P. 233 of Theorem List - Metamath Proof Explorer

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Theorem List for Metamath Proof Explorer - 23201-23300 *Has distinct variable group(s)

Type

Label

Description

Statement

Theorem

fvmptnn04ifd 23201*

The function value of a mapping from the nonnegative integers with four distinct cases for the forth case. (Contributed by AV, 10-Nov-2019.)

𝐺 = (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵)))) & (𝜑 → 𝑆 ∈ ℕ) & (𝜑 → 𝑁 ∈ ℕ₀) ⇒ ((𝜑 ∧ 𝑆 < 𝑁 ∧ ⦋𝑁 / 𝑛⦌𝐷 ∈ 𝑉) → (𝐺‘𝑁) = ⦋𝑁 / 𝑛⦌𝐷)

Theorem

chfacfisf 23202*

The "characteristic factor function" is a function from the nonnegative integers to polynomial matrices. (Contributed by AV, 8-Nov-2019.)

Theorem

chfacfisfcpmat 23203*

The "characteristic factor function" is a function from the nonnegative integers to constant polynomial matrices. (Contributed by AV, 19-Nov-2019.)

Theorem

chfacffsupp 23204*

The "characteristic factor function" is finitely supported. (Contributed by AV, 20-Nov-2019.) (Proof shortened by AV, 23-Dec-2019.)

Theorem

chfacfscmulcl 23205*

Closure of a scaled value of the "characteristic factor function". (Contributed by AV, 9-Nov-2019.)

𝐴 = (𝑁 Mat 𝑅) & 𝐵 = (Base‘𝐴) & 𝑃 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & × = (._r‘𝑌) & − = (-_g‘𝑌) & 0 = (0_g‘𝑌) & 𝑇 = (𝑁 matToPolyMat 𝑅) & 𝐺 = (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) & 𝑋 = (var₁‘𝑅) & · = ( ·_𝑠 ‘𝑌) & ↑ = (._g‘(mulGrp‘𝑃)) ⇒ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝐾 ∈ ℕ₀) → ((𝐾 ↑ 𝑋) · (𝐺‘𝐾)) ∈ (Base‘𝑌))

Theorem

chfacfscmul0 23206*

A scaled value of the "characteristic factor function" is zero almost always. (Contributed by AV, 9-Nov-2019.)

𝐴 = (𝑁 Mat 𝑅) & 𝐵 = (Base‘𝐴) & 𝑃 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & × = (._r‘𝑌) & − = (-_g‘𝑌) & 0 = (0_g‘𝑌) & 𝑇 = (𝑁 matToPolyMat 𝑅) & 𝐺 = (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) & 𝑋 = (var₁‘𝑅) & · = ( ·_𝑠 ‘𝑌) & ↑ = (._g‘(mulGrp‘𝑃)) ⇒ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝐾 ∈ (ℤ_≥‘(𝑠 + 2))) → ((𝐾 ↑ 𝑋) · (𝐺‘𝐾)) = 0 )

Theorem

chfacfscmulfsupp 23207*

A mapping of scaled values of the "characteristic factor function" is finitely supported. (Contributed by AV, 8-Nov-2019.)

𝐴 = (𝑁 Mat 𝑅) & 𝐵 = (Base‘𝐴) & 𝑃 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & × = (._r‘𝑌) & − = (-_g‘𝑌) & 0 = (0_g‘𝑌) & 𝑇 = (𝑁 matToPolyMat 𝑅) & 𝐺 = (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) & 𝑋 = (var₁‘𝑅) & · = ( ·_𝑠 ‘𝑌) & ↑ = (._g‘(mulGrp‘𝑃)) ⇒ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑖 ∈ ℕ₀ ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖))) finSupp 0 )

Theorem

chfacfscmulgsum 23208*

Breaking up a sum of values of the "characteristic factor function" scaled by a polynomial. (Contributed by AV, 9-Nov-2019.)

𝐴 = (𝑁 Mat 𝑅) & 𝐵 = (Base‘𝐴) & 𝑃 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & × = (._r‘𝑌) & − = (-_g‘𝑌) & 0 = (0_g‘𝑌) & 𝑇 = (𝑁 matToPolyMat 𝑅) & 𝐺 = (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) & 𝑋 = (var₁‘𝑅) & · = ( ·_𝑠 ‘𝑌) & ↑ = (._g‘(mulGrp‘𝑃)) & + = (+_g‘𝑌) ⇒ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ ℕ₀ ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))

Theorem

chfacfpmmulcl 23209*

Closure of the value of the "characteristic factor function" multiplied with a constant polynomial matrix. (Contributed by AV, 23-Nov-2019.)

Theorem

chfacfpmmul0 23210*

The value of the "characteristic factor function" multiplied with a constant polynomial matrix is zero almost always. (Contributed by AV, 23-Nov-2019.)

Theorem

chfacfpmmulfsupp 23211*

A mapping of values of the "characteristic factor function" multiplied with a constant polynomial matrix is finitely supported. (Contributed by AV, 23-Nov-2019.)

Theorem

chfacfpmmulgsum 23212*

Breaking up a sum of values of the "characteristic factor function" multiplied with a constant polynomial matrix. (Contributed by AV, 23-Nov-2019.)

𝐴 = (𝑁 Mat 𝑅) & 𝐵 = (Base‘𝐴) & 𝑃 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & × = (._r‘𝑌) & − = (-_g‘𝑌) & 0 = (0_g‘𝑌) & 𝑇 = (𝑁 matToPolyMat 𝑅) & 𝐺 = (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) & ↑ = (._g‘(mulGrp‘𝑌)) & + = (+_g‘𝑌) ⇒ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ ℕ₀ ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))

Theorem

chfacfpmmulgsum2 23213*

Breaking up a sum of values of the "characteristic factor function" multiplied with a constant polynomial matrix. (Contributed by AV, 23-Nov-2019.)

𝐴 = (𝑁 Mat 𝑅) & 𝐵 = (Base‘𝐴) & 𝑃 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & × = (._r‘𝑌) & − = (-_g‘𝑌) & 0 = (0_g‘𝑌) & 𝑇 = (𝑁 matToPolyMat 𝑅) & 𝐺 = (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) & ↑ = (._g‘(mulGrp‘𝑌)) & + = (+_g‘𝑌) ⇒ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ ℕ₀ ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) − (((𝑖 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑖)))))) + ((((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))

Theorem

cayhamlem1 23214*

Lemma 1 for cayleyhamilton 23238. (Contributed by AV, 11-Nov-2019.)

11.7.3 The Cayley-Hamilton theorem

In this section, a direct algebraic proof for the Cayley-Hamilton theorem is provided, according to Wikipedia ("Cayley-Hamilton theorem", 09-Nov-2019, https://en.wikipedia.org/wiki/Cayley%E2%80%93Hamilton_theorem, section "A direct algebraic proof" (this approach is also used for proving Lemma 1.9 in [Hefferon] p. 427):

"This proof uses just the kind of objects needed to formulate the Cayley-Hamilton theorem: matrices with polynomials as entries. The matrix (t * I_n - A) whose determinant is the characteristic polynomial of A is such a matrix, and since polynomials [over a commutative ring] form a commutative ring, it has an adjugate

(1) B = adj(t * I_n - A) .

Then, according to the right-hand fundamental relation of the adjugate, one has

(2) ( t * I_n - A ) x B = det(t * I_n - A) x I_n = p(t) * I_n .

Since B is also a matrix with polynomials in t as entries, one can, for each i, collect the coefficients of t^i in each entry to form a matrix B_i of numbers, such that one has

(3) B = sum_{i = 0 to (n-1)} t^i B_i .

(The way the entries of B are defined makes clear that no powers higher than t^(n-1) occur). While this looks like a polynomial with matrices as coefficients, we shall not consider such a notion; it is just a way to write a matrix with polynomial entries as a linear combination of n constant matrices, and the coefficient t^i has been written to the left of the matrix to stress this point of view.

Now, one can expand the matrix product in our equation by bilinearity

(4) p(t) * I_n = ( t * I_n - A ) x B
= ( t * I_n - A ) x sum_{i = 0 to (n-1)} t^i * B_i = sum_{i = 0 to (n-1)} t * I_n x t^i B_i - sum_{i = 0 to (n-1)} A * t^i B_i = sum_{i = 0 to (n-1)} t^(i+1) * B_i - sum_{i = 0 to (n-1)} t^i * A x B_i = t^n B_n-1 + sum_{i = 1 to (n-1)} t^i * ( B_i-1 - A x B_i ) - A x B₀ .

Writing

(5) p(t) I_n = t^n * I_n + t^(n-1) * c_(n-1) x I_n + ... + t * c₁ I_n + c₀ I_n ,

one obtains an equality of two matrices with polynomial entries, written as linear combinations of constant matrices with powers of t as coefficients. Such an equality can hold only if in any matrix position the entry that is multiplied by a given power t^i is the same on both sides; it follows that the constant matrices with coefficient t^i in both expressions must be equal. Writing these equations then for i from n down to 0, one finds

(6) B_n-1 = I_n , B_i-1 - A x B_i = c_i * I_n for 1 <= i <= n-1 , - A x B₀ = c₀ * I_n .

