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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | frlmrcl 20101 | If a free module is inhabited, this is sufficient to conclude that the ring expression defines a set. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝐹) ⇒ ⊢ (𝑋 ∈ 𝐵 → 𝑅 ∈ V) | ||
Theorem | frlmbasfsupp 20102 | Elements of the free module are finitely supported. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Revised by Thierry Arnoux, 21-Jun-2019.) (Proof shortened by AV, 20-Jul-2019.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐵 = (Base‘𝐹) ⇒ ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵) → 𝑋 finSupp 0 ) | ||
Theorem | frlmbasmap 20103 | Elements of the free module are set functions. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) & ⊢ 𝑁 = (Base‘𝑅) & ⊢ 𝐵 = (Base‘𝐹) ⇒ ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ (𝑁 ↑𝑚 𝐼)) | ||
Theorem | frlmbasf 20104 | Elements of the free module are functions. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) & ⊢ 𝑁 = (Base‘𝑅) & ⊢ 𝐵 = (Base‘𝐹) ⇒ ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵) → 𝑋:𝐼⟶𝑁) | ||
Theorem | frlmfibas 20105 | The base set of the finite free module as a set exponential. (Contributed by AV, 6-Dec-2018.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) & ⊢ 𝑁 = (Base‘𝑅) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ Fin) → (𝑁 ↑𝑚 𝐼) = (Base‘𝐹)) | ||
Theorem | elfrlmbasn0 20106 | If the dimension of a free module over a ring is not 0, every element of its base set is not empty. (Contributed by AV, 10-Feb-2019.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) & ⊢ 𝑁 = (Base‘𝑅) & ⊢ 𝐵 = (Base‘𝐹) ⇒ ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐼 ≠ ∅) → (𝑋 ∈ 𝐵 → 𝑋 ≠ ∅)) | ||
Theorem | frlmplusgval 20107 | Addition in a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.) |
⊢ 𝑌 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ + = (+g‘𝑅) & ⊢ ✚ = (+g‘𝑌) ⇒ ⊢ (𝜑 → (𝐹 ✚ 𝐺) = (𝐹 ∘𝑓 + 𝐺)) | ||
Theorem | frlmsubgval 20108 | Subtraction in a free module. (Contributed by Thierry Arnoux, 30-Jun-2019.) |
⊢ 𝑌 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ − = (-g‘𝑅) & ⊢ 𝑀 = (-g‘𝑌) ⇒ ⊢ (𝜑 → (𝐹𝑀𝐺) = (𝐹 ∘𝑓 − 𝐺)) | ||
Theorem | frlmvscafval 20109 | Scalar multiplication in a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.) |
⊢ 𝑌 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝐴 ∈ 𝐾) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ ∙ = ( ·𝑠 ‘𝑌) & ⊢ · = (.r‘𝑅) ⇒ ⊢ (𝜑 → (𝐴 ∙ 𝑋) = ((𝐼 × {𝐴}) ∘𝑓 · 𝑋)) | ||
Theorem | frlmvscaval 20110 | Scalar multiplication in a free module at a coordinate. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
⊢ 𝑌 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝐴 ∈ 𝐾) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝐽 ∈ 𝐼) & ⊢ ∙ = ( ·𝑠 ‘𝑌) & ⊢ · = (.r‘𝑅) ⇒ ⊢ (𝜑 → ((𝐴 ∙ 𝑋)‘𝐽) = (𝐴 · (𝑋‘𝐽))) | ||
Theorem | frlmgsum 20111* | Finite commutative sums in a free module are taken componentwise. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Mario Carneiro, 5-Jul-2015.) (Revised by AV, 23-Jun-2019.) |
⊢ 𝑌 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 0 = (0g‘𝑌) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐽 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐽) → (𝑥 ∈ 𝐼 ↦ 𝑈) ∈ 𝐵) & ⊢ (𝜑 → (𝑦 ∈ 𝐽 ↦ (𝑥 ∈ 𝐼 ↦ 𝑈)) finSupp 0 ) ⇒ ⊢ (𝜑 → (𝑌 Σg (𝑦 ∈ 𝐽 ↦ (𝑥 ∈ 𝐼 ↦ 𝑈))) = (𝑥 ∈ 𝐼 ↦ (𝑅 Σg (𝑦 ∈ 𝐽 ↦ 𝑈)))) | ||
Theorem | frlmsplit2 20112* | Restriction is homomorphic on free modules. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.) |
⊢ 𝑌 = (𝑅 freeLMod 𝑈) & ⊢ 𝑍 = (𝑅 freeLMod 𝑉) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝐶 = (Base‘𝑍) & ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝑥 ↾ 𝑉)) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → 𝐹 ∈ (𝑌 LMHom 𝑍)) | ||
Theorem | frlmsslss 20113* | A subset of a free module obtained by restricting the support set is a submodule. 𝐽 is the set of forbidden unit vectors. (Contributed by Stefan O'Rear, 4-Feb-2015.) |
⊢ 𝑌 = (𝑅 freeLMod 𝐼) & ⊢ 𝑈 = (LSubSp‘𝑌) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐶 = {𝑥 ∈ 𝐵 ∣ (𝑥 ↾ 𝐽) = (𝐽 × { 0 })} ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝐶 ∈ 𝑈) | ||
Theorem | frlmsslss2 20114* | A subset of a free module obtained by restricting the support set is a submodule. 𝐽 is the set of permitted unit vectors. (Contributed by Stefan O'Rear, 5-Feb-2015.) (Revised by AV, 23-Jun-2019.) |
⊢ 𝑌 = (𝑅 freeLMod 𝐼) & ⊢ 𝑈 = (LSubSp‘𝑌) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐶 = {𝑥 ∈ 𝐵 ∣ (𝑥 supp 0 ) ⊆ 𝐽} ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝐶 ∈ 𝑈) | ||
