P. 222 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 22101-22200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	evl1varpwval 22101	Value of a univariate polynomial evaluation mapping the exponentiation of a variable to the exponentiation of the evaluated variable. (Contributed by AV, 14-Sep-2019.)
⊢ 𝑄 = (eval1‘𝑅)    &   ⊢ 𝑊 = (Poly1‘𝑅)    &   ⊢ 𝐺 = (mulGrp‘𝑊)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ ↑ = (.g‘𝐺)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐵)    &   ⊢ 𝐻 = (mulGrp‘𝑅)    &   ⊢ 𝐸 = (.g‘𝐻)    ⇒   ⊢ (𝜑 → ((𝑄‘(𝑁 ↑ 𝑋))‘𝐶) = (𝑁𝐸𝐶))
Theorem	evl1scvarpw 22102	Univariate polynomial evaluation maps a multiple of an exponentiation of a variable to the multiple of an exponentiation of the evaluated variable. (Contributed by AV, 18-Sep-2019.)
⊢ 𝑄 = (eval1‘𝑅)    &   ⊢ 𝑊 = (Poly1‘𝑅)    &   ⊢ 𝐺 = (mulGrp‘𝑊)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ ↑ = (.g‘𝐺)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ × = ( ·𝑠 ‘𝑊)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ 𝑆 = (𝑅 ↑s 𝐵)    &   ⊢ ∙ = (.r‘𝑆)    &   ⊢ 𝑀 = (mulGrp‘𝑆)    &   ⊢ 𝐹 = (.g‘𝑀)    ⇒   ⊢ (𝜑 → (𝑄‘(𝐴 × (𝑁 ↑ 𝑋))) = ((𝐵 × {𝐴}) ∙ (𝑁𝐹(𝑄‘𝑋))))
Theorem	evl1scvarpwval 22103	Value of a univariate polynomial evaluation mapping a multiple of an exponentiation of a variable to the multiple of the exponentiation of the evaluated variable. (Contributed by AV, 18-Sep-2019.)
⊢ 𝑄 = (eval1‘𝑅)    &   ⊢ 𝑊 = (Poly1‘𝑅)    &   ⊢ 𝐺 = (mulGrp‘𝑊)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ ↑ = (.g‘𝐺)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ × = ( ·𝑠 ‘𝑊)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐵)    &   ⊢ 𝐻 = (mulGrp‘𝑅)    &   ⊢ 𝐸 = (.g‘𝐻)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝜑 → ((𝑄‘(𝐴 × (𝑁 ↑ 𝑋)))‘𝐶) = (𝐴 · (𝑁𝐸𝐶)))
Theorem	evl1gsummon 22104*	Value of a univariate polynomial evaluation mapping an additive group sum of a multiple of an exponentiation of a variable to a group sum of the multiple of the exponentiation of the evaluated variable. (Contributed by AV, 18-Sep-2019.)
⊢ 𝑄 = (eval1‘𝑅)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝑊 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝐻 = (mulGrp‘𝑅)    &   ⊢ 𝐸 = (.g‘𝐻)    &   ⊢ 𝐺 = (mulGrp‘𝑊)    &   ⊢ ↑ = (.g‘𝐺)    &   ⊢ × = ( ·𝑠 ‘𝑊)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝑀 𝐴 ∈ 𝐾)    &   ⊢ (𝜑 → 𝑀 ⊆ ℕ0)    &   ⊢ (𝜑 → 𝑀 ∈ Fin)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝑀 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐾)    ⇒   ⊢ (𝜑 → ((𝑄‘(𝑊 Σg (𝑥 ∈ 𝑀 ↦ (𝐴 × (𝑁 ↑ 𝑋)))))‘𝐶) = (𝑅 Σg (𝑥 ∈ 𝑀 ↦ (𝐴 · (𝑁𝐸𝐶)))))
11.4  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 21521) 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 21538) and scalar multiplication (see frlmvscafval 21540) for free modules. Actually, there is no 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 22106. Thus, 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 21523. However, for square matrices there is Definition df-mat 22128, 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 an 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 an 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 are excluded from the definitions of many authors, e.g., in [Lang] p. 503. It is shown in mat0dimbas0 22188 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). 22188 Its determinant is the ring unity, see mdet0fv0 22316.