Finally, multiply the equation of the coefficients of t^i from the left by A^i, and sum up:

(7) A^n B_n-1 + sum_{i
= 1 to (n-1)} ( A^i x B_i-1 - A^(i+1) x B_i ) - A x B₀ = A^n + c_n-1 * A^(n-1) + ... + c₁ * A + c₀ * I_n .

The left-hand sides form a telescoping sum and cancel completely; the right-hand sides add up to p(A):

(8) 0 = p(A) .

This completes the proof."

To formalize this approach, the steps mentioned in Wikipedia must be elaborated in more detail.

The first step is to formalize the preliminaries and the objective of the theorem. In Wikipedia, the Cayley-Hamilton theorem is stated as follows: "... the Cayley-Hamilton theorem ... states that every square matrix over a commutative ring ... satisfies its own characteristic equation." Or in more detail: "If A is a given n x n matrix and I_n is the n x n identity matrix, then the characteristic polynomial of A is defined as p(t) = det(t * I_n - A), where det is the determinant operation and t is a variable for a scalar element of the base ring. Since the entries of the matrix (t * I_n - A) are (linear or constant) polynomials in t, the determinant is also an n-th order monic polynomial in t. The Cayley-Hamilton theorem states that if one defines an analogous matrix equation, p(A), consisting of the replacement of the scalar eigenvalues t with the matrix A, then this polynomial in the matrix A results in the zero matrix,

p(A) = 0.

The powers of A, obtained by substitution from powers of t, are defined by repeated matrix multiplication; the constant term of p(t) gives a multiple of the power A^0, which is defined as the identity matrix. The theorem allows A^n to be expressed as a linear combination of the lower matrix powers of A. When the ring is a field, the Cayley-Hamilton theorem is equivalent to the statement that the minimal polynomial of a square matrix divides its characteristic polynomial."

Actually, the definition of the characteristic polynomial of a square matrix requires some attention. According to df-chpmat 23175, the characteristic polynomial of an 𝑁 x 𝑁 matrix 𝑀 over a ring 𝑅 is defined as

((𝑁 CharPlyMat 𝑅)‘𝑀) = (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑀))))

where 𝐷 = (𝑁 maDet 𝑃) is the function mapping an 𝑁 x 𝑁 matrix over the polynomial ring over the ring 𝑅 to its determinant, 𝑋 = (var₁‘𝑅) is the variable of the polynomials over 𝑅, 1 is the 𝑁 x 𝑁 identity matrix as matrix over the polynomial ring over the ring 𝑅 (not the 𝑁 x 𝑁 identity matrix of the matrices over the ring 𝑅!) and (𝑇‘𝑀) = ((𝑁 matToPolyMat 𝑅)‘𝑀) is the matrix 𝑀 over a ring 𝑅 transformed into a constant matrix over the polynomial ring over the ring 𝑅. Thus · is the multiplication of a polynomial matrix with a "scalar", i.e. a polynomial (see chpmatval 23179).

By this definition, it is ensured that ((𝑋 · 1 ) − (𝑇‘𝑀)), the matrix whose determinant is the characteristic polynomial of the matrix 𝑀, is actually a matrix over the polynomial ring over the ring 𝑅, as stated in Wikipedia ("matrix with polynomials as entries"). This matrix is called the characteristic matrix of matrix 𝑀 (see Wikipedia "Polynomial matrix", 16-Nov-2019, https://en.wikipedia.org/wiki/Polynomial_matrix 23179). Following the notation in Wikipedia, we denote the characteristic polynomial of the matrix 𝑀, formally defined by ((𝑁 CharPlyMat 𝑅)‘𝑀) as "p(M)" in the comments. The characteristic matrix ((𝑋 · 1 ) − (𝑇‘𝑀)) will be denoted by "c(M)", so that "p(M) = det( c(M) )".

After the preliminaries are clarified, the objective of the Cayley-Hamilton theorem must be considered. As described in Wikipedia, the matrix 𝑀 must be "inserted" into its characteristic polynomial in an appropriate way. Since every polynomial can be represented as infinite, but finitely supported sum of monomials scaled by the corresponding coefficients (see ply1coe 22646), also the characteristic polynomial can be written in this way:

p(M) = sum_i ( p_i * M^i )

Here, * is the scalar multiplication in the algebra of the polynomials over the ring 𝑅, and the coefficients are elements of the ring 𝑅.

By this, we can "define" the insertion of the matrix M into its characteristic polynomial by "p(M) = sum( p_i * M^i)", see also cayleyhamilton1 23240. Here, * is the scalar multiplication in the algebra of the matrices over the ring 𝑅.

To prove the Cayley-Hamilton theorem, we have to show that "p(M) = 0", where 0 is the zero matrix.

In this section, the following class variables and informal identifiers (acronyms in the form "A(B)" or "A_B") are used:

class variable/ informal identifier	definiens	explanation
𝑁		An arbitrary finite set, used as dimension for matrices
𝑅		An arbitrary (commutative) ring: 𝑅 ∈ CRing
B(R)	(Base‘𝑅)	Base set of (the ring) 𝑅
𝐴	(𝑁 Mat 𝑅)	Algebra of 𝑁 x 𝑁 matrices over (the ring) 𝑅
𝐵	(Base‘𝐴)	Base set of the algebra of 𝑁 x 𝑁 matrices, i .e. the set of all 𝑁 x 𝑁 matrices
𝑀		An arbitrary 𝑁 x 𝑁 matrix
𝑃	(Poly₁‘𝑅)	The algebra of polynomials over (the ring) 𝑅
B(P)	(Base‘𝑃)	Base set of the algebra of polynomials, i .e. the set of all polynomials
𝑋, X_R	(var₁‘𝑅)	The variable of polynomials over (the ring) 𝑅
𝑌, X_A	(var₁‘𝐴)	The variable of polynomials over matrices of the algebra 𝐴
↑	(._g‘(mulGrp‘𝑃))	The group exponentiation for polynomials over (the ring) 𝑅
^		Arbitrary group exponentiation
𝑈	(algSc‘𝑃)	The injection of scalars, i.e. elements of (the ring) 𝑅 into the base set of the algebra of polynomials over 𝑅
(𝑈‘𝑝), S(p)	((algSc‘𝑃)‘𝑝)	The element 𝑝 of (the ring) 𝑅 represented as polynomial over 𝑅
𝑌	(𝑁 Mat 𝑃)	Algebra of 𝑁 x 𝑁 matrices over (the polynomial ring) 𝑃 over the ring 𝑅
B(Y)	(Base‘𝑌)	Base set of the algebra of polynomial 𝑁 x 𝑁 matrices, i .e. the set of all polynomial 𝑁 x 𝑁 matrices
𝑄	(Poly₁‘𝐴)	Algebra of polynomials over the ring of 𝑁 x 𝑁 matrices over the ring 𝑅
B(Q)	(Base‘𝑄)	Base set of the algebra of polynomials over the ring of 𝑁 x 𝑁 matrices over the ring 𝑅, i .e. the set of all polynomials having 𝑁 x 𝑁 matrices as coefficients
+, +	(+_g‘𝑌)	The addition of polynomial matrices
−, -	(-_g‘𝑌)	The subtraction of polynomial matrices
·, *_Y	( ·_𝑠 ‘𝑌)	The multiplication of a polynomial matrix with a scalar ( i. e. a polynomial)
*_A	( ·_𝑠 ‘𝐴)	The multiplication of a matrix with a scalar ( i. e. an element of the underlying ring)
*_Q	( ·_𝑠 ‘𝑄)	The multiplication of a polynomial over matrices with a scalar ( i. e. a matrix)
×, x_Y	(._r‘𝑌)	The multiplication of polynomial matrices
x_A	(._r‘𝐴)	The multiplication of matrices
x_Q	(._r‘𝑄)	The multiplication of polynomials over matrices
1, 1_Y	(1_r‘𝑌)	The identity matrix in the algebra of polynomial matrices over 𝑅
1_A	(1_r‘𝐴)	The identity matrix in the algebra of matrices over 𝑅
0, 0_Y	(0_g‘𝑌)	The zero matrix in the algebra of matrices consisting of polynomials
𝑇	(𝑁 matToPolyMat 𝑅)	The transformation of an 𝑁 x 𝑁 matrix over 𝑅 into a polynomial 𝑁 x 𝑁 matrix over 𝑅
T₁(M)	(𝑇‘𝑀)	The matrix M transformed into a polynomial 𝑁 x 𝑁 matrix over 𝑅
U(M)	(𝑈‘𝑀)	The (constant) polynomial 𝑁 x 𝑁 matrix M transformed into a matrix over the ring 𝑅. Inverse function of 𝑇: (𝑇‘(𝑈‘𝑀)) = 𝑀 (see m2cpminvid2 23103 )
T₂(M)	((𝑁 pMatToMatPoly 𝑅)‘𝑀)	The polynomial 𝑁 x 𝑁 matrix M transformed into a polynomial over the 𝑁 x 𝑁 matrices over 𝑅
𝐼, c(M)	((𝑋 · 1 ) − (𝑇‘𝑀))	The characteristic matrix of a matrix 𝑀, i.e. the matrix whose determinant is the characteristic polynomial of the matrix 𝑀
𝐶	(𝑁 CharPlyMat 𝑅)	The function mapping a matrix over (a ring) 𝑅 to its characteristic polynomial
𝐾, p(M)	(𝐶‘𝑀)	The characteristic polynomial of a matrix over (a ring) 𝑅
𝐻	(𝐾 · 1 )	The scalar matrix (diagonal matrix) with the characteristic polynomial of a matrix as diagonal elements
𝐽	(𝑁 maAdju 𝑃)	The function mapping a matrix consisting of polynomials to its adjugate ("matrix of cofactors")
𝑊, adj(cm(M))	(𝐽‘𝐼)	The adjugate of the characteristic matrix of the matrix 𝑀

