P. 203 of Theorem List - Metamath Proof Explorer

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

Type	Label	Description
Statement
Theorem	islindf2 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)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐾 = (LSpan‘𝑊)    &   ⊢ 𝑆 = (Scalar‘𝑊)    &   ⊢ 𝑁 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑆)    ⇒   ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) → (𝐹 LIndF 𝑊 ↔ ∀𝑥 ∈ 𝐼 ∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹‘𝑥)) ∈ (𝐾‘(𝐹 “ (𝐼 ∖ {𝑥})))))
Theorem	lindff 20202	Functional property of a linearly independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ ((𝐹 LIndF 𝑊 ∧ 𝑊 ∈ 𝑌) → 𝐹:dom 𝐹⟶𝐵)
Theorem	lindfind 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.)
⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ 𝐿 = (Scalar‘𝑊)    &   ⊢ 0 = (0g‘𝐿)    &   ⊢ 𝐾 = (Base‘𝐿)    ⇒   ⊢ (((𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → ¬ (𝐴 · (𝐹‘𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸}))))
Theorem	lindsind 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.)
⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ 𝐿 = (Scalar‘𝑊)    &   ⊢ 0 = (0g‘𝐿)    &   ⊢ 𝐾 = (Base‘𝐿)    ⇒   ⊢ (((𝐹 ∈ (LIndS‘𝑊) ∧ 𝐸 ∈ 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐹 ∖ {𝐸})))
Theorem	lindfind2 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 𝑊 ∧ 𝐸 ∈ dom 𝐹) → ¬ (𝐹‘𝐸) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝐸}))))
Theorem	lindsind2 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) ∧ 𝐸 ∈ 𝐹) → ¬ 𝐸 ∈ (𝐾‘(𝐹 ∖ {𝐸})))
Theorem	lindff1 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 𝐹–1-1→𝐵)
Theorem	lindfrn 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))
Theorem	f1lindf 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 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (𝐹 ∘ 𝐺) LIndF 𝑊)
Theorem	lindfres 20210	Any restriction of an independent family is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → (𝐹 ↾ 𝑋) LIndF 𝑊)
Theorem	lindsss 20211	Any subset of an independent set is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ ((𝑊 ∈ LMod ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝐺 ⊆ 𝐹) → 𝐺 ∈ (LIndS‘𝑊))
Theorem	f1linds 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:𝐷–1-1→𝑆) → 𝐹 LIndF 𝑊)
Theorem	islindf3 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 𝐹–1-1→V ∧ ran 𝐹 ∈ (LIndS‘𝑊))))
Theorem	lindfmm 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:𝐵–1-1→𝐶 ∧ 𝐹:𝐼⟶𝐵) → (𝐹 LIndF 𝑆 ↔ (𝐺 ∘ 𝐹) LIndF 𝑇))
Theorem	lindsmm 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:𝐵–1-1→𝐶 ∧ 𝐹 ⊆ 𝐵) → (𝐹 ∈ (LIndS‘𝑆) ↔ (𝐺 “ 𝐹) ∈ (LIndS‘𝑇)))
Theorem	lindsmm2 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:𝐵–1-1→𝐶 ∧ 𝐹 ∈ (LIndS‘𝑆)) → (𝐺 “ 𝐹) ∈ (LIndS‘𝑇))
Theorem	lsslindf 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 = (𝑊 ↾s 𝑆)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) → (𝐹 LIndF 𝑋 ↔ 𝐹 LIndF 𝑊))
Theorem	lsslinds 20218	Linear independence is unchanged by working in a subspace. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ 𝑈 = (LSubSp‘𝑊)    &   ⊢ 𝑋 = (𝑊 ↾s 𝑆)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ 𝐹 ⊆ 𝑆) → (𝐹 ∈ (LIndS‘𝑋) ↔ 𝐹 ∈ (LIndS‘𝑊)))
Theorem	islbs4 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)    ⇒   ⊢ (𝑋 ∈ 𝐽 ↔ (𝑋 ∈ (LIndS‘𝑊) ∧ (𝐾‘𝑋) = 𝐵))
Theorem	lbslinds 20220	A basis is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ 𝐽 = (LBasis‘𝑊)    ⇒   ⊢ 𝐽 ⊆ (LIndS‘𝑊)
Theorem	islinds3 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 = (𝑊 ↾s (𝐾‘𝑌))    &   ⊢ 𝐽 = (LBasis‘𝑋)    ⇒   ⊢ (𝑊 ∈ LMod → (𝑌 ∈ (LIndS‘𝑊) ↔ 𝑌 ∈ 𝐽))
Theorem	islinds4 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) ↔ ∃𝑏 ∈ 𝐽 𝑌 ⊆ 𝑏))
11.1.5  Characterization of free modules
Theorem	lmimlbs 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) ∧ 𝐵 ∈ 𝐽) → (𝐹 “ 𝐵) ∈ 𝐾)
Theorem	lmiclbs 20224	Having a basis is an isomorphism invariant. (Contributed by Stefan O'Rear, 26-Feb-2015.)
