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Theorem List for Metamath Proof Explorer - 20501-20600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfrlmbasfsupp 20501 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 = (0g𝑅)    &   𝐵 = (Base‘𝐹)       ((𝐼𝑊𝑋𝐵) → 𝑋 finSupp 0 )

Theoremfrlmbasmap 20502 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)       ((𝐼𝑊𝑋𝐵) → 𝑋 ∈ (𝑁𝑚 𝐼))

Theoremfrlmbasf 20503 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)       ((𝐼𝑊𝑋𝐵) → 𝑋:𝐼𝑁)

Theoremfrlmlvec 20504 The free module over a division ring is a left vector space. (Contributed by Steven Nguyen, 29-Apr-2023.)
𝐹 = (𝑅 freeLMod 𝐼)       ((𝑅 ∈ DivRing ∧ 𝐼𝑊) → 𝐹 ∈ LVec)

Theoremfrlmfibas 20505 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)       ((𝑅𝑉𝐼 ∈ Fin) → (𝑁𝑚 𝐼) = (Base‘𝐹))

Theoremelfrlmbasn0 20506 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)       ((𝐼𝑉𝐼 ≠ ∅) → (𝑋𝐵𝑋 ≠ ∅))

Theoremfrlmplusgval 20507 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)    &   (𝜑𝑅𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &    + = (+g𝑅)    &    = (+g𝑌)       (𝜑 → (𝐹 𝐺) = (𝐹𝑓 + 𝐺))

Theoremfrlmsubgval 20508 Subtraction in a free module. (Contributed by Thierry Arnoux, 30-Jun-2019.)
𝑌 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝑌)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐼𝑊)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &    = (-g𝑅)    &   𝑀 = (-g𝑌)       (𝜑 → (𝐹𝑀𝐺) = (𝐹𝑓 𝐺))

Theoremfrlmvscafval 20509 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)    &   (𝜑𝐼𝑊)    &   (𝜑𝐴𝐾)    &   (𝜑𝑋𝐵)    &    = ( ·𝑠𝑌)    &    · = (.r𝑅)       (𝜑 → (𝐴 𝑋) = ((𝐼 × {𝐴}) ∘𝑓 · 𝑋))

Theoremfrlmvplusgvalc 20510 Coordinates of a sum with respect to a basis in a free module. (Contributed by AV, 16-Jan-2023.)
𝐹 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝐹)    &   (𝜑𝑅𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝐽𝐼)    &    + = (+g𝑅)    &    = (+g𝐹)       (𝜑 → ((𝑋 𝑌)‘𝐽) = ((𝑋𝐽) + (𝑌𝐽)))

Theoremfrlmvscaval 20511 Coordinates of a scalar multiple with respect to a basis in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.)
𝑌 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝑌)    &   𝐾 = (Base‘𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝐴𝐾)    &   (𝜑𝑋𝐵)    &   (𝜑𝐽𝐼)    &    = ( ·𝑠𝑌)    &    · = (.r𝑅)       (𝜑 → ((𝐴 𝑋)‘𝐽) = (𝐴 · (𝑋𝐽)))

Theoremfrlmplusgvalb 20512* Addition in a free module at the coordinates. (Contributed by AV, 16-Jan-2023.)
𝐹 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝐹)    &   (𝜑𝐼𝑊)    &   (𝜑𝑋𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑌𝐵)    &    + = (+g𝑅)    &    = (+g𝐹)       (𝜑 → (𝑍 = (𝑋 𝑌) ↔ ∀𝑖𝐼 (𝑍𝑖) = ((𝑋𝑖) + (𝑌𝑖))))

Theoremfrlmvscavalb 20513* Scalar multiplication in a free module at the coordinates. (Contributed by AV, 16-Jan-2023.)
𝐹 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝐹)    &   (𝜑𝐼𝑊)    &   (𝜑𝑋𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝑅 ∈ Ring)    &   𝐾 = (Base‘𝑅)    &   (𝜑𝐴𝐾)    &    = ( ·𝑠𝐹)    &    · = (.r𝑅)       (𝜑 → (𝑍 = (𝐴 𝑋) ↔ ∀𝑖𝐼 (𝑍𝑖) = (𝐴 · (𝑋𝑖))))