11.4.1  The matrix multiplication This section is about the multiplication of m x n matrices.
Syntax	cmmul 22105	Syntax for the matrix multiplication operator.
class maMul
Definition	df-mamu 22106*	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 22107*	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 22108*	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 22109*	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 22110	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 22111	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 22112	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 22113	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 22114	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 22115	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 22116	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)
Theorem	grpvlinv 22117	Tuple-wise left inverse in groups. (Contributed by Stefan O'Rear, 5-Sep-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝑁 = (invg‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑m 𝐼)) → ((𝑁 ∘ 𝑋) ∘f + 𝑋) = (𝐼 × { 0 }))
Theorem	grpvrinv 22118	Tuple-wise right inverse in groups. (Contributed by Mario Carneiro, 22-Sep-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝑁 = (invg‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑m 𝐼)) → (𝑋 ∘f + (𝑁 ∘ 𝑋)) = (𝐼 × { 0 }))
Theorem	mhmvlin 22119	Tuple extension of monoid homomorphisms. (Contributed by Stefan O'Rear, 5-Sep-2015.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ + = (+g‘𝑀)    &   ⊢ ⨣ = (+g‘𝑁)    ⇒   ⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (𝐵 ↑m 𝐼) ∧ 𝑌 ∈ (𝐵 ↑m 𝐼)) → (𝐹 ∘ (𝑋 ∘f + 𝑌)) = ((𝐹 ∘ 𝑋) ∘f ⨣ (𝐹 ∘ 𝑌)))
Theorem	ringvcl 22120	Tuple-wise multiplication closure in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ (𝐵 ↑m 𝐼) ∧ 𝑌 ∈ (𝐵 ↑m 𝐼)) → (𝑋 ∘f · 𝑌) ∈ (𝐵 ↑m 𝐼))
Theorem	mamucl 22121	Operation closure of matrix multiplication. (Contributed by Stefan O'Rear, 2-Sep-2015.) (Proof shortened by AV, 23-Jul-2019.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)    &   ⊢ (𝜑 → 𝑀 ∈ Fin)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑃 ∈ Fin)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁)))    &   ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑m (𝑁 × 𝑃)))    ⇒   ⊢ (𝜑 → (𝑋𝐹𝑌) ∈ (𝐵 ↑m (𝑀 × 𝑃)))
Theorem	mamuass 22122	Matrix multiplication is associative, see also statement in [Lang] p. 505. (Contributed by Stefan O'Rear, 5-Sep-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑀 ∈ Fin)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑂 ∈ Fin)    &   ⊢ (𝜑 → 𝑃 ∈ Fin)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁)))    &   ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑m (𝑁 × 𝑂)))    &   ⊢ (𝜑 → 𝑍 ∈ (𝐵 ↑m (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)))
Theorem	mamudi 22123	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)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑂⟩)    &   ⊢ (𝜑 → 𝑀 ∈ Fin)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑂 ∈ Fin)    &   ⊢ + = (+g‘𝑅)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁)))    &   ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑m (𝑀 × 𝑁)))    &   ⊢ (𝜑 → 𝑍 ∈ (𝐵 ↑m (𝑁 × 𝑂)))    ⇒   ⊢ (𝜑 → ((𝑋 ∘f + 𝑌)𝐹𝑍) = ((𝑋𝐹𝑍) ∘f + (𝑌𝐹𝑍)))
Theorem	mamudir 22124	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)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑂⟩)    &   ⊢ (𝜑 → 𝑀 ∈ Fin)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑂 ∈ Fin)    &   ⊢ + = (+g‘𝑅)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁)))    &   ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑m (𝑁 × 𝑂)))    &   ⊢ (𝜑 → 𝑍 ∈ (𝐵 ↑m (𝑁 × 𝑂)))    ⇒   ⊢ (𝜑 → (𝑋𝐹(𝑌 ∘f + 𝑍)) = ((𝑋𝐹𝑌) ∘f + (𝑋𝐹𝑍)))
Theorem	mamuvs1 22125	Matrix multiplication distributes over scalar multiplication on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑂⟩)    &   ⊢ (𝜑 → 𝑀 ∈ Fin)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑂 ∈ Fin)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑m (𝑀 × 𝑁)))    &   ⊢ (𝜑 → 𝑍 ∈ (𝐵 ↑m (𝑁 × 𝑂)))    ⇒   ⊢ (𝜑 → ((((𝑀 × 𝑁) × {𝑋}) ∘f · 𝑌)𝐹𝑍) = (((𝑀 × 𝑂) × {𝑋}) ∘f · (𝑌𝐹𝑍)))
Theorem	mamuvs2 22126	Matrix multiplication distributes over scalar multiplication on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 22-Jul-2019.)
⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑂⟩)    &   ⊢ (𝜑 → 𝑀 ∈ Fin)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑂 ∈ Fin)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁)))    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ (𝐵 ↑m (𝑁 × 𝑂)))    ⇒   ⊢ (𝜑 → (𝑋𝐹(((𝑁 × 𝑂) × {𝑌}) ∘f · 𝑍)) = (((𝑀 × 𝑂) × {𝑌}) ∘f · (𝑋𝐹𝑍)))
11.4.2  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 22151. 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.
Syntax	cmat 22127	Syntax for the square matrix algebra.
class Mat
Definition	df-mat 22128*	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⟩)⟩))
Theorem	matbas0pc 22129	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)) = ∅)
Theorem	matbas0 22130	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)) = ∅)
Theorem	matval 22131	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 ∧ 𝑅 ∈ 𝑉) → 𝐴 = (𝐺 sSet ⟨(.r‘ndx), · ⟩))
Theorem	matrcl 22132	Reverse closure for the matrix algebra. (Contributed by Stefan O'Rear, 5-Sep-2015.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    ⇒   ⊢ (𝑋 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
Theorem	matbas 22133	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 ∧ 𝑅 ∈ 𝑉) → (Base‘𝐺) = (Base‘𝐴))
Theorem	matplusg 22134	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 ∧ 𝑅 ∈ 𝑉) → (+g‘𝐺) = (+g‘𝐴))
Theorem	matsca 22135	The matrix ring has the same scalars as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.) (Proof shortened by AV, 12-Nov-2024.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (Scalar‘𝐺) = (Scalar‘𝐴))
Theorem	matscaOLD 22136	Obsolete proof of matsca 22135 as of 12-Nov-2024. The matrix ring has the same scalars as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (Scalar‘𝐺) = (Scalar‘𝐴))
Theorem	matvsca 22137	The matrix ring has the same scalar multiplication as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.) (Proof shortened by AV, 12-Nov-2024.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → ( ·𝑠 ‘𝐺) = ( ·𝑠 ‘𝐴))
Theorem	matvscaOLD 22138	Obsolete proof of matvsca 22137 as of 12-Nov-2024. The matrix ring has the same scalar multiplication as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → ( ·𝑠 ‘𝐺) = ( ·𝑠 ‘𝐴))
Theorem	mat0 22139	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 ∧ 𝑅 ∈ 𝑉) → (0g‘𝐺) = (0g‘𝐴))
Theorem	matinvg 22140	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 ∧ 𝑅 ∈ 𝑉) → (invg‘𝐺) = (invg‘𝐴))
Theorem	mat0op 22141*	Value of a zero matrix as operation. (Contributed by AV, 2-Dec-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g‘𝐴) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 0 ))
Theorem	matsca2 22142	The scalars of the matrix ring are the underlying ring. (Contributed by Stefan O'Rear, 5-Sep-2015.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑅 = (Scalar‘𝐴))
Theorem	matbas2 22143	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 ∧ 𝑅 ∈ 𝑉) → (𝐾 ↑m (𝑁 × 𝑁)) = (Base‘𝐴))
Theorem	matbas2i 22144	A matrix is a function. (Contributed by Stefan O'Rear, 11-Sep-2015.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    ⇒   ⊢ (𝑀 ∈ 𝐵 → 𝑀 ∈ (𝐾 ↑m (𝑁 × 𝑁)))
Theorem	matbas2d 22145*	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)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝐶 ∈ 𝐾)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 𝐶) ∈ 𝐵)
Theorem	eqmat 22146*	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)    ⇒   ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 = 𝑌 ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑋𝑗) = (𝑖𝑌𝑗)))