Theorem | frlmbas3 20115 | An element of the base set of a finite free module with a Cartesian product as index set as operation value. (Contributed by AV, 14-Feb-2019.) |
⊢ 𝐹 = (𝑅 freeLMod (𝑁 × 𝑀)) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑉 = (Base‘𝐹) ⇒ ⊢ (((𝑅 ∈ 𝑊 ∧ 𝑋 ∈ 𝑉) ∧ (𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑀)) → (𝐼𝑋𝐽) ∈ 𝐵) | ||
Theorem | mpt2frlmd 20116* | Elements of the free module are mappings with two arguments defined by their operation values. (Contributed by AV, 20-Feb-2019.) |
⊢ 𝐹 = (𝑅 freeLMod (𝑁 × 𝑀)) & ⊢ 𝑉 = (Base‘𝐹) & ⊢ ((𝑖 = 𝑎 ∧ 𝑗 = 𝑏) → 𝐴 = 𝐵) & ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑀) → 𝐴 ∈ 𝑋) & ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑀) → 𝐵 ∈ 𝑌) & ⊢ (𝜑 → (𝑁 ∈ 𝑈 ∧ 𝑀 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉)) ⇒ ⊢ (𝜑 → (𝑍 = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑀 ↦ 𝐵) ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑀 (𝑖𝑍𝑗) = 𝐴)) | ||
Theorem | frlmip 20117* | The inner product of a free module. (Contributed by Thierry Arnoux, 20-Jun-2019.) |
⊢ 𝑌 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (𝑓 ∈ (𝐵 ↑𝑚 𝐼), 𝑔 ∈ (𝐵 ↑𝑚 𝐼) ↦ (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥) · (𝑔‘𝑥))))) = (·𝑖‘𝑌)) | ||
Theorem | frlmipval 20118 | The inner product of a free module. (Contributed by Thierry Arnoux, 21-Jun-2019.) |
⊢ 𝑌 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑉 = (Base‘𝑌) & ⊢ , = (·𝑖‘𝑌) ⇒ ⊢ (((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑋) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑉)) → (𝐹 , 𝐺) = (𝑅 Σg (𝐹 ∘𝑓 · 𝐺))) | ||
Theorem | frlmphllem 20119* | Lemma for frlmphl 20120. (Contributed by AV, 21-Jul-2019.) |
⊢ 𝑌 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑉 = (Base‘𝑌) & ⊢ , = (·𝑖‘𝑌) & ⊢ 𝑂 = (0g‘𝑌) & ⊢ 0 = (0g‘𝑅) & ⊢ ∗ = (*𝑟‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Field) & ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ (𝑔 , 𝑔) = 0 ) → 𝑔 = 𝑂) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( ∗ ‘𝑥) = 𝑥) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) ⇒ ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥))) finSupp 0 ) | ||
Theorem | frlmphl 20120* | Conditions for a free module to be a pre-Hilbert space. (Contributed by Thierry Arnoux, 21-Jun-2019.) (Proof shortened by AV, 21-Jul-2019.) |
⊢ 𝑌 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑉 = (Base‘𝑌) & ⊢ , = (·𝑖‘𝑌) & ⊢ 𝑂 = (0g‘𝑌) & ⊢ 0 = (0g‘𝑅) & ⊢ ∗ = (*𝑟‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Field) & ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ (𝑔 , 𝑔) = 0 ) → 𝑔 = 𝑂) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( ∗ ‘𝑥) = 𝑥) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) ⇒ ⊢ (𝜑 → 𝑌 ∈ PreHil) | ||
According to Wikipedia ("Standard basis", 16-Mar-2019, https://en.wikipedia.org/wiki/Standard_basis) "In mathematics, the standard basis (also called natural basis) for a Euclidean space is the set of unit vectors pointing in the direction of the axes of a Cartesian coordinate system.", and ("Unit vector", 16-Mar-2019, https://en.wikipedia.org/wiki/Unit_vector) "In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1.". In the following, the term "unit vector" (or more specific "basic unit vector") is used for the (special) unit vectors forming the standard basis of free modules. However, the length of the unit vectors is not considered here, so it is not required to regard normed spaces. | ||
Syntax | cuvc 20121 | Class of basic unit vectors for an explicit free module. |
class unitVec | ||
Definition | df-uvc 20122* | ((𝑅 unitVec 𝐼)‘𝑗) is the unit vector in (𝑅 freeLMod 𝐼) along the 𝑗 axis. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ unitVec = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ (𝑘 ∈ 𝑖 ↦ if(𝑘 = 𝑗, (1r‘𝑟), (0g‘𝑟))))) | ||
Theorem | uvcfval 20123* | Value of the unit-vector generator for a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ 𝑈 = (𝑅 unitVec 𝐼) & ⊢ 1 = (1r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑈 = (𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 )))) | ||
Theorem | uvcval 20124* | Value of a single unit vector in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
⊢ 𝑈 = (𝑅 unitVec 𝐼) & ⊢ 1 = (1r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → (𝑈‘𝐽) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 ))) | ||
Theorem | uvcvval 20125 | Value of a unit vector coordinate in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
⊢ 𝑈 = (𝑅 unitVec 𝐼) & ⊢ 1 = (1r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) ∧ 𝐾 ∈ 𝐼) → ((𝑈‘𝐽)‘𝐾) = if(𝐾 = 𝐽, 1 , 0 )) | ||
