P. 339 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 33801-33900   *Has distinct variable group(s)

Type	Label	Description
Statement
21.3.13  Matrices
21.3.13.1  Submatrices
Syntax	csmat 33801	Syntax for a function generating submatrices.
class subMat1
Definition	df-smat 33802*	Define a function generating submatrices of an integer-indexed matrix. The function maps an index in ((1...𝑀) × (1...𝑁)) into a new index in ((1...(𝑀 − 1)) × (1...(𝑁 − 1))). A submatrix is obtained by deleting a row and a column of the original matrix. Because this function re-indexes the matrix, the resulting submatrix still has the same index set for rows and columns, and its determinent is defined, unlike the current df-subma 22490. (Contributed by Thierry Arnoux, 18-Aug-2020.)
⊢ subMat1 = (𝑚 ∈ V ↦ (𝑘 ∈ ℕ, 𝑙 ∈ ℕ ↦ (𝑚 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝑘, 𝑖, (𝑖 + 1)), if(𝑗 < 𝑙, 𝑗, (𝑗 + 1))⟩))))
Theorem	smatfval 33803*	Value of the submatrix. (Contributed by Thierry Arnoux, 19-Aug-2020.)
⊢ ((𝐾 ∈ ℕ ∧ 𝐿 ∈ ℕ ∧ 𝑀 ∈ 𝑉) → (𝐾(subMat1‘𝑀)𝐿) = (𝑀 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
Theorem	smatrcl 33804	Closure of the rectangular submatrix. (Contributed by Thierry Arnoux, 19-Aug-2020.)
⊢ 𝑆 = (𝐾(subMat1‘𝐴)𝐿)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐾 ∈ (1...𝑀))    &   ⊢ (𝜑 → 𝐿 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐵 ↑m ((1...𝑀) × (1...𝑁))))    ⇒   ⊢ (𝜑 → 𝑆 ∈ (𝐵 ↑m ((1...(𝑀 − 1)) × (1...(𝑁 − 1)))))
Theorem	smatlem 33805	Lemma for the next theorems. (Contributed by Thierry Arnoux, 19-Aug-2020.)
⊢ 𝑆 = (𝐾(subMat1‘𝐴)𝐿)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐾 ∈ (1...𝑀))    &   ⊢ (𝜑 → 𝐿 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐵 ↑m ((1...𝑀) × (1...𝑁))))    &   ⊢ (𝜑 → 𝐼 ∈ ℕ)    &   ⊢ (𝜑 → 𝐽 ∈ ℕ)    &   ⊢ (𝜑 → if(𝐼 < 𝐾, 𝐼, (𝐼 + 1)) = 𝑋)    &   ⊢ (𝜑 → if(𝐽 < 𝐿, 𝐽, (𝐽 + 1)) = 𝑌)    ⇒   ⊢ (𝜑 → (𝐼𝑆𝐽) = (𝑋𝐴𝑌))
Theorem	smattl 33806	Entries of a submatrix, top left. (Contributed by Thierry Arnoux, 19-Aug-2020.)
⊢ 𝑆 = (𝐾(subMat1‘𝐴)𝐿)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐾 ∈ (1...𝑀))    &   ⊢ (𝜑 → 𝐿 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐵 ↑m ((1...𝑀) × (1...𝑁))))    &   ⊢ (𝜑 → 𝐼 ∈ (1..^𝐾))    &   ⊢ (𝜑 → 𝐽 ∈ (1..^𝐿))    ⇒   ⊢ (𝜑 → (𝐼𝑆𝐽) = (𝐼𝐴𝐽))
Theorem	smattr 33807	Entries of a submatrix, top right. (Contributed by Thierry Arnoux, 19-Aug-2020.)
⊢ 𝑆 = (𝐾(subMat1‘𝐴)𝐿)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐾 ∈ (1...𝑀))    &   ⊢ (𝜑 → 𝐿 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐵 ↑m ((1...𝑀) × (1...𝑁))))    &   ⊢ (𝜑 → 𝐼 ∈ (𝐾...𝑀))    &   ⊢ (𝜑 → 𝐽 ∈ (1..^𝐿))    ⇒   ⊢ (𝜑 → (𝐼𝑆𝐽) = ((𝐼 + 1)𝐴𝐽))
Theorem	smatbl 33808	Entries of a submatrix, bottom left. (Contributed by Thierry Arnoux, 19-Aug-2020.)
⊢ 𝑆 = (𝐾(subMat1‘𝐴)𝐿)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐾 ∈ (1...𝑀))    &   ⊢ (𝜑 → 𝐿 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐵 ↑m ((1...𝑀) × (1...𝑁))))    &   ⊢ (𝜑 → 𝐼 ∈ (1..^𝐾))    &   ⊢ (𝜑 → 𝐽 ∈ (𝐿...𝑁))    ⇒   ⊢ (𝜑 → (𝐼𝑆𝐽) = (𝐼𝐴(𝐽 + 1)))
Theorem	smatbr 33809	Entries of a submatrix, bottom right. (Contributed by Thierry Arnoux, 19-Aug-2020.)
⊢ 𝑆 = (𝐾(subMat1‘𝐴)𝐿)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐾 ∈ (1...𝑀))    &   ⊢ (𝜑 → 𝐿 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐵 ↑m ((1...𝑀) × (1...𝑁))))    &   ⊢ (𝜑 → 𝐼 ∈ (𝐾...𝑀))    &   ⊢ (𝜑 → 𝐽 ∈ (𝐿...𝑁))    ⇒   ⊢ (𝜑 → (𝐼𝑆𝐽) = ((𝐼 + 1)𝐴(𝐽 + 1)))
Theorem	smatcl 33810	Closure of the square submatrix: if 𝑀 is a square matrix of dimension 𝑁 with indices in (1...𝑁), then a submatrix of 𝑀 is of dimension (𝑁 − 1). (Contributed by Thierry Arnoux, 19-Aug-2020.)