Using this notation, we have:

"c(M) e. B(Y)", or 𝐼 ∈ (Base‘𝑌), see chmatcl 23176
"p(M) e. B(P)", or 𝐾 ∈ (Base‘𝑃), see chpmatply1 23180
"T(M) e. B(Y)", or (𝑇‘𝑀) ∈ (Base‘𝑌), see mat2pmatbas 23074
𝐽:(Base‘𝑌)⟶(Base‘𝑌), see maduf 22986
"adj(cm(M)) e. B(Y)", or 𝑊 ∈ (Base‘𝑌)

Following the proof shown in Wikipedia, the following steps are performed:

Write 𝑊, the adjugate of the characteristic matrix, as a finite sum of scaled monomials, see pmatcollpw3fi1 23136:
adj(cm(M)) = sum_{i=0 to s} ( X_R ^i *_Y T₁(b(i)) )
where b(i) are matrices over the ring 𝑅, so T₁(b(i)) are constant polynomial matrices.
This step corresponds to (3) in Wikipedia. In contrast to Wikipedia, we write 𝑊 as finite sum of not exactly determined number of summands, which may be greater than needed (including summands of value 0). This will be sufficient to provide a representation of (𝐼 × 𝑊) as infinite, but finitely supported sum, see step 3.
Write (𝐼 × 𝑊), the product of the characteristic matrix and its adjugate as finite sum of scaled monomials, see cpmadugsumfi 23225. This representation is obtained by replacing 𝑊 by the representation resulting from step 1. and performing calculation rules available for the associative algebra of matrices over polynomials over a commutative ring:
cm(M) *_Y adj(cm(M)) = sum_{i=0 to s} ( X_R ^i *_Y ( T₁(b(i-1)) - T₁(M) x_Y T₁(b(i)) ) ) + X_R ^(s+1) *_Y ( T₁(b(s)) - T₁(M) x_Y T₁(b(0))
where b(i) are matrices over 𝑅, so T₁(b(i)) are constant polynomial matrices:
cm(M) *_Y adj(cm(M))
= cm(M) *_Y sum_{i=0 to s} ( X_R ^i *_Y T₁(b(i)) ) [see pmatcollpw3fi1 23136 (step 1.)]
= ( ( X_A *_Y 1_Y ) - T₁(M) ) *_Y sum_{i=0 to s} ( X_R ^i *_Y T₁(b(i)) ) [def. of cm(M)]
= ( X_A *_Y 1_Y ) *_Y sum_{i=0 to s} ( X_R ^i *_Y T₁(b(i)) ) - T₁(M) *_Y sum_{i=0 to s} ( X_R ^i *_Y T₁(b(i)) ) [see ringsubdir 20569]
= sum_{i=0 to s} ( X_R ^i *_Y ( T₁(b(i-1)) - T₁(M) x_Y T₁(b(i)) ) ) + X_R ^(s+1) *_Y ( T₁(b(s)) - T₁(M) x_Y T₁(b(0)) [see cpmadugsumlemF 23224]
This step corresponds partially to (4) in Wikipedia.
Write (𝐼 × 𝑊) as infinite, but finitely supported sum of scaled monomials, see cpmadugsum 23226:
cm(M) * adj(cm(M)) = sum_i ( X_R ^i *_Y G(i) )
This representation is obtained by defining a function G for the coefficients, which we call "characteristic factor function", see chfacfisf 23202, which covers the special terms and the padding with 0. G(i) is a constant polynomial matrix (see chfacfisfcpmat 23203). This step corresponds partially to (4) in Wikipedia, with summands of value 0 added.
Write 𝐻 = (𝐾 · 1 ), the scalar matrix (diagonal matrix) with the characteristic polynomial of a matrix as diagonal elements, as infinite, but finitely supported sum of scaled monomials. See cpmidgsum 23216:
p(m) *_Y I_Y = sum_i ( X_R ^i *_Y ( S(p_i) *_Y I_Y ) )
The proof of cpmidgsum 23216 is making use of pmatcollpwscmat 23139, because 𝐻 = (𝐾 · 1 ) is a scalar/diagonal polynomial matrix with the characteristic polynomial "p(M)" as diagonal entries (since p_i is an element of the ring 𝑅, S(p_i) is a (constant) polynomial). This corresponds to (5) in Wikipedia, with summands of value 0 added.
Transform the sum representation of (𝐼 × 𝑊) from step 3. into polynomials over matrices:
T₂(cm(M) * adj(cm(M))) = sum_i ( U(G(i)) *_Q X_A ^i ) [see cpmadumatpoly 23231]
where U(G(i)) is a matrix over the ring 𝑅.
Transform the sum representation of 𝐻 from step 4. into polynomials over matrices:
T₂(p(m) *_Y I_Y) = sum_i ( p_i *_A I_A ) *_Q X_A ^i ) [see cpmidpmat 23221]
Equate the sum representations resulting from steps 5. and 6. by using cpmadurid 23215 to obtain the equation
sum_i ( U(G(i)) *_Q X_A ^i ) = sum_i ( p_i *_A I_A ) *_Q X_A ^i ):
sum_i ( U(G(i)) *_Q X_A ^i )
= T₂(cm(M) * adj(cm(M))) [see step 5.]
= T₂(p(m) *_Y I_Y) [see cpmadurid 23215]
= sum_i ( p_i *_A I_A ) *_Q X_A ^i ) [see step 6.]
Note that this step is contained in the proof of chcoeffeq 23234, see step 9. This step corresponds to the conclusion from (4) and (5) in Wikipedia, with summands of value 0 added.
Compare the sum representations of step 7. to obtain the equations U(G(i)) = p_i *_A I_A , see chcoeffeqlem 23233. This corresponds to (6) in Wikipedia. Since the coefficients of the transformed representations and the original representations are identical, the equations of the coefficients are also valid for the original representations of steps 3. and 4.
Multiply the equations of the coefficients from step 8. from the left by M^i, and sum up, see chcoeffeq 23234:
sum_i ( M^i x_A U(G(i)) ) = sum_i ( M^i x_A ( p_i *_A I_A) )
This corresponds to (7) in Wikipedia.
Transform the right hand side of the equation in step 9. into an appropriate form, see cayhamlem3 23235:
sum_i ( p_i *_A M^i )
= sum_i ( M^i x_A ( p_i *_A I_A) ) [see cayhamlem2 23232]
= sum_i ( M^i x_A U(G(i)) ) [see chcoeffeq 23234]
Apply the theorem for telescoping sums, see telgsumfz 20234, to the sum sum_i ( T₁(M)^i x_Y G(i) ), which results in an equation to 0:
sum_i ( T₁(M)^i x_Y G(i) ) = 0_Y, see cayhamlem1 23214:
sum_i ( T₁(M)^i x_Y G(i) )
= sum_{i=1 to s} ( T₁(M)^i x_Y T₁(b(i-1)) - T₁(M)^(i+1) x_Y T₁(b(i)) )
+ ( T₁(M)^(s+1) x_Y T₁(b(s)) - T₁(M) x_Y T₁(b(0)) ) [see chfacfpmmulgsum2 23213]
= ( T₁(M) x_Y T₁(b(0)) - T₁(M)^(s+1) x_Y T₁(b(s)) ) + ( T₁ M)^(s+1) x_Y T₁(b(s)) - T₁(M) x_Y T₁(b(0)) ) [see telgsumfz 20234]
= 0_Y [see grpnpncan0 19275] This step corresponds partially to (8) in Wikipedia.
Since 𝑇 is a ring homomorphism (see mat2pmatrhm 23082), the left hand side of the equation in step 10. can be transformed into a representation appropriate to apply the result of step 11., see cayhamlem4 23236:
sum_i ( p_i *_A M^i )
= sum_i ( M^i x_A U(G(i)) ) [see cayhamlem3 23235 (step 10.)]
= U(T₁(sum_i ( M^i x_A U(G(i)) ))) [see m2cpminvid 23101]
= U(sum_i T₁( M^i x_A U(G(i)) )) [see gsummptmhm 20184]
= U(sum_i ( T₁(M^i) x_Y T₁(U(G(i))) )) [see rhmmul 20750]
= U(sum_i ( T₁(M)^i x_Y T₁(U(G(i))) )) [see mhmmulg 19354]
= U(sum_i ( T₁(M)^i x_Y G(i) )) [see m2cpminvid2 23103 ]
Finally, combine the results of steps 11. and 12., and use the fact that 𝑇 (and therefore also its inverse 𝑈) is an injective ring homomorphism (see mat2pmatf1 23077 and mat2pmatrhm 23082) to transform the equality resulting from steps 11. and 12. into the desired equation sum_i ( p_i *_A M^i ) = 0_A , see cayleyhamilton 23238 resp. cayleyhamilton0 23237:
sum_i ( p_i *_A M^i )
= U(sum_i ( T₁(M)^i x_Y G(i) )) [see cayhamlem4 23236 (step 12.)]
= U(0_Y ) [see cayhamlem1 23214 (step 11.)]
= 0_A [see m2cpminv0 23109]

