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Theorem List for Metamath Proof Explorer - 19001-19100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Syntaxclmim 19001 The class of left module isomorphism sets.
class LMIso
 
Syntaxclmic 19002 The class of the left module isomorphism relation.
class 𝑚
 
Definitiondf-lmhm 19003* A homomorphism of left modules is a group homomorphism which additionally preserves the scalar product. This requires both structures to be left modules over the same ring. (Contributed by Stefan O'Rear, 31-Dec-2014.)
LMHom = (𝑠 ∈ LMod, 𝑡 ∈ LMod ↦ {𝑓 ∈ (𝑠 GrpHom 𝑡) ∣ [(Scalar‘𝑠) / 𝑤]((Scalar‘𝑡) = 𝑤 ∧ ∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥( ·𝑠𝑠)𝑦)) = (𝑥( ·𝑠𝑡)(𝑓𝑦)))})
 
Definitiondf-lmim 19004* An isomorphism of modules is a homomorphism which is also a bijection, i.e. it preserves equality as well as the group and scalar operations. (Contributed by Stefan O'Rear, 21-Jan-2015.)
LMIso = (𝑠 ∈ LMod, 𝑡 ∈ LMod ↦ {𝑔 ∈ (𝑠 LMHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)})
 
Definitiondf-lmic 19005 Two modules are said to be isomorphic iff they are connected by at least one isomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.)
𝑚 = ( LMIso “ (V ∖ 1𝑜))
 
Theoremreldmlmhm 19006 Lemma for module homomorphisms. (Contributed by Stefan O'Rear, 31-Dec-2014.)
Rel dom LMHom
 
Theoremlmimfn 19007 Lemma for module isomorphisms. (Contributed by Stefan O'Rear, 23-Aug-2015.)
LMIso Fn (LMod × LMod)
 
Theoremislmhm 19008* Property of being a homomorphism of left modules. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Proof shortened by Mario Carneiro, 30-Apr-2015.)
𝐾 = (Scalar‘𝑆)    &   𝐿 = (Scalar‘𝑇)    &   𝐵 = (Base‘𝐾)    &   𝐸 = (Base‘𝑆)    &    · = ( ·𝑠𝑆)    &    × = ( ·𝑠𝑇)       (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ ∀𝑥𝐵𝑦𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹𝑦)))))
 
Theoremislmhm3 19009* Property of a module homomorphism, similar to ismhm 17318. (Contributed by Stefan O'Rear, 7-Mar-2015.)
𝐾 = (Scalar‘𝑆)    &   𝐿 = (Scalar‘𝑇)    &   𝐵 = (Base‘𝐾)    &   𝐸 = (Base‘𝑆)    &    · = ( ·𝑠𝑆)    &    × = ( ·𝑠𝑇)       ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) → (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ ∀𝑥𝐵𝑦𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹𝑦)))))
 
Theoremlmhmlem 19010 Non-quantified consequences of a left module homomorphism. (Contributed by Stefan O'Rear, 1-Jan-2015.)
𝐾 = (Scalar‘𝑆)    &   𝐿 = (Scalar‘𝑇)       (𝐹 ∈ (𝑆 LMHom 𝑇) → ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾)))
 
Theoremlmhmsca 19011 A homomorphism of left modules constrains both modules to the same ring of scalars. (Contributed by Stefan O'Rear, 1-Jan-2015.)
𝐾 = (Scalar‘𝑆)    &   𝐿 = (Scalar‘𝑇)       (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐿 = 𝐾)
 
Theoremlmghm 19012 A homomorphism of left modules is a homomorphism of groups. (Contributed by Stefan O'Rear, 1-Jan-2015.)
(𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
 
Theoremlmhmlmod2 19013 A homomorphism of left modules has a left module as codomain. (Contributed by Stefan O'Rear, 1-Jan-2015.)
(𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
 
Theoremlmhmlmod1 19014 A homomorphism of left modules has a left module as domain. (Contributed by Stefan O'Rear, 1-Jan-2015.)
(𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
 
Theoremlmhmf 19015 A homomorphism of left modules is a function. (Contributed by Stefan O'Rear, 1-Jan-2015.)
𝐵 = (Base‘𝑆)    &   𝐶 = (Base‘𝑇)       (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:𝐵𝐶)
 
Theoremlmhmlin 19016 A homomorphism of left modules is 𝐾-linear. (Contributed by Stefan O'Rear, 1-Jan-2015.)
𝐾 = (Scalar‘𝑆)    &   𝐵 = (Base‘𝐾)    &   𝐸 = (Base‘𝑆)    &    · = ( ·𝑠𝑆)    &    × = ( ·𝑠𝑇)       ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝐵𝑌𝐸) → (𝐹‘(𝑋 · 𝑌)) = (𝑋 × (𝐹𝑌)))
 