⊢ 𝐴 = ((1...𝑁) Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐶 = (Base‘((1...(𝑁 − 1)) Mat 𝑅))    &   ⊢ 𝑆 = (𝐾(subMat1‘𝑀)𝐿)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐾 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐿 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝑀 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝑆 ∈ 𝐶)
Theorem	matmpo 33811*	Write a square matrix as a mapping operation. (Contributed by Thierry Arnoux, 16-Aug-2020.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    ⇒   ⊢ (𝑀 ∈ 𝐵 → 𝑀 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑖𝑀𝑗)))
Theorem	1smat1 33812	The submatrix of the identity matrix obtained by removing the ith row and the ith column is an identity matrix. Cf. 1marepvsma1 22496. (Contributed by Thierry Arnoux, 19-Aug-2020.)
⊢ 1 = (1r‘((1...𝑁) Mat 𝑅))    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐼 ∈ (1...𝑁))    ⇒   ⊢ (𝜑 → (𝐼(subMat1‘ 1 )𝐼) = (1r‘((1...(𝑁 − 1)) Mat 𝑅)))
Theorem	submat1n 33813	One case where the submatrix with integer indices, subMat1, and the general submatrix subMat, agree. (Contributed by Thierry Arnoux, 22-Aug-2020.)
⊢ 𝐴 = ((1...𝑁) Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    ⇒   ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ 𝐵) → (𝑁(subMat1‘𝑀)𝑁) = (𝑁(((1...𝑁) subMat 𝑅)‘𝑀)𝑁))
Theorem	submatres 33814	Special case where the submatrix is a restriction of the initial matrix, and no renumbering occurs. (Contributed by Thierry Arnoux, 26-Aug-2020.)
⊢ 𝐴 = ((1...𝑁) Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    ⇒   ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ 𝐵) → (𝑁(subMat1‘𝑀)𝑁) = (𝑀 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))))
Theorem	submateqlem1 33815	Lemma for submateq 33817. (Contributed by Thierry Arnoux, 25-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐾 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝑀 ∈ (1...(𝑁 − 1)))    &   ⊢ (𝜑 → 𝐾 ≤ 𝑀)    ⇒   ⊢ (𝜑 → (𝑀 ∈ (𝐾...𝑁) ∧ (𝑀 + 1) ∈ ((1...𝑁) ∖ {𝐾})))
Theorem	submateqlem2 33816	Lemma for submateq 33817. (Contributed by Thierry Arnoux, 26-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐾 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝑀 ∈ (1...(𝑁 − 1)))    &   ⊢ (𝜑 → 𝑀 < 𝐾)    ⇒   ⊢ (𝜑 → (𝑀 ∈ (1..^𝐾) ∧ 𝑀 ∈ ((1...𝑁) ∖ {𝐾})))
Theorem	submateq 33817*	Sufficient condition for two submatrices to be equal. (Contributed by Thierry Arnoux, 25-Aug-2020.)
⊢ 𝐴 = ((1...𝑁) Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐼 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐽 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐸 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑗 ∈ ((1...𝑁) ∖ {𝐽})) → (𝑖𝐸𝑗) = (𝑖𝐹𝑗))    ⇒   ⊢ (𝜑 → (𝐼(subMat1‘𝐸)𝐽) = (𝐼(subMat1‘𝐹)𝐽))
Theorem	submatminr1 33818	If we take a submatrix by removing the row 𝐼 and column 𝐽, then the result is the same on the matrix with row 𝐼 and column 𝐽 modified by the minMatR1 operator. (Contributed by Thierry Arnoux, 25-Aug-2020.)
⊢ 𝐴 = ((1...𝑁) Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐼 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐽 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑀 ∈ 𝐵)    &   ⊢ 𝐸 = (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)    ⇒   ⊢ (𝜑 → (𝐼(subMat1‘𝑀)𝐽) = (𝐼(subMat1‘𝐸)𝐽))
21.3.13.2  Matrix literals
Syntax	clmat 33819	Extend class notation with the literal matrix conversion function.
class litMat
Definition	df-lmat 33820*	Define a function converting words of words into matrices. (Contributed by Thierry Arnoux, 28-Aug-2020.)
⊢ litMat = (𝑚 ∈ V ↦ (𝑖 ∈ (1...(♯‘𝑚)), 𝑗 ∈ (1...(♯‘(𝑚‘0))) ↦ ((𝑚‘(𝑖 − 1))‘(𝑗 − 1))))
Theorem	lmatval 33821*	Value of the literal matrix conversion function. (Contributed by Thierry Arnoux, 28-Aug-2020.)
⊢ (𝑀 ∈ 𝑉 → (litMat‘𝑀) = (𝑖 ∈ (1...(♯‘𝑀)), 𝑗 ∈ (1...(♯‘(𝑀‘0))) ↦ ((𝑀‘(𝑖 − 1))‘(𝑗 − 1))))
Theorem	lmatfval 33822*	Entries of a literal matrix. (Contributed by Thierry Arnoux, 28-Aug-2020.)
⊢ 𝑀 = (litMat‘𝑊)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑊 ∈ Word Word 𝑉)    &   ⊢ (𝜑 → (♯‘𝑊) = 𝑁)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (♯‘(𝑊‘𝑖)) = 𝑁)    &   ⊢ (𝜑 → 𝐼 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐽 ∈ (1...𝑁))    ⇒   ⊢ (𝜑 → (𝐼𝑀𝐽) = ((𝑊‘(𝐼 − 1))‘(𝐽 − 1)))
Theorem	lmatfvlem 33823*	Useful lemma to extract literal matrix entries. Suggested by Mario Carneiro. (Contributed by Thierry Arnoux, 3-Sep-2020.)