Theorem | uvcvvcl 20126 | A coodinate of a unit vector is either 0 or 1. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
⊢ 𝑈 = (𝑅 unitVec 𝐼) & ⊢ 1 = (1r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) ∧ 𝐾 ∈ 𝐼) → ((𝑈‘𝐽)‘𝐾) ∈ { 0 , 1 }) | ||
Theorem | uvcvvcl2 20127 | A unit vector coordinate is a ring element. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
⊢ 𝑈 = (𝑅 unitVec 𝐼) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝐽 ∈ 𝐼) & ⊢ (𝜑 → 𝐾 ∈ 𝐼) ⇒ ⊢ (𝜑 → ((𝑈‘𝐽)‘𝐾) ∈ 𝐵) | ||
Theorem | uvcvv1 20128 | The unit vector is one at its designated coordinate. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
⊢ 𝑈 = (𝑅 unitVec 𝐼) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝐽 ∈ 𝐼) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝜑 → ((𝑈‘𝐽)‘𝐽) = 1 ) | ||
Theorem | uvcvv0 20129 | The unit vector is zero at its designated coordinate. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
⊢ 𝑈 = (𝑅 unitVec 𝐼) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝐽 ∈ 𝐼) & ⊢ (𝜑 → 𝐾 ∈ 𝐼) & ⊢ (𝜑 → 𝐽 ≠ 𝐾) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝜑 → ((𝑈‘𝐽)‘𝐾) = 0 ) | ||
Theorem | uvcff 20130 | Domain and range of the unit vector generator; ring condition required to be sure 1 and 0 are actually in the ring. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.) |
⊢ 𝑈 = (𝑅 unitVec 𝐼) & ⊢ 𝑌 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝑌) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝑈:𝐼⟶𝐵) | ||
Theorem | uvcf1 20131 | In a nonzero ring, each unit vector is different. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.) |
⊢ 𝑈 = (𝑅 unitVec 𝐼) & ⊢ 𝑌 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝑌) ⇒ ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊) → 𝑈:𝐼–1-1→𝐵) | ||
Theorem | uvcresum 20132 | Any element of a free module can be expressed as a finite linear combination of unit vectors. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Proof shortened by Mario Carneiro, 5-Jul-2015.) |
⊢ 𝑈 = (𝑅 unitVec 𝐼) & ⊢ 𝑌 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ · = ( ·𝑠 ‘𝑌) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵) → 𝑋 = (𝑌 Σg (𝑋 ∘𝑓 · 𝑈))) | ||
Theorem | frlmssuvc1 20133* | A scalar multiple of a unit vector included in a support-restriction subspace is included in the subspace. (Contributed by Stefan O'Rear, 5-Feb-2015.) (Revised by AV, 24-Jun-2019.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) & ⊢ 𝑈 = (𝑅 unitVec 𝐼) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ · = ( ·𝑠 ‘𝐹) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐶 = {𝑥 ∈ 𝐵 ∣ (𝑥 supp 0 ) ⊆ 𝐽} & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐽 ⊆ 𝐼) & ⊢ (𝜑 → 𝐿 ∈ 𝐽) & ⊢ (𝜑 → 𝑋 ∈ 𝐾) ⇒ ⊢ (𝜑 → (𝑋 · (𝑈‘𝐿)) ∈ 𝐶) | ||
Theorem | frlmssuvc2 20134* | A nonzero scalar multiple of a unit vector not included in a support-restriction subspace is not included in the subspace. (Contributed by Stefan O'Rear, 5-Feb-2015.) (Revised by AV, 24-Jun-2019.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) & ⊢ 𝑈 = (𝑅 unitVec 𝐼) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ · = ( ·𝑠 ‘𝐹) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐶 = {𝑥 ∈ 𝐵 ∣ (𝑥 supp 0 ) ⊆ 𝐽} & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐽 ⊆ 𝐼) & ⊢ (𝜑 → 𝐿 ∈ (𝐼 ∖ 𝐽)) & ⊢ (𝜑 → 𝑋 ∈ (𝐾 ∖ { 0 })) ⇒ ⊢ (𝜑 → ¬ (𝑋 · (𝑈‘𝐿)) ∈ 𝐶) | ||
Theorem | frlmsslsp 20135* | A subset of a free module obtained by restricting the support set is spanned by the relevant unit vectors. (Contributed by Stefan O'Rear, 6-Feb-2015.) (Revised by AV, 24-Jun-2019.) |
⊢ 𝑌 = (𝑅 freeLMod 𝐼) & ⊢ 𝑈 = (𝑅 unitVec 𝐼) & ⊢ 𝐾 = (LSpan‘𝑌) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐶 = {𝑥 ∈ 𝐵 ∣ (𝑥 supp 0 ) ⊆ 𝐽} ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (𝐾‘(𝑈 “ 𝐽)) = 𝐶) | ||
Theorem | frlmlbs 20136 | The unit vectors comprise a basis for a free module. (Contributed by Stefan O'Rear, 6-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) & ⊢ 𝑈 = (𝑅 unitVec 𝐼) & ⊢ 𝐽 = (LBasis‘𝐹) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → ran 𝑈 ∈ 𝐽) | ||
Theorem | frlmup1 20137* | Any assignment of unit vectors to target vectors can be extended (uniquely) to a homomorphism from a free module to an arbitrary other module on the same base ring. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ 𝐶 = (Base‘𝑇) & ⊢ · = ( ·𝑠 ‘𝑇) & ⊢ 𝐸 = (𝑥 ∈ 𝐵 ↦ (𝑇 Σg (𝑥 ∘𝑓 · 𝐴))) & ⊢ (𝜑 → 𝑇 ∈ LMod) & ⊢ (𝜑 → 𝐼 ∈ 𝑋) & ⊢ (𝜑 → 𝑅 = (Scalar‘𝑇)) & ⊢ (𝜑 → 𝐴:𝐼⟶𝐶) ⇒ ⊢ (𝜑 → 𝐸 ∈ (𝐹 LMHom 𝑇)) | ||
