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Theorem List for Metamath Proof Explorer - 20201-20300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremislindf2 20201* 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 20202 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 20203 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 20204 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 20205 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 20206 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 20207 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 20208 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 20209 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 20210 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 20211 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 20212 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 20213 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 20214 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 20215 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 20216 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 20217 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 20218 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 20219 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 20220 A basis is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.)
𝐽 = (LBasis‘𝑊)       𝐽 ⊆ (LIndS‘𝑊)

Theoremislinds3 20221 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 20222* 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 20223 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 20224 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 20225* 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 20226* 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 20227* 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 20228 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 20229* 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 19213 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 20230* 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 20231 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 20232 Module isomorphism is transitive. (Contributed by AV, 10-Mar-2019.)
((𝑅𝑚 𝑆𝑆𝑚 𝑇) → 𝑅𝑚 𝑇)

Theoremuvcf1o 20233 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 20234 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 20235 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 20236 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 20139) 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 20155) and scalar multiplication (see frlmvscafval 20157) 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 20238. 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 20141.

However, for square matrices there is the definition df-mat 20262, 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 20320 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 20446.

11.2.1  The matrix multiplication

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

Syntaxcmmul 20237 Syntax for the matrix multiplication operator.
class maMul

Definitiondf-mamu 20238* 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 20239* 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 20240* 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 20241* 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 20242 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 20243 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 20244 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))

Theoremmndvcl 20245 Tuple-wise additive closure in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑀)    &    + = (+g𝑀)       ((𝑀 ∈ Mnd ∧ 𝑋 ∈ (𝐵𝑚 𝐼) ∧ 𝑌 ∈ (𝐵𝑚 𝐼)) → (𝑋𝑓 + 𝑌) ∈ (𝐵𝑚 𝐼))

Theoremmndvass 20246 Tuple-wise associativity in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑀)    &    + = (+g𝑀)       ((𝑀 ∈ Mnd ∧ (𝑋 ∈ (𝐵𝑚 𝐼) ∧ 𝑌 ∈ (𝐵𝑚 𝐼) ∧ 𝑍 ∈ (𝐵𝑚 𝐼))) → ((𝑋𝑓 + 𝑌) ∘𝑓 + 𝑍) = (𝑋𝑓 + (𝑌𝑓 + 𝑍)))

Theoremmndvlid 20247 Tuple-wise left identity in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑀)    &    + = (+g𝑀)    &    0 = (0g𝑀)       ((𝑀 ∈ Mnd ∧ 𝑋 ∈ (𝐵𝑚 𝐼)) → ((𝐼 × { 0 }) ∘𝑓 + 𝑋) = 𝑋)

Theoremmndvrid 20248 Tuple-wise right identity in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑀)    &    + = (+g𝑀)    &    0 = (0g𝑀)       ((𝑀 ∈ Mnd ∧ 𝑋 ∈ (𝐵𝑚 𝐼)) → (𝑋𝑓 + (𝐼 × { 0 })) = 𝑋)

Theoremgrpvlinv 20249 Tuple-wise left inverse in groups. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝑁 = (invg𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵𝑚 𝐼)) → ((𝑁𝑋) ∘𝑓 + 𝑋) = (𝐼 × { 0 }))

Theoremgrpvrinv 20250 Tuple-wise right inverse in groups. (Contributed by Mario Carneiro, 22-Sep-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝑁 = (invg𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵𝑚 𝐼)) → (𝑋𝑓 + (𝑁𝑋)) = (𝐼 × { 0 }))

Theoremmhmvlin 20251 Tuple extension of monoid homomorphisms. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑀)    &    + = (+g𝑀)    &    = (+g𝑁)       ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (𝐵𝑚 𝐼) ∧ 𝑌 ∈ (𝐵𝑚 𝐼)) → (𝐹 ∘ (𝑋𝑓 + 𝑌)) = ((𝐹𝑋) ∘𝑓 (𝐹𝑌)))