Theoremfrlmvplusgscavalb 20514* Addition combined with scalar multiplication in a free module at the coordinates. (Contributed by AV, 16-Jan-2023.)
𝐹 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝐹)    &   (𝜑𝐼𝑊)    &   (𝜑𝑋𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝑅 ∈ Ring)    &   𝐾 = (Base‘𝑅)    &   (𝜑𝐴𝐾)    &    = ( ·𝑠𝐹)    &    · = (.r𝑅)    &   (𝜑𝑌𝐵)    &    + = (+g𝑅)    &    = (+g𝐹)    &   (𝜑𝐶𝐾)       (𝜑 → (𝑍 = ((𝐴 𝑋) (𝐶 𝑌)) ↔ ∀𝑖𝐼 (𝑍𝑖) = ((𝐴 · (𝑋𝑖)) + (𝐶 · (𝑌𝑖)))))

Theoremfrlmgsum 20515* 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 = (0g𝑌)    &   (𝜑𝐼𝑉)    &   (𝜑𝐽𝑊)    &   (𝜑𝑅 ∈ Ring)    &   ((𝜑𝑦𝐽) → (𝑥𝐼𝑈) ∈ 𝐵)    &   (𝜑 → (𝑦𝐽 ↦ (𝑥𝐼𝑈)) finSupp 0 )       (𝜑 → (𝑌 Σg (𝑦𝐽 ↦ (𝑥𝐼𝑈))) = (𝑥𝐼 ↦ (𝑅 Σg (𝑦𝐽𝑈))))

Theoremfrlmsplit2 20516* 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 = (𝑥𝐵 ↦ (𝑥𝑉))       ((𝑅 ∈ Ring ∧ 𝑈𝑋𝑉𝑈) → 𝐹 ∈ (𝑌 LMHom 𝑍))

Theoremfrlmsslss 20517* 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 = (0g𝑅)    &   𝐶 = {𝑥𝐵 ∣ (𝑥𝐽) = (𝐽 × { 0 })}       ((𝑅 ∈ Ring ∧ 𝐼𝑉𝐽𝐼) → 𝐶𝑈)

Theoremfrlmsslss2 20518* 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 = (0g𝑅)    &   𝐶 = {𝑥𝐵 ∣ (𝑥 supp 0 ) ⊆ 𝐽}       ((𝑅 ∈ Ring ∧ 𝐼𝑉𝐽𝐼) → 𝐶𝑈)

Theoremfrlmbas3 20519 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)       (((𝑅𝑊𝑋𝑉) ∧ (𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ (𝐼𝑁𝐽𝑀)) → (𝐼𝑋𝐽) ∈ 𝐵)

Theoremmpt2frlmd 20520* Elements of the free module are mappings with two arguments defined by their operation values. (Contributed by AV, 20-Feb-2019.) (Proof shortened by AV, 3-Jul-2022.)
𝐹 = (𝑅 freeLMod (𝑁 × 𝑀))    &   𝑉 = (Base‘𝐹)    &   ((𝑖 = 𝑎𝑗 = 𝑏) → 𝐴 = 𝐵)    &   ((𝜑𝑖𝑁𝑗𝑀) → 𝐴𝑋)    &   ((𝜑𝑎𝑁𝑏𝑀) → 𝐵𝑌)    &   (𝜑 → (𝑁𝑈𝑀𝑊𝑍𝑉))       (𝜑 → (𝑍 = (𝑎𝑁, 𝑏𝑀𝐵) ↔ ∀𝑖𝑁𝑗𝑀 (𝑖𝑍𝑗) = 𝐴))

Theoremfrlmip 20521* The inner product of a free module. (Contributed by Thierry Arnoux, 20-Jun-2019.)
𝑌 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝐼𝑊𝑅𝑉) → (𝑓 ∈ (𝐵𝑚 𝐼), 𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑅 Σg (𝑥𝐼 ↦ ((𝑓𝑥) · (𝑔𝑥))))) = (·𝑖𝑌))

Theoremfrlmipval 20522 The inner product of a free module. (Contributed by Thierry Arnoux, 21-Jun-2019.)
𝑌 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝑉 = (Base‘𝑌)    &    , = (·𝑖𝑌)       (((𝐼𝑊𝑅𝑋) ∧ (𝐹𝑉𝐺𝑉)) → (𝐹 , 𝐺) = (𝑅 Σg (𝐹𝑓 · 𝐺)))

