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Theorem List for Metamath Proof Explorer - 18601-18700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlmodring 18601 The scalar component of a left module is a ring. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ LMod → 𝐹 ∈ Ring)
 
Theoremlmodfgrp 18602 The scalar component of a left module is an additive group. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ LMod → 𝐹 ∈ Grp)
 
Theoremlmodbn0 18603 The base set of a left module is nonempty. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝐵 = (Base‘𝑊)       (𝑊 ∈ LMod → 𝐵 ≠ ∅)
 
Theoremlmodacl 18604 Closure of ring addition for a left module. (Contributed by NM, 14-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    + = (+g𝐹)       ((𝑊 ∈ LMod ∧ 𝑋𝐾𝑌𝐾) → (𝑋 + 𝑌) ∈ 𝐾)
 
Theoremlmodmcl 18605 Closure of ring multiplication for a left module. (Contributed by NM, 14-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    · = (.r𝐹)       ((𝑊 ∈ LMod ∧ 𝑋𝐾𝑌𝐾) → (𝑋 · 𝑌) ∈ 𝐾)
 
Theoremlmodsn0 18606 The set of scalars in a left module is nonempty. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝐹 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝐹)       (𝑊 ∈ LMod → 𝐵 ≠ ∅)
 
Theoremlmodvacl 18607 Closure of vector addition for a left module. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑉𝑌𝑉) → (𝑋 + 𝑌) ∈ 𝑉)
 
Theoremlmodass 18608 Left module vector sum is associative. (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)       ((𝑊 ∈ LMod ∧ (𝑋𝑉𝑌𝑉𝑍𝑉)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
 
Theoremlmodlcan 18609 Left cancellation law for vector sum. (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)       ((𝑊 ∈ LMod ∧ (𝑋𝑉𝑌𝑉𝑍𝑉)) → ((𝑍 + 𝑋) = (𝑍 + 𝑌) ↔ 𝑋 = 𝑌))
 
Theoremlmodvscl 18610 Closure of scalar product for a left module. (hvmulcl 27042 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ LMod ∧ 𝑅𝐾𝑋𝑉) → (𝑅 · 𝑋) ∈ 𝑉)
 
Theoremscaffval 18611* The scalar multiplication operation as a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝐵 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    = ( ·sf𝑊)    &    · = ( ·𝑠𝑊)        = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦))
 
Theoremscafval 18612 The scalar multiplication operation as a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝐵 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    = ( ·sf𝑊)    &    · = ( ·𝑠𝑊)       ((𝑋𝐾𝑌𝐵) → (𝑋 𝑌) = (𝑋 · 𝑌))
 
Theoremscafeq 18613 If the scalar multiplication operation is already a function, the functionalization of it is equal to the original operation. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝐵 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    = ( ·sf𝑊)    &    · = ( ·𝑠𝑊)       ( · Fn (𝐾 × 𝐵) → = · )
 
Theoremscaffn 18614 The scalar multiplication operation is a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝐵 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    = ( ·sf𝑊)        Fn (𝐾 × 𝐵)
 
Theoremlmodscaf 18615 The scalar multiplication operation is a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝐵 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    = ( ·sf𝑊)       (𝑊 ∈ LMod → :(𝐾 × 𝐵)⟶𝐵)
 
Theoremlmodvsdi 18616 Distributive law for scalar product (left-distributivity). (ax-hvdistr1 27037 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ LMod ∧ (𝑅𝐾𝑋𝑉𝑌𝑉)) → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌)))
 
Theoremlmodvsdir 18617 Distributive law for scalar product (right-distributivity). (ax-hvdistr1 27037 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)    &    = (+g𝐹)       ((𝑊 ∈ LMod ∧ (𝑄𝐾𝑅𝐾𝑋𝑉)) → ((𝑄 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋)))
 
Theoremlmodvsass 18618 Associative law for scalar product. (ax-hvmulass 27036 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)    &    × = (.r𝐹)       ((𝑊 ∈ LMod ∧ (𝑄𝐾𝑅𝐾𝑋𝑉)) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)))
 
Theoremlmod0cl 18619 The ring zero in a left module belongs to the ring base set. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    0 = (0g𝐹)       (𝑊 ∈ LMod → 0𝐾)
 
Theoremlmod1cl 18620 The ring unit in a left module belongs to the ring base set. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    1 = (1r𝐹)       (𝑊 ∈ LMod → 1𝐾)
 
Theoremlmodvs1 18621 Scalar product with ring unit. (ax-hvmulid 27035 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &    1 = (1r𝐹)       ((𝑊 ∈ LMod ∧ 𝑋𝑉) → ( 1 · 𝑋) = 𝑋)
 
Theoremlmod0vcl 18622 The zero vector is a vector. (ax-hv0cl 27032 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)       (𝑊 ∈ LMod → 0𝑉)
 
Theoremlmod0vlid 18623 Left identity law for the zero vector. (hvaddid2 27052 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑉) → ( 0 + 𝑋) = 𝑋)
 
Theoremlmod0vrid 18624 Right identity law for the zero vector. (ax-hvaddid 27033 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑉) → (𝑋 + 0 ) = 𝑋)
 