Theoremlmodvsinv 19017 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)    &   𝑀 = (invg𝐹)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ LMod ∧ 𝑅𝐾𝑋𝐵) → ((𝑀𝑅) · 𝑋) = (𝑁‘(𝑅 · 𝑋)))
 
Theoremlmodvsinv2 19018 Multiplying a negated vector by a scalar. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝑁 = (invg𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ LMod ∧ 𝑅𝐾𝑋𝐵) → (𝑅 · (𝑁𝑋)) = (𝑁‘(𝑅 · 𝑋)))
 
Theoremislmhm2 19019* A one-equation proof of linearity of a left module homomorphism, similar to df-lss 18914. (Contributed by Mario Carneiro, 7-Oct-2015.)
𝐵 = (Base‘𝑆)    &   𝐶 = (Base‘𝑇)    &   𝐾 = (Scalar‘𝑆)    &   𝐿 = (Scalar‘𝑇)    &   𝐸 = (Base‘𝐾)    &    + = (+g𝑆)    &    = (+g𝑇)    &    · = ( ·𝑠𝑆)    &    × = ( ·𝑠𝑇)       ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) → (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ (𝐹:𝐵𝐶𝐿 = 𝐾 ∧ ∀𝑥𝐸𝑦𝐵𝑧𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹𝑦)) (𝐹𝑧)))))
 
Theoremislmhmd 19020* Deduction for a module homomorphism. (Contributed by Stefan O'Rear, 4-Feb-2015.)
𝑋 = (Base‘𝑆)    &    · = ( ·𝑠𝑆)    &    × = ( ·𝑠𝑇)    &   𝐾 = (Scalar‘𝑆)    &   𝐽 = (Scalar‘𝑇)    &   𝑁 = (Base‘𝐾)    &   (𝜑𝑆 ∈ LMod)    &   (𝜑𝑇 ∈ LMod)    &   (𝜑𝐽 = 𝐾)    &   (𝜑𝐹 ∈ (𝑆 GrpHom 𝑇))    &   ((𝜑 ∧ (𝑥𝑁𝑦𝑋)) → (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹𝑦)))       (𝜑𝐹 ∈ (𝑆 LMHom 𝑇))
 
Theorem0lmhm 19021 The constant zero linear function between two modules. (Contributed by Stefan O'Rear, 5-Sep-2015.)
0 = (0g𝑁)    &   𝐵 = (Base‘𝑀)    &   𝑆 = (Scalar‘𝑀)    &   𝑇 = (Scalar‘𝑁)       ((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) → (𝐵 × { 0 }) ∈ (𝑀 LMHom 𝑁))
 
Theoremidlmhm 19022 The identity function on a module is linear. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐵 = (Base‘𝑀)       (𝑀 ∈ LMod → ( I ↾ 𝐵) ∈ (𝑀 LMHom 𝑀))
 
Theoreminvlmhm 19023 The negative function on a module is linear. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐼 = (invg𝑀)       (𝑀 ∈ LMod → 𝐼 ∈ (𝑀 LMHom 𝑀))
 
Theoremlmhmco 19024 The composition of two module-linear functions is module-linear. (Contributed by Stefan O'Rear, 4-Sep-2015.)
((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → (𝐹𝐺) ∈ (𝑀 LMHom 𝑂))
 
Theoremlmhmplusg 19025 The pointwise sum of two linear functions is linear. (Contributed by Stefan O'Rear, 5-Sep-2015.)
+ = (+g𝑁)       ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → (𝐹𝑓 + 𝐺) ∈ (𝑀 LMHom 𝑁))
 
Theoremlmhmvsca 19026 The pointwise scalar product of a linear function and a constant is linear, over a commutative ring. (Contributed by Mario Carneiro, 22-Sep-2015.)
𝑉 = (Base‘𝑀)    &    · = ( ·𝑠𝑁)    &   𝐽 = (Scalar‘𝑁)    &   𝐾 = (Base‘𝐽)       ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑉 × {𝐴}) ∘𝑓 · 𝐹) ∈ (𝑀 LMHom 𝑁))
 
Theoremlmhmf1o 19027 A bijective module homomorphism is also converse homomorphic. (Contributed by Stefan O'Rear, 25-Jan-2015.)
𝑋 = (Base‘𝑆)    &   𝑌 = (Base‘𝑇)       (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝐹:𝑋1-1-onto𝑌𝐹 ∈ (𝑇 LMHom 𝑆)))
 