Theoremfrlmphllem 20523* Lemma for frlmphl 20524. (Contributed by AV, 21-Jul-2019.)
𝑌 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝑉 = (Base‘𝑌)    &    , = (·𝑖𝑌)    &   𝑂 = (0g𝑌)    &    0 = (0g𝑅)    &    = (*𝑟𝑅)    &   (𝜑𝑅 ∈ Field)    &   ((𝜑𝑔𝑉 ∧ (𝑔 , 𝑔) = 0 ) → 𝑔 = 𝑂)    &   ((𝜑𝑥𝐵) → ( 𝑥) = 𝑥)    &   (𝜑𝐼𝑊)       ((𝜑𝑔𝑉𝑉) → (𝑥𝐼 ↦ ((𝑔𝑥) · (𝑥))) finSupp 0 )

Theoremfrlmphl 20524* 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)    &    · = (.r𝑅)    &   𝑉 = (Base‘𝑌)    &    , = (·𝑖𝑌)    &   𝑂 = (0g𝑌)    &    0 = (0g𝑅)    &    = (*𝑟𝑅)    &   (𝜑𝑅 ∈ Field)    &   ((𝜑𝑔𝑉 ∧ (𝑔 , 𝑔) = 0 ) → 𝑔 = 𝑂)    &   ((𝜑𝑥𝐵) → ( 𝑥) = 𝑥)    &   (𝜑𝐼𝑊)       (𝜑𝑌 ∈ PreHil)

11.1.3  Standard basis (unit vectors)

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.

Syntaxcuvc 20525 Class of basic unit vectors for an explicit free module.
class unitVec

Definitiondf-uvc 20526* ((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 ↦ (𝑗𝑖 ↦ (𝑘𝑖 ↦ if(𝑘 = 𝑗, (1r𝑟), (0g𝑟)))))

Theoremuvcfval 20527* Value of the unit-vector generator for a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
𝑈 = (𝑅 unitVec 𝐼)    &    1 = (1r𝑅)    &    0 = (0g𝑅)       ((𝑅𝑉𝐼𝑊) → 𝑈 = (𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))))

Theoremuvcval 20528* Value of a single unit vector in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.)
𝑈 = (𝑅 unitVec 𝐼)    &    1 = (1r𝑅)    &    0 = (0g𝑅)       ((𝑅𝑉𝐼𝑊𝐽𝐼) → (𝑈𝐽) = (𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )))

Theoremuvcvval 20529 Value of a unit vector coordinate in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.)
𝑈 = (𝑅 unitVec 𝐼)    &    1 = (1r𝑅)    &    0 = (0g𝑅)       (((𝑅𝑉𝐼𝑊𝐽𝐼) ∧ 𝐾𝐼) → ((𝑈𝐽)‘𝐾) = if(𝐾 = 𝐽, 1 , 0 ))

Theoremuvcvvcl 20530 A coodinate of a unit vector is either 0 or 1. (Contributed by Stefan O'Rear, 3-Feb-2015.)
𝑈 = (𝑅 unitVec 𝐼)    &    1 = (1r𝑅)    &    0 = (0g𝑅)       (((𝑅𝑉𝐼𝑊𝐽𝐼) ∧ 𝐾𝐼) → ((𝑈𝐽)‘𝐾) ∈ { 0 , 1 })

Theoremuvcvvcl2 20531 A unit vector coordinate is a ring element. (Contributed by Stefan O'Rear, 3-Feb-2015.)
𝑈 = (𝑅 unitVec 𝐼)    &   𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐼𝑊)    &   (𝜑𝐽𝐼)    &   (𝜑𝐾𝐼)       (𝜑 → ((𝑈𝐽)‘𝐾) ∈ 𝐵)

Theoremuvcvv1 20532 The unit vector is one at its designated coordinate. (Contributed by Stefan O'Rear, 3-Feb-2015.)
𝑈 = (𝑅 unitVec 𝐼)    &   (𝜑𝑅𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝐽𝐼)    &    1 = (1r𝑅)       (𝜑 → ((𝑈𝐽)‘𝐽) = 1 )

