)
𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑈𝑆𝑋𝑈) → (𝑁‘{𝑋}) ⊆ 𝑈)

Theoremlspsnel3 19390 A member of the span of the singleton of a vector is a member of a subspace containing the vector. (elspansn3 29007 analog.) (Contributed by NM, 4-Jul-2014.)
𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝑆)    &   (𝜑𝑋𝑈)    &   (𝜑𝑌 ∈ (𝑁‘{𝑋}))       (𝜑𝑌𝑈)

Theoremlspprss 19391 The span of a pair of vectors in a subspace belongs to the subspace. (Contributed by NM, 12-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝑆)    &   (𝜑𝑋𝑈)    &   (𝜑𝑌𝑈)       (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ 𝑈)

Theoremlspsnid 19392 A vector belongs to the span of its singleton. (spansnid 28998 analog.) (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑉) → 𝑋 ∈ (𝑁‘{𝑋}))

Theoremlspsnel6 19393 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 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝑆)       (𝜑 → (𝑋𝑈 ↔ (𝑋𝑉 ∧ (𝑁‘{𝑋}) ⊆ 𝑈)))

Theoremlspsnel5 19394 Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 8-Aug-2014.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝑆)    &   (𝜑𝑋𝑉)       (𝜑 → (𝑋𝑈 ↔ (𝑁‘{𝑋}) ⊆ 𝑈))

Theoremlspsnel5a 19395 Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 20-Feb-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝑆)    &   (𝜑𝑋𝑈)       (𝜑 → (𝑁‘{𝑋}) ⊆ 𝑈)

Theoremlspprid1 19396 A member of a pair of vectors belongs to their span. (Contributed by NM, 14-May-2015.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑𝑋 ∈ (𝑁‘{𝑋, 𝑌}))

Theoremlspprid2 19397 A member of a pair of vectors belongs to their span. (Contributed by NM, 14-May-2015.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑𝑌 ∈ (𝑁‘{𝑋, 𝑌}))

Theoremlspprvacl 19398 The sum of two vectors belongs to their span. (Contributed by NM, 20-May-2015.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → (𝑋 + 𝑌) ∈ (𝑁‘{𝑋, 𝑌}))

Theoremlssats2 19399* A way to express atomisticity (a subspace is the union of its atoms). (Contributed by NM, 3-Feb-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝑆)       (𝜑𝑈 = 𝑥𝑈 (𝑁‘{𝑥}))

Theoremlspsneli 19400 A scalar product with a vector belongs to the span of its singleton. (spansnmul 28999 analog.) (Contributed by NM, 2-Jul-2014.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐴𝐾)    &   (𝜑𝑋𝑉)       (𝜑 → (𝐴 · 𝑋) ∈ (𝑁‘{𝑋}))

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