Theoremringvcl 20252 Tuple-wise multiplication closure in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋 ∈ (𝐵𝑚 𝐼) ∧ 𝑌 ∈ (𝐵𝑚 𝐼)) → (𝑋𝑓 · 𝑌) ∈ (𝐵𝑚 𝐼))

Theoremgsumcom3 20253* A commutative law for finitely supported iterated sums. (Contributed by Stefan O'Rear, 2-Nov-2015.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝐶𝑊)    &   ((𝜑 ∧ (𝑗𝐴𝑘𝐶)) → 𝑋𝐵)    &   (𝜑𝑈 ∈ Fin)    &   ((𝜑 ∧ ((𝑗𝐴𝑘𝐶) ∧ ¬ 𝑗𝑈𝑘)) → 𝑋 = 0 )       (𝜑 → (𝐺 Σg (𝑗𝐴 ↦ (𝐺 Σg (𝑘𝐶𝑋)))) = (𝐺 Σg (𝑘𝐶 ↦ (𝐺 Σg (𝑗𝐴𝑋)))))

Theoremgsumcom3fi 20254* A commutative law for finite iterated sums. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐶 ∈ Fin)    &   ((𝜑 ∧ (𝑗𝐴𝑘𝐶)) → 𝑋𝐵)       (𝜑 → (𝐺 Σg (𝑗𝐴 ↦ (𝐺 Σg (𝑘𝐶𝑋)))) = (𝐺 Σg (𝑘𝐶 ↦ (𝐺 Σg (𝑗𝐴𝑋)))))

Theoremmamucl 20255 Operation closure of matrix multiplication. (Contributed by Stefan O'Rear, 2-Sep-2015.) (Proof shortened by AV, 23-Jul-2019.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑃 ∈ Fin)    &   (𝜑𝑋 ∈ (𝐵𝑚 (𝑀 × 𝑁)))    &   (𝜑𝑌 ∈ (𝐵𝑚 (𝑁 × 𝑃)))       (𝜑 → (𝑋𝐹𝑌) ∈ (𝐵𝑚 (𝑀 × 𝑃)))

Theoremmamuass 20256 Matrix multiplication is associative, see also statement in [Lang] p. 505. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑂 ∈ Fin)    &   (𝜑𝑃 ∈ Fin)    &   (𝜑𝑋 ∈ (𝐵𝑚 (𝑀 × 𝑁)))    &   (𝜑𝑌 ∈ (𝐵𝑚 (𝑁 × 𝑂)))    &   (𝜑𝑍 ∈ (𝐵𝑚 (𝑂 × 𝑃)))    &   𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑂⟩)    &   𝐺 = (𝑅 maMul ⟨𝑀, 𝑂, 𝑃⟩)    &   𝐻 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)    &   𝐼 = (𝑅 maMul ⟨𝑁, 𝑂, 𝑃⟩)       (𝜑 → ((𝑋𝐹𝑌)𝐺𝑍) = (𝑋𝐻(𝑌𝐼𝑍)))

Theoremmamudi 20257 Matrix multiplication distributes over addition on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 23-Jul-2019.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑂⟩)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑂 ∈ Fin)    &    + = (+g𝑅)    &   (𝜑𝑋 ∈ (𝐵𝑚 (𝑀 × 𝑁)))    &   (𝜑𝑌 ∈ (𝐵𝑚 (𝑀 × 𝑁)))    &   (𝜑𝑍 ∈ (𝐵𝑚 (𝑁 × 𝑂)))       (𝜑 → ((𝑋𝑓 + 𝑌)𝐹𝑍) = ((𝑋𝐹𝑍) ∘𝑓 + (𝑌𝐹𝑍)))

Theoremmamudir 20258 Matrix multiplication distributes over addition on the right. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 23-Jul-2019.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑂⟩)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑂 ∈ Fin)    &    + = (+g𝑅)    &   (𝜑𝑋 ∈ (𝐵𝑚 (𝑀 × 𝑁)))    &   (𝜑𝑌 ∈ (𝐵𝑚 (𝑁 × 𝑂)))    &   (𝜑𝑍 ∈ (𝐵𝑚 (𝑁 × 𝑂)))       (𝜑 → (𝑋𝐹(𝑌𝑓 + 𝑍)) = ((𝑋𝐹𝑌) ∘𝑓 + (𝑋𝐹𝑍)))