Theoremuvcvv0 20533 The unit vector is zero at its designated coordinate. (Contributed by Stefan O'Rear, 3-Feb-2015.)
𝑈 = (𝑅 unitVec 𝐼)    &   (𝜑𝑅𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝐽𝐼)    &   (𝜑𝐾𝐼)    &   (𝜑𝐽𝐾)    &    0 = (0g𝑅)       (𝜑 → ((𝑈𝐽)‘𝐾) = 0 )

Theoremuvcff 20534 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 ∧ 𝐼𝑊) → 𝑈:𝐼𝐵)

Theoremuvcf1 20535 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 ∧ 𝐼𝑊) → 𝑈:𝐼1-1𝐵)

Theoremuvcresum 20536 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)    &    · = ( ·𝑠𝑌)       ((𝑅 ∈ Ring ∧ 𝐼𝑊𝑋𝐵) → 𝑋 = (𝑌 Σg (𝑋𝑓 · 𝑈)))

Theoremfrlmssuvc1 20537* 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)    &    · = ( ·𝑠𝐹)    &    0 = (0g𝑅)    &   𝐶 = {𝑥𝐵 ∣ (𝑥 supp 0 ) ⊆ 𝐽}    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐼𝑉)    &   (𝜑𝐽𝐼)    &   (𝜑𝐿𝐽)    &   (𝜑𝑋𝐾)       (𝜑 → (𝑋 · (𝑈𝐿)) ∈ 𝐶)

Theoremfrlmssuvc2 20538* 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)    &    · = ( ·𝑠𝐹)    &    0 = (0g𝑅)    &   𝐶 = {𝑥𝐵 ∣ (𝑥 supp 0 ) ⊆ 𝐽}    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐼𝑉)    &   (𝜑𝐽𝐼)    &   (𝜑𝐿 ∈ (𝐼𝐽))    &   (𝜑𝑋 ∈ (𝐾 ∖ { 0 }))       (𝜑 → ¬ (𝑋 · (𝑈𝐿)) ∈ 𝐶)

Theoremfrlmsslsp 20539* 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 = (0g𝑅)    &   𝐶 = {𝑥𝐵 ∣ (𝑥 supp 0 ) ⊆ 𝐽}       ((𝑅 ∈ Ring ∧ 𝐼𝑉𝐽𝐼) → (𝐾‘(𝑈𝐽)) = 𝐶)

Theoremfrlmlbs 20540 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 ∧ 𝐼𝑉) → ran 𝑈𝐽)

Theoremfrlmup1 20541* 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)    &    · = ( ·𝑠𝑇)    &   𝐸 = (𝑥𝐵 ↦ (𝑇 Σg (𝑥𝑓 · 𝐴)))    &   (𝜑𝑇 ∈ LMod)    &   (𝜑𝐼𝑋)    &   (𝜑𝑅 = (Scalar‘𝑇))    &   (𝜑𝐴:𝐼𝐶)       (𝜑𝐸 ∈ (𝐹 LMHom 𝑇))

Theoremfrlmup2 20542* 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)    &    · = ( ·𝑠𝑇)    &   𝐸 = (𝑥𝐵 ↦ (𝑇 Σg (𝑥𝑓 · 𝐴)))    &   (𝜑𝑇 ∈ LMod)    &   (𝜑𝐼𝑋)    &   (𝜑𝑅 = (Scalar‘𝑇))    &   (𝜑𝐴:𝐼𝐶)    &   (𝜑𝑌𝐼)    &   𝑈 = (𝑅 unitVec 𝐼)       (𝜑 → (𝐸‘(𝑈𝑌)) = (𝐴𝑌))

Theoremfrlmup3 20543* 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)    &    · = ( ·𝑠𝑇)    &   𝐸 = (𝑥𝐵 ↦ (𝑇 Σg (𝑥𝑓 · 𝐴)))    &   (𝜑𝑇 ∈ LMod)    &   (𝜑𝐼𝑋)    &   (𝜑𝑅 = (Scalar‘𝑇))    &   (𝜑𝐴:𝐼𝐶)    &   𝐾 = (LSpan‘𝑇)       (𝜑 → ran 𝐸 = (𝐾‘ran 𝐴))