The transformations in steps 5., 6., 10., 12. and 13. are not mentioned in the proof provided in Wikipedia, since it makes no distinction between a matrix over a ring 𝑅 and its representation as matrix over the polynomial ring over the ring 𝑅 in general!

Theorem

cpmadurid 23215

The right-hand fundamental relation of the adjugate (see madurid 22989) applied to the characteristic matrix of a matrix. (Contributed by AV, 25-Oct-2019.)

𝐴 = (𝑁 Mat 𝑅) & 𝐵 = (Base‘𝐴) & 𝐶 = (𝑁 CharPlyMat 𝑅) & 𝑃 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & 𝑋 = (var₁‘𝑅) & 𝑇 = (𝑁 matToPolyMat 𝑅) & − = (-_g‘𝑌) & · = ( ·_𝑠 ‘𝑌) & 1 = (1_r‘𝑌) & 𝐼 = ((𝑋 · 1 ) − (𝑇‘𝑀)) & 𝐽 = (𝑁 maAdju 𝑃) & × = (._r‘𝑌) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐼 × (𝐽‘𝐼)) = ((𝐶‘𝑀) · 1 ))

Theorem

cpmidgsum 23216*

Representation of the identity matrix multiplied with the characteristic polynomial of a matrix as group sum. (Contributed by AV, 7-Nov-2019.)

𝐴 = (𝑁 Mat 𝑅) & 𝐵 = (Base‘𝐴) & 𝑃 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & 𝑋 = (var₁‘𝑅) & ↑ = (._g‘(mulGrp‘𝑃)) & · = ( ·_𝑠 ‘𝑌) & 1 = (1_r‘𝑌) & 𝑈 = (algSc‘𝑃) & 𝐶 = (𝑁 CharPlyMat 𝑅) & 𝐾 = (𝐶‘𝑀) & 𝐻 = (𝐾 · 1 ) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐻 = (𝑌 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛 ↑ 𝑋) · ((𝑈‘((coe₁‘𝐾)‘𝑛)) · 1 )))))

Theorem

cpmidgsumm2pm 23217*

Representation of the identity matrix multiplied with the characteristic polynomial of a matrix as group sum with a matrix to polynomial matrix transformation. (Contributed by AV, 13-Nov-2019.)

𝐴 = (𝑁 Mat 𝑅) & 𝐵 = (Base‘𝐴) & 𝑃 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & 𝑋 = (var₁‘𝑅) & ↑ = (._g‘(mulGrp‘𝑃)) & · = ( ·_𝑠 ‘𝑌) & 1 = (1_r‘𝑌) & 𝑈 = (algSc‘𝑃) & 𝐶 = (𝑁 CharPlyMat 𝑅) & 𝐾 = (𝐶‘𝑀) & 𝐻 = (𝐾 · 1 ) & 𝑂 = (1_r‘𝐴) & ∗ = ( ·_𝑠 ‘𝐴) & 𝑇 = (𝑁 matToPolyMat 𝑅) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐻 = (𝑌 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛 ↑ 𝑋) · (𝑇‘(((coe₁‘𝐾)‘𝑛) ∗ 𝑂))))))

Theorem

cpmidpmatlem1 23218*

Lemma 1 for cpmidpmat 23221. (Contributed by AV, 13-Nov-2019.)

𝐴 = (𝑁 Mat 𝑅) & 𝐵 = (Base‘𝐴) & 𝑃 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & 𝑋 = (var₁‘𝑅) & ↑ = (._g‘(mulGrp‘𝑃)) & · = ( ·_𝑠 ‘𝑌) & 1 = (1_r‘𝑌) & 𝑈 = (algSc‘𝑃) & 𝐶 = (𝑁 CharPlyMat 𝑅) & 𝐾 = (𝐶‘𝑀) & 𝐻 = (𝐾 · 1 ) & 𝑂 = (1_r‘𝐴) & ∗ = ( ·_𝑠 ‘𝐴) & 𝑇 = (𝑁 matToPolyMat 𝑅) & 𝐺 = (𝑘 ∈ ℕ₀ ↦ (((coe₁‘𝐾)‘𝑘) ∗ 𝑂)) ⇒ (𝐿 ∈ ℕ₀ → (𝐺‘𝐿) = (((coe₁‘𝐾)‘𝐿) ∗ 𝑂))

Theorem

cpmidpmatlem2 23219*

Lemma 2 for cpmidpmat 23221. (Contributed by AV, 14-Nov-2019.) (Proof shortened by AV, 7-Dec-2019.)

𝐴 = (𝑁 Mat 𝑅) & 𝐵 = (Base‘𝐴) & 𝑃 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & 𝑋 = (var₁‘𝑅) & ↑ = (._g‘(mulGrp‘𝑃)) & · = ( ·_𝑠 ‘𝑌) & 1 = (1_r‘𝑌) & 𝑈 = (algSc‘𝑃) & 𝐶 = (𝑁 CharPlyMat 𝑅) & 𝐾 = (𝐶‘𝑀) & 𝐻 = (𝐾 · 1 ) & 𝑂 = (1_r‘𝐴) & ∗ = ( ·_𝑠 ‘𝐴) & 𝑇 = (𝑁 matToPolyMat 𝑅) & 𝐺 = (𝑘 ∈ ℕ₀ ↦ (((coe₁‘𝐾)‘𝑘) ∗ 𝑂)) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐺 ∈ (𝐵 ↑_m ℕ₀))

Theorem

cpmidpmatlem3 23220*

Lemma 3 for cpmidpmat 23221. (Contributed by AV, 14-Nov-2019.) (Proof shortened by AV, 7-Dec-2019.)

𝐴 = (𝑁 Mat 𝑅) & 𝐵 = (Base‘𝐴) & 𝑃 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & 𝑋 = (var₁‘𝑅) & ↑ = (._g‘(mulGrp‘𝑃)) & · = ( ·_𝑠 ‘𝑌) & 1 = (1_r‘𝑌) & 𝑈 = (algSc‘𝑃) & 𝐶 = (𝑁 CharPlyMat 𝑅) & 𝐾 = (𝐶‘𝑀) & 𝐻 = (𝐾 · 1 ) & 𝑂 = (1_r‘𝐴) & ∗ = ( ·_𝑠 ‘𝐴) & 𝑇 = (𝑁 matToPolyMat 𝑅) & 𝐺 = (𝑘 ∈ ℕ₀ ↦ (((coe₁‘𝐾)‘𝑘) ∗ 𝑂)) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐺 finSupp (0_g‘𝐴))

Theorem

cpmidpmat 23221*

Representation of the identity matrix multiplied with the characteristic polynomial of a matrix as polynomial over the ring of matrices. (Contributed by AV, 14-Nov-2019.) (Revised by AV, 7-Dec-2019.)

𝐴 = (𝑁 Mat 𝑅) & 𝐵 = (Base‘𝐴) & 𝑃 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & 𝑋 = (var₁‘𝑅) & ↑ = (._g‘(mulGrp‘𝑃)) & · = ( ·_𝑠 ‘𝑌) & 1 = (1_r‘𝑌) & 𝑈 = (algSc‘𝑃) & 𝐶 = (𝑁 CharPlyMat 𝑅) & 𝐾 = (𝐶‘𝑀) & 𝐻 = (𝐾 · 1 ) & 𝑂 = (1_r‘𝐴) & ∗ = ( ·_𝑠 ‘𝐴) & 𝑇 = (𝑁 matToPolyMat 𝑅) & 𝑊 = (Base‘𝑌) & 𝑄 = (Poly₁‘𝐴) & 𝑍 = (var₁‘𝐴) & ∙ = ( ·_𝑠 ‘𝑄) & 𝐸 = (._g‘(mulGrp‘𝑄)) & 𝐼 = (𝑁 pMatToMatPoly 𝑅) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐼‘𝐻) = (𝑄 Σ_g (𝑛 ∈ ℕ₀ ↦ ((((coe₁‘𝐾)‘𝑛) ∗ 𝑂) ∙ (𝑛𝐸𝑍)))))

Theorem

cpmadugsumlemB 23222*

Lemma B for cpmadugsum 23226. (Contributed by AV, 2-Nov-2019.)