⊢ 𝑀 = (litMat‘𝑊)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑊 ∈ Word Word 𝑉)    &   ⊢ (𝜑 → (♯‘𝑊) = 𝑁)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (♯‘(𝑊‘𝑖)) = 𝑁)    &   ⊢ 𝐾 ∈ ℕ0    &   ⊢ 𝐿 ∈ ℕ0    &   ⊢ 𝐼 ≤ 𝑁    &   ⊢ 𝐽 ≤ 𝑁    &   ⊢ (𝐾 + 1) = 𝐼    &   ⊢ (𝐿 + 1) = 𝐽    &   ⊢ (𝑊‘𝐾) = 𝑋    &   ⊢ (𝜑 → (𝑋‘𝐿) = 𝑌)    ⇒   ⊢ (𝜑 → (𝐼𝑀𝐽) = 𝑌)
Theorem	lmatcl 33824*	Closure of the literal matrix. (Contributed by Thierry Arnoux, 12-Sep-2020.)
⊢ 𝑀 = (litMat‘𝑊)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑊 ∈ Word Word 𝑉)    &   ⊢ (𝜑 → (♯‘𝑊) = 𝑁)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (♯‘(𝑊‘𝑖)) = 𝑁)    &   ⊢ 𝑉 = (Base‘𝑅)    &   ⊢ 𝑂 = ((1...𝑁) Mat 𝑅)    &   ⊢ 𝑃 = (Base‘𝑂)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑋)    ⇒   ⊢ (𝜑 → 𝑀 ∈ 𝑃)
Theorem	lmat22lem 33825*	Lemma for lmat22e11 33826 and co. (Contributed by Thierry Arnoux, 28-Aug-2020.)
⊢ 𝑀 = (litMat‘⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    ⇒   ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^2)) → (♯‘(⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩‘𝑖)) = 2)
Theorem	lmat22e11 33826	Entry of a 2x2 literal matrix. (Contributed by Thierry Arnoux, 28-Aug-2020.)
⊢ 𝑀 = (litMat‘⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (1𝑀1) = 𝐴)
Theorem	lmat22e12 33827	Entry of a 2x2 literal matrix. (Contributed by Thierry Arnoux, 12-Sep-2020.)
⊢ 𝑀 = (litMat‘⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (1𝑀2) = 𝐵)
Theorem	lmat22e21 33828	Entry of a 2x2 literal matrix. (Contributed by Thierry Arnoux, 12-Sep-2020.)
⊢ 𝑀 = (litMat‘⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (2𝑀1) = 𝐶)
Theorem	lmat22e22 33829	Entry of a 2x2 literal matrix. (Contributed by Thierry Arnoux, 12-Sep-2020.)
⊢ 𝑀 = (litMat‘⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (2𝑀2) = 𝐷)
Theorem	lmat22det 33830	The determinant of a literal 2x2 complex matrix. (Contributed by Thierry Arnoux, 1-Sep-2020.)
⊢ 𝑀 = (litMat‘⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    &   ⊢ · = (.r‘𝑅)    &   ⊢ − = (-g‘𝑅)    &   ⊢ 𝑉 = (Base‘𝑅)    &   ⊢ 𝐽 = ((1...2) maDet 𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    ⇒   ⊢ (𝜑 → (𝐽‘𝑀) = ((𝐴 · 𝐷) − (𝐶 · 𝐵)))
21.3.13.3  Laplace expansion of determinants
Theorem	mdetpmtr1 33831*	The determinant of a matrix with permuted rows is the determinant of the original matrix multiplied by the sign of the permutation. (Contributed by Thierry Arnoux, 22-Aug-2020.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐺 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝑆 = (pmSgn‘𝑁)    &   ⊢ 𝑍 = (ℤRHom‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝐸 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑃‘𝑖)𝑀𝑗))    ⇒   ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (𝐷‘𝑀) = (((𝑍 ∘ 𝑆)‘𝑃) · (𝐷‘𝐸)))
Theorem	mdetpmtr2 33832*	The determinant of a matrix with permuted columns is the determinant of the original matrix multiplied by the sign of the permutation. (Contributed by Thierry Arnoux, 22-Aug-2020.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐺 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝑆 = (pmSgn‘𝑁)    &   ⊢ 𝑍 = (ℤRHom‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝐸 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑖𝑀(𝑃‘𝑗)))    ⇒   ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (𝐷‘𝑀) = (((𝑍 ∘ 𝑆)‘𝑃) · (𝐷‘𝐸)))
Theorem	mdetpmtr12 33833*	The determinant of a matrix with permuted rows and columns is the determinant of the original matrix multiplied by the product of the signs of the permutations. (Contributed by Thierry Arnoux, 22-Aug-2020.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐺 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝑆 = (pmSgn‘𝑁)    &   ⊢ 𝑍 = (ℤRHom‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝐸 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑃‘𝑖)𝑀(𝑄‘𝑗)))    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑀 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑃 ∈ 𝐺)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐺)    ⇒   ⊢ (𝜑 → (𝐷‘𝑀) = ((𝑍‘((𝑆‘𝑃) · (𝑆‘𝑄))) · (𝐷‘𝐸)))
Theorem	mdetlap1 33834*	A Laplace expansion of the determinant of a matrix, using the adjunct (cofactor) matrix. (Contributed by Thierry Arnoux, 16-Aug-2020.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐾 = (𝑁 maAdju 𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝐼 ∈ 𝑁) → (𝐷‘𝑀) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑀𝑗) · (𝑗(𝐾‘𝑀)𝐼)))))
Theorem	madjusmdetlem1 33835*	Lemma for madjusmdet 33839. (Contributed by Thierry Arnoux, 22-Aug-2020.)
⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐴 = ((1...𝑁) Mat 𝑅)    &   ⊢ 𝐷 = ((1...𝑁) maDet 𝑅)    &   ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑍 = (ℤRHom‘𝑅)    &   ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐼 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐽 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝑀 ∈ 𝐵)    &   ⊢ 𝐺 = (Base‘(SymGrp‘(1...𝑁)))    &   ⊢ 𝑆 = (pmSgn‘(1...𝑁))    &   ⊢ 𝑈 = (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)    &   ⊢ 𝑊 = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ ((𝑃‘𝑖)𝑈(𝑄‘𝑗)))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐺)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐺)    &   ⊢ (𝜑 → (𝑃‘𝑁) = 𝐼)    &   ⊢ (𝜑 → (𝑄‘𝑁) = 𝐽)    &   ⊢ (𝜑 → (𝐼(subMat1‘𝑈)𝐽) = (𝑁(subMat1‘𝑊)𝑁))    ⇒   ⊢ (𝜑 → (𝐽(𝐾‘𝑀)𝐼) = ((𝑍‘((𝑆‘𝑃) · (𝑆‘𝑄))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽))))
Theorem	madjusmdetlem2 33836*	Lemma for madjusmdet 33839. (Contributed by Thierry Arnoux, 26-Aug-2020.)
⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐴 = ((1...𝑁) Mat 𝑅)    &   ⊢ 𝐷 = ((1...𝑁) maDet 𝑅)    &   ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑍 = (ℤRHom‘𝑅)    &   ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐼 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐽 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝑀 ∈ 𝐵)    &   ⊢ 𝑃 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖)))    &   ⊢ 𝑆 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝑁, if(𝑖 ≤ 𝑁, (𝑖 − 1), 𝑖)))    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → if(𝑋 < 𝐼, 𝑋, (𝑋 + 1)) = ((𝑃 ∘ ◡𝑆)‘𝑋))
Theorem	madjusmdetlem3 33837*	Lemma for madjusmdet 33839. (Contributed by Thierry Arnoux, 27-Aug-2020.)
⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐴 = ((1...𝑁) Mat 𝑅)    &   ⊢ 𝐷 = ((1...𝑁) maDet 𝑅)    &   ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑍 = (ℤRHom‘𝑅)    &   ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐼 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐽 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝑀 ∈ 𝐵)    &   ⊢ 𝑃 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖)))    &   ⊢ 𝑆 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝑁, if(𝑖 ≤ 𝑁, (𝑖 − 1), 𝑖)))    &   ⊢ 𝑄 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝐽, if(𝑗 ≤ 𝐽, (𝑗 − 1), 𝑗)))    &   ⊢ 𝑇 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝑁, if(𝑗 ≤ 𝑁, (𝑗 − 1), 𝑗)))    &   ⊢ 𝑊 = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ (((𝑃 ∘ ◡𝑆)‘𝑖)𝑈((𝑄 ∘ ◡𝑇)‘𝑗)))    &   ⊢ (𝜑 → 𝑈 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐼(subMat1‘𝑈)𝐽) = (𝑁(subMat1‘𝑊)𝑁))
Theorem	madjusmdetlem4 33838*	Lemma for madjusmdet 33839. (Contributed by Thierry Arnoux, 22-Aug-2020.)
⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐴 = ((1...𝑁) Mat 𝑅)    &   ⊢ 𝐷 = ((1...𝑁) maDet 𝑅)    &   ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑍 = (ℤRHom‘𝑅)    &   ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐼 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐽 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝑀 ∈ 𝐵)    &   ⊢ 𝑃 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖)))    &   ⊢ 𝑆 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝑁, if(𝑖 ≤ 𝑁, (𝑖 − 1), 𝑖)))    &   ⊢ 𝑄 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝐽, if(𝑗 ≤ 𝐽, (𝑗 − 1), 𝑗)))    &   ⊢ 𝑇 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝑁, if(𝑗 ≤ 𝑁, (𝑗 − 1), 𝑗)))    ⇒   ⊢ (𝜑 → (𝐽(𝐾‘𝑀)𝐼) = ((𝑍‘(-1↑(𝐼 + 𝐽))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽))))
Theorem	madjusmdet 33839	Express the cofactor of the matrix, i.e. the entries of its adjunct matrix, using determinant of submatrices. (Contributed by Thierry Arnoux, 23-Aug-2020.)
⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐴 = ((1...𝑁) Mat 𝑅)    &   ⊢ 𝐷 = ((1...𝑁) maDet 𝑅)    &   ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑍 = (ℤRHom‘𝑅)    &   ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐼 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐽 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝑀 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐽(𝐾‘𝑀)𝐼) = ((𝑍‘(-1↑(𝐼 + 𝐽))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽))))
Theorem	mdetlap 33840*	Laplace expansion of the determinant of a square matrix. (Contributed by Thierry Arnoux, 19-Aug-2020.)
⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐴 = ((1...𝑁) Mat 𝑅)    &   ⊢ 𝐷 = ((1...𝑁) maDet 𝑅)    &   ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑍 = (ℤRHom‘𝑅)    &   ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐼 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐽 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝑀 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐷‘𝑀) = (𝑅 Σg (𝑗 ∈ (1...𝑁) ↦ ((𝑍‘(-1↑(𝐼 + 𝑗))) · ((𝐼𝑀𝑗) · (𝐸‘(𝐼(subMat1‘𝑀)𝑗)))))))
21.3.14  Topology
Theorem	ist0cld 33841*	The predicate "is a T0 space", using closed sets. (Contributed by Thierry Arnoux, 16-Aug-2020.)
⊢ (𝜑 → 𝐵 = ∪ 𝐽)    &   ⊢ (𝜑 → 𝐷 = (Clsd‘𝐽))    ⇒   ⊢ (𝜑 → (𝐽 ∈ Kol2 ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (∀𝑑 ∈ 𝐷 (𝑥 ∈ 𝑑 ↔ 𝑦 ∈ 𝑑) → 𝑥 = 𝑦))))
21.3.14.1  Open maps
Theorem	txomap 33842*	Given two open maps 𝐹 and 𝐺, 𝐻 mapping pairs of sets, is also an open map for the product topology. (Contributed by Thierry Arnoux, 29-Dec-2019.)