Theoremfrlmup4 20544* Universal property of the free module by existential uniqueness. (Contributed by Stefan O'Rear, 7-Mar-2015.)
𝑅 = (Scalar‘𝑇)    &   𝐹 = (𝑅 freeLMod 𝐼)    &   𝑈 = (𝑅 unitVec 𝐼)    &   𝐶 = (Base‘𝑇)       ((𝑇 ∈ LMod ∧ 𝐼𝑋𝐴:𝐼𝐶) → ∃!𝑚 ∈ (𝐹 LMHom 𝑇)(𝑚𝑈) = 𝐴)

Theoremellspd 20545* 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 = (0g𝑆)    &    · = ( ·𝑠𝑀)    &   (𝜑𝐹:𝐼𝐵)    &   (𝜑𝑀 ∈ LMod)    &   (𝜑𝐼 ∈ V)       (𝜑 → (𝑋 ∈ (𝑁‘(𝐹𝐼)) ↔ ∃𝑓 ∈ (𝐾𝑚 𝐼)(𝑓 finSupp 0𝑋 = (𝑀 Σg (𝑓𝑓 · 𝐹)))))

Theoremelfilspd 20546* Simplified version of ellspd 20545 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 = (0g𝑆)    &    · = ( ·𝑠𝑀)    &   (𝜑𝐹:𝐼𝐵)    &   (𝜑𝑀 ∈ LMod)    &   (𝜑𝐼 ∈ Fin)       (𝜑 → (𝑋 ∈ (𝑁‘(𝐹𝐼)) ↔ ∃𝑓 ∈ (𝐾𝑚 𝐼)𝑋 = (𝑀 Σg (𝑓𝑓 · 𝐹))))

11.1.4  Independent sets and families

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∈Saxx 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∈Iaixi = 0, then ai = 0 for all i ∈ I." These definitions correspond to the definitions df-linds 20550 and df-lindf 20549 respectively, 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.

Syntaxclindf 20547 The class relationship of independent families in a module.
class LIndF

Syntaxclinds 20548 The class generator of independent sets in a module.
class LIndS

Definitiondf-lindf 20549* 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 20569, 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 20581) and only one representation for each element of the range (islindf5 20582). (Contributed by Stefan O'Rear, 24-Feb-2015.)

LIndF = {⟨𝑓, 𝑤⟩ ∣ (𝑓:dom 𝑓⟶(Base‘𝑤) ∧ [(Scalar‘𝑤) / 𝑠]𝑥 ∈ dom 𝑓𝑘 ∈ ((Base‘𝑠) ∖ {(0g𝑠)}) ¬ (𝑘( ·𝑠𝑤)(𝑓𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))))}

Definitiondf-linds 20550* An independent set is a set which is independent as a family. See also islinds3 20577 and islinds4 20578. (Contributed by Stefan O'Rear, 24-Feb-2015.)
LIndS = (𝑤 ∈ V ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ( I ↾ 𝑠) LIndF 𝑤})

Theoremrellindf 20551 The independent-family predicate is a proper relation and can be used with brrelex1i 5406. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Rel LIndF

Theoremislinds 20552 Property of an independent set of vectors in terms of an independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
𝐵 = (Base‘𝑊)       (𝑊𝑉 → (𝑋 ∈ (LIndS‘𝑊) ↔ (𝑋𝐵 ∧ ( I ↾ 𝑋) LIndF 𝑊)))

Theoremlinds1 20553 An independent set of vectors is a set of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.)
𝐵 = (Base‘𝑊)       (𝑋 ∈ (LIndS‘𝑊) → 𝑋𝐵)

Theoremlinds2 20554 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)

Theoremislindf 20555* Property of an independent family of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.)
𝐵 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (LSpan‘𝑊)    &   𝑆 = (Scalar‘𝑊)    &   𝑁 = (Base‘𝑆)    &    0 = (0g𝑆)       ((𝑊𝑌𝐹𝑋) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹𝐵 ∧ ∀𝑥 ∈ dom 𝐹𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))))