𝐴 = (𝑁 Mat 𝑅) & 𝐵 = (Base‘𝐴) & 𝑃 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & 𝑇 = (𝑁 matToPolyMat 𝑅) & 𝑋 = (var₁‘𝑅) & ↑ = (._g‘(mulGrp‘𝑃)) & · = ( ·_𝑠 ‘𝑌) & × = (._r‘𝑌) & 1 = (1_r‘𝑌) ⇒ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑋 · 1 ) × (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))) = (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))))

Theorem

cpmadugsumlemC 23223*

Lemma C for cpmadugsum 23226. (Contributed by AV, 2-Nov-2019.)

𝐴 = (𝑁 Mat 𝑅) & 𝐵 = (Base‘𝐴) & 𝑃 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & 𝑇 = (𝑁 matToPolyMat 𝑅) & 𝑋 = (var₁‘𝑅) & ↑ = (._g‘(mulGrp‘𝑃)) & · = ( ·_𝑠 ‘𝑌) & × = (._r‘𝑌) & 1 = (1_r‘𝑌) ⇒ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑇‘𝑀) × (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))) = (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))))

Theorem

cpmadugsumlemF 23224*

Lemma F for cpmadugsum 23226. (Contributed by AV, 7-Nov-2019.)

𝐴 = (𝑁 Mat 𝑅) & 𝐵 = (Base‘𝐴) & 𝑃 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & 𝑇 = (𝑁 matToPolyMat 𝑅) & 𝑋 = (var₁‘𝑅) & ↑ = (._g‘(mulGrp‘𝑃)) & · = ( ·_𝑠 ‘𝑌) & × = (._r‘𝑌) & 1 = (1_r‘𝑌) & + = (+_g‘𝑌) & − = (-_g‘𝑌) ⇒ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (((𝑋 · 1 ) × (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))) − ((𝑇‘𝑀) × (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))

Theorem

cpmadugsumfi 23225*

The product of the characteristic matrix of a given matrix and its adjunct represented as finite sum. (Contributed by AV, 7-Nov-2019.) (Proof shortened by AV, 29-Nov-2019.)

𝐴 = (𝑁 Mat 𝑅) & 𝐵 = (Base‘𝐴) & 𝑃 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & 𝑇 = (𝑁 matToPolyMat 𝑅) & 𝑋 = (var₁‘𝑅) & ↑ = (._g‘(mulGrp‘𝑃)) & · = ( ·_𝑠 ‘𝑌) & × = (._r‘𝑌) & 1 = (1_r‘𝑌) & + = (+_g‘𝑌) & − = (-_g‘𝑌) & 𝐼 = ((𝑋 · 1 ) − (𝑇‘𝑀)) & 𝐽 = (𝑁 maAdju 𝑃) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑_m (0...𝑠))(𝐼 × (𝐽‘𝐼)) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))

Theorem

cpmadugsum 23226*

The product of the characteristic matrix of a given matrix and its adjunct represented as an infinite sum. (Contributed by AV, 10-Nov-2019.)

𝐴 = (𝑁 Mat 𝑅) & 𝐵 = (Base‘𝐴) & 𝑃 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & 𝑇 = (𝑁 matToPolyMat 𝑅) & 𝑋 = (var₁‘𝑅) & ↑ = (._g‘(mulGrp‘𝑃)) & · = ( ·_𝑠 ‘𝑌) & × = (._r‘𝑌) & 1 = (1_r‘𝑌) & + = (+_g‘𝑌) & − = (-_g‘𝑌) & 𝐼 = ((𝑋 · 1 ) − (𝑇‘𝑀)) & 𝐽 = (𝑁 maAdju 𝑃) & 0 = (0_g‘𝑌) & 𝐺 = (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑_m (0...𝑠))(𝐼 × (𝐽‘𝐼)) = (𝑌 Σ_g (𝑖 ∈ ℕ₀ ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))))

Theorem

cpmidgsum2 23227*

Representation of the identity matrix multiplied with the characteristic polynomial of a matrix as another group sum. (Contributed by AV, 10-Nov-2019.)

𝐴 = (𝑁 Mat 𝑅) & 𝐵 = (Base‘𝐴) & 𝑃 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & 𝑇 = (𝑁 matToPolyMat 𝑅) & 𝑋 = (var₁‘𝑅) & ↑ = (._g‘(mulGrp‘𝑃)) & · = ( ·_𝑠 ‘𝑌) & × = (._r‘𝑌) & 1 = (1_r‘𝑌) & + = (+_g‘𝑌) & − = (-_g‘𝑌) & 𝐼 = ((𝑋 · 1 ) − (𝑇‘𝑀)) & 𝐽 = (𝑁 maAdju 𝑃) & 0 = (0_g‘𝑌) & 𝐺 = (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) & 𝐶 = (𝑁 CharPlyMat 𝑅) & 𝐾 = (𝐶‘𝑀) & 𝐻 = (𝐾 · 1 ) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑_m (0...𝑠))𝐻 = (𝑌 Σ_g (𝑖 ∈ ℕ₀ ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))))

Theorem

cpmidg2sum 23228*

Equality of two sums representing the identity matrix multiplied with the characteristic polynomial of a matrix. (Contributed by AV, 11-Nov-2019.)

𝐴 = (𝑁 Mat 𝑅) & 𝐵 = (Base‘𝐴) & 𝑃 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & 𝑇 = (𝑁 matToPolyMat 𝑅) & 𝑋 = (var₁‘𝑅) & ↑ = (._g‘(mulGrp‘𝑃)) & · = ( ·_𝑠 ‘𝑌) & × = (._r‘𝑌) & 1 = (1_r‘𝑌) & + = (+_g‘𝑌) & − = (-_g‘𝑌) & 𝐼 = ((𝑋 · 1 ) − (𝑇‘𝑀)) & 𝐽 = (𝑁 maAdju 𝑃) & 0 = (0_g‘𝑌) & 𝐺 = (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) & 𝐶 = (𝑁 CharPlyMat 𝑅) & 𝐾 = (𝐶‘𝑀) & 𝑈 = (algSc‘𝑃) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑_m (0...𝑠))(𝑌 Σ_g (𝑖 ∈ ℕ₀ ↦ ((𝑖 ↑ 𝑋) · ((𝑈‘((coe₁‘𝐾)‘𝑖)) · 1 )))) = (𝑌 Σ_g (𝑖 ∈ ℕ₀ ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))))

Theorem

cpmadumatpolylem1 23229*

Lemma 1 for cpmadumatpoly 23231. (Contributed by AV, 20-Nov-2019.) (Revised by AV, 15-Dec-2019.)

𝐴 = (𝑁 Mat 𝑅) & 𝐵 = (Base‘𝐴) & 𝑃 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & 𝑇 = (𝑁 matToPolyMat 𝑅) & × = (._r‘𝑌) & − = (-_g‘𝑌) & 0 = (0_g‘𝑌) & 𝐺 = (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) & 𝑆 = (𝑁 ConstPolyMat 𝑅) & · = ( ·_𝑠 ‘𝑌) & 1 = (1_r‘𝑌) & 𝑍 = (var₁‘𝑅) & 𝐷 = ((𝑍 · 1 ) − (𝑇‘𝑀)) & 𝐽 = (𝑁 maAdju 𝑃) & 𝑊 = (Base‘𝑌) & 𝑄 = (Poly₁‘𝐴) & 𝑋 = (var₁‘𝐴) & ∗ = ( ·_𝑠 ‘𝑄) & ↑ = (._g‘(mulGrp‘𝑄)) & 𝑈 = (𝑁 cPolyMatToMat 𝑅) ⇒ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → (𝑈 ∘ 𝐺) ∈ (𝐵 ↑_m ℕ₀))

Theorem

cpmadumatpolylem2 23230*

Lemma 2 for cpmadumatpoly 23231. (Contributed by AV, 20-Nov-2019.) (Revised by AV, 15-Dec-2019.)

Theorem

cpmadumatpoly 23231*

The product of the characteristic matrix of a given matrix and its adjunct represented as a polynomial over matrices. (Contributed by AV, 20-Nov-2019.) (Revised by AV, 7-Dec-2019.)