Theorem	matecl 22147	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)    ⇒   ⊢ ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝑀 ∈ (Base‘𝐴)) → (𝐼𝑀𝐽) ∈ 𝐾)
Theorem	matecld 22148	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)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑁)    &   ⊢ (𝜑 → 𝐽 ∈ 𝑁)    &   ⊢ (𝜑 → 𝑀 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐼𝑀𝐽) ∈ 𝐾)
Theorem	matplusg2 22149	Addition in the matrix ring is cell-wise. (Contributed by Stefan O'Rear, 5-Sep-2015.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ ✚ = (+g‘𝐴)    &   ⊢ + = (+g‘𝑅)    ⇒   ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ✚ 𝑌) = (𝑋 ∘f + 𝑌))
Theorem	matvsca2 22150	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)    &   ⊢ · = ( ·𝑠 ‘𝐴)    &   ⊢ × = (.r‘𝑅)    &   ⊢ 𝐶 = (𝑁 × 𝑁)    ⇒   ⊢ ((𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (𝑋 · 𝑌) = ((𝐶 × {𝑋}) ∘f × 𝑌))
Theorem	matlmod 22151	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)
Theorem	matgrp 22152	The matrix ring is a group. (Contributed by AV, 21-Dec-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Grp)
Theorem	matvscl 22153	Closure of the scalar multiplication in the matrix ring. (lmodvscl 20632 analog.) (Contributed by AV, 27-Nov-2019.)
⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ · = ( ·𝑠 ‘𝐴)    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵)) → (𝐶 · 𝑋) ∈ 𝐵)
Theorem	matsubg 22154	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 ∧ 𝑅 ∈ 𝑉) → (-g‘𝐺) = (-g‘𝐴))
Theorem	matplusgcell 22155	Addition in the matrix ring is cell-wise. (Contributed by AV, 2-Aug-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ ✚ = (+g‘𝐴)    &   ⊢ + = (+g‘𝑅)    ⇒   ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝑋 ✚ 𝑌)𝐽) = ((𝐼𝑋𝐽) + (𝐼𝑌𝐽)))
Theorem	matsubgcell 22156	Subtraction in the matrix ring is cell-wise. (Contributed by AV, 2-Aug-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑆 = (-g‘𝐴)    &   ⊢ − = (-g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝑋𝑆𝑌)𝐽) = ((𝐼𝑋𝐽) − (𝐼𝑌𝐽)))
Theorem	matinvgcell 22157	Additive inversion in the matrix ring is cell-wise. (Contributed by AV, 17-Nov-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑉 = (invg‘𝑅)    &   ⊢ 𝑊 = (invg‘𝐴)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝑊‘𝑋)𝐽) = (𝑉‘(𝐼𝑋𝐽)))
Theorem	matvscacell 22158	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)    &   ⊢ · = ( ·𝑠 ‘𝐴)    &   ⊢ × = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝑋 · 𝑌)𝐽) = (𝑋 × (𝐼𝑌𝐽)))
Theorem	matgsum 22159*	Finite commutative sums in a matrix algebra are taken componentwise. (Contributed by AV, 26-Sep-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 0 = (0g‘𝐴)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝐽 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐽) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈) ∈ 𝐵)    &   ⊢ (𝜑 → (𝑦 ∈ 𝐽 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈)) finSupp 0 )    ⇒   ⊢ (𝜑 → (𝐴 Σg (𝑦 ∈ 𝐽 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑅 Σg (𝑦 ∈ 𝐽 ↦ 𝑈))))
11.4.3  The matrix algebra The main result of this subsection are the theorems showing that (𝑁 Mat 𝑅) is a ring (see matring 22165) and an associative algebra (see matassa 22166). Additionally, theorems for the identity matrix and transposed matrices are provided.