⊢ (𝜑 → 𝐹:𝑋⟶𝑍)    &   ⊢ (𝜑 → 𝐺:𝑌⟶𝑇)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))    &   ⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑇))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ∈ 𝐿)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → (𝐺 “ 𝑦) ∈ 𝑀)    &   ⊢ (𝜑 → 𝐴 ∈ (𝐽 ×t 𝐾))    &   ⊢ 𝐻 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩)    ⇒   ⊢ (𝜑 → (𝐻 “ 𝐴) ∈ (𝐿 ×t 𝑀))
21.3.14.2  Topology of the unit circle
Theorem	qtopt1 33843*	If every equivalence class is closed, then the quotient space is T1 . (Contributed by Thierry Arnoux, 5-Jan-2020.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ (𝜑 → 𝐽 ∈ Fre)    &   ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (◡𝐹 “ {𝑥}) ∈ (Clsd‘𝐽))    ⇒   ⊢ (𝜑 → (𝐽 qTop 𝐹) ∈ Fre)
Theorem	qtophaus 33844*	If an open map's graph in the product space (𝐽 ×t 𝐽) is closed, then its quotient topology is Hausdorff. (Contributed by Thierry Arnoux, 4-Jan-2020.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ ∼ = (◡𝐹 ∘ 𝐹)    &   ⊢ 𝐻 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)    &   ⊢ (𝜑 → 𝐽 ∈ Haus)    &   ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ∈ (𝐽 qTop 𝐹))    &   ⊢ (𝜑 → ∼ ∈ (Clsd‘(𝐽 ×t 𝐽)))    ⇒   ⊢ (𝜑 → (𝐽 qTop 𝐹) ∈ Haus)
Theorem	circtopn 33845*	The topology of the unit circle is generated by open intervals of the polar coordinate. (Contributed by Thierry Arnoux, 4-Jan-2020.)
⊢ 𝐼 = (0[,](2 · π))    &   ⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ (exp‘(i · 𝑥)))    &   ⊢ 𝐶 = (◡abs “ {1})    ⇒   ⊢ (𝐽 qTop 𝐹) = (TopOpen‘(𝐹 “s ℝfld))
Theorem	circcn 33846*	The function gluing the real line into the unit circle is continuous. (Contributed by Thierry Arnoux, 5-Jan-2020.)
⊢ 𝐼 = (0[,](2 · π))    &   ⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ (exp‘(i · 𝑥)))    &   ⊢ 𝐶 = (◡abs “ {1})    ⇒   ⊢ 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹))
21.3.14.3  Refinements
Theorem	reff 33847*	For any cover refinement, there exists a function associating with each set in the refinement a set in the original cover containing it. This is sometimes used as a definition of refinement. Note that this definition uses the axiom of choice through ac6sg 10376. (Contributed by Thierry Arnoux, 12-Jan-2020.)
⊢ (𝐴 ∈ 𝑉 → (𝐴Ref𝐵 ↔ (∪ 𝐵 ⊆ ∪ 𝐴 ∧ ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣)))))
Theorem	locfinreflem 33848*	A locally finite refinement of an open cover induces a locally finite open cover with the original index set. This is fact 2 of http://at.yorku.ca/p/a/c/a/02.pdf, it is expressed by exposing a function 𝑓 from the original cover 𝑈, which is taken as the index set. The solution is constructed by building unions, so the same method can be used to prove a similar theorem about closed covers. (Contributed by Thierry Arnoux, 29-Jan-2020.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ (𝜑 → 𝑈 ⊆ 𝐽)    &   ⊢ (𝜑 → 𝑋 = ∪ 𝑈)    &   ⊢ (𝜑 → 𝑉 ⊆ 𝐽)    &   ⊢ (𝜑 → 𝑉Ref𝑈)    &   ⊢ (𝜑 → 𝑉 ∈ (LocFin‘𝐽))    ⇒   ⊢ (𝜑 → ∃𝑓((Fun 𝑓 ∧ dom 𝑓 ⊆ 𝑈 ∧ ran 𝑓 ⊆ 𝐽) ∧ (ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽))))
Theorem	locfinref 33849*	A locally finite refinement of an open cover induces a locally finite open cover with the original index set. This is fact 2 of http://at.yorku.ca/p/a/c/a/02.pdf, it is expressed by exposing a function 𝑓 from the original cover 𝑈, which is taken as the index set. (Contributed by Thierry Arnoux, 31-Jan-2020.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ (𝜑 → 𝑈 ⊆ 𝐽)    &   ⊢ (𝜑 → 𝑋 = ∪ 𝑈)    &   ⊢ (𝜑 → 𝑉 ⊆ 𝐽)    &   ⊢ (𝜑 → 𝑉Ref𝑈)    &   ⊢ (𝜑 → 𝑉 ∈ (LocFin‘𝐽))    ⇒   ⊢ (𝜑 → ∃𝑓(𝑓:𝑈⟶𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))
21.3.14.4  Open cover refinement property
Syntax	ccref 33850	The "every open cover has an 𝐴 refinement" predicate.
class CovHasRef𝐴
Definition	df-cref 33851*	Define a statement "every open cover has an 𝐴 refinement" , where 𝐴 is a property for refinements like "finite", "countable", "point finite" or "locally finite". (Contributed by Thierry Arnoux, 7-Jan-2020.)
⊢ CovHasRef𝐴 = {𝑗 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦)}
Theorem	iscref 33852*	The property that every open cover has an 𝐴 refinement for the topological space 𝐽. (Contributed by Thierry Arnoux, 7-Jan-2020.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ CovHasRef𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦)))
Theorem	crefeq 33853	Equality theorem for the "every open cover has an A refinement" predicate. (Contributed by Thierry Arnoux, 7-Jan-2020.)