Theoremlmhmima 19028 The image of a subspace under a homomorphism. (Contributed by Stefan O'Rear, 1-Jan-2015.)
𝑋 = (LSubSp‘𝑆)    &   𝑌 = (LSubSp‘𝑇)       ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) → (𝐹𝑈) ∈ 𝑌)
 
Theoremlmhmpreima 19029 The inverse image of a subspace under a homomorphism. (Contributed by Stefan O'Rear, 1-Jan-2015.)
𝑋 = (LSubSp‘𝑆)    &   𝑌 = (LSubSp‘𝑇)       ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → (𝐹𝑈) ∈ 𝑋)
 
Theoremlmhmlsp 19030 Homomorphisms preserve spans. (Contributed by Stefan O'Rear, 1-Jan-2015.)
𝑉 = (Base‘𝑆)    &   𝐾 = (LSpan‘𝑆)    &   𝐿 = (LSpan‘𝑇)       ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑉) → (𝐹 “ (𝐾𝑈)) = (𝐿‘(𝐹𝑈)))
 
Theoremlmhmrnlss 19031 The range of a homomorphism is a submodule. (Contributed by Stefan O'Rear, 1-Jan-2015.)
(𝐹 ∈ (𝑆 LMHom 𝑇) → ran 𝐹 ∈ (LSubSp‘𝑇))
 
Theoremlmhmkerlss 19032 The kernel of a homomorphism is a submodule. (Contributed by Stefan O'Rear, 1-Jan-2015.)
𝐾 = (𝐹 “ { 0 })    &    0 = (0g𝑇)    &   𝑈 = (LSubSp‘𝑆)       (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐾𝑈)
 
Theoremreslmhm 19033 Restriction of a homomorphism to a subspace. (Contributed by Stefan O'Rear, 1-Jan-2015.)
𝑈 = (LSubSp‘𝑆)    &   𝑅 = (𝑆s 𝑋)       ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → (𝐹𝑋) ∈ (𝑅 LMHom 𝑇))
 
Theoremreslmhm2 19034 Expansion of the codomain of a homomorphism. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
𝑈 = (𝑇s 𝑋)    &   𝐿 = (LSubSp‘𝑇)       ((𝐹 ∈ (𝑆 LMHom 𝑈) ∧ 𝑇 ∈ LMod ∧ 𝑋𝐿) → 𝐹 ∈ (𝑆 LMHom 𝑇))
 
Theoremreslmhm2b 19035 Expansion of the codomain of a homomorphism. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
𝑈 = (𝑇s 𝑋)    &   𝐿 = (LSubSp‘𝑇)       ((𝑇 ∈ LMod ∧ 𝑋𝐿 ∧ ran 𝐹𝑋) → (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ 𝐹 ∈ (𝑆 LMHom 𝑈)))
 
Theoremlmhmeql 19036 The equalizer of two module homomorphisms is a subspace. (Contributed by Stefan O'Rear, 7-Mar-2015.)
𝑈 = (LSubSp‘𝑆)       ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) → dom (𝐹𝐺) ∈ 𝑈)
 
Theoremlspextmo 19037* A linear function is completely determined (or overdetermined) by its values on a spanning subset. (Contributed by Stefan O'Rear, 7-Mar-2015.) (Revised by NM, 17-Jun-2017.)
𝐵 = (Base‘𝑆)    &   𝐾 = (LSpan‘𝑆)       ((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) → ∃*𝑔 ∈ (𝑆 LMHom 𝑇)(𝑔𝑋) = 𝐹)
 
Theorempwsdiaglmhm 19038* Diagonal homomorphism into a structure power. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝑌 = (𝑅s 𝐼)    &   𝐵 = (Base‘𝑅)    &   𝐹 = (𝑥𝐵 ↦ (𝐼 × {𝑥}))       ((𝑅 ∈ LMod ∧ 𝐼𝑊) → 𝐹 ∈ (𝑅 LMHom 𝑌))
 
Theorempwssplit0 19039* Splitting for structure powers, part 0: restriction is a function. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝑌 = (𝑊s 𝑈)    &   𝑍 = (𝑊s 𝑉)    &   𝐵 = (Base‘𝑌)    &   𝐶 = (Base‘𝑍)    &   𝐹 = (𝑥𝐵 ↦ (𝑥𝑉))       ((𝑊𝑇𝑈𝑋𝑉𝑈) → 𝐹:𝐵𝐶)
 