Theoremmamuvs1 20259 Matrix multiplication distributes over scalar multiplication on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑂⟩)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑂 ∈ Fin)    &    · = (.r𝑅)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌 ∈ (𝐵𝑚 (𝑀 × 𝑁)))    &   (𝜑𝑍 ∈ (𝐵𝑚 (𝑁 × 𝑂)))       (𝜑 → ((((𝑀 × 𝑁) × {𝑋}) ∘𝑓 · 𝑌)𝐹𝑍) = (((𝑀 × 𝑂) × {𝑋}) ∘𝑓 · (𝑌𝐹𝑍)))

Theoremmamuvs2 20260 Matrix multiplication distributes over scalar multiplication on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 22-Jul-2019.)
(𝜑𝑅 ∈ CRing)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑂⟩)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑂 ∈ Fin)    &   (𝜑𝑋 ∈ (𝐵𝑚 (𝑀 × 𝑁)))    &   (𝜑𝑌𝐵)    &   (𝜑𝑍 ∈ (𝐵𝑚 (𝑁 × 𝑂)))       (𝜑 → (𝑋𝐹(((𝑁 × 𝑂) × {𝑌}) ∘𝑓 · 𝑍)) = (((𝑀 × 𝑂) × {𝑌}) ∘𝑓 · (𝑋𝐹𝑍)))

11.2.2  Square matrices

In the following, the square matrix algebra is defined as extensible structure Mat. In this subsection, however, only square matrices and their basic properties are regarded. This includes showing that (𝑁 Mat 𝑅) is a left module, see matlmod 20283. That (𝑁 Mat 𝑅) is a ring and an associative algebra is shown in the next subsection, after theorems about the identity matrix are available. Nevertheless, (𝑁 Mat 𝑅) is called "matrix ring" or "matrix algebra" already in this subsection.

Syntaxcmat 20261 Syntax for the square matrix algebra.
class Mat

Definitiondf-mat 20262* Define the algebra of n x n matrices over a ring r. (Contributed by Stefan O'Rear, 31-Aug-2015.)
Mat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ((𝑟 freeLMod (𝑛 × 𝑛)) sSet ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩))

Theoremmatbas0pc 20263 There is no matrix with a proper class either as dimension or as underlying ring. (Contributed by AV, 28-Dec-2018.)
(¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (Base‘(𝑁 Mat 𝑅)) = ∅)

Theoremmatbas0 20264 There is no matrix for a not finite dimension or a proper class as the underlying ring. (Contributed by AV, 28-Dec-2018.)
(¬ (𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (Base‘(𝑁 Mat 𝑅)) = ∅)

Theoremmatval 20265 Value of the matrix algebra. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))    &    · = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝐴 = (𝐺 sSet ⟨(.r‘ndx), · ⟩))

Theoremmatrcl 20266 Reverse closure for the matrix algebra. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)       (𝑋𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))

Theoremmatbas 20267 The matrix ring has the same base set as its underlying group. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (Base‘𝐺) = (Base‘𝐴))

Theoremmatplusg 20268 The matrix ring has the same addition as its underlying group. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (+g𝐺) = (+g𝐴))

Theoremmatsca 20269 The matrix ring has the same scalars as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (Scalar‘𝐺) = (Scalar‘𝐴))

Theoremmatvsca 20270 The matrix ring has the same scalar multiplication as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → ( ·𝑠𝐺) = ( ·𝑠𝐴))

Theoremmat0 20271 The matrix ring has the same zero as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (0g𝐺) = (0g𝐴))

Theoremmatinvg 20272 The matrix ring has the same additive inverse as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (invg𝐺) = (invg𝐴))