⊢ (𝐴 = 𝐵 → CovHasRef𝐴 = CovHasRef𝐵)
Theorem	creftop 33854	A space where every open cover has an 𝐴 refinement is a topological space. (Contributed by Thierry Arnoux, 7-Jan-2020.)
⊢ (𝐽 ∈ CovHasRef𝐴 → 𝐽 ∈ Top)
Theorem	crefi 33855*	The property that every open cover has an 𝐴 refinement for the topological space 𝐽. (Contributed by Thierry Arnoux, 7-Jan-2020.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ CovHasRef𝐴 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝐶)
Theorem	crefdf 33856*	A formulation of crefi 33855 easier to use for definitions. (Contributed by Thierry Arnoux, 7-Jan-2020.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐵 = CovHasRef𝐴    &   ⊢ (𝑧 ∈ 𝐴 → 𝜑)    ⇒   ⊢ ((𝐽 ∈ 𝐵 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → ∃𝑧 ∈ 𝒫 𝐽(𝜑 ∧ 𝑧Ref𝐶))
Theorem	crefss 33857	The "every open cover has an 𝐴 refinement" predicate respects inclusion. (Contributed by Thierry Arnoux, 7-Jan-2020.)
⊢ (𝐴 ⊆ 𝐵 → CovHasRef𝐴 ⊆ CovHasRef𝐵)
Theorem	cmpcref 33858	Equivalent definition of compact space in terms of open cover refinements. Compact spaces are topologies with finite open cover refinements. (Contributed by Thierry Arnoux, 7-Jan-2020.)
⊢ Comp = CovHasRefFin
Theorem	cmpfiref 33859*	Every open cover of a Compact space has a finite refinement. (Contributed by Thierry Arnoux, 1-Feb-2020.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Comp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ∈ Fin ∧ 𝑣Ref𝑈))
21.3.14.5  Lindelöf spaces
Syntax	cldlf 33860	Extend class notation with the class of all Lindelöf spaces.
class Ldlf
Definition	df-ldlf 33861	Definition of a Lindelöf space. A Lindelöf space is a topological space in which every open cover has a countable subcover. Definition 1 of [BourbakiTop2] p. 195. (Contributed by Thierry Arnoux, 30-Jan-2020.)
⊢ Ldlf = CovHasRef{𝑥 ∣ 𝑥 ≼ ω}
Theorem	ldlfcntref 33862*	Every open cover of a Lindelöf space has a countable refinement. (Contributed by Thierry Arnoux, 1-Feb-2020.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Ldlf ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ≼ ω ∧ 𝑣Ref𝑈))
21.3.14.6  Paracompact spaces
Syntax	cpcmp 33863	Extend class notation with the class of all paracompact topologies.
class Paracomp
Definition	df-pcmp 33864	Definition of a paracompact topology. A topology is said to be paracompact iff every open cover has an open refinement that is locally finite. The definition 6 of [BourbakiTop1] p. I.69. also requires the topology to be Hausdorff, but this is dropped here. (Contributed by Thierry Arnoux, 7-Jan-2020.)
⊢ Paracomp = {𝑗 ∣ 𝑗 ∈ CovHasRef(LocFin‘𝑗)}
Theorem	ispcmp 33865	The predicate "is a paracompact topology". (Contributed by Thierry Arnoux, 7-Jan-2020.)
⊢ (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽))
Theorem	cmppcmp 33866	Every compact space is paracompact. (Contributed by Thierry Arnoux, 7-Jan-2020.)
⊢ (𝐽 ∈ Comp → 𝐽 ∈ Paracomp)
Theorem	dispcmp 33867	Every discrete space is paracompact. (Contributed by Thierry Arnoux, 7-Jan-2020.)
⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ Paracomp)
Theorem	pcmplfin 33868*	Given a paracompact topology 𝐽 and an open cover 𝑈, there exists an open refinement 𝑣 that is locally finite. (Contributed by Thierry Arnoux, 31-Jan-2020.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈))
Theorem	pcmplfinf 33869*	Given a paracompact topology 𝐽 and an open cover 𝑈, there exists an open refinement ran 𝑓 that is locally finite, using the same index as the original cover 𝑈. (Contributed by Thierry Arnoux, 31-Jan-2020.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑓(𝑓:𝑈⟶𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))
21.3.14.7  Spectrum of a ring The prime ideals of a ring 𝑅 can be endowed with the Zariski topology. This is done by defining a function 𝑉 which maps ideals of 𝑅 to closed sets (see for example zarcls0 33876 for the definition of 𝑉). The closed sets of the topology are in the range of 𝑉 (see zartopon 33885). The correspondence with the open sets is made in zarcls 33882. As proved in zart0 33887, the Zariski topology is T0 , but generally not T1 .
Syntax	crspec 33870	Extend class notation with the spectrum of a ring.
class Spec
Definition	df-rspec 33871	Define the spectrum of a ring. (Contributed by Thierry Arnoux, 21-Jan-2024.)
⊢ Spec = (𝑟 ∈ Ring ↦ ((IDLsrg‘𝑟) ↾s (PrmIdeal‘𝑟)))
Theorem	rspecval 33872	Value of the spectrum of the ring 𝑅. Notation 1.1.1 of [EGA] p. 80. (Contributed by Thierry Arnoux, 2-Jun-2024.)
⊢ (𝑅 ∈ Ring → (Spec‘𝑅) = ((IDLsrg‘𝑅) ↾s (PrmIdeal‘𝑅)))
Theorem	rspecbas 33873	The prime ideals form the base of the spectrum of a ring. (Contributed by Thierry Arnoux, 2-Jun-2024.)
