HomeHome Metamath Proof Explorer
Theorem List (p. 197 of 449)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-28623)
  Hilbert Space Explorer  Hilbert Space Explorer
(28624-30146)
  Users' Mathboxes  Users' Mathboxes
(30147-44804)
 

Theorem List for Metamath Proof Explorer - 19601-19700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlmodfopnelem1 19601 Lemma 1 for lmodfopne 19603. (Contributed by AV, 2-Oct-2021.)
· = ( ·sf𝑊)    &    + = (+𝑓𝑊)    &   𝑉 = (Base‘𝑊)    &   𝑆 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑆)       ((𝑊 ∈ LMod ∧ + = · ) → 𝑉 = 𝐾)
 
Theoremlmodfopnelem2 19602 Lemma 2 for lmodfopne 19603. (Contributed by AV, 2-Oct-2021.)
· = ( ·sf𝑊)    &    + = (+𝑓𝑊)    &   𝑉 = (Base‘𝑊)    &   𝑆 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑆)    &    0 = (0g𝑆)    &    1 = (1r𝑆)       ((𝑊 ∈ LMod ∧ + = · ) → ( 0𝑉1𝑉))
 
Theoremlmodfopne 19603 The (functionalized) operations of a left module (over a nonzero ring) cannot be identical. (Contributed by NM, 31-May-2008.) (Revised by AV, 2-Oct-2021.)
· = ( ·sf𝑊)    &    + = (+𝑓𝑊)    &   𝑉 = (Base‘𝑊)    &   𝑆 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑆)    &    0 = (0g𝑆)    &    1 = (1r𝑆)       ((𝑊 ∈ LMod ∧ 10 ) → +· )
 
Theoremlcomf 19604 A linear-combination sum is a function. (Contributed by Stefan O'Rear, 28-Feb-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    · = ( ·𝑠𝑊)    &   𝐵 = (Base‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐺:𝐼𝐾)    &   (𝜑𝐻:𝐼𝐵)    &   (𝜑𝐼𝑉)       (𝜑 → (𝐺f · 𝐻):𝐼𝐵)
 
Theoremlcomfsupp 19605 A linear-combination sum is finitely supported if the coefficients are. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by AV, 15-Jul-2019.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    · = ( ·𝑠𝑊)    &   𝐵 = (Base‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐺:𝐼𝐾)    &   (𝜑𝐻:𝐼𝐵)    &   (𝜑𝐼𝑉)    &    0 = (0g𝑊)    &   𝑌 = (0g𝐹)    &   (𝜑𝐺 finSupp 𝑌)       (𝜑 → (𝐺f · 𝐻) finSupp 0 )
 
Theoremlmodvnegcl 19606 Closure of vector negative. (Contributed by NM, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (invg𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑉) → (𝑁𝑋) ∈ 𝑉)
 
Theoremlmodvnegid 19607 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 19608 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 19609 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 19610 Closure of vector subtraction. (hvsubcl 28722 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 19611 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 19612 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 19613 A left module is a commutative monoid under addition. (Contributed by NM, 7-Jan-2015.)
(𝑊 ∈ LMod → 𝑊 ∈ CMnd)
 
Theoremlmodnegadd 19614 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 19615 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 19616 Relationship between vector subtraction and addition. (hvsubadd 28782 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 19617 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 19618 Addition/subtraction cancellation law for vectors. (hvpncan 28744 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 19619 Cancellation law for vector subtraction (npcan 10884 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 19620 Value of vector subtraction in terms of addition. (hvsubval 28721 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 19621 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 19622 Scalar multiplication distributive law for subtraction. (hvsubdistr1 28754 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 19623 Scalar multiplication distributive law for subtraction. (hvsubdistr2 28755 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 19624 If the difference between two vectors is zero, they are equal. (hvsubeq0 28773 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 19625 Subtraction of a vector from itself. (hvsubid 28731 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 19626* 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 19627* If two structures have the same components (properties), one is a left module iff the other one is. This version of lmodpropd 19628 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 19628* 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 19629* Pull a scalar multiplication out of a sum of vectors. This theorem properly generalizes gsummulc2 19288, 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 19630* 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 19631* A function mapping to a scalar product in which one factor is finitely supported is finitely supported. Formerly part of proof for ply1coe 20394. (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 19632* Lemma for rmodislmod 19633. This is the part of the proof of rmodislmod 19633 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 19633* 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 19567 of a left module, see also islmod 19569. (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)
 