Theoremislinds2 20556* Expanded property of an independent set of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.)
𝐵 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (LSpan‘𝑊)    &   𝑆 = (Scalar‘𝑊)    &   𝑁 = (Base‘𝑆)    &    0 = (0g𝑆)       (𝑊𝑌 → (𝐹 ∈ (LIndS‘𝑊) ↔ (𝐹𝐵 ∧ ∀𝑥𝐹𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · 𝑥) ∈ (𝐾‘(𝐹 ∖ {𝑥})))))

Theoremislindf2 20557* Property of an independent family of vectors with prior constrained domain and codomain. (Contributed by Stefan O'Rear, 26-Feb-2015.)
𝐵 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (LSpan‘𝑊)    &   𝑆 = (Scalar‘𝑊)    &   𝑁 = (Base‘𝑆)    &    0 = (0g𝑆)       ((𝑊𝑌𝐼𝑋𝐹:𝐼𝐵) → (𝐹 LIndF 𝑊 ↔ ∀𝑥𝐼𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹𝑥)) ∈ (𝐾‘(𝐹 “ (𝐼 ∖ {𝑥})))))

Theoremlindff 20558 Functional property of a linearly independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
𝐵 = (Base‘𝑊)       ((𝐹 LIndF 𝑊𝑊𝑌) → 𝐹:dom 𝐹𝐵)

Theoremlindfind 20559 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.)
· = ( ·𝑠𝑊)    &   𝑁 = (LSpan‘𝑊)    &   𝐿 = (Scalar‘𝑊)    &    0 = (0g𝐿)    &   𝐾 = (Base‘𝐿)       (((𝐹 LIndF 𝑊𝐸 ∈ dom 𝐹) ∧ (𝐴𝐾𝐴0 )) → ¬ (𝐴 · (𝐹𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸}))))

Theoremlindsind 20560 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.)
· = ( ·𝑠𝑊)    &   𝑁 = (LSpan‘𝑊)    &   𝐿 = (Scalar‘𝑊)    &    0 = (0g𝐿)    &   𝐾 = (Base‘𝐿)       (((𝐹 ∈ (LIndS‘𝑊) ∧ 𝐸𝐹) ∧ (𝐴𝐾𝐴0 )) → ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐹 ∖ {𝐸})))

Theoremlindfind2 20561 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 𝑊𝐸 ∈ dom 𝐹) → ¬ (𝐹𝐸) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝐸}))))

Theoremlindsind2 20562 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) ∧ 𝐸𝐹) → ¬ 𝐸 ∈ (𝐾‘(𝐹 ∖ {𝐸})))

Theoremlindff1 20563 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 𝐹1-1𝐵)

Theoremlindfrn 20564 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))

Theoremf1lindf 20565 Rearranging and deleting elements from an independent family gives an independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → (𝐹𝐺) LIndF 𝑊)

Theoremlindfres 20566 Any restriction of an independent family is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.)
((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → (𝐹𝑋) LIndF 𝑊)

Theoremlindsss 20567 Any subset of an independent set is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.)
((𝑊 ∈ LMod ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝐺𝐹) → 𝐺 ∈ (LIndS‘𝑊))

Theoremf1linds 20568 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:𝐷1-1𝑆) → 𝐹 LIndF 𝑊)

Theoremislindf3 20569 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 𝐹1-1→V ∧ ran 𝐹 ∈ (LIndS‘𝑊))))

Theoremlindfmm 20570 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:𝐵1-1𝐶𝐹:𝐼𝐵) → (𝐹 LIndF 𝑆 ↔ (𝐺𝐹) LIndF 𝑇))

Theoremlindsmm 20571 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:𝐵1-1𝐶𝐹𝐵) → (𝐹 ∈ (LIndS‘𝑆) ↔ (𝐺𝐹) ∈ (LIndS‘𝑇)))

Theoremlindsmm2 20572 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:𝐵1-1𝐶𝐹 ∈ (LIndS‘𝑆)) → (𝐺𝐹) ∈ (LIndS‘𝑇))

Theoremlsslindf 20573 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 = (𝑊s 𝑆)       ((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) → (𝐹 LIndF 𝑋𝐹 LIndF 𝑊))