Theorempwssplit1 19040* Splitting for structure powers, part 1: restriction is an onto function. The only actual monoid law we need here is that the base set is nonempty. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝑌 = (𝑊s 𝑈)    &   𝑍 = (𝑊s 𝑉)    &   𝐵 = (Base‘𝑌)    &   𝐶 = (Base‘𝑍)    &   𝐹 = (𝑥𝐵 ↦ (𝑥𝑉))       ((𝑊 ∈ Mnd ∧ 𝑈𝑋𝑉𝑈) → 𝐹:𝐵onto𝐶)
 
Theorempwssplit2 19041* Splitting for structure powers, part 2: restriction is a group homomorphism. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝑌 = (𝑊s 𝑈)    &   𝑍 = (𝑊s 𝑉)    &   𝐵 = (Base‘𝑌)    &   𝐶 = (Base‘𝑍)    &   𝐹 = (𝑥𝐵 ↦ (𝑥𝑉))       ((𝑊 ∈ Grp ∧ 𝑈𝑋𝑉𝑈) → 𝐹 ∈ (𝑌 GrpHom 𝑍))
 
Theorempwssplit3 19042* Splitting for structure powers, part 3: restriction is a module homomorphism. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝑌 = (𝑊s 𝑈)    &   𝑍 = (𝑊s 𝑉)    &   𝐵 = (Base‘𝑌)    &   𝐶 = (Base‘𝑍)    &   𝐹 = (𝑥𝐵 ↦ (𝑥𝑉))       ((𝑊 ∈ LMod ∧ 𝑈𝑋𝑉𝑈) → 𝐹 ∈ (𝑌 LMHom 𝑍))
 
Theoremislmim 19043 An isomorphism of left modules is a bijective homomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.)
𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)       (𝐹 ∈ (𝑅 LMIso 𝑆) ↔ (𝐹 ∈ (𝑅 LMHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶))
 
Theoremlmimf1o 19044 An isomorphism of left modules is a bijection. (Contributed by Stefan O'Rear, 21-Jan-2015.)
𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)       (𝐹 ∈ (𝑅 LMIso 𝑆) → 𝐹:𝐵1-1-onto𝐶)
 
Theoremlmimlmhm 19045 An isomorphism of modules is a homomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.)
(𝐹 ∈ (𝑅 LMIso 𝑆) → 𝐹 ∈ (𝑅 LMHom 𝑆))
 
Theoremlmimgim 19046 An isomorphism of modules is an isomorphism of groups. (Contributed by Stefan O'Rear, 21-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
(𝐹 ∈ (𝑅 LMIso 𝑆) → 𝐹 ∈ (𝑅 GrpIso 𝑆))
 
Theoremislmim2 19047 An isomorphism of left modules is a homomorphism whose converse is a homomorphism. (Contributed by Mario Carneiro, 6-May-2015.)
(𝐹 ∈ (𝑅 LMIso 𝑆) ↔ (𝐹 ∈ (𝑅 LMHom 𝑆) ∧ 𝐹 ∈ (𝑆 LMHom 𝑅)))
 
Theoremlmimcnv 19048 The converse of a bijective module homomorphism is a bijective module homomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
(𝐹 ∈ (𝑆 LMIso 𝑇) → 𝐹 ∈ (𝑇 LMIso 𝑆))
 
Theorembrlmic 19049 The relation "is isomorphic to" for modules. (Contributed by Stefan O'Rear, 25-Jan-2015.)
(𝑅𝑚 𝑆 ↔ (𝑅 LMIso 𝑆) ≠ ∅)
 
Theorembrlmici 19050 Prove isomorphic by an explicit isomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.)
(𝐹 ∈ (𝑅 LMIso 𝑆) → 𝑅𝑚 𝑆)
 
Theoremlmiclcl 19051 Isomorphism implies the left side is a module. (Contributed by Stefan O'Rear, 25-Jan-2015.)
(𝑅𝑚 𝑆𝑅 ∈ LMod)
 
Theoremlmicrcl 19052 Isomorphism implies the right side is a module. (Contributed by Mario Carneiro, 6-May-2015.)
(𝑅𝑚 𝑆𝑆 ∈ LMod)
 
Theoremlmicsym 19053 Module isomorphism is symmetric. (Contributed by Stefan O'Rear, 26-Feb-2015.)
(𝑅𝑚 𝑆𝑆𝑚 𝑅)
 