Theoremmat0op 20273* Value of a zero matrix as operation. (Contributed by AV, 2-Dec-2018.)
𝐴 = (𝑁 Mat 𝑅)    &    0 = (0g𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g𝐴) = (𝑖𝑁, 𝑗𝑁0 ))

Theoremmatsca2 20274 The scalars of the matrix ring are the underlying ring. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑅 = (Scalar‘𝐴))

Theoremmatbas2 20275 The base set of the matrix ring as a set exponential. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 16-Dec-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐾 = (Base‘𝑅)       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝐾𝑚 (𝑁 × 𝑁)) = (Base‘𝐴))

Theoremmatbas2i 20276 A matrix is a function. (Contributed by Stefan O'Rear, 11-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐾 = (Base‘𝑅)    &   𝐵 = (Base‘𝐴)       (𝑀𝐵𝑀 ∈ (𝐾𝑚 (𝑁 × 𝑁)))

Theoremmatbas2d 20277* The base set of the matrix ring as a mapping operation. (Contributed by Stefan O'Rear, 11-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐾 = (Base‘𝑅)    &   𝐵 = (Base‘𝐴)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑅𝑉)    &   ((𝜑𝑥𝑁𝑦𝑁) → 𝐶𝐾)       (𝜑 → (𝑥𝑁, 𝑦𝑁𝐶) ∈ 𝐵)

Theoremeqmat 20278* Two square matrices of the same dimension are equal if they have the same entries. (Contributed by AV, 25-Sep-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)       ((𝑋𝐵𝑌𝐵) → (𝑋 = 𝑌 ↔ ∀𝑖𝑁𝑗𝑁 (𝑖𝑋𝑗) = (𝑖𝑌𝑗)))

Theoremmatecl 20279 Each entry (according to Wikipedia "Matrix (mathematics)", 30-Dec-2018, https://en.wikipedia.org/wiki/Matrix_(mathematics)#Definition (or element or component or coefficient or cell) of a matrix is an element of the underlying ring. (Contributed by AV, 16-Dec-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐾 = (Base‘𝑅)       ((𝐼𝑁𝐽𝑁𝑀 ∈ (Base‘𝐴)) → (𝐼𝑀𝐽) ∈ 𝐾)

Theoremmatecld 20280 Each entry (according to Wikipedia "Matrix (mathematics)", 30-Dec-2018, https://en.wikipedia.org/wiki/Matrix_(mathematics)#Definition (or element or component or coefficient or cell) of a matrix is an element of the underlying ring, deduction form. (Contributed by AV, 27-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐾 = (Base‘𝑅)    &   𝐵 = (Base‘𝐴)    &   (𝜑𝐼𝑁)    &   (𝜑𝐽𝑁)    &   (𝜑𝑀𝐵)       (𝜑 → (𝐼𝑀𝐽) ∈ 𝐾)

Theoremmatplusg2 20281 Addition in the matrix ring is cell-wise. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    = (+g𝐴)    &    + = (+g𝑅)       ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑋𝑓 + 𝑌))

Theoremmatvsca2 20282 Scalar multiplication in the matrix ring is cell-wise. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐾 = (Base‘𝑅)    &    · = ( ·𝑠𝐴)    &    × = (.r𝑅)    &   𝐶 = (𝑁 × 𝑁)       ((𝑋𝐾𝑌𝐵) → (𝑋 · 𝑌) = ((𝐶 × {𝑋}) ∘𝑓 × 𝑌))

Theoremmatlmod 20283 The matrix ring is a linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ LMod)

Theoremmatgrp 20284 The matrix ring is a group. (Contributed by AV, 21-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Grp)

Theoremmatvscl 20285 Closure of the scalar multiplication in the matrix ring. (lmodvscl 18928 analog.) (Contributed by AV, 27-Nov-2019.)
𝐾 = (Base‘𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    · = ( ·𝑠𝐴)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐶𝐾𝑋𝐵)) → (𝐶 · 𝑋) ∈ 𝐵)