Theorem | frlmup2 20138* | The evaluation map has the intended behavior on the unit vectors. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ 𝐶 = (Base‘𝑇) & ⊢ · = ( ·𝑠 ‘𝑇) & ⊢ 𝐸 = (𝑥 ∈ 𝐵 ↦ (𝑇 Σg (𝑥 ∘𝑓 · 𝐴))) & ⊢ (𝜑 → 𝑇 ∈ LMod) & ⊢ (𝜑 → 𝐼 ∈ 𝑋) & ⊢ (𝜑 → 𝑅 = (Scalar‘𝑇)) & ⊢ (𝜑 → 𝐴:𝐼⟶𝐶) & ⊢ (𝜑 → 𝑌 ∈ 𝐼) & ⊢ 𝑈 = (𝑅 unitVec 𝐼) ⇒ ⊢ (𝜑 → (𝐸‘(𝑈‘𝑌)) = (𝐴‘𝑌)) | ||
Theorem | frlmup3 20139* | The range of such an evaluation map is the finite linear combinations of the target vectors and also the span of the target vectors. (Contributed by Stefan O'Rear, 6-Feb-2015.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ 𝐶 = (Base‘𝑇) & ⊢ · = ( ·𝑠 ‘𝑇) & ⊢ 𝐸 = (𝑥 ∈ 𝐵 ↦ (𝑇 Σg (𝑥 ∘𝑓 · 𝐴))) & ⊢ (𝜑 → 𝑇 ∈ LMod) & ⊢ (𝜑 → 𝐼 ∈ 𝑋) & ⊢ (𝜑 → 𝑅 = (Scalar‘𝑇)) & ⊢ (𝜑 → 𝐴:𝐼⟶𝐶) & ⊢ 𝐾 = (LSpan‘𝑇) ⇒ ⊢ (𝜑 → ran 𝐸 = (𝐾‘ran 𝐴)) | ||
Theorem | frlmup4 20140* | Universal property of the free module by existential uniquenes. (Contributed by Stefan O'Rear, 7-Mar-2015.) |
⊢ 𝑅 = (Scalar‘𝑇) & ⊢ 𝐹 = (𝑅 freeLMod 𝐼) & ⊢ 𝑈 = (𝑅 unitVec 𝐼) & ⊢ 𝐶 = (Base‘𝑇) ⇒ ⊢ ((𝑇 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐴:𝐼⟶𝐶) → ∃!𝑚 ∈ (𝐹 LMHom 𝑇)(𝑚 ∘ 𝑈) = 𝐴) | ||
Theorem | ellspd 20141* | The elements of the span of an indexed collection of basic vectors are those vectors which can be written as finite linear combinations of basic vectors. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Revised by AV, 24-Jun-2019.) |
⊢ 𝑁 = (LSpan‘𝑀) & ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 𝑆 = (Scalar‘𝑀) & ⊢ 0 = (0g‘𝑆) & ⊢ · = ( ·𝑠 ‘𝑀) & ⊢ (𝜑 → 𝐹:𝐼⟶𝐵) & ⊢ (𝜑 → 𝑀 ∈ LMod) & ⊢ (𝜑 → 𝐼 ∈ V) ⇒ ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(𝐹 “ 𝐼)) ↔ ∃𝑓 ∈ (𝐾 ↑𝑚 𝐼)(𝑓 finSupp 0 ∧ 𝑋 = (𝑀 Σg (𝑓 ∘𝑓 · 𝐹))))) | ||
Theorem | elfilspd 20142* | Simplified version of ellspd 20141 when the spanning set is finite: all linear combinations are then acceptable. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.) |
⊢ 𝑁 = (LSpan‘𝑀) & ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 𝑆 = (Scalar‘𝑀) & ⊢ 0 = (0g‘𝑆) & ⊢ · = ( ·𝑠 ‘𝑀) & ⊢ (𝜑 → 𝐹:𝐼⟶𝐵) & ⊢ (𝜑 → 𝑀 ∈ LMod) & ⊢ (𝜑 → 𝐼 ∈ Fin) ⇒ ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(𝐹 “ 𝐼)) ↔ ∃𝑓 ∈ (𝐾 ↑𝑚 𝐼)𝑋 = (𝑀 Σg (𝑓 ∘𝑓 · 𝐹)))) | ||
According to the definition in [Lang] p. 129: "A subset S of a module M is said
to be linearly independent (over A) if whenever we have a linear
combination ∑x ∈ S axx which is equal to 0, then
ax = 0 for all x ∈ S.", and according to the Definition in
[Lang] p. 130: "a familiy {xi}i ∈ I of elements of M
is said to be linearly independent (over A) if whenever we have a linear
combination ∑i ∈ I aixi = 0, then
ai = 0 for all i.". These definitions correspond to the definitions
df-linds 20146 resp. df-lindf 20145, where it is claimed that a nonzero summand can be
extracted ( ∑i ∈ {I \ { j } }aixi =
-ajxj ) and be represented as a linear combination of the
remaining elements of the family.
| ||
Syntax | clindf 20143 | The class relationship of independent families in a module. |
class LIndF | ||
Syntax | clinds 20144 | The class generator of independent sets in a module. |
class LIndS | ||
Definition | df-lindf 20145* |
An independent family is a family of vectors, no nonzero multiple of
which can be expressed as a linear combination of other elements of the
family. This is almost, but not quite, the same as a function into an
independent set.
This is a defined concept because it matters in many cases whether independence is taken at a set or family level. For instance, a number is transcedental iff its nonzero powers are linearly independent. Is 1 transcedental? It has only one nonzero power. We can almost define family independence as a family of unequal elements with independent range, as islindf3 20165, but taking that as primitive would lead to unpleasant corner case behavior with the zero ring. This is equivalent to the common definition of having no nontrivial representations of zero (islindf4 20177) and only one representation for each element of the range (islindf5 20178). (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ LIndF = {〈𝑓, 𝑤〉 ∣ (𝑓:dom 𝑓⟶(Base‘𝑤) ∧ [(Scalar‘𝑤) / 𝑠]∀𝑥 ∈ dom 𝑓∀𝑘 ∈ ((Base‘𝑠) ∖ {(0g‘𝑠)}) ¬ (𝑘( ·𝑠 ‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))))} | ||
Definition | df-linds 20146* | An independent set is a set which is independent as a family. See also islinds3 20173 and islinds4 20174. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ LIndS = (𝑤 ∈ V ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ( I ↾ 𝑠) LIndF 𝑤}) | ||
Theorem | rellindf 20147 | The independent-family predicate is a proper relation and can be used with brrelexi 5158. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ Rel LIndF | ||