Theoremlmhmpropd 19054* Module homomorphism depends only on the module attributes of structures. (Contributed by Mario Carneiro, 8-Oct-2015.)
(𝜑𝐵 = (Base‘𝐽))    &   (𝜑𝐶 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   (𝜑𝐶 = (Base‘𝑀))    &   (𝜑𝐹 = (Scalar‘𝐽))    &   (𝜑𝐺 = (Scalar‘𝐾))    &   (𝜑𝐹 = (Scalar‘𝐿))    &   (𝜑𝐺 = (Scalar‘𝑀))    &   𝑃 = (Base‘𝐹)    &   𝑄 = (Base‘𝐺)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐽)𝑦) = (𝑥(+g𝐿)𝑦))    &   ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝑀)𝑦))    &   ((𝜑 ∧ (𝑥𝑃𝑦𝐵)) → (𝑥( ·𝑠𝐽)𝑦) = (𝑥( ·𝑠𝐿)𝑦))    &   ((𝜑 ∧ (𝑥𝑄𝑦𝐶)) → (𝑥( ·𝑠𝐾)𝑦) = (𝑥( ·𝑠𝑀)𝑦))       (𝜑 → (𝐽 LMHom 𝐾) = (𝐿 LMHom 𝑀))
 
10.6.4  Subspace sum; bases for a left module
 
Syntaxclbs 19055 Extend class notation with the set of bases for a vector space.
class LBasis
 
Definitiondf-lbs 19056* Define the set of bases to a left module or left vector space. (Contributed by Mario Carneiro, 24-Jun-2014.)
LBasis = (𝑤 ∈ V ↦ {𝑏 ∈ 𝒫 (Base‘𝑤) ∣ [(LSpan‘𝑤) / 𝑛][(Scalar‘𝑤) / 𝑠]((𝑛𝑏) = (Base‘𝑤) ∧ ∀𝑥𝑏𝑦 ∈ ((Base‘𝑠) ∖ {(0g𝑠)}) ¬ (𝑦( ·𝑠𝑤)𝑥) ∈ (𝑛‘(𝑏 ∖ {𝑥})))})
 
Theoremislbs 19057* The predicate "𝐵 is a basis for the left module or vector space 𝑊". A subset of the base set is a basis if zero is not in the set, it spans the set, and no nonzero multiple of an element of the basis is in the span of the rest of the family. (Contributed by Mario Carneiro, 24-Jun-2014.) (Revised by Mario Carneiro, 14-Jan-2015.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)    &   𝐽 = (LBasis‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &    0 = (0g𝐹)       (𝑊𝑋 → (𝐵𝐽 ↔ (𝐵𝑉 ∧ (𝑁𝐵) = 𝑉 ∧ ∀𝑥𝐵𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})))))
 
Theoremlbsss 19058 A basis is a set of vectors. (Contributed by Mario Carneiro, 24-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝐽 = (LBasis‘𝑊)       (𝐵𝐽𝐵𝑉)
 
Theoremlbsel 19059 An element of a basis is a vector. (Contributed by Mario Carneiro, 24-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝐽 = (LBasis‘𝑊)       ((𝐵𝐽𝐸𝐵) → 𝐸𝑉)
 
Theoremlbssp 19060 The span of a basis is the whole space. (Contributed by Mario Carneiro, 24-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝐽 = (LBasis‘𝑊)    &   𝑁 = (LSpan‘𝑊)       (𝐵𝐽 → (𝑁𝐵) = 𝑉)
 
Theoremlbsind 19061 A basis is linearly independent; that is, every element has a span which trivially intersects the span of the remainder of the basis. (Contributed by Mario Carneiro, 12-Jan-2015.)
𝑉 = (Base‘𝑊)    &   𝐽 = (LBasis‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)    &    0 = (0g𝐹)       (((𝐵𝐽𝐸𝐵) ∧ (𝐴𝐾𝐴0 )) → ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸})))
 
Theoremlbsind2 19062 A basis is linearly independent; that is, every element is not in the span of the remainder of the basis. (Contributed by Mario Carneiro, 25-Jun-2014.) (Revised by Mario Carneiro, 12-Jan-2015.)
𝐽 = (LBasis‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    1 = (1r𝐹)    &    0 = (0g𝐹)       (((𝑊 ∈ LMod ∧ 10 ) ∧ 𝐵𝐽𝐸𝐵) → ¬ 𝐸 ∈ (𝑁‘(𝐵 ∖ {𝐸})))
 
Theoremlbspss 19063 No proper subset of a basis spans the space. (Contributed by Mario Carneiro, 25-Jun-2014.)
𝐽 = (LBasis‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    1 = (1r𝐹)    &    0 = (0g𝐹)    &   𝑉 = (Base‘𝑊)       (((𝑊 ∈ LMod ∧ 10 ) ∧ 𝐵𝐽𝐶𝐵) → (𝑁𝐶) ≠ 𝑉)
 