Theoremmatsubg 20286 The matrix ring has the same addition as its underlying group. (Contributed by AV, 2-Aug-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (-g𝐺) = (-g𝐴))

Theoremmatplusgcell 20287 Addition in the matrix ring is cell-wise. (Contributed by AV, 2-Aug-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    = (+g𝐴)    &    + = (+g𝑅)       (((𝑋𝐵𝑌𝐵) ∧ (𝐼𝑁𝐽𝑁)) → (𝐼(𝑋 𝑌)𝐽) = ((𝐼𝑋𝐽) + (𝐼𝑌𝐽)))

Theoremmatsubgcell 20288 Subtraction in the matrix ring is cell-wise. (Contributed by AV, 2-Aug-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑆 = (-g𝐴)    &    = (-g𝑅)       ((𝑅 ∈ Ring ∧ (𝑋𝐵𝑌𝐵) ∧ (𝐼𝑁𝐽𝑁)) → (𝐼(𝑋𝑆𝑌)𝐽) = ((𝐼𝑋𝐽) (𝐼𝑌𝐽)))

Theoremmatinvgcell 20289 Additive inversion in the matrix ring is cell-wise. (Contributed by AV, 17-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑉 = (invg𝑅)    &   𝑊 = (invg𝐴)       ((𝑅 ∈ Ring ∧ 𝑋𝐵 ∧ (𝐼𝑁𝐽𝑁)) → (𝐼(𝑊𝑋)𝐽) = (𝑉‘(𝐼𝑋𝐽)))

Theoremmatvscacell 20290 Scalar multiplication in the matrix ring is cell-wise. (Contributed by AV, 7-Aug-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐾 = (Base‘𝑅)    &    · = ( ·𝑠𝐴)    &    × = (.r𝑅)       ((𝑅 ∈ Ring ∧ (𝑋𝐾𝑌𝐵) ∧ (𝐼𝑁𝐽𝑁)) → (𝐼(𝑋 · 𝑌)𝐽) = (𝑋 × (𝐼𝑌𝐽)))

Theoremmatgsum 20291* Finite commutative sums in a matrix algebra are taken componentwise. (Contributed by AV, 26-Sep-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    0 = (0g𝐴)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝐽𝑊)    &   (𝜑𝑅 ∈ Ring)    &   ((𝜑𝑦𝐽) → (𝑖𝑁, 𝑗𝑁𝑈) ∈ 𝐵)    &   (𝜑 → (𝑦𝐽 ↦ (𝑖𝑁, 𝑗𝑁𝑈)) finSupp 0 )       (𝜑 → (𝐴 Σg (𝑦𝐽 ↦ (𝑖𝑁, 𝑗𝑁𝑈))) = (𝑖𝑁, 𝑗𝑁 ↦ (𝑅 Σg (𝑦𝐽𝑈))))

11.2.3  The matrix algebra

The main result of this subsection are the theorems showing that (𝑁 Mat 𝑅) is a ring (see matring 20297) and an associative algebra (see matassa 20298). Additionally, theorems for the identity matrix and transposed matrices are provided.

Theoremmatmulr 20292 Multiplication in the matrix algebra. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &    · = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → · = (.r𝐴))

Theoremmamumat1cl 20293* The identity matrix (as operation in maps-to notation) is a matrix. (Contributed by Stefan O'Rear, 2-Sep-2015.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &    1 = (1r𝑅)    &    0 = (0g𝑅)    &   𝐼 = (𝑖𝑀, 𝑗𝑀 ↦ if(𝑖 = 𝑗, 1 , 0 ))    &   (𝜑𝑀 ∈ Fin)       (𝜑𝐼 ∈ (𝐵𝑚 (𝑀 × 𝑀)))

Theoremmat1comp 20294* The components of the identity matrix (as operation in maps-to notation). (Contributed by AV, 22-Jul-2019.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &    1 = (1r𝑅)    &    0 = (0g𝑅)    &   𝐼 = (𝑖𝑀, 𝑗𝑀 ↦ if(𝑖 = 𝑗, 1 , 0 ))    &   (𝜑𝑀 ∈ Fin)       ((𝐴𝑀𝐽𝑀) → (𝐴𝐼𝐽) = if(𝐴 = 𝐽, 1 , 0 ))