𝐴 = (𝑁 Mat 𝑅) & 𝐵 = (Base‘𝐴) & 𝑃 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & 𝑇 = (𝑁 matToPolyMat 𝑅) & × = (._r‘𝑌) & − = (-_g‘𝑌) & 0 = (0_g‘𝑌) & 𝐺 = (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) & 𝑆 = (𝑁 ConstPolyMat 𝑅) & · = ( ·_𝑠 ‘𝑌) & 1 = (1_r‘𝑌) & 𝑍 = (var₁‘𝑅) & 𝐷 = ((𝑍 · 1 ) − (𝑇‘𝑀)) & 𝐽 = (𝑁 maAdju 𝑃) & 𝑊 = (Base‘𝑌) & 𝑄 = (Poly₁‘𝐴) & 𝑋 = (var₁‘𝐴) & ∗ = ( ·_𝑠 ‘𝑄) & ↑ = (._g‘(mulGrp‘𝑄)) & 𝑈 = (𝑁 cPolyMatToMat 𝑅) & 𝐼 = (𝑁 pMatToMatPoly 𝑅) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑_m (0...𝑠))(𝐼‘(𝐷 × (𝐽‘𝐷))) = (𝑄 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑈‘(𝐺‘𝑛)) ∗ (𝑛 ↑ 𝑋)))))

Theorem

cayhamlem2 23232

Lemma for cayhamlem3 23235. (Contributed by AV, 24-Nov-2019.)

𝐾 = (Base‘𝑅) & 𝐴 = (𝑁 Mat 𝑅) & 𝐵 = (Base‘𝐴) & 1 = (1_r‘𝐴) & ∗ = ( ·_𝑠 ‘𝐴) & ↑ = (._g‘(mulGrp‘𝐴)) & · = (._r‘𝐴) ⇒ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐻 ∈ (𝐾 ↑_m ℕ₀) ∧ 𝐿 ∈ ℕ₀)) → ((𝐻‘𝐿) ∗ (𝐿 ↑ 𝑀)) = ((𝐿 ↑ 𝑀) · ((𝐻‘𝐿) ∗ 1 )))

Theorem

chcoeffeqlem 23233*

Lemma for chcoeffeq 23234. (Contributed by AV, 21-Nov-2019.) (Proof shortened by AV, 7-Dec-2019.) (Revised by AV, 15-Dec-2019.)

𝐴 = (𝑁 Mat 𝑅) & 𝐵 = (Base‘𝐴) & 𝑃 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & × = (._r‘𝑌) & − = (-_g‘𝑌) & 0 = (0_g‘𝑌) & 𝑇 = (𝑁 matToPolyMat 𝑅) & 𝐶 = (𝑁 CharPlyMat 𝑅) & 𝐾 = (𝐶‘𝑀) & 𝐺 = (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) & 𝑊 = (Base‘𝑌) & 1 = (1_r‘𝐴) & ∗ = ( ·_𝑠 ‘𝐴) & 𝑈 = (𝑁 cPolyMatToMat 𝑅) ⇒ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (((Poly₁‘𝐴) Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑈‘(𝐺‘𝑛))( ·_𝑠 ‘(Poly₁‘𝐴))(𝑛(._g‘(mulGrp‘(Poly₁‘𝐴)))(var₁‘𝐴))))) = ((Poly₁‘𝐴) Σ_g (𝑛 ∈ ℕ₀ ↦ ((((coe₁‘𝐾)‘𝑛) ∗ 1 )( ·_𝑠 ‘(Poly₁‘𝐴))(𝑛(._g‘(mulGrp‘(Poly₁‘𝐴)))(var₁‘𝐴))))) → ∀𝑛 ∈ ℕ₀ (𝑈‘(𝐺‘𝑛)) = (((coe₁‘𝐾)‘𝑛) ∗ 1 )))

Theorem

chcoeffeq 23234*

The coefficients of the characteristic polynomial multiplied with the identity matrix represented by (transformed) ring elements obtained from the adjunct of the characteristic matrix. (Contributed by AV, 21-Nov-2019.) (Proof shortened by AV, 8-Dec-2019.) (Revised by AV, 15-Dec-2019.)

Theorem

cayhamlem3 23235*

Lemma for cayhamlem4 23236. (Contributed by AV, 24-Nov-2019.) (Revised by AV, 15-Dec-2019.)

𝐴 = (𝑁 Mat 𝑅) & 𝐵 = (Base‘𝐴) & 𝑃 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & × = (._r‘𝑌) & − = (-_g‘𝑌) & 0 = (0_g‘𝑌) & 𝑇 = (𝑁 matToPolyMat 𝑅) & 𝐶 = (𝑁 CharPlyMat 𝑅) & 𝐾 = (𝐶‘𝑀) & 𝐺 = (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) & 𝑊 = (Base‘𝑌) & 1 = (1_r‘𝐴) & ∗ = ( ·_𝑠 ‘𝐴) & 𝑈 = (𝑁 cPolyMatToMat 𝑅) & ↑ = (._g‘(mulGrp‘𝐴)) & · = (._r‘𝐴) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑_m (0...𝑠))(𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ (((coe₁‘𝐾)‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛 ↑ 𝑀) · (𝑈‘(𝐺‘𝑛))))))

Theorem

cayhamlem4 23236*

Lemma for cayleyhamilton 23238. (Contributed by AV, 25-Nov-2019.) (Revised by AV, 15-Dec-2019.)

𝐴 = (𝑁 Mat 𝑅) & 𝐵 = (Base‘𝐴) & 𝑃 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & × = (._r‘𝑌) & − = (-_g‘𝑌) & 0 = (0_g‘𝑌) & 𝑇 = (𝑁 matToPolyMat 𝑅) & 𝐶 = (𝑁 CharPlyMat 𝑅) & 𝐾 = (𝐶‘𝑀) & 𝐺 = (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) & 𝑊 = (Base‘𝑌) & 1 = (1_r‘𝐴) & ∗ = ( ·_𝑠 ‘𝐴) & 𝑈 = (𝑁 cPolyMatToMat 𝑅) & ↑ = (._g‘(mulGrp‘𝐴)) & 𝐸 = (._g‘(mulGrp‘𝑌)) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑_m (0...𝑠))(𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ (((coe₁‘𝐾)‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝑈‘(𝑌 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛))))))

Theorem

cayleyhamilton0 23237*

The Cayley-Hamilton theorem: A matrix over a commutative ring "satisfies its own characteristic equation". This version of cayleyhamilton 23238 provides definitions not used in the theorem itself, but in its proof to make it clearer, more readable and shorter compared with a proof without them (see cayleyhamiltonALT 23239)! (Contributed by AV, 25-Nov-2019.) (Revised by AV, 15-Dec-2019.)

𝐴 = (𝑁 Mat 𝑅) & 𝐵 = (Base‘𝐴) & 0 = (0_g‘𝐴) & 1 = (1_r‘𝐴) & ∗ = ( ·_𝑠 ‘𝐴) & ↑ = (._g‘(mulGrp‘𝐴)) & 𝐶 = (𝑁 CharPlyMat 𝑅) & 𝐾 = (coe₁‘(𝐶‘𝑀)) & 𝑃 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & × = (._r‘𝑌) & − = (-_g‘𝑌) & 𝑍 = (0_g‘𝑌) & 𝑊 = (Base‘𝑌) & 𝐸 = (._g‘(mulGrp‘𝑌)) & 𝑇 = (𝑁 matToPolyMat 𝑅) & 𝐺 = (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, (𝑍 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 𝑍, ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) & 𝑈 = (𝑁 cPolyMatToMat 𝑅) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = 0 )

Theorem

cayleyhamilton 23238*

The Cayley-Hamilton theorem: A matrix over a commutative ring "satisfies its own characteristic equation", see theorem 7.8 in [Roman] p. 170 (without proof!), or theorem 3.1 in [Lang] p. 561. In other words, a matrix over a commutative ring "inserted" into its characteristic polynomial results in zero. This is Metamath 100 proof #49. (Contributed by Alexander van der Vekens, 25-Nov-2019.)

𝐴 = (𝑁 Mat 𝑅) & 𝐵 = (Base‘𝐴) & 0 = (0_g‘𝐴) & 𝐶 = (𝑁 CharPlyMat 𝑅) & 𝐾 = (coe₁‘(𝐶‘𝑀)) & ∗ = ( ·_𝑠 ‘𝐴) & ↑ = (._g‘(mulGrp‘𝐴)) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = 0 )

Theorem

cayleyhamiltonALT 23239*

Alternate proof of cayleyhamilton 23238, the Cayley-Hamilton theorem. This proof does not use cayleyhamilton0 23237 directly, but has the same structure as the proof of cayleyhamilton0 23237. In contrast to the proof of cayleyhamilton0 23237, only the definitions required to formulate the theorem itself are used, causing the definitions used in the lemmas being expanded, which makes the proof longer and more difficult to read. (Contributed by AV, 25-Nov-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

Theorem

cayleyhamilton1 23240*

The Cayley-Hamilton theorem: A matrix over a commutative ring "satisfies its own characteristic equation", or, in other words, a matrix over a commutative ring "inserted" into its characteristic polynomial results in zero. In this variant of cayleyhamilton 23238, the meaning of "inserted" is made more transparent: If the characteristic polynomial is a polynomial with coefficients (𝐹‘𝑛), then a matrix over a commutative ring "inserted" into its characteristic polynomial is the sum of these coefficients multiplied with the corresponding power of the matrix. (Contributed by AV, 25-Nov-2019.)