Theoremlsmcl 19064 The sum of two subspaces is a subspace. (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
𝑆 = (LSubSp‘𝑊)    &    = (LSSum‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑇𝑆𝑈𝑆) → (𝑇 𝑈) ∈ 𝑆)
 
Theoremlsmspsn 19065* Member of subspace sum of spans of singletons. (Contributed by NM, 8-Apr-2015.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    · = ( ·𝑠𝑊)    &    = (LSSum‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → (𝑈 ∈ ((𝑁‘{𝑋}) (𝑁‘{𝑌})) ↔ ∃𝑗𝐾𝑘𝐾 𝑈 = ((𝑗 · 𝑋) + (𝑘 · 𝑌))))
 
Theoremlsmelval2 19066* Subspace sum membership in terms of a sum of 1-dim subspaces (atoms), which can be useful for treating subspaces as projective lattice elements. (Contributed by NM, 9-Aug-2014.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &    = (LSSum‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)       (𝜑 → (𝑋 ∈ (𝑇 𝑈) ↔ (𝑋𝑉 ∧ ∃𝑦𝑇𝑧𝑈 (𝑁‘{𝑋}) ⊆ ((𝑁‘{𝑦}) (𝑁‘{𝑧})))))
 
Theoremlsmsp 19067 Subspace sum in terms of span. (Contributed by NM, 6-Feb-2014.) (Proof shortened by Mario Carneiro, 21-Jun-2014.)
𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &    = (LSSum‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑇𝑆𝑈𝑆) → (𝑇 𝑈) = (𝑁‘(𝑇𝑈)))
 
Theoremlsmsp2 19068 Subspace sum of spans of subsets is the span of their union. (spanuni 28373 analog.) (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &    = (LSSum‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑇𝑉𝑈𝑉) → ((𝑁𝑇) (𝑁𝑈)) = (𝑁‘(𝑇𝑈)))
 
Theoremlsmssspx 19069 Subspace sum (in its extended domain) is a subset of the span of the union of its arguments. (Contributed by NM, 6-Aug-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &    = (LSSum‘𝑊)    &   (𝜑𝑇𝑉)    &   (𝜑𝑈𝑉)    &   (𝜑𝑊 ∈ LMod)       (𝜑 → (𝑇 𝑈) ⊆ (𝑁‘(𝑇𝑈)))
 
Theoremlsmpr 19070 The span of a pair of vectors equals the sum of the spans of their singletons. (Contributed by NM, 13-Jan-2015.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &    = (LSSum‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → (𝑁‘{𝑋, 𝑌}) = ((𝑁‘{𝑋}) (𝑁‘{𝑌})))
 
Theoremlsppreli 19071 A vector expressed as a sum belongs to the span of its components. (Contributed by NM, 9-Apr-2015.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐴𝐾)    &   (𝜑𝐵𝐾)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → ((𝐴 · 𝑋) + (𝐵 · 𝑌)) ∈ (𝑁‘{𝑋, 𝑌}))
 
Theoremlsmelpr 19072 Two ways to say that a vector belongs to the span of a pair of vectors. (Contributed by NM, 14-Jan-2015.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &    = (LSSum‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑍𝑉)       (𝜑 → (𝑋 ∈ (𝑁‘{𝑌, 𝑍}) ↔ (𝑁‘{𝑋}) ⊆ ((𝑁‘{𝑌}) (𝑁‘{𝑍}))))
 
Theoremlsppr0 19073 The span of a vector paired with zero equals the span of the singleton of the vector. (Contributed by NM, 29-Aug-2014.)
𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝑉)       (𝜑 → (𝑁‘{𝑋, 0 }) = (𝑁‘{𝑋}))
 
Theoremlsppr 19074* Span of a pair of vectors. (Contributed by NM, 22-Aug-2014.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    · = ( ·𝑠𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → (𝑁‘{𝑋, 𝑌}) = {𝑣 ∣ ∃𝑘𝐾𝑙𝐾 𝑣 = ((𝑘 · 𝑋) + (𝑙 · 𝑌))})
 
Theoremlspprel 19075* Member of the span of a pair of vectors. (Contributed by NM, 10-Apr-2015.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    · = ( ·𝑠𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → (𝑍 ∈ (𝑁‘{𝑋, 𝑌}) ↔ ∃𝑘𝐾𝑙𝐾 𝑍 = ((𝑘 · 𝑋) + (𝑙 · 𝑌))))
 