Theorem	matmulr 22160	Multiplication in the matrix algebra. (Contributed by Stefan O'Rear, 4-Sep-2015.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ · = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → · = (.r‘𝐴))
Theorem	mamumat1cl 22161*	The identity matrix (as operation in maps-to notation) is a matrix. (Contributed by Stefan O'Rear, 2-Sep-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐼 = (𝑖 ∈ 𝑀, 𝑗 ∈ 𝑀 ↦ if(𝑖 = 𝑗, 1 , 0 ))    &   ⊢ (𝜑 → 𝑀 ∈ Fin)    ⇒   ⊢ (𝜑 → 𝐼 ∈ (𝐵 ↑m (𝑀 × 𝑀)))
Theorem	mat1comp 22162*	The components of the identity matrix (as operation in maps-to notation). (Contributed by AV, 22-Jul-2019.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐼 = (𝑖 ∈ 𝑀, 𝑗 ∈ 𝑀 ↦ if(𝑖 = 𝑗, 1 , 0 ))    &   ⊢ (𝜑 → 𝑀 ∈ Fin)    ⇒   ⊢ ((𝐴 ∈ 𝑀 ∧ 𝐽 ∈ 𝑀) → (𝐴𝐼𝐽) = if(𝐴 = 𝐽, 1 , 0 ))
Theorem	mamulid 22163*	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)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐼 = (𝑖 ∈ 𝑀, 𝑗 ∈ 𝑀 ↦ if(𝑖 = 𝑗, 1 , 0 ))    &   ⊢ (𝜑 → 𝑀 ∈ Fin)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ 𝐹 = (𝑅 maMul ⟨𝑀, 𝑀, 𝑁⟩)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁)))    ⇒   ⊢ (𝜑 → (𝐼𝐹𝑋) = 𝑋)
Theorem	mamurid 22164*	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)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐼 = (𝑖 ∈ 𝑀, 𝑗 ∈ 𝑀 ↦ if(𝑖 = 𝑗, 1 , 0 ))    &   ⊢ (𝜑 → 𝑀 ∈ Fin)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ 𝐹 = (𝑅 maMul ⟨𝑁, 𝑀, 𝑀⟩)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑁 × 𝑀)))    ⇒   ⊢ (𝜑 → (𝑋𝐹𝐼) = 𝑋)
Theorem	matring 22165	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)
Theorem	matassa 22166	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)
Theorem	matmulcell 22167*	Multiplication in the matrix ring for a single cell of a matrix. (Contributed by AV, 17-Nov-2019.) (Revised by AV, 3-Jul-2022.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ × = (.r‘𝐴)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝑋 × 𝑌)𝐽) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝐽)))))
Theorem	mpomatmul 22168*	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)    &   ⊢ × = (.r‘𝐴)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ 𝑋 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝐶)    &   ⊢ 𝑌 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝐸)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝐶 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝐸 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ (𝑘 = 𝑖 ∧ 𝑚 = 𝑗)) → 𝐷 = 𝐶)    &   ⊢ ((𝜑 ∧ (𝑚 = 𝑖 ∧ 𝑙 = 𝑗)) → 𝐹 = 𝐸)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑁 ∧ 𝑚 ∈ 𝑁) → 𝐷 ∈ 𝑈)    &   ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → 𝐹 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝑋 × 𝑌) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑅 Σg (𝑚 ∈ 𝑁 ↦ (𝐷 · 𝐹)))))
Theorem	mat1 22169*	Value of an identity matrix, see also the statement in [Lang] p. 504: "The unit element of the ring of n x n matrices is the matrix In ... whose components are equal to 0 except on the diagonal, in which case they are equal to 1.". (Contributed by Stefan O'Rear, 7-Sep-2015.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r‘𝐴) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, 1 , 0 )))
Theorem	mat1ov 22170	Entries of an identity matrix, deduction form. (Contributed by Stefan O'Rear, 10-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑁)    &   ⊢ (𝜑 → 𝐽 ∈ 𝑁)    &   ⊢ 𝑈 = (1r‘𝐴)    ⇒   ⊢ (𝜑 → (𝐼𝑈𝐽) = if(𝐼 = 𝐽, 1 , 0 ))
Theorem	mat1bas 22171	The identity matrix is a matrix. (Contributed by AV, 15-Feb-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 1 = (1r‘(𝑁 Mat 𝑅))    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 1 ∈ 𝐵)
Theorem	matsc 22172*	The identity matrix multiplied with a scalar. (Contributed by Stefan O'Rear, 16-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝐴)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐿 ∈ 𝐾) → (𝐿 · (1r‘𝐴)) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, 𝐿, 0 )))
Theorem	ofco2 22173	Distribution law for the function operation and the composition of functions. (Contributed by Stefan O'Rear, 17-Jul-2018.)
⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → ((𝐹 ∘f 𝑅𝐺) ∘ 𝐻) = ((𝐹 ∘ 𝐻) ∘f 𝑅(𝐺 ∘ 𝐻)))
Theorem	oftpos 22174	The transposition of the value of a function operation for two functions is the value of the function operation for the two functions transposed. (Contributed by Stefan O'Rear, 17-Jul-2018.)
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → tpos (𝐹 ∘f 𝑅𝐺) = (tpos 𝐹 ∘f 𝑅tpos 𝐺))
Theorem	mattposcl 22175	The transpose of a square matrix is a square matrix of the same size. (Contributed by SO, 9-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    ⇒   ⊢ (𝑀 ∈ 𝐵 → tpos 𝑀 ∈ 𝐵)
Theorem	mattpostpos 22176	The transpose of the transpose of a square matrix is the square matrix itself. (Contributed by SO, 17-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    ⇒   ⊢ (𝑀 ∈ 𝐵 → tpos tpos 𝑀 = 𝑀)
Theorem	mattposvs 22177	The transposition of a matrix multiplied with a scalar equals the transposed matrix multiplied with the scalar, see also the statement in [Lang] p. 505. (Contributed by Stefan O'Rear, 17-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝐴)    ⇒   ⊢ ((𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → tpos (𝑋 · 𝑌) = (𝑋 · tpos 𝑌))
Theorem	mattpos1 22178	The transposition of the identity matrix is the identity matrix. (Contributed by Stefan O'Rear, 17-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 1 = (1r‘𝐴)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → tpos 1 = 1 )
Theorem	tposmap 22179	The transposition of an I X J -matrix is a J X I -matrix, see also the statement in [Lang] p. 505. (Contributed by Stefan O'Rear, 9-Jul-2018.)
⊢ (𝐴 ∈ (𝐵 ↑m (𝐼 × 𝐽)) → tpos 𝐴 ∈ (𝐵 ↑m (𝐽 × 𝐼)))
Theorem	mamutpos 22180	Behavior of transposes in matrix products, see also the statement in [Lang] p. 505. (Contributed by Stefan O'Rear, 9-Jul-2018.)