10.5.2  Subspaces and spans in a left module
 
Syntaxclss 19634 Extend class notation with linear subspaces of a left module or left vector space.
class LSubSp
 
Definitiondf-lss 19635* 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 19636* 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 19637* 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 19638* 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 19639 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 19640 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 19641 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 19642 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 19643 A subspace is not empty. (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 8-Jan-2015.)
𝑆 = (LSubSp‘𝑊)       (𝑈𝑆𝑈 ≠ ∅)
 
Theorem00lss 19644 The empty structure has no subspaces (for use with fvco4i 6756). (Contributed by Stefan O'Rear, 31-Mar-2015.)
∅ = (LSubSp‘∅)
 
Theoremlsscl 19645 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 19646 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 19647 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 19839. Can it be used along with lspsnne1 19820, lspsnne2 19821 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 19648 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 19649 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 19650 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 19651 The zero subspace is included in every subspace. (sh0le 29145 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 19652 No subspace is smaller than the zero subspace. (shle0 29147 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 19653* A nonzero subspace has a nonzero vector. (shne0i 29153 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 19654 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 19655 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 }))
 
Theoremlssssr 19656* Conclude subspace ordering from nonzero vector membership. (ssrdv 3972 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)
 
Theoremlssvacl 19657 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 19658 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 19659 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 19660 All subspaces are subgroups. (Contributed by Stefan O'Rear, 11-Dec-2014.)
𝑆 = (LSubSp‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑈𝑆) → 𝑈 ∈ (SubGrp‘𝑊))
 
Theoremlsssssubg 19661 All subspaces are subgroups. (Contributed by Mario Carneiro, 19-Apr-2016.)
𝑆 = (LSubSp‘𝑊)       (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊))
 
Theoremislss3 19662 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 19663 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 19664 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 19665* 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 19666* 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 19667 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 19668 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 19669 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 19670 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 19671* 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 19672* 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 19673 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 19674 Extend class notation with span of a set of vectors.
class LSpan
 
Definitiondf-lsp 19675* 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 19676* The span function for a left vector space (or a left module). (df-span 29014 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 19677 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 19678* The span of a set of vectors (in a left module). (spanval 29038 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 19679 The span of a set of vectors is a subspace. (spancl 29041 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 19680 The span of a singleton is a subspace (frequently used special case of lspcl 19679). (Contributed by NM, 17-Jul-2014.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑉) → (𝑁‘{𝑋}) ∈ 𝑆)
 
Theoremlspprcl 19681 The span of a pair is a subspace (frequently used special case of lspcl 19679). (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 19682 The span of an unordered triple is a subspace (frequently used special case of lspcl 19679). (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 19683 The span of a singleton is an additive subgroup (frequently used special case of lspcl 19679). (Contributed by Mario Carneiro, 21-Apr-2016.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑉) → (𝑁‘{𝑋}) ∈ (SubGrp‘𝑊))
 
Theorem00lsp 19684 fvco4i 6756 lemma for linear spans. (Contributed by Stefan O'Rear, 4-Apr-2015.)
∅ = (LSpan‘∅)
 
Theoremlspid 19685 The span of a subspace is itself. (spanid 29052 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 19686 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 19687 Span preserves subset ordering. (spanss 29053 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 19688 A set of vectors is a subset of its span. (spanss2 29050 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 19689 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 19690 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 19691 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 19692 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 19693 The span of the singleton of a subspace member is included in the subspace. (spansnss 29276 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 19694 A member of the span of the singleton of a vector is a member of a subspace containing the vector. (elspansn3 29277 analog.) (Contributed by NM, 4-Jul-2014.)
𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝑆)    &   (𝜑𝑋𝑈)    &   (𝜑𝑌 ∈ (𝑁‘{𝑋}))       (𝜑𝑌𝑈)
 
Theoremlspprss 19695 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 19696 A vector belongs to the span of its singleton. (spansnid 29268 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 19697 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 19698 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 19699 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 19700 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}))
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 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 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 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 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44804
  Copyright terms: Public domain < Previous  Next >