Theoremlspprabs 19076 Absorption of vector sum into span of pair. (Contributed by NM, 27-Apr-2015.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → (𝑁‘{𝑋, (𝑋 + 𝑌)}) = (𝑁‘{𝑋, 𝑌}))
 
Theoremlspvadd 19077 The span of a vector sum is included in the span of its arguments. (Contributed by NM, 22-Feb-2014.) (Proof shortened by Mario Carneiro, 21-Jun-2014.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑉𝑌𝑉) → (𝑁‘{(𝑋 + 𝑌)}) ⊆ (𝑁‘{𝑋, 𝑌}))
 
Theoremlspsntri 19078 Triangle-type inequality for span of a singleton. (Contributed by NM, 24-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &   𝑁 = (LSpan‘𝑊)    &    = (LSSum‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑉𝑌𝑉) → (𝑁‘{(𝑋 + 𝑌)}) ⊆ ((𝑁‘{𝑋}) (𝑁‘{𝑌})))
 
Theoremlspsntrim 19079 Triangle-type inequality for span of a singleton of vector difference. (Contributed by NM, 25-Apr-2014.) (Revised by Mario Carneiro, 21-Jun-2014.)
𝑉 = (Base‘𝑊)    &    = (-g𝑊)    &    = (LSSum‘𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑉𝑌𝑉) → (𝑁‘{(𝑋 𝑌)}) ⊆ ((𝑁‘{𝑋}) (𝑁‘{𝑌})))
 
Theoremlbspropd 19080* If two structures have the same components (properties), they have the same set of bases. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   (𝜑𝐵𝑊)    &   ((𝜑 ∧ (𝑥𝑊𝑦𝑊)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))    &   ((𝜑 ∧ (𝑥𝑃𝑦𝐵)) → (𝑥( ·𝑠𝐾)𝑦) ∈ 𝑊)    &   ((𝜑 ∧ (𝑥𝑃𝑦𝐵)) → (𝑥( ·𝑠𝐾)𝑦) = (𝑥( ·𝑠𝐿)𝑦))    &   𝐹 = (Scalar‘𝐾)    &   𝐺 = (Scalar‘𝐿)    &   (𝜑𝑃 = (Base‘𝐹))    &   (𝜑𝑃 = (Base‘𝐺))    &   ((𝜑 ∧ (𝑥𝑃𝑦𝑃)) → (𝑥(+g𝐹)𝑦) = (𝑥(+g𝐺)𝑦))    &   (𝜑𝐾 ∈ V)    &   (𝜑𝐿 ∈ V)       (𝜑 → (LBasis‘𝐾) = (LBasis‘𝐿))
 
Theorempj1lmhm 19081 The left projection function is a linear operator. (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
𝐿 = (LSubSp‘𝑊)    &    = (LSSum‘𝑊)    &    0 = (0g𝑊)    &   𝑃 = (proj1𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑇𝐿)    &   (𝜑𝑈𝐿)    &   (𝜑 → (𝑇𝑈) = { 0 })       (𝜑 → (𝑇𝑃𝑈) ∈ ((𝑊s (𝑇 𝑈)) LMHom 𝑊))
 
Theorempj1lmhm2 19082 The left projection function is a linear operator. (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
𝐿 = (LSubSp‘𝑊)    &    = (LSSum‘𝑊)    &    0 = (0g𝑊)    &   𝑃 = (proj1𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑇𝐿)    &   (𝜑𝑈𝐿)    &   (𝜑 → (𝑇𝑈) = { 0 })       (𝜑 → (𝑇𝑃𝑈) ∈ ((𝑊s (𝑇 𝑈)) LMHom (𝑊s 𝑇)))
 
10.7  Vector spaces
 
10.7.1  Definition and basic properties
 
Syntaxclvec 19083 Extend class notation with class of all left vector spaces.
class LVec
 
Definitiondf-lvec 19084 Define the class of all left vector spaces. A left vector space over a division ring is an Abelian group (vectors) together with a division ring (scalars) and a left scalar product connecting them. Some authors call this a "left module over a division ring", reserving "vector space" for those where the division ring multiplication is commutative i.e. a field. (Contributed by NM, 11-Nov-2013.)
LVec = {𝑓 ∈ LMod ∣ (Scalar‘𝑓) ∈ DivRing}
 
Theoremislvec 19085 The predicate "is a left vector space". (Contributed by NM, 11-Nov-2013.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ LVec ↔ (𝑊 ∈ LMod ∧ 𝐹 ∈ DivRing))
 
Theoremlvecdrng 19086 The set of scalars of a left vector space is a division ring. (Contributed by NM, 17-Apr-2014.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ LVec → 𝐹 ∈ DivRing)
 