𝐴 = (𝑁 Mat 𝑅) & 𝐵 = (Base‘𝐴) & 0 = (0_g‘𝐴) & 𝐶 = (𝑁 CharPlyMat 𝑅) & 𝐾 = (coe₁‘(𝐶‘𝑀)) & ∗ = ( ·_𝑠 ‘𝐴) & ↑ = (._g‘(mulGrp‘𝐴)) & 𝐿 = (Base‘𝑅) & 𝑋 = (var₁‘𝑅) & 𝑃 = (Poly₁‘𝑅) & · = ( ·_𝑠 ‘𝑃) & 𝐸 = (._g‘(mulGrp‘𝑃)) & 𝑍 = (0_g‘𝑅) ⇒ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑_m ℕ₀) ∧ 𝐹 finSupp 𝑍)) → ((𝐶‘𝑀) = (𝑃 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋)))) → (𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝐹‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = 0 ))

PART 12 BASIC TOPOLOGY

12.1 Topology

12.1.1 Topological spaces

A topology on a set is a set of subsets of that set, called open sets, which satisfy certain conditions. One condition is that the whole set be an open set. Therefore, a set is recoverable from a topology on it (as its union, see toponuni 23262), and it may sometimes be more convenient to consider topologies without reference to the underlying set. This is why we define successively the class of topologies (df-top 23242), then the function which associates with a set the set of topologies on it (df-topon 23259), and finally the class of topological spaces, as extensible structures having an underlying set and a topology on it (df-topsp 23281). Of course, a topology is the same thing as a topology on a set (see toprntopon 23273).

12.1.1.1 Topologies

Syntax

ctop 23241

Syntax for the class of topologies.

class Top

Definition

df-top 23242*

Define the class of topologies. It is a proper class (see topnex 23344). See istopg 23243 and istop2g 23244 for the corresponding characterizations, using respectively binary intersections like in this definition and nonempty finite intersections.

The final form of the definition is due to Bourbaki (Def. 1 of [BourbakiTop1] p. I.1), while the idea of defining a topology in terms of its open sets is due to Aleksandrov. For the convoluted history of the definitions of these notions, see

Gregory H. Moore, The emergence of open sets, closed sets, and limit points in analysis and topology, Historia Mathematica 35 (2008) 220--241.

(Contributed by NM, 3-Mar-2006.) (Revised by BJ, 20-Oct-2018.)

Top = {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥∪ 𝑦 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥)}

Theorem

istopg 23243*

Express the predicate "𝐽 is a topology". See istop2g 23244 for another characterization using nonempty finite intersections instead of binary intersections.

Note: In the literature, a topology is often represented by a calligraphic letter T, which resembles the letter J. This confusion may have led to J being used by some authors (e.g., K. D. Joshi, Introduction to General Topology (1983), p. 114) and it is convenient for us since we later use 𝑇 to represent linear transformations (operators). (Contributed by Stefan Allan, 3-Mar-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)

(𝐽 ∈ 𝐴 → (𝐽 ∈ Top ↔ (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽)))

Theorem

istop2g 23244*

Express the predicate "𝐽 is a topology" using nonempty finite intersections instead of binary intersections as in istopg 23243. (Contributed by NM, 19-Jul-2006.)

(𝐽 ∈ 𝐴 → (𝐽 ∈ Top ↔ (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) ∧ ∀𝑥((𝑥 ⊆ 𝐽 ∧ 𝑥 ≠ ∅ ∧ 𝑥 ∈ Fin) → ∩ 𝑥 ∈ 𝐽))))

Theorem

uniopn 23245

The union of a subset of a topology (that is, the union of any family of open sets of a topology) is an open set. (Contributed by Stefan Allan, 27-Feb-2006.)

((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐽) → ∪ 𝐴 ∈ 𝐽)

Theorem

iunopn 23246*

The indexed union of a subset of a topology is an open set. (Contributed by NM, 5-Oct-2006.)

((𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐽)

Theorem

inopn 23247

The intersection of two open sets of a topology is an open set. (Contributed by NM, 17-Jul-2006.)

((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → (𝐴 ∩ 𝐵) ∈ 𝐽)

Theorem

fitop 23248

A topology is closed under finite intersections. (Contributed by Jeff Hankins, 7-Oct-2009.)

(𝐽 ∈ Top → (fi‘𝐽) = 𝐽)

Theorem

fiinopn 23249

The intersection of a nonempty finite family of open sets is open. (Contributed by FL, 20-Apr-2012.)

(𝐽 ∈ Top → ((𝐴 ⊆ 𝐽 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin) → ∩ 𝐴 ∈ 𝐽))

Theorem

iinopn 23250*

The intersection of a nonempty finite family of open sets is open. (Contributed by Mario Carneiro, 14-Sep-2014.)

((𝐽 ∈ Top ∧ (𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽)) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐽)

Theorem

unopn 23251

The union of two open sets is open. (Contributed by Jeff Madsen, 2-Sep-2009.)

((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → (𝐴 ∪ 𝐵) ∈ 𝐽)

Theorem

0opn 23252

The empty set is an open subset of any topology. (Contributed by Stefan Allan, 27-Feb-2006.)

(𝐽 ∈ Top → ∅ ∈ 𝐽)

Theorem

0ntop 23253

The empty set is not a topology. (Contributed by FL, 1-Jun-2008.)

¬ ∅ ∈ Top

Theorem

topopn 23254

The underlying set of a topology is an open set. (Contributed by NM, 17-Jul-2006.)

𝑋 = ∪ 𝐽 ⇒ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)

Theorem

eltopss 23255

A member of a topology is a subset of its underlying set. (Contributed by NM, 12-Sep-2006.)

𝑋 = ∪ 𝐽 ⇒ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ 𝑋)

Theorem

riinopn 23256*

A finite indexed relative intersection of open sets is open. (Contributed by Mario Carneiro, 22-Aug-2015.)

𝑋 = ∪ 𝐽 ⇒ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽) → (𝑋 ∩ ∩ 𝑥 ∈ 𝐴 𝐵) ∈ 𝐽)

Theorem

rintopn 23257

A finite relative intersection of open sets is open. (Contributed by Mario Carneiro, 22-Aug-2015.)

𝑋 = ∪ 𝐽 ⇒ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐽 ∧ 𝐴 ∈ Fin) → (𝑋 ∩ ∩ 𝐴) ∈ 𝐽)

12.1.1.2 Topologies on sets

Syntax

ctopon 23258

Syntax for the function of topologies on sets.

class TopOn

Definition

df-topon 23259*

Define the function that associates with a set the set of topologies on it. (Contributed by Stefan O'Rear, 31-Jan-2015.)

TopOn = (𝑏 ∈ V ↦ {𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗})

Theorem

istopon 23260

Property of being a topology with a given base set. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by Mario Carneiro, 13-Aug-2015.)

(𝐽 ∈ (TopOn‘𝐵) ↔ (𝐽 ∈ Top ∧ 𝐵 = ∪ 𝐽))

Theorem

topontop 23261

A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)

(𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top)

Theorem

toponuni 23262

The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.)

(𝐽 ∈ (TopOn‘𝐵) → 𝐵 = ∪ 𝐽)

Theorem

topontopi 23263

A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)

𝐽 ∈ (TopOn‘𝐵) ⇒ 𝐽 ∈ Top

Theorem

toponunii 23264

The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.)

𝐽 ∈ (TopOn‘𝐵) ⇒ 𝐵 = ∪ 𝐽

Theorem

toptopon 23265

Alternative definition of Top in terms of TopOn. (Contributed by Mario Carneiro, 13-Aug-2015.)

𝑋 = ∪ 𝐽 ⇒ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))

Theorem

toptopon2 23266

A topology is the same thing as a topology on the union of its open sets. (Contributed by BJ, 27-Apr-2021.)

(𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))

Theorem

topontopon 23267

A topology on a set is a topology on the union of its open sets. (Contributed by BJ, 27-Apr-2021.)

(𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ (TopOn‘∪ 𝐽))

Theorem

funtopon 23268

The class TopOn is a function. (Contributed by BJ, 29-Apr-2021.)

Fun TopOn

Theorem

toponrestid 23269

Given a topology on a set, restricting it to that same set has no effect. (Contributed by Jim Kingdon, 6-Jul-2022.)

𝐴 ∈ (TopOn‘𝐵) ⇒ 𝐴 = (𝐴 ↾_t 𝐵)

Theorem

toponsspwpw 23270

The set of topologies on a set is included in the double power set of that set. (Contributed by BJ, 29-Apr-2021.)

(TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴

Theorem

dmtopon 23271

The domain of TopOn is the universal class V. (Contributed by BJ, 29-Apr-2021.)

dom TopOn = V

Theorem

fntopon 23272

The class TopOn is a function with domain the universal class V. Analogue for topologies of fnmre 17790 for Moore collections. (Contributed by BJ, 29-Apr-2021.)