Theorem | islinds 20148 | Property of an independent set of vectors in terms of an independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ 𝐵 = (Base‘𝑊) ⇒ ⊢ (𝑊 ∈ 𝑉 → (𝑋 ∈ (LIndS‘𝑊) ↔ (𝑋 ⊆ 𝐵 ∧ ( I ↾ 𝑋) LIndF 𝑊))) | ||
Theorem | linds1 20149 | An independent set of vectors is a set of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ 𝐵 = (Base‘𝑊) ⇒ ⊢ (𝑋 ∈ (LIndS‘𝑊) → 𝑋 ⊆ 𝐵) | ||
Theorem | linds2 20150 | An independent set of vectors is independent as a family. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ (𝑋 ∈ (LIndS‘𝑊) → ( I ↾ 𝑋) LIndF 𝑊) | ||
Theorem | islindf 20151* | Property of an independent family of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐾 = (LSpan‘𝑊) & ⊢ 𝑆 = (Scalar‘𝑊) & ⊢ 𝑁 = (Base‘𝑆) & ⊢ 0 = (0g‘𝑆) ⇒ ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐹 ∈ 𝑋) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹⟶𝐵 ∧ ∀𝑥 ∈ dom 𝐹∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹‘𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥})))))) | ||
Theorem | islinds2 20152* | Expanded property of an independent set of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐾 = (LSpan‘𝑊) & ⊢ 𝑆 = (Scalar‘𝑊) & ⊢ 𝑁 = (Base‘𝑆) & ⊢ 0 = (0g‘𝑆) ⇒ ⊢ (𝑊 ∈ 𝑌 → (𝐹 ∈ (LIndS‘𝑊) ↔ (𝐹 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐹 ∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · 𝑥) ∈ (𝐾‘(𝐹 ∖ {𝑥}))))) | ||
Theorem | islindf2 20153* | Property of an independent family of vectors with prior constrained domain and codomain. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐾 = (LSpan‘𝑊) & ⊢ 𝑆 = (Scalar‘𝑊) & ⊢ 𝑁 = (Base‘𝑆) & ⊢ 0 = (0g‘𝑆) ⇒ ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) → (𝐹 LIndF 𝑊 ↔ ∀𝑥 ∈ 𝐼 ∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹‘𝑥)) ∈ (𝐾‘(𝐹 “ (𝐼 ∖ {𝑥}))))) | ||
Theorem | lindff 20154 | Functional property of a linearly independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ 𝐵 = (Base‘𝑊) ⇒ ⊢ ((𝐹 LIndF 𝑊 ∧ 𝑊 ∈ 𝑌) → 𝐹:dom 𝐹⟶𝐵) | ||
Theorem | lindfind 20155 | A linearly independent family is independent: no nonzero element multiple can be expressed as a linear combination of the others. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 𝐿 = (Scalar‘𝑊) & ⊢ 0 = (0g‘𝐿) & ⊢ 𝐾 = (Base‘𝐿) ⇒ ⊢ (((𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → ¬ (𝐴 · (𝐹‘𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸})))) | ||
Theorem | lindsind 20156 | A linearly independent set is independent: no nonzero element multiple can be expressed as a linear combination of the others. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 𝐿 = (Scalar‘𝑊) & ⊢ 0 = (0g‘𝐿) & ⊢ 𝐾 = (Base‘𝐿) ⇒ ⊢ (((𝐹 ∈ (LIndS‘𝑊) ∧ 𝐸 ∈ 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐹 ∖ {𝐸}))) | ||
Theorem | lindfind2 20157 | In a linearly independent family in a module over a nonzero ring, no element is contained in the span of any non-containing set. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ 𝐾 = (LSpan‘𝑊) & ⊢ 𝐿 = (Scalar‘𝑊) ⇒ ⊢ (((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) ∧ 𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) → ¬ (𝐹‘𝐸) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝐸})))) | ||
Theorem | lindsind2 20158 | In a linearly independent set in a module over a nonzero ring, no element is contained in the span of any non-containing set. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ 𝐾 = (LSpan‘𝑊) & ⊢ 𝐿 = (Scalar‘𝑊) ⇒ ⊢ (((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝐸 ∈ 𝐹) → ¬ 𝐸 ∈ (𝐾‘(𝐹 ∖ {𝐸}))) | ||
Theorem | lindff1 20159 | A linearly independent family over a nonzero ring has no repeated elements. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐿 = (Scalar‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing ∧ 𝐹 LIndF 𝑊) → 𝐹:dom 𝐹–1-1→𝐵) | ||
Theorem | lindfrn 20160 | The range of an independent family is an independent set. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → ran 𝐹 ∈ (LIndS‘𝑊)) | ||
Theorem | f1lindf 20161 | Rearranging and deleting elements from an independent family gives an independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (𝐹 ∘ 𝐺) LIndF 𝑊) | ||
Theorem | lindfres 20162 | Any restriction of an independent family is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → (𝐹 ↾ 𝑋) LIndF 𝑊) | ||
Theorem | lindsss 20163 | Any subset of an independent set is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ ((𝑊 ∈ LMod ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝐺 ⊆ 𝐹) → 𝐺 ∈ (LIndS‘𝑊)) | ||
Theorem | f1linds 20164 | A family constructed from non-repeated elements of an independent set is independent. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ (LIndS‘𝑊) ∧ 𝐹:𝐷–1-1→𝑆) → 𝐹 LIndF 𝑊) | ||