Theoremlmod0vid 18625 Identity equivalent to the value of the zero vector. Provides a convenient way to compute the value. (Contributed by NM, 9-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑉) → ((𝑋 + 𝑋) = 𝑋0 = 𝑋))
 
Theoremlmod0vs 18626 Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (ax-hvmul0 27039 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝑂 = (0g𝐹)    &    0 = (0g𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑉) → (𝑂 · 𝑋) = 0 )
 
Theoremlmodvs0 18627 Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (hvmul0 27053 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)    &    0 = (0g𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝐾) → (𝑋 · 0 ) = 0 )
 
Theoremlmodvsmmulgdi 18628 Distributive law for a group multiple of a scalar multiplication. (Contributed by AV, 2-Sep-2019.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)    &    = (.g𝑊)    &   𝐸 = (.g𝐹)       ((𝑊 ∈ LMod ∧ (𝐶𝐾𝑁 ∈ ℕ0𝑋𝑉)) → (𝑁 (𝐶 · 𝑋)) = ((𝑁𝐸𝐶) · 𝑋))
 
Theoremlcomf 18629 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 18630 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 18631 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 18632 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 18633 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 18634 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 18635 Closure of vector subtraction. (hvsubcl 27046 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 18636 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 18637 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 18638 A left module is a commutative monoid under addition. (Contributed by NM, 7-Jan-2015.)
(𝑊 ∈ LMod → 𝑊 ∈ CMnd)
 
Theoremlmodnegadd 18639 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 18640 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 18641 Relationship between vector subtraction and addition. (hvsubadd 27106 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 18642 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 18643 Addition/subtraction cancellation law for vectors. (hvpncan 27068 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 18644 Cancellation law for vector subtraction (npcan 10041 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 18645 Value of vector subtraction in terms of addition. (hvsubval 27045 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 18646 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 18647 Scalar multiplication distributive law for subtraction. (hvsubdistr1 27078 analog, 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 18648 Scalar multiplication distributive law for subtraction. (hvsubdistr2 27079 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 18649 If the difference between two vectors is zero, they are equal. (hvsubeq0 27097 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 18650 Subtraction of a vector from itself. (hvsubid 27055 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 18651* Scalar multiplication of the vector space by a fixed scalar is an automorphism 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 18652* If two structures have the same components (properties), one is a left module iff the other one is. This version of lmodpropd 18653 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 18653* 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 18654* Pull a scalar multiplication out of a sum of vectors. This theorem properly generalizes gsummulc2 18337, 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 18655* 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 18656* A function mapping to a scalar product in which one factor is finitely supported is finitely supported. Formerly part of proof for ply1coe 19391. (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))
 
10.6.2  Subspaces and spans in a left module
 
Syntaxclss 18657 Extend class notation with linear subspaces of a left module or left vector space.
class LSubSp
 
Definitiondf-lss 18658* 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 18659* 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 18660* 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 18661* 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 18662 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 18663 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 18664 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 18665 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 18666 A subspace is not empty. (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 8-Jan-2015.)
𝑆 = (LSubSp‘𝑊)       (𝑈𝑆𝑈 ≠ ∅)
 
Theorem00lss 18667 The empty structure has no subspaces (for use with fvco4i 6070). (Contributed by Stefan O'Rear, 31-Mar-2015.)
∅ = (LSubSp‘∅)
 
Theoremlsscl 18668 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 18669 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 18670 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 18861. Can it be used along with lspsnne1 18842, lspsnne2 18843 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 18671 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 18672 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 18673 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 18674 The zero subspace is included in every subspace. (sh0le 27471 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 18675 No subspace is smaller than the zero subspace. (shle0 27473 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 18676* A nonzero subspace has a nonzero vector. (shne0i 27479 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 ))
 
Theoremlssneln0 18677 A vector which doesn't belong to a subspace is nonzero. (Contributed by NM, 14-May-2015.)
𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝑆)    &   (𝜑𝑋𝑉)    &   (𝜑 → ¬ 𝑋𝑈)       (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))
 
Theoremlssssr 18678* Conclude subspace ordering from nonzero vector membership. (ssrdv 3478 analog.) (Contributed by NM, 17-Aug-2014.)
𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑇𝑉)    &   (𝜑𝑈𝑆)    &   ((𝜑𝑥 ∈ (𝑉 ∖ { 0 })) → (𝑥𝑇𝑥𝑈))       (𝜑𝑇𝑈)
 
Theoremlssvacl 18679 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 18680 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 18681 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 18682 All subspaces are subgroups. (Contributed by Stefan O'Rear, 11-Dec-2014.)
𝑆 = (LSubSp‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑈𝑆) → 𝑈 ∈ (SubGrp‘𝑊))
 
Theoremlsssssubg 18683 All subspaces are subgroups. (Contributed by Mario Carneiro, 19-Apr-2016.)
𝑆 = (LSubSp‘𝑊)       (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊))
 
Theoremislss3 18684 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 18685 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 18686 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 18687* 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 18688* 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 18689 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 18690 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 18691 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 18692 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 18693* 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 18694* 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 18695 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 18696 Extend class notation with span of a set of vectors.
class LSpan
 
Definitiondf-lsp 18697* 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 18698* The span function for a left vector space (or a left module). (df-span 27340 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 18699 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 18700* The span of a set of vectors (in a left module). (spanval 27364 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})
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