⊢ 𝑆 = (Spec‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (PrmIdeal‘𝑅) = (Base‘𝑆))
Theorem	rspectset 33874*	Topology component of the spectrum of a ring. (Contributed by Thierry Arnoux, 2-Jun-2024.)
⊢ 𝑆 = (Spec‘𝑅)    &   ⊢ 𝐼 = (LIdeal‘𝑅)    &   ⊢ 𝐽 = ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗})    ⇒   ⊢ (𝑅 ∈ Ring → 𝐽 = (TopSet‘𝑆))
Theorem	rspectopn 33875*	The topology component of the spectrum of a ring. (Contributed by Thierry Arnoux, 4-Jun-2024.)
⊢ 𝑆 = (Spec‘𝑅)    &   ⊢ 𝐼 = (LIdeal‘𝑅)    &   ⊢ 𝑃 = (PrmIdeal‘𝑅)    &   ⊢ 𝐽 = ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗})    ⇒   ⊢ (𝑅 ∈ Ring → 𝐽 = (TopOpen‘𝑆))
Theorem	zarcls0 33876*	The closure of the identity ideal in the Zariski topology. Proposition 1.1.2(i) of [EGA] p. 80. (Contributed by Thierry Arnoux, 16-Jun-2024.)
⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})    &   ⊢ 𝑃 = (PrmIdeal‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (𝑉‘{ 0 }) = 𝑃)
Theorem	zarcls1 33877*	The unit ideal 𝐵 is the only ideal whose closure in the Zariski topology is the empty set. Stronger form of the Proposition 1.1.2(i) of [EGA] p. 80. (Contributed by Thierry Arnoux, 16-Jun-2024.)
⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → ((𝑉‘𝐼) = ∅ ↔ 𝐼 = 𝐵))
Theorem	zarclsun 33878*	The union of two closed sets of the Zariski topology is closed. (Contributed by Thierry Arnoux, 16-Jun-2024.)
⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ ran 𝑉 ∧ 𝑌 ∈ ran 𝑉) → (𝑋 ∪ 𝑌) ∈ ran 𝑉)
Theorem	zarclsiin 33879*	In a Zariski topology, the intersection of the closures of a family of ideals is the closure of the span of their union. (Contributed by Thierry Arnoux, 16-Jun-2024.)
⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})    &   ⊢ 𝐾 = (RSpan‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) → ∩ 𝑙 ∈ 𝑇 (𝑉‘𝑙) = (𝑉‘(𝐾‘∪ 𝑇)))
Theorem	zarclsint 33880*	The intersection of a family of closed sets is closed in the Zariski topology. (Contributed by Thierry Arnoux, 16-Jun-2024.)
⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉 ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ ran 𝑉)
Theorem	zarclssn 33881*	The closed points of Zariski topology are the maximal ideals. (Contributed by Thierry Arnoux, 16-Jun-2024.)
⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})    &   ⊢ 𝐵 = (LIdeal‘𝑅)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ({𝑀} = (𝑉‘𝑀) ↔ 𝑀 ∈ (MaxIdeal‘𝑅)))
Theorem	zarcls 33882*	The open sets of the Zariski topology are the complements of the closed sets. (Contributed by Thierry Arnoux, 16-Jun-2024.)
⊢ 𝑆 = (Spec‘𝑅)    &   ⊢ 𝐽 = (TopOpen‘𝑆)    &   ⊢ 𝑃 = (PrmIdeal‘𝑅)    &   ⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗})    ⇒   ⊢ (𝑅 ∈ Ring → 𝐽 = {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉})
Theorem	zartopn 33883*	The Zariski topology is a topology, and its closed sets are images by 𝑉 of the ideals of 𝑅. (Contributed by Thierry Arnoux, 16-Jun-2024.)
⊢ 𝑆 = (Spec‘𝑅)    &   ⊢ 𝐽 = (TopOpen‘𝑆)    &   ⊢ 𝑃 = (PrmIdeal‘𝑅)    &   ⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗})    ⇒   ⊢ (𝑅 ∈ CRing → (𝐽 ∈ (TopOn‘𝑃) ∧ ran 𝑉 = (Clsd‘𝐽)))
Theorem	zartop 33884	The Zariski topology is a topology. Proposition 1.1.2 of [EGA] p. 80. (Contributed by Thierry Arnoux, 16-Jun-2024.)
⊢ 𝑆 = (Spec‘𝑅)    &   ⊢ 𝐽 = (TopOpen‘𝑆)    ⇒   ⊢ (𝑅 ∈ CRing → 𝐽 ∈ Top)
Theorem	zartopon 33885	The points of the Zariski topology are the prime ideals. (Contributed by Thierry Arnoux, 16-Jun-2024.)
⊢ 𝑆 = (Spec‘𝑅)    &   ⊢ 𝐽 = (TopOpen‘𝑆)    &   ⊢ 𝑃 = (PrmIdeal‘𝑅)    ⇒   ⊢ (𝑅 ∈ CRing → 𝐽 ∈ (TopOn‘𝑃))
Theorem	zar0ring 33886	The Zariski Topology of the trivial ring. (Contributed by Thierry Arnoux, 1-Jul-2024.)
⊢ 𝑆 = (Spec‘𝑅)    &   ⊢ 𝐽 = (TopOpen‘𝑆)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → 𝐽 = {∅})
Theorem	zart0 33887	The Zariski topology is T0 . Corollary 1.1.8 of [EGA] p. 81. (Contributed by Thierry Arnoux, 16-Jun-2024.)
⊢ 𝑆 = (Spec‘𝑅)    &   ⊢ 𝐽 = (TopOpen‘𝑆)    ⇒   ⊢ (𝑅 ∈ CRing → 𝐽 ∈ Kol2)
Theorem	zarmxt1 33888	The Zariski topology restricted to maximal ideals is T1 . (Contributed by Thierry Arnoux, 16-Jun-2024.)