⊢ 𝐽 = (LBasis‘𝑆)    &   ⊢ 𝐾 = (LBasis‘𝑇)    ⇒   ⊢ (𝑆 ≃𝑚 𝑇 → (𝐽 ≠ ∅ → 𝐾 ≠ ∅))
Theorem	islindf4 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)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑌 = (0g‘𝑅)    &   ⊢ 𝐿 = (Base‘(𝑅 freeLMod 𝐼))    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) → (𝐹 LIndF 𝑊 ↔ ∀𝑥 ∈ 𝐿 ((𝑊 Σg (𝑥 ∘𝑓 · 𝐹)) = 0 → 𝑥 = (𝐼 × {𝑌}))))
Theorem	islindf5 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)    &   ⊢ · = ( ·𝑠 ‘𝑇)    &   ⊢ 𝐸 = (𝑥 ∈ 𝐵 ↦ (𝑇 Σg (𝑥 ∘𝑓 · 𝐴)))    &   ⊢ (𝜑 → 𝑇 ∈ LMod)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑅 = (Scalar‘𝑇))    &   ⊢ (𝜑 → 𝐴:𝐼⟶𝐶)    ⇒   ⊢ (𝜑 → (𝐴 LIndF 𝑇 ↔ 𝐸:𝐵–1-1→𝐶))
Theorem	indlcim 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)    &   ⊢ · = ( ·𝑠 ‘𝑇)    &   ⊢ 𝑁 = (LSpan‘𝑇)    &   ⊢ 𝐸 = (𝑥 ∈ 𝐵 ↦ (𝑇 Σg (𝑥 ∘𝑓 · 𝐴)))    &   ⊢ (𝜑 → 𝑇 ∈ LMod)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑅 = (Scalar‘𝑇))    &   ⊢ (𝜑 → 𝐴:𝐼–onto→𝐽)    &   ⊢ (𝜑 → 𝐴 LIndF 𝑇)    &   ⊢ (𝜑 → (𝑁‘𝐽) = 𝐶)    ⇒   ⊢ (𝜑 → 𝐸 ∈ (𝐹 LMIso 𝑇))
Theorem	lbslcic 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 ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) → 𝑊 ≃𝑚 (𝐹 freeLMod 𝐼))
Theorem	lmisfree 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 𝑊 ≃𝑚 (𝐹 freeLMod 𝑘)))
Theorem	lvecisfrlm 20230*	Every vector space is isomorphic to a free module. (Contributed by AV, 7-Mar-2019.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ (𝑊 ∈ LVec → ∃𝑘 𝑊 ≃𝑚 (𝐹 freeLMod 𝑘))
Theorem	lmimco 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)) → (𝐹 ∘ 𝐺) ∈ (𝑅 LMIso 𝑇))
Theorem	lmictra 20232	Module isomorphism is transitive. (Contributed by AV, 10-Mar-2019.)
⊢ ((𝑅 ≃𝑚 𝑆 ∧ 𝑆 ≃𝑚 𝑇) → 𝑅 ≃𝑚 𝑇)
Theorem	uvcf1o 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 ∧ 𝐼 ∈ 𝑊) → 𝑈:𝐼–1-1-onto→ran 𝑈)
Theorem	uvcendim 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 ∧ 𝐼 ∈ 𝑊) → 𝐼 ≈ ran 𝑈)
Theorem	frlmisfrlm 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 ∧ 𝐼 ∈ 𝑌 ∧ 𝐼 ≈ 𝐽) → (𝑅 freeLMod 𝐼) ≃𝑚 (𝑅 freeLMod 𝐽))
Theorem	frlmiscvec 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 ∧ 𝐼 ∈ 𝑌) → (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.
Syntax	cmmul 20237	Syntax for the matrix multiplication operator.