Theoremlveclmod 19087 A left vector space is a left module. (Contributed by NM, 9-Dec-2013.)
(𝑊 ∈ LVec → 𝑊 ∈ LMod)
 
Theoremlsslvec 19088 A vector subspace is a vector space. (Contributed by NM, 14-Mar-2015.)
𝑋 = (𝑊s 𝑈)    &   𝑆 = (LSubSp‘𝑊)       ((𝑊 ∈ LVec ∧ 𝑈𝑆) → 𝑋 ∈ LVec)
 
Theoremlvecvs0or 19089 If a scalar product is zero, one of its factors must be zero. (hvmul0or 27852 analog.) (Contributed by NM, 2-Jul-2014.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &   𝑂 = (0g𝐹)    &    0 = (0g𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐴𝐾)    &   (𝜑𝑋𝑉)       (𝜑 → ((𝐴 · 𝑋) = 0 ↔ (𝐴 = 𝑂𝑋 = 0 )))
 
Theoremlvecvsn0 19090 A scalar product is nonzero iff both of its factors are nonzero. (Contributed by NM, 3-Jan-2015.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &   𝑂 = (0g𝐹)    &    0 = (0g𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐴𝐾)    &   (𝜑𝑋𝑉)       (𝜑 → ((𝐴 · 𝑋) ≠ 0 ↔ (𝐴𝑂𝑋0 )))
 
Theoremlssvs0or 19091 If a scalar product belongs to a subspace, either the scalar component is zero or the vector component also belongs to the subspace. (Contributed by NM, 5-Apr-2015.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    0 = (0g𝐹)    &   𝑆 = (LSubSp‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝑆)    &   (𝜑𝑋𝑉)    &   (𝜑𝐴𝐾)       (𝜑 → ((𝐴 · 𝑋) ∈ 𝑈 ↔ (𝐴 = 0𝑋𝑈)))
 
Theoremlvecvscan 19092 Cancellation law for scalar multiplication. (hvmulcan 27899 analog.) (Contributed by NM, 2-Jul-2014.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    0 = (0g𝐹)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐴𝐾)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝐴0 )       (𝜑 → ((𝐴 · 𝑋) = (𝐴 · 𝑌) ↔ 𝑋 = 𝑌))
 
Theoremlvecvscan2 19093 Cancellation law for scalar multiplication. (hvmulcan2 27900 analog.) (Contributed by NM, 2-Jul-2014.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    0 = (0g𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐴𝐾)    &   (𝜑𝐵𝐾)    &   (𝜑𝑋𝑉)    &   (𝜑𝑋0 )       (𝜑 → ((𝐴 · 𝑋) = (𝐵 · 𝑋) ↔ 𝐴 = 𝐵))
 
Theoremlvecinv 19094 Invert coefficient of scalar product. (Contributed by NM, 11-Apr-2015.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    0 = (0g𝐹)    &   𝐼 = (invr𝐹)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐴 ∈ (𝐾 ∖ { 0 }))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → (𝑋 = (𝐴 · 𝑌) ↔ 𝑌 = ((𝐼𝐴) · 𝑋)))
 
Theoremlspsnvs 19095 A nonzero scalar product does not change the span of a singleton. (spansncol 28397 analog.) (Contributed by NM, 23-Apr-2014.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)    &    0 = (0g𝐹)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LVec ∧ (𝑅𝐾𝑅0 ) ∧ 𝑋𝑉) → (𝑁‘{(𝑅 · 𝑋)}) = (𝑁‘{𝑋}))
 
Theoremlspsneleq 19096 Membership relation that implies equality of spans. (spansneleq 28399 analog.) (Contributed by NM, 4-Jul-2014.)
𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌 ∈ (𝑁‘{𝑋}))    &   (𝜑𝑌0 )       (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑋}))
 
Theoremlspsncmp 19097 Comparable spans of nonzero singletons are equal. (Contributed by NM, 27-Apr-2015.)
𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌𝑉)       (𝜑 → ((𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌}) ↔ (𝑁‘{𝑋}) = (𝑁‘{𝑌})))
 
Theoremlspsnne1 19098 Two ways to express that vectors have different spans. (Contributed by NM, 28-May-2015.)
𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌𝑉)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))       (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌}))
 
Theoremlspsnne2 19099 Two ways to express that vectors have different spans. (Contributed by NM, 20-May-2015.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌}))       (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
 
Theoremlspsnnecom 19100 Swap two vectors with different spans. (Contributed by NM, 20-May-2015.)
𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌}))       (𝜑 → ¬ 𝑌 ∈ (𝑁‘{𝑋}))
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