⊢ 𝑆 = (Spec‘𝑅)    &   ⊢ 𝐽 = (TopOpen‘𝑆)    &   ⊢ 𝑀 = (MaxIdeal‘𝑅)    &   ⊢ 𝑇 = (𝐽 ↾t 𝑀)    ⇒   ⊢ (𝑅 ∈ CRing → 𝑇 ∈ Fre)
Theorem	zarcmplem 33889*	Lemma for zarcmp 33890. (Contributed by Thierry Arnoux, 2-Jul-2024.)
⊢ 𝑆 = (Spec‘𝑅)    &   ⊢ 𝐽 = (TopOpen‘𝑆)    &   ⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})    ⇒   ⊢ (𝑅 ∈ CRing → 𝐽 ∈ Comp)
Theorem	zarcmp 33890	The Zariski topology is compact. Proposition 1.1.10(ii) of [EGA], p. 82. (Contributed by Thierry Arnoux, 2-Jul-2024.)
⊢ 𝑆 = (Spec‘𝑅)    &   ⊢ 𝐽 = (TopOpen‘𝑆)    ⇒   ⊢ (𝑅 ∈ CRing → 𝐽 ∈ Comp)
Theorem	rspectps 33891	The spectrum of a ring 𝑅 is a topological space. (Contributed by Thierry Arnoux, 16-Jun-2024.)
⊢ 𝑆 = (Spec‘𝑅)    ⇒   ⊢ (𝑅 ∈ CRing → 𝑆 ∈ TopSp)
Theorem	rhmpreimacnlem 33892*	Lemma for rhmpreimacn 33893. (Contributed by Thierry Arnoux, 7-Jul-2024.)
⊢ 𝑇 = (Spec‘𝑅)    &   ⊢ 𝑈 = (Spec‘𝑆)    &   ⊢ 𝐴 = (PrmIdeal‘𝑅)    &   ⊢ 𝐵 = (PrmIdeal‘𝑆)    &   ⊢ 𝐽 = (TopOpen‘𝑇)    &   ⊢ 𝐾 = (TopOpen‘𝑈)    &   ⊢ 𝐺 = (𝑖 ∈ 𝐵 ↦ (◡𝐹 “ 𝑖))    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆))    &   ⊢ (𝜑 → ran 𝐹 = (Base‘𝑆))    &   ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))    &   ⊢ 𝑉 = (𝑗 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑗 ⊆ 𝑘})    &   ⊢ 𝑊 = (𝑗 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑗 ⊆ 𝑘})    ⇒   ⊢ (𝜑 → (𝑊‘(𝐹 “ 𝐼)) = (◡𝐺 “ (𝑉‘𝐼)))
Theorem	rhmpreimacn 33893*	The function mapping a prime ideal to its preimage by a surjective ring homomorphism is continuous, when considering the Zariski topology. Corollary 1.2.3 of [EGA], p. 83. Notice that the direction of the continuous map 𝐺 is reverse: the original ring homomorphism 𝐹 goes from 𝑅 to 𝑆, but the continuous map 𝐺 goes from 𝐵 to 𝐴. This mapping is also called "induced map on prime spectra" or "pullback on primes". (Contributed by Thierry Arnoux, 8-Jul-2024.)
⊢ 𝑇 = (Spec‘𝑅)    &   ⊢ 𝑈 = (Spec‘𝑆)    &   ⊢ 𝐴 = (PrmIdeal‘𝑅)    &   ⊢ 𝐵 = (PrmIdeal‘𝑆)    &   ⊢ 𝐽 = (TopOpen‘𝑇)    &   ⊢ 𝐾 = (TopOpen‘𝑈)    &   ⊢ 𝐺 = (𝑖 ∈ 𝐵 ↦ (◡𝐹 “ 𝑖))    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆))    &   ⊢ (𝜑 → ran 𝐹 = (Base‘𝑆))    ⇒   ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))
21.3.14.8  Pseudometrics
Syntax	cmetid 33894	Extend class notation with the class of metric identifications.
class ~Met
Syntax	cpstm 33895	Extend class notation with the metric induced by a pseudometric.
class pstoMet
Definition	df-metid 33896*	Define the metric identification relation for a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
⊢ ~Met = (𝑑 ∈ ∪ ran PsMet ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ dom dom 𝑑 ∧ 𝑦 ∈ dom dom 𝑑) ∧ (𝑥𝑑𝑦) = 0)})
Definition	df-pstm 33897*	Define the metric induced by a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
⊢ pstoMet = (𝑑 ∈ ∪ ran PsMet ↦ (𝑎 ∈ (dom dom 𝑑 / (~Met‘𝑑)), 𝑏 ∈ (dom dom 𝑑 / (~Met‘𝑑)) ↦ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝑑𝑦)}))
Theorem	metidval 33898*	Value of the metric identification relation. (Contributed by Thierry Arnoux, 7-Feb-2018.)
⊢ (𝐷 ∈ (PsMet‘𝑋) → (~Met‘𝐷) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝑥𝐷𝑦) = 0)})
Theorem	metidss 33899	As a relation, the metric identification is a subset of a Cartesian product. (Contributed by Thierry Arnoux, 7-Feb-2018.)
⊢ (𝐷 ∈ (PsMet‘𝑋) → (~Met‘𝐷) ⊆ (𝑋 × 𝑋))
Theorem	metidv 33900	𝐴 and 𝐵 identify by the metric 𝐷 if their distance is zero. (Contributed by Thierry Arnoux, 7-Feb-2018.)
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴(~Met‘𝐷)𝐵 ↔ (𝐴𝐷𝐵) = 0))