TopOn Fn V

Theorem

toprntopon 23273

A topology is the same thing as a topology on a set (variable-free version). (Contributed by BJ, 27-Apr-2021.)

Top = ∪ ran TopOn

Theorem

toponmax 23274

The base set of a topology is an open set. (Contributed by Mario Carneiro, 13-Aug-2015.)

(𝐽 ∈ (TopOn‘𝐵) → 𝐵 ∈ 𝐽)

Theorem

toponss 23275

A member of a topology is a subset of its underlying set. (Contributed by Mario Carneiro, 21-Aug-2015.)

((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ 𝑋)

Theorem

toponcom 23276

If 𝐾 is a topology on the base set of topology 𝐽, then 𝐽 is a topology on the base of 𝐾. (Contributed by Mario Carneiro, 22-Aug-2015.)

((𝐽 ∈ Top ∧ 𝐾 ∈ (TopOn‘∪ 𝐽)) → 𝐽 ∈ (TopOn‘∪ 𝐾))

Theorem

toponcomb 23277

Biconditional form of toponcom 23276. (Contributed by BJ, 5-Dec-2021.)

((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ∈ (TopOn‘∪ 𝐾) ↔ 𝐾 ∈ (TopOn‘∪ 𝐽)))

Theorem

topgele 23278

The topologies over the same set have the greatest element (the discrete topology) and the least element (the indiscrete topology). (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 16-Sep-2015.)

(𝐽 ∈ (TopOn‘𝑋) → ({∅, 𝑋} ⊆ 𝐽 ∧ 𝐽 ⊆ 𝒫 𝑋))

Theorem

topsn 23279

The only topology on a singleton is the discrete topology (which is also the indiscrete topology by pwsn 4830). (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 16-Sep-2015.)

(𝐽 ∈ (TopOn‘{𝐴}) → 𝐽 = 𝒫 {𝐴})

12.1.1.3 Topological spaces

Syntax

ctps 23280

Syntax for the class of topological spaces.

class TopSp

Definition

df-topsp 23281

Define the class of topological spaces (as extensible structures). (Contributed by Stefan O'Rear, 13-Aug-2015.)

TopSp = {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))}

Theorem

istps 23282

Express the predicate "is a topological space." (Contributed by Mario Carneiro, 13-Aug-2015.)

𝐴 = (Base‘𝐾) & 𝐽 = (TopOpen‘𝐾) ⇒ (𝐾 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐴))

Theorem

istps2 23283

Express the predicate "is a topological space." (Contributed by NM, 20-Oct-2012.)

𝐴 = (Base‘𝐾) & 𝐽 = (TopOpen‘𝐾) ⇒ (𝐾 ∈ TopSp ↔ (𝐽 ∈ Top ∧ 𝐴 = ∪ 𝐽))

Theorem

tpsuni 23284

The base set of a topological space. (Contributed by FL, 27-Jun-2014.)

𝐴 = (Base‘𝐾) & 𝐽 = (TopOpen‘𝐾) ⇒ (𝐾 ∈ TopSp → 𝐴 = ∪ 𝐽)

Theorem

tpstop 23285

The topology extractor on a topological space is a topology. (Contributed by FL, 27-Jun-2014.)

𝐽 = (TopOpen‘𝐾) ⇒ (𝐾 ∈ TopSp → 𝐽 ∈ Top)

Theorem

tpspropd 23286

A topological space depends only on the base and topology components. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 13-Aug-2015.)

(𝜑 → (Base‘𝐾) = (Base‘𝐿)) & (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿)) ⇒ (𝜑 → (𝐾 ∈ TopSp ↔ 𝐿 ∈ TopSp))

Theorem

tpsprop2d 23287

A topological space depends only on the base and topology components. (Contributed by Mario Carneiro, 13-Aug-2015.)

(𝜑 → (Base‘𝐾) = (Base‘𝐿)) & (𝜑 → (TopSet‘𝐾) = (TopSet‘𝐿)) ⇒ (𝜑 → (𝐾 ∈ TopSp ↔ 𝐿 ∈ TopSp))

Theorem

topontopn 23288

Express the predicate "is a topological space." (Contributed by Mario Carneiro, 13-Aug-2015.)

𝐴 = (Base‘𝐾) & 𝐽 = (TopSet‘𝐾) ⇒ (𝐽 ∈ (TopOn‘𝐴) → 𝐽 = (TopOpen‘𝐾))

Theorem

tsettps 23289

If the topology component is already correctly truncated, then it forms a topological space (with the topology extractor function coming out the same as the component). (Contributed by Mario Carneiro, 13-Aug-2015.)

𝐴 = (Base‘𝐾) & 𝐽 = (TopSet‘𝐾) ⇒ (𝐽 ∈ (TopOn‘𝐴) → 𝐾 ∈ TopSp)

Theorem

istpsi 23290

Properties that determine a topological space. (Contributed by NM, 20-Oct-2012.)

(Base‘𝐾) = 𝐴 & (TopOpen‘𝐾) = 𝐽 & 𝐴 = ∪ 𝐽 & 𝐽 ∈ Top ⇒ 𝐾 ∈ TopSp

Theorem

eltpsg 23291

Properties that determine a topological space from a construction (using no explicit indices). (Contributed by Mario Carneiro, 13-Aug-2015.) (Revised by AV, 31-Oct-2024.)

𝐾 = {⟨(Base‘ndx), 𝐴⟩, ⟨(TopSet‘ndx), 𝐽⟩} ⇒ (𝐽 ∈ (TopOn‘𝐴) → 𝐾 ∈ TopSp)

Theorem

eltpsi 23292

Properties that determine a topological space from a construction (using no explicit indices). (Contributed by NM, 20-Oct-2012.) (Revised by Mario Carneiro, 13-Aug-2015.)

𝐾 = {⟨(Base‘ndx), 𝐴⟩, ⟨(TopSet‘ndx), 𝐽⟩} & 𝐴 = ∪ 𝐽 & 𝐽 ∈ Top ⇒ 𝐾 ∈ TopSp

12.1.2 Topological bases

Syntax

ctb 23293

Syntax for the class of topological bases.

class TopBases

Definition

df-bases 23294*

Define the class of topological bases. Equivalent to definition of basis in [Munkres] p. 78 (see isbasis2g 23296). Note that "bases" is the plural of "basis". (Contributed by NM, 17-Jul-2006.)

TopBases = {𝑥 ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ⊆ ∪ (𝑥 ∩ 𝒫 (𝑦 ∩ 𝑧))}

Theorem

isbasisg 23295*

Express the predicate "the set 𝐵 is a basis for a topology". (Contributed by NM, 17-Jul-2006.)

(𝐵 ∈ 𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))))

Theorem

isbasis2g 23296*

Express the predicate "the set 𝐵 is a basis for a topology". (Contributed by NM, 17-Jul-2006.)

(𝐵 ∈ 𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))))

Theorem

isbasis3g 23297*

Express the predicate "the set 𝐵 is a basis for a topology". Definition of basis in [Munkres] p. 78. (Contributed by NM, 17-Jul-2006.)

(𝐵 ∈ 𝐶 → (𝐵 ∈ TopBases ↔ (∀𝑥 ∈ 𝐵 𝑥 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ ∪ 𝐵∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)))))

Theorem

basis1 23298

Property of a basis. (Contributed by NM, 16-Jul-2006.)

((𝐵 ∈ TopBases ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵) → (𝐶 ∩ 𝐷) ⊆ ∪ (𝐵 ∩ 𝒫 (𝐶 ∩ 𝐷)))

Theorem

basis2 23299*

Property of a basis. (Contributed by NM, 17-Jul-2006.)

(((𝐵 ∈ TopBases ∧ 𝐶 ∈ 𝐵) ∧ (𝐷 ∈ 𝐵 ∧ 𝐴 ∈ (𝐶 ∩ 𝐷))) → ∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷)))

Theorem

fiinbas 23300*

If a set is closed under finite intersection, then it is a basis for a topology. (Contributed by Jeff Madsen, 2-Sep-2009.)

((𝐵 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ∈ 𝐵) → 𝐵 ∈ TopBases)

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26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-47000 471 47001-47100 472 47101-47200 473 47201-47300 474 47301-47400 475 47401-47500 476 47501-47600 477 47601-47700 478 47701-47800 479 47801-47900 480 47901-48000 481 48001-48100 482 48101-48200 483 48201-48300 484 48301-48400 485 48401-48500 486 48501-48600 487 48601-48700 488 48701-48800 489 48801-48900 490 48901-49000 491 49001-49100 492 49101-49200 493 49201-49300 494 49301-49400 495 49401-49500 496 49501-49600 497 49601-49700 498 49701-49800 499 49801-49900 500 49901-50000 501 50001-50100 502 50101-50200 503 50201-50300 504 50301-50400 505 50401-50500 506 50501-50600 507 50601-50700 508 50701-50800 509 50801-50900 510 50901-51000 511 51001-51100 512 51101-51200 513 51201-51270

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