⊢ 𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)    &   ⊢ 𝐺 = (𝑅 maMul ⟨𝑃, 𝑁, 𝑀⟩)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑀 ∈ Fin)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑃 ∈ Fin)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁)))    &   ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑m (𝑁 × 𝑃)))    ⇒   ⊢ (𝜑 → tpos (𝑋𝐹𝑌) = (tpos 𝑌𝐺tpos 𝑋))
Theorem	mattposm 22181	Multiplying two transposed matrices results in the transposition of the product of the two matrices. (Contributed by Stefan O'Rear, 17-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ · = (.r‘𝐴)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → tpos (𝑋 · 𝑌) = (tpos 𝑌 · tpos 𝑋))
Theorem	matgsumcl 22182*	Closure of a group sum over the diagonal coefficients of a square matrix over a commutative ring. (Contributed by AV, 29-Dec-2018.) (Proof shortened by AV, 23-Jul-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑈 = (mulGrp‘𝑅)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑈 Σg (𝑟 ∈ 𝑁 ↦ (𝑟𝑀𝑟))) ∈ (Base‘𝑅))
Theorem	madetsumid 22183*	The identity summand in the Leibniz' formula of a determinant for a square matrix over a commutative ring. (Contributed by AV, 29-Dec-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑈 = (mulGrp‘𝑅)    &   ⊢ 𝑌 = (ℤRHom‘𝑅)    &   ⊢ 𝑆 = (pmSgn‘𝑁)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑃 = ( I ↾ 𝑁)) → (((𝑌 ∘ 𝑆)‘𝑃) · (𝑈 Σg (𝑟 ∈ 𝑁 ↦ ((𝑃‘𝑟)𝑀𝑟)))) = (𝑈 Σg (𝑟 ∈ 𝑁 ↦ (𝑟𝑀𝑟))))
Theorem	matepmcl 22184*	Each entry of a matrix with an index as permutation of the other is an element of the underlying ring. (Contributed by AV, 1-Jan-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃 ∧ 𝑀 ∈ 𝐵) → ∀𝑛 ∈ 𝑁 ((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅))
Theorem	matepm2cl 22185*	Each entry of a matrix with an index as permutation of the other is an element of the underlying ring. (Contributed by AV, 1-Jan-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃 ∧ 𝑀 ∈ 𝐵) → ∀𝑛 ∈ 𝑁 (𝑛𝑀(𝑄‘𝑛)) ∈ (Base‘𝑅))
Theorem	madetsmelbas 22186*	A summand of the determinant of a matrix belongs to the underlying ring. (Contributed by AV, 1-Jan-2019.)
⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝑆 = (pmSgn‘𝑁)    &   ⊢ 𝑌 = (ℤRHom‘𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐺 = (mulGrp‘𝑅)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃) → (((𝑌 ∘ 𝑆)‘𝑄)(.r‘𝑅)(𝐺 Σg (𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛)))) ∈ (Base‘𝑅))
Theorem	madetsmelbas2 22187*	A summand of the determinant of a matrix belongs to the underlying ring. (Contributed by AV, 1-Jan-2019.)
⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝑆 = (pmSgn‘𝑁)    &   ⊢ 𝑌 = (ℤRHom‘𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐺 = (mulGrp‘𝑅)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃) → (((𝑌 ∘ 𝑆)‘𝑄)(.r‘𝑅)(𝐺 Σg (𝑛 ∈ 𝑁 ↦ (𝑛𝑀(𝑄‘𝑛))))) ∈ (Base‘𝑅))
11.4.4  Matrices of dimension 0 and 1 As already mentioned before, and shown in mat0dimbas0 22188, the empty set is the sole zero-dimensional matrix (also called "empty matrix", see Wikipedia https://en.wikipedia.org/wiki/Matrix_(mathematics)#Empty_matrices). 22188 In the following, some properties of the empty matrix are shown, especially that the empty matrix over an arbitrary ring forms a commutative ring, see mat0dimcrng 22192. For the one-dimensional case, it can be shown that a ring of matrices with dimension 1 is isomorphic to the underlying ring, see mat1ric 22209.
Theorem	mat0dimbas0 22188	The empty set is the one and only matrix of dimension 0, called "the empty matrix". (Contributed by AV, 27-Feb-2019.)
⊢ (𝑅 ∈ 𝑉 → (Base‘(∅ Mat 𝑅)) = {∅})
Theorem	mat0dim0 22189	The zero of the algebra of matrices with dimension 0. (Contributed by AV, 6-Aug-2019.)
⊢ 𝐴 = (∅ Mat 𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (0g‘𝐴) = ∅)
Theorem	mat0dimid 22190	The identity of the algebra of matrices with dimension 0. (Contributed by AV, 6-Aug-2019.)