Theoremlsslinds 20574 Linear independence is unchanged by working in a subspace. (Contributed by Stefan O'Rear, 24-Feb-2015.)
𝑈 = (LSubSp‘𝑊)    &   𝑋 = (𝑊s 𝑆)       ((𝑊 ∈ LMod ∧ 𝑆𝑈𝐹𝑆) → (𝐹 ∈ (LIndS‘𝑋) ↔ 𝐹 ∈ (LIndS‘𝑊)))

Theoremislbs4 20575 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)       (𝑋𝐽 ↔ (𝑋 ∈ (LIndS‘𝑊) ∧ (𝐾𝑋) = 𝐵))

Theoremlbslinds 20576 A basis is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.)
𝐽 = (LBasis‘𝑊)       𝐽 ⊆ (LIndS‘𝑊)

Theoremislinds3 20577 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 = (𝑊s (𝐾𝑌))    &   𝐽 = (LBasis‘𝑋)       (𝑊 ∈ LMod → (𝑌 ∈ (LIndS‘𝑊) ↔ 𝑌𝐽))

Theoremislinds4 20578* 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) ↔ ∃𝑏𝐽 𝑌𝑏))

11.1.5  Characterization of free modules

Theoremlmimlbs 20579 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) ∧ 𝐵𝐽) → (𝐹𝐵) ∈ 𝐾)

Theoremlmiclbs 20580 Having a basis is an isomorphism invariant. (Contributed by Stefan O'Rear, 26-Feb-2015.)
𝐽 = (LBasis‘𝑆)    &   𝐾 = (LBasis‘𝑇)       (𝑆𝑚 𝑇 → (𝐽 ≠ ∅ → 𝐾 ≠ ∅))

Theoremislindf4 20581* 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)    &    · = ( ·𝑠𝑊)    &    0 = (0g𝑊)    &   𝑌 = (0g𝑅)    &   𝐿 = (Base‘(𝑅 freeLMod 𝐼))       ((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) → (𝐹 LIndF 𝑊 ↔ ∀𝑥𝐿 ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0𝑥 = (𝐼 × {𝑌}))))

Theoremislindf5 20582* 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)    &    · = ( ·𝑠𝑇)    &   𝐸 = (𝑥𝐵 ↦ (𝑇 Σg (𝑥𝑓 · 𝐴)))    &   (𝜑𝑇 ∈ LMod)    &   (𝜑𝐼𝑋)    &   (𝜑𝑅 = (Scalar‘𝑇))    &   (𝜑𝐴:𝐼𝐶)       (𝜑 → (𝐴 LIndF 𝑇𝐸:𝐵1-1𝐶))

Theoremindlcim 20583* 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)    &    · = ( ·𝑠𝑇)    &   𝑁 = (LSpan‘𝑇)    &   𝐸 = (𝑥𝐵 ↦ (𝑇 Σg (𝑥𝑓 · 𝐴)))    &   (𝜑𝑇 ∈ LMod)    &   (𝜑𝐼𝑋)    &   (𝜑𝑅 = (Scalar‘𝑇))    &   (𝜑𝐴:𝐼onto𝐽)    &   (𝜑𝐴 LIndF 𝑇)    &   (𝜑 → (𝑁𝐽) = 𝐶)       (𝜑𝐸 ∈ (𝐹 LMIso 𝑇))

Theoremlbslcic 20584 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 ∧ 𝐵𝐽𝐼𝐵) → 𝑊𝑚 (𝐹 freeLMod 𝐼))

Theoremlmisfree 20585* 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 19562 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 𝑊𝑚 (𝐹 freeLMod 𝑘)))

Theoremlvecisfrlm 20586* Every vector space is isomorphic to a free module. (Contributed by AV, 7-Mar-2019.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ LVec → ∃𝑘 𝑊𝑚 (𝐹 freeLMod 𝑘))

Theoremlmimco 20587 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)) → (𝐹𝐺) ∈ (𝑅 LMIso 𝑇))

Theoremlmictra 20588 Module isomorphism is transitive. (Contributed by AV, 10-Mar-2019.)
((𝑅𝑚 𝑆𝑆𝑚 𝑇) → 𝑅𝑚 𝑇)

Theoremuvcf1o 20589 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 ∧ 𝐼𝑊) → 𝑈:𝐼1-1-onto→ran 𝑈)