class maMul
Definition	df-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 ‘(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 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)    &   ⊢ · = (.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 20240*	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 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)    &   ⊢ · = (.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 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⟩)    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑀 ∈ Fin ∧ 𝑁 ∈ Fin ∧ 𝑃 ∈ Fin)) → dom × = (𝐵 × 𝐶))
Theorem	mamufacex 20243	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 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)    &   ⊢ (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 20245	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 20246	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 20247	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 20248	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 20249	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 20250	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 20251	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 20252	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	gsumcom3 20253*	A commutative law for finitely supported iterated sums. (Contributed by Stefan O'Rear, 2-Nov-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶)) → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑈 ∈ Fin)    &   ⊢ ((𝜑 ∧ ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶) ∧ ¬ 𝑗𝑈𝑘)) → 𝑋 = 0 )    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝑗 ∈ 𝐴 ↦ (𝐺 Σg (𝑘 ∈ 𝐶 ↦ 𝑋)))) = (𝐺 Σg (𝑘 ∈ 𝐶 ↦ (𝐺 Σg (𝑗 ∈ 𝐴 ↦ 𝑋)))))
Theorem	gsumcom3fi 20254*	A commutative law for finite iterated sums. (Contributed by Stefan O'Rear, 5-Sep-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐶 ∈ Fin)    &   ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶)) → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝑗 ∈ 𝐴 ↦ (𝐺 Σg (𝑘 ∈ 𝐶 ↦ 𝑋)))) = (𝐺 Σg (𝑘 ∈ 𝐶 ↦ (𝐺 Σg (𝑗 ∈ 𝐴 ↦ 𝑋)))))
Theorem	mamucl 20255	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 20256	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 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)    &   ⊢ (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 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)    &   ⊢ (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 20259	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 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.)
⊢ (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)))
11.2.2  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.
Syntax	cmat 20261	Syntax for the square matrix algebra.
class Mat
Definition	df-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⟩)⟩))
Theorem	matbas0pc 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)) = ∅)
Theorem	matbas0 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)) = ∅)
Theorem	matval 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 ∧ 𝑅 ∈ 𝑉) → 𝐴 = (𝐺 sSet ⟨(.r‘ndx), · ⟩))
Theorem	matrcl 20266	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 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 ∧ 𝑅 ∈ 𝑉) → (Base‘𝐺) = (Base‘𝐴))
Theorem	matplusg 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 ∧ 𝑅 ∈ 𝑉) → (+g‘𝐺) = (+g‘𝐴))
Theorem	matsca 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 ∧ 𝑅 ∈ 𝑉) → (Scalar‘𝐺) = (Scalar‘𝐴))
Theorem	matvsca 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 ∧ 𝑅 ∈ 𝑉) → ( ·𝑠 ‘𝐺) = ( ·𝑠 ‘𝐴))
Theorem	mat0 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 ∧ 𝑅 ∈ 𝑉) → (0g‘𝐺) = (0g‘𝐴))
Theorem	matinvg 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 ∧ 𝑅 ∈ 𝑉) → (invg‘𝐺) = (invg‘𝐴))
Theorem	mat0op 20273*	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 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 ∧ 𝑅 ∈ 𝑉) → 𝑅 = (Scalar‘𝐴))
Theorem	matbas2 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 ∧ 𝑅 ∈ 𝑉) → (𝐾 ↑𝑚 (𝑁 × 𝑁)) = (Base‘𝐴))
Theorem	matbas2i 20276	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 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)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝐶 ∈ 𝐾)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 𝐶) ∈ 𝐵)
Theorem	eqmat 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)    ⇒   ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 = 𝑌 ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑋𝑗) = (𝑖𝑌𝑗)))
Theorem	matecl 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)    ⇒   ⊢ ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝑀 ∈ (Base‘𝐴)) → (𝐼𝑀𝐽) ∈ 𝐾)
Theorem	matecld 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)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑁)    &   ⊢ (𝜑 → 𝐽 ∈ 𝑁)    &   ⊢ (𝜑 → 𝑀 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐼𝑀𝐽) ∈ 𝐾)
Theorem	matplusg2 20281	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 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)    &   ⊢ · = ( ·𝑠 ‘𝐴)    &   ⊢ × = (.r‘𝑅)    &   ⊢ 𝐶 = (𝑁 × 𝑁)    ⇒   ⊢ ((𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (𝑋 · 𝑌) = ((𝐶 × {𝑋}) ∘𝑓 × 𝑌))
Theorem	matlmod 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)
Theorem	matgrp 20284	The matrix ring is a group. (Contributed by AV, 21-Dec-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Grp)
Theorem	matvscl 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)    &   ⊢ · = ( ·𝑠 ‘𝐴)    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵)) → (𝐶 · 𝑋) ∈ 𝐵)
Theorem	matsubg 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 ∧ 𝑅 ∈ 𝑉) → (-g‘𝐺) = (-g‘𝐴))
Theorem	matplusgcell 20287	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 20288	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 20289	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 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)    &   ⊢ · = ( ·𝑠 ‘𝐴)    &   ⊢ × = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝑋 · 𝑌)𝐽) = (𝑋 × (𝐼𝑌𝐽)))
Theorem	matgsum 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 = (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))))
11.2.3  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.
Theorem	matmulr 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 ∧ 𝑅 ∈ 𝑉) → · = (.r‘𝐴))
Theorem	mamumat1cl 20293*	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 20294*	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 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)    &   ⊢ (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 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)    &   ⊢ (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 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)
Theorem	matassa 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)
Theorem	matmulcell 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)    &   ⊢ · = (.r‘𝑅)    &   ⊢ × = (.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	mpt2matmul 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)    &   ⊢ × = (.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)))))