⊢ 𝐴 = (∅ Mat 𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (1r‘𝐴) = ∅)
Theorem	mat0dimscm 22191	The scalar multiplication in the algebra of matrices with dimension 0. (Contributed by AV, 6-Aug-2019.)
⊢ 𝐴 = (∅ Mat 𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ (Base‘𝑅)) → (𝑋( ·𝑠 ‘𝐴)∅) = ∅)
Theorem	mat0dimcrng 22192	The algebra of matrices with dimension 0 (over an arbitrary ring!) is a commutative ring. (Contributed by AV, 10-Aug-2019.)
⊢ 𝐴 = (∅ Mat 𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → 𝐴 ∈ CRing)
Theorem	mat1dimelbas 22193*	A matrix with dimension 1 is an ordered pair with an ordered pair (of the one and only pair of indices) as first component. (Contributed by AV, 15-Aug-2019.)
⊢ 𝐴 = ({𝐸} Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑂 = ⟨𝐸, 𝐸⟩    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → (𝑀 ∈ (Base‘𝐴) ↔ ∃𝑟 ∈ 𝐵 𝑀 = {⟨𝑂, 𝑟⟩}))
Theorem	mat1dimbas 22194	A matrix with dimension 1 is an ordered pair with an ordered pair (of the one and only pair of indices) as first component. (Contributed by AV, 15-Aug-2019.)
⊢ 𝐴 = ({𝐸} Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑂 = ⟨𝐸, 𝐸⟩    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → {⟨𝑂, 𝑋⟩} ∈ (Base‘𝐴))
Theorem	mat1dim0 22195	The zero of the algebra of matrices with dimension 1. (Contributed by AV, 15-Aug-2019.)
⊢ 𝐴 = ({𝐸} Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑂 = ⟨𝐸, 𝐸⟩    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → (0g‘𝐴) = {⟨𝑂, (0g‘𝑅)⟩})
Theorem	mat1dimid 22196	The identity of the algebra of matrices with dimension 1. (Contributed by AV, 15-Aug-2019.)
⊢ 𝐴 = ({𝐸} Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑂 = ⟨𝐸, 𝐸⟩    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → (1r‘𝐴) = {⟨𝑂, (1r‘𝑅)⟩})
Theorem	mat1dimscm 22197	The scalar multiplication in the algebra of matrices with dimension 1. (Contributed by AV, 16-Aug-2019.)
⊢ 𝐴 = ({𝐸} Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑂 = ⟨𝐸, 𝐸⟩    ⇒   ⊢ (((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋( ·𝑠 ‘𝐴){⟨𝑂, 𝑌⟩}) = {⟨𝑂, (𝑋(.r‘𝑅)𝑌)⟩})
Theorem	mat1dimmul 22198	The ring multiplication in the algebra of matrices with dimension 1. (Contributed by AV, 16-Aug-2019.) (Proof shortened by AV, 18-Apr-2021.)
⊢ 𝐴 = ({𝐸} Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑂 = ⟨𝐸, 𝐸⟩    ⇒   ⊢ (((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ({⟨𝑂, 𝑋⟩} (.r‘𝐴){⟨𝑂, 𝑌⟩}) = {⟨𝑂, (𝑋(.r‘𝑅)𝑌)⟩})
Theorem	mat1dimcrng 22199	The algebra of matrices with dimension 1 over a commutative ring is a commutative ring. (Contributed by AV, 16-Aug-2019.)
⊢ 𝐴 = ({𝐸} Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑂 = ⟨𝐸, 𝐸⟩    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝐸 ∈ 𝑉) → 𝐴 ∈ CRing)
Theorem	mat1f1o 22200*	There is a 1-1 function from a ring onto the ring of matrices with dimension 1 over this ring. (Contributed by AV, 22-Dec-2019.)
⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝐴 = ({𝐸} Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑂 = ⟨𝐸, 𝐸⟩    &   ⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ {⟨𝑂, 𝑥⟩})    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → 𝐹:𝐾–1-1-onto→𝐵)