Theoremuvcendim 20590 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 ∧ 𝐼𝑊) → 𝐼 ≈ ran 𝑈)

Theoremfrlmisfrlm 20591 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 ∧ 𝐼𝑌𝐼𝐽) → (𝑅 freeLMod 𝐼) ≃𝑚 (𝑅 freeLMod 𝐽))

Theoremfrlmiscvec 20592 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 ∧ 𝐼𝑌) → (𝑅 freeLMod 𝐼) ≃𝑚 (𝑅 freeLMod (𝐼 × {∅})))

11.2  Matrices

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 20490) 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 20507) and scalar multiplication (see frlmvscafval 20509) 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 20594. 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 20492.

However, for square matrices there is the definition df-mat 20618, 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 20677 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 20803.

11.2.1  The matrix multiplication

This section is about the multiplication of m x n matrices.

Syntaxcmmul 20593 Syntax for the matrix multiplication operator.
class maMul

Definitiondf-mamu 20594* 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 ‘(1st𝑜)) / 𝑚(2nd ‘(1st𝑜)) / 𝑛(2nd𝑜) / 𝑝(𝑥 ∈ ((Base‘𝑟) ↑𝑚 (𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑𝑚 (𝑛 × 𝑝)) ↦ (𝑖𝑚, 𝑘𝑝 ↦ (𝑟 Σg (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘)))))))

Theoremmamufval 20595* Functional value of the matrix multiplication operator. (Contributed by Stefan O'Rear, 2-Sep-2015.)
𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅𝑉)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑃 ∈ Fin)       (𝜑𝐹 = (𝑥 ∈ (𝐵𝑚 (𝑀 × 𝑁)), 𝑦 ∈ (𝐵𝑚 (𝑁 × 𝑃)) ↦ (𝑖𝑀, 𝑘𝑃 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑗𝑦𝑘)))))))

Theoremmamuval 20596* Multiplication of two matrices. (Contributed by Stefan O'Rear, 2-Sep-2015.)
𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅𝑉)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑃 ∈ Fin)    &   (𝜑𝑋 ∈ (𝐵𝑚 (𝑀 × 𝑁)))    &   (𝜑𝑌 ∈ (𝐵𝑚 (𝑁 × 𝑃)))       (𝜑 → (𝑋𝐹𝑌) = (𝑖𝑀, 𝑘𝑃 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘))))))

Theoremmamufv 20597* 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)    &    · = (.r𝑅)    &   (𝜑𝑅𝑉)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑃 ∈ Fin)    &   (𝜑𝑋 ∈ (𝐵𝑚 (𝑀 × 𝑁)))    &   (𝜑𝑌 ∈ (𝐵𝑚 (𝑁 × 𝑃)))    &   (𝜑𝐼𝑀)    &   (𝜑𝐾𝑃)       (𝜑 → (𝐼(𝑋𝐹𝑌)𝐾) = (𝑅 Σg (𝑗𝑁 ↦ ((𝐼𝑋𝑗) · (𝑗𝑌𝐾)))))

Theoremmamudm 20598 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⟩)       ((𝑅𝑉 ∧ (𝑀 ∈ Fin ∧ 𝑁 ∈ Fin ∧ 𝑃 ∈ Fin)) → dom × = (𝐵 × 𝐶))

Theoremmamufacex 20599 Every solution of the equation 𝐴𝑋 = 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 ≠ ∅) ∧ (𝑅𝑉𝑌𝐷) ∧ (𝑀 ∈ Fin ∧ 𝑁 ∈ Fin ∧ 𝑃 ∈ Fin)) → ((𝑋 × 𝑍) = 𝑌𝑍𝐶))

Theoremmamures 20600 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)    &   (𝜑𝑅𝑉)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑃 ∈ Fin)    &   (𝜑𝐼𝑀)    &   (𝜑𝑋 ∈ (𝐵𝑚 (𝑀 × 𝑁)))    &   (𝜑𝑌 ∈ (𝐵𝑚 (𝑁 × 𝑃)))       (𝜑 → ((𝑋𝐹𝑌) ↾ (𝐼 × 𝑃)) = ((𝑋 ↾ (𝐼 × 𝑁))𝐺𝑌))

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