Theoremmamulid 20295* The identity matrix (as operation in maps-to notation) is a left identity (for any matrix with the same number of rows). (Contributed by Stefan O'Rear, 3-Sep-2015.) (Proof shortened by AV, 22-Jul-2019.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &    1 = (1r𝑅)    &    0 = (0g𝑅)    &   𝐼 = (𝑖𝑀, 𝑗𝑀 ↦ if(𝑖 = 𝑗, 1 , 0 ))    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   𝐹 = (𝑅 maMul ⟨𝑀, 𝑀, 𝑁⟩)    &   (𝜑𝑋 ∈ (𝐵𝑚 (𝑀 × 𝑁)))       (𝜑 → (𝐼𝐹𝑋) = 𝑋)

Theoremmamurid 20296* The identity matrix (as operation in maps-to notation) is a right identity (for any matrix with the same number of columns). (Contributed by Stefan O'Rear, 3-Sep-2015.) (Proof shortened by AV, 22-Jul-2019.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &    1 = (1r𝑅)    &    0 = (0g𝑅)    &   𝐼 = (𝑖𝑀, 𝑗𝑀 ↦ if(𝑖 = 𝑗, 1 , 0 ))    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   𝐹 = (𝑅 maMul ⟨𝑁, 𝑀, 𝑀⟩)    &   (𝜑𝑋 ∈ (𝐵𝑚 (𝑁 × 𝑀)))       (𝜑 → (𝑋𝐹𝐼) = 𝑋)

Theoremmatring 20297 Existence of the matrix ring, see also the statement in [Lang] p. 504: "For a given integer n > 0 the set of square n x n matrices form a ring." (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)

Theoremmatassa 20298 Existence of the matrix algebra, see also the statement in [Lang] p. 505:"Then Matn(R) is an algebra over R" . (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ AssAlg)

Theoremmatmulcell 20299* Multiplication in the matrix ring for a single cell of a matrix. (Contributed by AV, 17-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    · = (.r𝑅)    &    × = (.r𝐴)       ((𝑅 ∈ Ring ∧ (𝑋𝐵𝑌𝐵) ∧ (𝐼𝑁𝐽𝑁)) → (𝐼(𝑋 × 𝑌)𝐽) = (𝑅 Σg (𝑗𝑁 ↦ ((𝐼𝑋𝑗)(.r𝑅)(𝑗𝑌𝐽)))))

Theoremmpt2matmul 20300* Multiplication of two N x N matrices given in maps-to notation. (Contributed by AV, 29-Oct-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝑅)    &    × = (.r𝐴)    &    · = (.r𝑅)    &   (𝜑𝑅𝑉)    &   (𝜑𝑁 ∈ Fin)    &   𝑋 = (𝑖𝑁, 𝑗𝑁𝐶)    &   𝑌 = (𝑖𝑁, 𝑗𝑁𝐸)    &   ((𝜑𝑖𝑁𝑗𝑁) → 𝐶𝐵)    &   ((𝜑𝑖𝑁𝑗𝑁) → 𝐸𝐵)    &   ((𝜑 ∧ (𝑘 = 𝑖𝑚 = 𝑗)) → 𝐷 = 𝐶)    &   ((𝜑 ∧ (𝑚 = 𝑖𝑙 = 𝑗)) → 𝐹 = 𝐸)    &   ((𝜑𝑘𝑁𝑚𝑁) → 𝐷𝑈)    &   ((𝜑𝑚𝑁𝑙𝑁) → 𝐹𝑊)       (𝜑 → (𝑋 × 𝑌) = (𝑘𝑁, 𝑙𝑁 ↦ (𝑅 Σg (𝑚𝑁 ↦ (𝐷 · 𝐹)))))

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