Theorem | islindf3 20165 | In a nonzero ring, independent families can be equivalently characterized as renamings of independent sets. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
⊢ 𝐿 = (Scalar‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹–1-1→V ∧ ran 𝐹 ∈ (LIndS‘𝑊)))) | ||
Theorem | lindfmm 20166 | Linear independence of a family is unchanged by injective linear functions. (Contributed by Stefan O'Rear, 26-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.) |
⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝐶 = (Base‘𝑇) ⇒ ⊢ ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶 ∧ 𝐹:𝐼⟶𝐵) → (𝐹 LIndF 𝑆 ↔ (𝐺 ∘ 𝐹) LIndF 𝑇)) | ||
Theorem | lindsmm 20167 | Linear independence of a set is unchanged by injective linear functions. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝐶 = (Base‘𝑇) ⇒ ⊢ ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶 ∧ 𝐹 ⊆ 𝐵) → (𝐹 ∈ (LIndS‘𝑆) ↔ (𝐺 “ 𝐹) ∈ (LIndS‘𝑇))) | ||
Theorem | lindsmm2 20168 | The monomorphic image of an independent set is independent. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝐶 = (Base‘𝑇) ⇒ ⊢ ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶 ∧ 𝐹 ∈ (LIndS‘𝑆)) → (𝐺 “ 𝐹) ∈ (LIndS‘𝑇)) | ||
Theorem | lsslindf 20169 | Linear independence is unchanged by working in a subspace. (Contributed by Stefan O'Rear, 24-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.) |
⊢ 𝑈 = (LSubSp‘𝑊) & ⊢ 𝑋 = (𝑊 ↾s 𝑆) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) → (𝐹 LIndF 𝑋 ↔ 𝐹 LIndF 𝑊)) | ||
Theorem | lsslinds 20170 | Linear independence is unchanged by working in a subspace. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ 𝑈 = (LSubSp‘𝑊) & ⊢ 𝑋 = (𝑊 ↾s 𝑆) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ 𝐹 ⊆ 𝑆) → (𝐹 ∈ (LIndS‘𝑋) ↔ 𝐹 ∈ (LIndS‘𝑊))) | ||
Theorem | islbs4 20171 | A basis is an independent spanning set. This could have been used as alternative definition of a basis: LBasis = (𝑤 ∈ V ↦ {𝑏 ∈ 𝒫 (Base‘𝑤) ∣ (((LSpan‘𝑤) ‘𝑏) = (Base‘𝑤) ∧ 𝑏 ∈ (LIndS‘𝑤))}). (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐽 = (LBasis‘𝑊) & ⊢ 𝐾 = (LSpan‘𝑊) ⇒ ⊢ (𝑋 ∈ 𝐽 ↔ (𝑋 ∈ (LIndS‘𝑊) ∧ (𝐾‘𝑋) = 𝐵)) | ||
Theorem | lbslinds 20172 | A basis is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ 𝐽 = (LBasis‘𝑊) ⇒ ⊢ 𝐽 ⊆ (LIndS‘𝑊) | ||
Theorem | islinds3 20173 | A subset is linearly independent iff it is a basis of its span. (Contributed by Stefan O'Rear, 25-Feb-2015.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐾 = (LSpan‘𝑊) & ⊢ 𝑋 = (𝑊 ↾s (𝐾‘𝑌)) & ⊢ 𝐽 = (LBasis‘𝑋) ⇒ ⊢ (𝑊 ∈ LMod → (𝑌 ∈ (LIndS‘𝑊) ↔ 𝑌 ∈ 𝐽)) | ||
Theorem | islinds4 20174* | A set is independent in a vector space iff it is a subset of some basis. (AC equivalent) (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ 𝐽 = (LBasis‘𝑊) ⇒ ⊢ (𝑊 ∈ LVec → (𝑌 ∈ (LIndS‘𝑊) ↔ ∃𝑏 ∈ 𝐽 𝑌 ⊆ 𝑏)) | ||
Theorem | lmimlbs 20175 | The isomorphic image of a basis is a basis. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
⊢ 𝐽 = (LBasis‘𝑆) & ⊢ 𝐾 = (LBasis‘𝑇) ⇒ ⊢ ((𝐹 ∈ (𝑆 LMIso 𝑇) ∧ 𝐵 ∈ 𝐽) → (𝐹 “ 𝐵) ∈ 𝐾) | ||
Theorem | lmiclbs 20176 | Having a basis is an isomorphism invariant. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
⊢ 𝐽 = (LBasis‘𝑆) & ⊢ 𝐾 = (LBasis‘𝑇) ⇒ ⊢ (𝑆 ≃𝑚 𝑇 → (𝐽 ≠ ∅ → 𝐾 ≠ ∅)) | ||
Theorem | islindf4 20177* | A family is independent iff it has no nontrivial representations of zero. (Contributed by Stefan O'Rear, 28-Feb-2015.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑌 = (0g‘𝑅) & ⊢ 𝐿 = (Base‘(𝑅 freeLMod 𝐼)) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) → (𝐹 LIndF 𝑊 ↔ ∀𝑥 ∈ 𝐿 ((𝑊 Σg (𝑥 ∘𝑓 · 𝐹)) = 0 → 𝑥 = (𝐼 × {𝑌})))) | ||
Theorem | islindf5 20178* | A family is independent iff the linear combinations homomorphism is injective. (Contributed by Stefan O'Rear, 28-Feb-2015.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ 𝐶 = (Base‘𝑇) & ⊢ · = ( ·𝑠 ‘𝑇) & ⊢ 𝐸 = (𝑥 ∈ 𝐵 ↦ (𝑇 Σg (𝑥 ∘𝑓 · 𝐴))) & ⊢ (𝜑 → 𝑇 ∈ LMod) & ⊢ (𝜑 → 𝐼 ∈ 𝑋) & ⊢ (𝜑 → 𝑅 = (Scalar‘𝑇)) & ⊢ (𝜑 → 𝐴:𝐼⟶𝐶) ⇒ ⊢ (𝜑 → (𝐴 LIndF 𝑇 ↔ 𝐸:𝐵–1-1→𝐶)) | ||
Theorem | indlcim 20179* | An independent, spanning family extends to an isomorphism from a free module. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ 𝐶 = (Base‘𝑇) & ⊢ · = ( ·𝑠 ‘𝑇) & ⊢ 𝑁 = (LSpan‘𝑇) & ⊢ 𝐸 = (𝑥 ∈ 𝐵 ↦ (𝑇 Σg (𝑥 ∘𝑓 · 𝐴))) & ⊢ (𝜑 → 𝑇 ∈ LMod) & ⊢ (𝜑 → 𝐼 ∈ 𝑋) & ⊢ (𝜑 → 𝑅 = (Scalar‘𝑇)) & ⊢ (𝜑 → 𝐴:𝐼–onto→𝐽) & ⊢ (𝜑 → 𝐴 LIndF 𝑇) & ⊢ (𝜑 → (𝑁‘𝐽) = 𝐶) ⇒ ⊢ (𝜑 → 𝐸 ∈ (𝐹 LMIso 𝑇)) | ||
Theorem | lbslcic 20180 | A module with a basis is isomorphic to a free module with the same cardinality. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐽 = (LBasis‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) → 𝑊 ≃𝑚 (𝐹 freeLMod 𝐼)) | ||
Theorem | lmisfree 20181* | A module has a basis iff it is isomorphic to a free module. In settings where isomorphic objects are not distinguished, it is common to define "free module" as any module with a basis; thus for instance lbsex 19165 might be described as "every vector space is free." (Contributed by Stefan O'Rear, 26-Feb-2015.) |
⊢ 𝐽 = (LBasis‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) ⇒ ⊢ (𝑊 ∈ LMod → (𝐽 ≠ ∅ ↔ ∃𝑘 𝑊 ≃𝑚 (𝐹 freeLMod 𝑘))) | ||
Theorem | lvecisfrlm 20182* | Every vector space is isomorphic to a free module. (Contributed by AV, 7-Mar-2019.) |
⊢ 𝐹 = (Scalar‘𝑊) ⇒ ⊢ (𝑊 ∈ LVec → ∃𝑘 𝑊 ≃𝑚 (𝐹 freeLMod 𝑘)) | ||
Theorem | lmimco 20183 | The composition of two isomorphisms of modules is an isomorphism of modules. (Contributed by AV, 10-Mar-2019.) |
⊢ ((𝐹 ∈ (𝑆 LMIso 𝑇) ∧ 𝐺 ∈ (𝑅 LMIso 𝑆)) → (𝐹 ∘ 𝐺) ∈ (𝑅 LMIso 𝑇)) | ||
Theorem | lmictra 20184 | Module isomorphism is transitive. (Contributed by AV, 10-Mar-2019.) |
⊢ ((𝑅 ≃𝑚 𝑆 ∧ 𝑆 ≃𝑚 𝑇) → 𝑅 ≃𝑚 𝑇) | ||
Theorem | uvcf1o 20185 | In a nonzero ring, the mapping of the index set of a free module onto the unit vectors of the free module is a 1-1 onto function. (Contributed by AV, 10-Mar-2019.) |
⊢ 𝑈 = (𝑅 unitVec 𝐼) ⇒ ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊) → 𝑈:𝐼–1-1-onto→ran 𝑈) | ||
Theorem | uvcendim 20186 | In a nonzero ring, the number of unit vectors of a free module corresponds to the dimension of the free module. (Contributed by AV, 10-Mar-2019.) |
⊢ 𝑈 = (𝑅 unitVec 𝐼) ⇒ ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊) → 𝐼 ≈ ran 𝑈) | ||
Theorem | frlmisfrlm 20187 | A free module is isomorphic to a free module over the same (nonzero) ring, with the same cardinality. (Contributed by AV, 10-Mar-2019.) |
⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑌 ∧ 𝐼 ≈ 𝐽) → (𝑅 freeLMod 𝐼) ≃𝑚 (𝑅 freeLMod 𝐽)) | ||
Theorem | frlmiscvec 20188 | Every free module is isomorphic to the free module of "column vectors" of the same dimension over the same (nonzero) ring. (Contributed by AV, 10-Mar-2019.) |
⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑌) → (𝑅 freeLMod 𝐼) ≃𝑚 (𝑅 freeLMod (𝐼 × {∅}))) | ||
According to Wikipedia ("Matrix (mathemetics)", 02-Apr-2019, https://en.wikipedia.org/wiki/Matrix_(mathematics)) "A matrix is a rectangular array of numbers or other mathematical objects for which operations such as addition and multiplication are defined. Most commonly, a matrix over a field F is a rectangular array of scalars each of which is a member of F. The numbers, symbols or expressions in the matrix are called its entries or its elements. The horizontal and vertical lines of entries in a matrix are called rows and columns, respectively.", and in the definition of [Lang] p. 503 "By an m x n matrix in [a commutative ring] R one means a doubly indexed family of elements of R, (aij), (i= 1,..., m and j = 1,... n) ... We call the elements aij the coefficients or components of the matrix. A 1 x n matrix is called a row vector (of dimension, or size, n) and a m x 1 matrix is called a column vector (of dimension, or size, m). In general, we say that (m,n) is the size of the matrix, ...". In contrast to these definitions, we denote any free module over a (not necessarily commutative) ring (in the meaning of df-frlm 20091) with a Cartesian product as index set as "matrix". The two sets of the Cartesian product even need neither to be ordered or a range of (nonnegative/positive) integers nor finite. By this, the addition and scalar multiplication for matrices correspond to the addition (see frlmplusgval 20107) and scalar multiplication (see frlmvscafval 20109) for free modules. Actually, there isn't a definition for (arbitrary) matrices: Even the (general) matrix multiplication can be defined using functions from Cartesian products into a ring (which are elements of the base set of free modules), see df-mamu 20190. By this, a statement like "Then the set of m x n matrices in R is a module (i.e. an R-module)" as in [Lang] p. 504 follows immediately from frlmlmod 20093. However, for square matrices there is the definition df-mat 20214, defining the algebras of square matrices (of the same size over the same ring), extending the structure of the corresponding free module by the matrix multiplication as ring multiplication. A "usual" matrix (aij), (i= 1,..., m and j = 1,... n) would be represented as element of (the base set of) (𝑅 freeLMod ((1...𝑚) × (1...𝑛))), and a square matrix (aij), (i= 1,..., n and j = 1,... n) would be represented as element of (the base set of) ((1...𝑛) Mat 𝑅). Finally, it should be mentioned that our definitions of matrices include the zero-dimensional cases, which is excluded in the definition of many authors, e.g. in [Lang] p. 503. It is shown in mat0dimbas0 20272 that the empty set is the sole zero-dimensional matrix (also called "empty matrix", see Wikipedia https://en.wikipedia.org/wiki/Matrix_(mathematics)#Empty_matrices). The determinant is also defined for such an empty matrix, see mdet0pr 20398. | ||
This section is about the multiplication of m x n matrices. | ||
Syntax | cmmul 20189 | Syntax for the matrix multiplication operator. |
class maMul | ||
Definition | df-mamu 20190* | The operator which multiplies an m x n matrix with an n x p matrix, see also the definition in [Lang] p. 504. Note that it is not generally possible to recover the dimensions from the matrix, since all n x 0 and all 0 x n matrices are represented by the empty set. (Contributed by Stefan O'Rear, 4-Sep-2015.) |
⊢ maMul = (𝑟 ∈ V, 𝑜 ∈ V ↦ ⦋(1st ‘(1st ‘𝑜)) / 𝑚⦌⦋(2nd ‘(1st ‘𝑜)) / 𝑛⦌⦋(2nd ‘𝑜) / 𝑝⦌(𝑥 ∈ ((Base‘𝑟) ↑𝑚 (𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑𝑚 (𝑛 × 𝑝)) ↦ (𝑖 ∈ 𝑚, 𝑘 ∈ 𝑝 ↦ (𝑟 Σg (𝑗 ∈ 𝑛 ↦ ((𝑖𝑥𝑗)(.r‘𝑟)(𝑗𝑦𝑘))))))) | ||
Theorem | mamufval 20191* | Functional value of the matrix multiplication operator. (Contributed by Stefan O'Rear, 2-Sep-2015.) |
⊢ 𝐹 = (𝑅 maMul 〈𝑀, 𝑁, 𝑃〉) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ Fin) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑃 ∈ Fin) ⇒ ⊢ (𝜑 → 𝐹 = (𝑥 ∈ (𝐵 ↑𝑚 (𝑀 × 𝑁)), 𝑦 ∈ (𝐵 ↑𝑚 (𝑁 × 𝑃)) ↦ (𝑖 ∈ 𝑀, 𝑘 ∈ 𝑃 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑗𝑦𝑘))))))) | ||
Theorem | mamuval 20192* | Multiplication of two matrices. (Contributed by Stefan O'Rear, 2-Sep-2015.) |
⊢ 𝐹 = (𝑅 maMul 〈𝑀, 𝑁, 𝑃〉) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ Fin) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑃 ∈ Fin) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑𝑚 (𝑀 × 𝑁))) & ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑𝑚 (𝑁 × 𝑃))) ⇒ ⊢ (𝜑 → (𝑋𝐹𝑌) = (𝑖 ∈ 𝑀, 𝑘 ∈ 𝑃 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘)))))) | ||
Theorem | mamufv 20193* | A cell in the multiplication of two matrices. (Contributed by Stefan O'Rear, 2-Sep-2015.) |
⊢ 𝐹 = (𝑅 maMul 〈𝑀, 𝑁, 𝑃〉) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ Fin) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑃 ∈ Fin) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑𝑚 (𝑀 × 𝑁))) & ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑𝑚 (𝑁 × 𝑃))) & ⊢ (𝜑 → 𝐼 ∈ 𝑀) & ⊢ (𝜑 → 𝐾 ∈ 𝑃) ⇒ ⊢ (𝜑 → (𝐼(𝑋𝐹𝑌)𝐾) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑗𝑌𝐾))))) | ||
Theorem | mamudm 20194 | The domain of the matrix multiplication function. (Contributed by AV, 10-Feb-2019.) |
⊢ 𝐸 = (𝑅 freeLMod (𝑀 × 𝑁)) & ⊢ 𝐵 = (Base‘𝐸) & ⊢ 𝐹 = (𝑅 freeLMod (𝑁 × 𝑃)) & ⊢ 𝐶 = (Base‘𝐹) & ⊢ × = (𝑅 maMul 〈𝑀, 𝑁, 𝑃〉) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑀 ∈ Fin ∧ 𝑁 ∈ Fin ∧ 𝑃 ∈ Fin)) → dom × = (𝐵 × 𝐶)) | ||
Theorem | mamufacex 20195 | Every solution of the equation 𝐴∗𝑋 = 𝐵 for matrices 𝐴 and 𝐵 is a matrix. (Contributed by AV, 10-Feb-2019.) |
⊢ 𝐸 = (𝑅 freeLMod (𝑀 × 𝑁)) & ⊢ 𝐵 = (Base‘𝐸) & ⊢ 𝐹 = (𝑅 freeLMod (𝑁 × 𝑃)) & ⊢ 𝐶 = (Base‘𝐹) & ⊢ × = (𝑅 maMul 〈𝑀, 𝑁, 𝑃〉) & ⊢ 𝐺 = (𝑅 freeLMod (𝑀 × 𝑃)) & ⊢ 𝐷 = (Base‘𝐺) ⇒ ⊢ (((𝑀 ≠ ∅ ∧ 𝑃 ≠ ∅) ∧ (𝑅 ∈ 𝑉 ∧ 𝑌 ∈ 𝐷) ∧ (𝑀 ∈ Fin ∧ 𝑁 ∈ Fin ∧ 𝑃 ∈ Fin)) → ((𝑋 × 𝑍) = 𝑌 → 𝑍 ∈ 𝐶)) | ||
Theorem | mamures 20196 | Rows in a matrix product are functions only of the corresponding rows in the left argument. (Contributed by SO, 9-Jul-2018.) |
⊢ 𝐹 = (𝑅 maMul 〈𝑀, 𝑁, 𝑃〉) & ⊢ 𝐺 = (𝑅 maMul 〈𝐼, 𝑁, 𝑃〉) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ Fin) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑃 ∈ Fin) & ⊢ (𝜑 → 𝐼 ⊆ 𝑀) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑𝑚 (𝑀 × 𝑁))) & ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑𝑚 (𝑁 × 𝑃))) ⇒ ⊢ (𝜑 → ((𝑋𝐹𝑌) ↾ (𝐼 × 𝑃)) = ((𝑋 ↾ (𝐼 × 𝑁))𝐺𝑌)) | ||
Theorem | mndvcl 20197 | Tuple-wise additive closure in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) ⇒ ⊢ ((𝑀 ∈ Mnd ∧ 𝑋 ∈ (𝐵 ↑𝑚 𝐼) ∧ 𝑌 ∈ (𝐵 ↑𝑚 𝐼)) → (𝑋 ∘𝑓 + 𝑌) ∈ (𝐵 ↑𝑚 𝐼)) | ||
Theorem | mndvass 20198 | Tuple-wise associativity in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) ⇒ ⊢ ((𝑀 ∈ Mnd ∧ (𝑋 ∈ (𝐵 ↑𝑚 𝐼) ∧ 𝑌 ∈ (𝐵 ↑𝑚 𝐼) ∧ 𝑍 ∈ (𝐵 ↑𝑚 𝐼))) → ((𝑋 ∘𝑓 + 𝑌) ∘𝑓 + 𝑍) = (𝑋 ∘𝑓 + (𝑌 ∘𝑓 + 𝑍))) | ||
Theorem | mndvlid 20199 | Tuple-wise left identity in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) & ⊢ 0 = (0g‘𝑀) ⇒ ⊢ ((𝑀 ∈ Mnd ∧ 𝑋 ∈ (𝐵 ↑𝑚 𝐼)) → ((𝐼 × { 0 }) ∘𝑓 + 𝑋) = 𝑋) | ||
Theorem | mndvrid 20200 | Tuple-wise right identity in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) & ⊢ 0 = (0g‘𝑀) ⇒ ⊢ ((𝑀 ∈ Mnd ∧ 𝑋 ∈ (𝐵 ↑𝑚 𝐼)) → (𝑋 ∘𝑓 + (𝐼 × { 0 })) = 𝑋) |
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