P. 230 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 22901-23000   *Has distinct variable group(s)

Type	Label	Description
Statement
Definition	df-mdet 22901*	Determinant of a square matrix. This definition is based on Leibniz' Formula (see mdetleib 22903). The properties of the axiomatic definition of a determinant according to [Weierstrass] p. 272 are derived from this definition as theorems: "The determinant function is the unique multilinear, alternating and normalized function from the algebra of square matrices of the same dimension over a commutative ring to this ring". Functionality is shown by mdetf 22911. Multilineary means "linear for each row" - the additivity is shown by mdetrlin 22918, the homogeneity by mdetrsca 22919. Furthermore, it is shown that the determinant function is alternating (see mdetralt 22924) and normalized (see mdet1 22917). Finally, uniqueness is shown by mdetuni 22938. As a consequence, the "determinant of a square matrix" is the function value of the determinant function for this square matrix, see mdetleib 22903. (Contributed by Stefan O'Rear, 9-Sep-2015.) (Revised by SO, 10-Jul-2018.)
⊢ maDet = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑟 Σg (𝑝 ∈ (Base‘(SymGrp‘𝑛)) ↦ ((((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)(.r‘𝑟)((mulGrp‘𝑟) Σg (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥))))))))
Theorem	mdetfval 22902*	First substitution for the determinant definition. (Contributed by Stefan O'Rear, 9-Sep-2015.) (Revised by SO, 9-Jul-2018.)
⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝑌 = (ℤRHom‘𝑅)    &   ⊢ 𝑆 = (pmSgn‘𝑁)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑈 = (mulGrp‘𝑅)    ⇒   ⊢ 𝐷 = (𝑚 ∈ 𝐵 ↦ (𝑅 Σg (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥)))))))
Theorem	mdetleib 22903*	Full substitution of our determinant definition (also known as Leibniz' Formula, expanding by columns). Proposition 4.6 in [Lang] p. 514. (Contributed by Stefan O'Rear, 3-Oct-2015.) (Revised by SO, 9-Jul-2018.)
⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝑌 = (ℤRHom‘𝑅)    &   ⊢ 𝑆 = (pmSgn‘𝑁)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑈 = (mulGrp‘𝑅)    ⇒   ⊢ (𝑀 ∈ 𝐵 → (𝐷‘𝑀) = (𝑅 Σg (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑀𝑥)))))))
Theorem	mdetleib2 22904*	Leibniz' formula can also be expanded by rows. (Contributed by Stefan O'Rear, 9-Jul-2018.) (Proof shortened by AV, 23-Jul-2019.)
⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝑌 = (ℤRHom‘𝑅)    &   ⊢ 𝑆 = (pmSgn‘𝑁)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑈 = (mulGrp‘𝑅)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐷‘𝑀) = (𝑅 Σg (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ (𝑥𝑀(𝑝‘𝑥))))))))
Theorem	nfimdetndef 22905	The determinant is not defined for an infinite matrix. (Contributed by AV, 27-Dec-2018.)
⊢ 𝐷 = (𝑁 maDet 𝑅)    ⇒   ⊢ (𝑁 ∉ Fin → 𝐷 = ∅)
Theorem	mdetfval1 22906*	First substitution of an alternative determinant definition. (Contributed by Stefan O'Rear, 9-Sep-2015.) (Revised by AV, 27-Dec-2018.)
⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝑌 = (ℤRHom‘𝑅)    &   ⊢ 𝑆 = (pmSgn‘𝑁)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑈 = (mulGrp‘𝑅)    ⇒   ⊢ 𝐷 = (𝑚 ∈ 𝐵 ↦ (𝑅 Σg (𝑝 ∈ 𝑃 ↦ ((𝑌‘(𝑆‘𝑝)) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥)))))))
Theorem	mdetleib1 22907*	Full substitution of an alternative determinant definition (also known as Leibniz' Formula). (Contributed by Stefan O'Rear, 3-Oct-2015.) (Revised by AV, 26-Dec-2018.)
⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝑌 = (ℤRHom‘𝑅)    &   ⊢ 𝑆 = (pmSgn‘𝑁)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑈 = (mulGrp‘𝑅)    ⇒   ⊢ (𝑀 ∈ 𝐵 → (𝐷‘𝑀) = (𝑅 Σg (𝑝 ∈ 𝑃 ↦ ((𝑌‘(𝑆‘𝑝)) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑀𝑥)))))))
Theorem	mdet0pr 22908	The determinant function for 0-dimensional matrices on a given ring is the function mapping the empty set to the unity element of that ring. (Contributed by AV, 28-Feb-2019.)
⊢ (𝑅 ∈ Ring → (∅ maDet 𝑅) = {⟨∅, (1r‘𝑅)⟩})
Theorem	mdet0f1o 22909	The determinant function for 0-dimensional matrices on a given ring is a bijection from the singleton containing the empty set (empty matrix) onto the singleton containing the unity element of that ring. (Contributed by AV, 28-Feb-2019.)
⊢ (𝑅 ∈ Ring → (∅ maDet 𝑅):{∅}–1-1-onto→{(1r‘𝑅)})
Theorem	mdet0fv0 22910	The determinant of the empty matrix on a given ring is the unity element of that ring. (Contributed by AV, 28-Feb-2019.)
⊢ (𝑅 ∈ Ring → ((∅ maDet 𝑅)‘∅) = (1r‘𝑅))
Theorem	mdetf 22911	Functionality of the determinant, see also definition in [Lang] p. 513. (Contributed by Stefan O'Rear, 9-Jul-2018.) (Proof shortened by AV, 23-Jul-2019.)
⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐾 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ CRing → 𝐷:𝐵⟶𝐾)
Theorem	mdetcl 22912	The determinant evaluates to an element of the base ring. (Contributed by Stefan O'Rear, 9-Sep-2015.) (Revised by AV, 7-Feb-2019.)
⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐾 = (Base‘𝑅)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐷‘𝑀) ∈ 𝐾)
Theorem	m1detdiag 22913	The determinant of a 1-dimensional matrix equals its (single) entry. (Contributed by AV, 6-Aug-2019.)
⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ (𝑁 = {𝐼} ∧ 𝐼 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → (𝐷‘𝑀) = (𝐼𝑀𝐼))
Theorem	mdetdiaglem 22914*	Lemma for mdetdiag 22915. Previously part of proof for mdet1 22917. (Contributed by SO, 10-Jul-2018.) (Revised by AV, 17-Aug-2019.)
⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐻 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝑍 = (ℤRHom‘𝑅)    &   ⊢ 𝑆 = (pmSgn‘𝑁)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 ) ∧ (𝑃 ∈ 𝐻 ∧ 𝑃 ≠ ( I ↾ 𝑁))) → (((𝑍 ∘ 𝑆)‘𝑃) · (𝐺 Σg (𝑘 ∈ 𝑁 ↦ ((𝑃‘𝑘)𝑀𝑘)))) = 0 )
Theorem	mdetdiag 22915*	The determinant of a diagonal matrix is the product of the entries in the diagonal. (Contributed by AV, 17-Aug-2019.)
⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐵) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 ) → (𝐷‘𝑀) = (𝐺 Σg (𝑘 ∈ 𝑁 ↦ (𝑘𝑀𝑘)))))
Theorem	mdetdiagid 22916*	The determinant of a diagonal matrix with identical entries is the power of the entry in the diagonal. (Contributed by AV, 17-Aug-2019.)
⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑅)    &   ⊢ · = (.g‘𝐺)    ⇒   ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑋 ∈ 𝐶)) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑋, 0 ) → (𝐷‘𝑀) = ((♯‘𝑁) · 𝑋)))
Theorem	mdet1 22917	The determinant of the identity matrix is 1, i.e. the determinant function is normalized, see also definition in [Lang] p. 513. (Contributed by SO, 10-Jul-2018.) (Proof shortened by AV, 25-Nov-2019.)
⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐼 = (1r‘𝐴)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → (𝐷‘𝐼) = 1 )
Theorem	mdetrlin 22918	The determinant function is additive for each row: The matrices X, Y, Z are identical except for the I's row, and the I's row of the matrix X is the componentwise sum of the I's row of the matrices Y and Z. In this case the determinant of X is the sum of the determinants of Y and Z. (Contributed by SO, 9-Jul-2018.) (Proof shortened by AV, 23-Jul-2019.)
⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ + = (+g‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑁)    &   ⊢ (𝜑 → (𝑋 ↾ ({𝐼} × 𝑁)) = ((𝑌 ↾ ({𝐼} × 𝑁)) ∘f + (𝑍 ↾ ({𝐼} × 𝑁))))    &   ⊢ (𝜑 → (𝑋 ↾ ((𝑁 ∖ {𝐼}) × 𝑁)) = (𝑌 ↾ ((𝑁 ∖ {𝐼}) × 𝑁)))    &   ⊢ (𝜑 → (𝑋 ↾ ((𝑁 ∖ {𝐼}) × 𝑁)) = (𝑍 ↾ ((𝑁 ∖ {𝐼}) × 𝑁)))    ⇒   ⊢ (𝜑 → (𝐷‘𝑋) = ((𝐷‘𝑌) + (𝐷‘𝑍)))
Theorem	mdetrsca 22919	The determinant function is homogeneous for each row: If the matrices 𝑋 and 𝑍 are identical except for the 𝐼-th row, and the 𝐼-th row of the matrix 𝑋 is the componentwise product of the 𝐼-th row of the matrix 𝑍 and the scalar 𝑌, then the determinant of 𝑋 is the determinant of 𝑍 multiplied by 𝑌. (Contributed by SO, 9-Jul-2018.) (Proof shortened by AV, 23-Jul-2019.)
⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐾)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑁)    &   ⊢ (𝜑 → (𝑋 ↾ ({𝐼} × 𝑁)) = ((({𝐼} × 𝑁) × {𝑌}) ∘f · (𝑍 ↾ ({𝐼} × 𝑁))))    &   ⊢ (𝜑 → (𝑋 ↾ ((𝑁 ∖ {𝐼}) × 𝑁)) = (𝑍 ↾ ((𝑁 ∖ {𝐼}) × 𝑁)))    ⇒   ⊢ (𝜑 → (𝐷‘𝑋) = (𝑌 · (𝐷‘𝑍)))
Theorem	mdetrsca2 22920*	The determinant function is homogeneous for each row (matrices are given explicitly by their entries). (Contributed by SO, 16-Jul-2018.)
⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑋 ∈ 𝐾)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑌 ∈ 𝐾)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐾)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑁)    ⇒   ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, (𝐹 · 𝑋), 𝑌))) = (𝐹 · (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, 𝑌)))))
Theorem	mdetr0 22921*	The determinant of a matrix with a row containing only 0's is 0. (Contributed by SO, 16-Jul-2018.)
⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑋 ∈ 𝐾)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑁)    ⇒   ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 0 , 𝑋))) = 0 )
Theorem	mdet0 22922	The determinant of the zero matrix (of dimension greater 0!) is 0. (Contributed by AV, 17-Aug-2019.) (Revised by AV, 3-Jul-2022.)
⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝑍 = (0g‘𝐴)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑁 ≠ ∅) → (𝐷‘𝑍) = 0 )
Theorem	mdetrlin2 22923*	The determinant function is additive for each row (matrices are given explicitly by their entries). (Contributed by SO, 16-Jul-2018.)
⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑋 ∈ 𝐾)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑌 ∈ 𝐾)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑍 ∈ 𝐾)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑁)    ⇒   ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, (𝑋 + 𝑌), 𝑍))) = ((𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, 𝑍))) + (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑌, 𝑍)))))
Theorem	mdetralt 22924*	The determinant function is alternating regarding rows: if a matrix has two identical rows, its determinant is 0. Corollary 4.9 in [Lang] p. 515. (Contributed by SO, 10-Jul-2018.) (Proof shortened by AV, 23-Jul-2018.)
⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑁)    &   ⊢ (𝜑 → 𝐽 ∈ 𝑁)    &   ⊢ (𝜑 → 𝐼 ≠ 𝐽)    &   ⊢ (𝜑 → ∀𝑎 ∈ 𝑁 (𝐼𝑋𝑎) = (𝐽𝑋𝑎))    ⇒   ⊢ (𝜑 → (𝐷‘𝑋) = 0 )
Theorem	mdetralt2 22925*	The determinant function is alternating regarding rows (matrix is given explicitly by its entries). (Contributed by SO, 16-Jul-2018.)
⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑁) → 𝑋 ∈ 𝐾)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑌 ∈ 𝐾)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑁)    &   ⊢ (𝜑 → 𝐽 ∈ 𝑁)    &   ⊢ (𝜑 → 𝐼 ≠ 𝐽)    ⇒   ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)))) = 0 )
Theorem	mdetero 22926*	The determinant function is multilinear (additive and homogeneous for each row (matrices are given explicitly by their entries). Corollary 4.9 in [Lang] p. 515. (Contributed by SO, 16-Jul-2018.)
⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑁) → 𝑋 ∈ 𝐾)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑁) → 𝑌 ∈ 𝐾)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑍 ∈ 𝐾)    &   ⊢ (𝜑 → 𝑊 ∈ 𝐾)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑁)    &   ⊢ (𝜑 → 𝐽 ∈ 𝑁)    &   ⊢ (𝜑 → 𝐼 ≠ 𝐽)    ⇒   ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, (𝑋 + (𝑊 · 𝑌)), if(𝑖 = 𝐽, 𝑌, 𝑍)))) = (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑌, 𝑍)))))
Theorem	mdettpos 22927	Determinant is invariant under transposition. Proposition 4.8 in [Lang] p. 514. (Contributed by Stefan O'Rear, 9-Jul-2018.)
⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐷‘tpos 𝑀) = (𝐷‘𝑀))
Theorem	mdetunilem1 22928*	Lemma for mdetuni 22938. (Contributed by SO, 14-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐷:𝐵⟶𝐾)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑁 ∀𝑧 ∈ 𝑁 ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷‘𝑥) = 0 ))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧))))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐾 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧))))    ⇒   ⊢ (((𝜑 ∧ 𝐸 ∈ 𝐵 ∧ ∀𝑤 ∈ 𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹 ∈ 𝑁 ∧ 𝐺 ∈ 𝑁 ∧ 𝐹 ≠ 𝐺)) → (𝐷‘𝐸) = 0 )
Theorem	mdetunilem2 22929*	Lemma for mdetuni 22938. (Contributed by SO, 15-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐷:𝐵⟶𝐾)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑁 ∀𝑧 ∈ 𝑁 ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷‘𝑥) = 0 ))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧))))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐾 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧))))    &   ⊢ (𝜓 → 𝜑)    &   ⊢ (𝜓 → (𝐸 ∈ 𝑁 ∧ 𝐺 ∈ 𝑁 ∧ 𝐸 ≠ 𝐺))    &   ⊢ ((𝜓 ∧ 𝑏 ∈ 𝑁) → 𝐹 ∈ 𝐾)    &   ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝐻 ∈ 𝐾)    ⇒   ⊢ (𝜓 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐹, if(𝑎 = 𝐺, 𝐹, 𝐻)))) = 0 )
Theorem	mdetunilem3 22930*	Lemma for mdetuni 22938. (Contributed by SO, 15-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐷:𝐵⟶𝐾)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑁 ∀𝑧 ∈ 𝑁 ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷‘𝑥) = 0 ))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧))))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐾 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧))))    ⇒   ⊢ (((𝜑 ∧ 𝐸 ∈ 𝐵 ∧ 𝐹 ∈ 𝐵) ∧ (𝐺 ∈ 𝐵 ∧ 𝐻 ∈ 𝑁 ∧ (𝐸 ↾ ({𝐻} × 𝑁)) = ((𝐹 ↾ ({𝐻} × 𝑁)) ∘f + (𝐺 ↾ ({𝐻} × 𝑁)))) ∧ ((𝐸 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)) = (𝐹 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)) ∧ (𝐸 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)) = (𝐺 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)))) → (𝐷‘𝐸) = ((𝐷‘𝐹) + (𝐷‘𝐺)))
Theorem	mdetunilem4 22931*	Lemma for mdetuni 22938. (Contributed by SO, 15-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐷:𝐵⟶𝐾)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑁 ∀𝑧 ∈ 𝑁 ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷‘𝑥) = 0 ))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧))))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐾 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧))))    ⇒   ⊢ ((𝜑 ∧ (𝐸 ∈ 𝐵 ∧ 𝐹 ∈ 𝐾 ∧ 𝐺 ∈ 𝐵) ∧ (𝐻 ∈ 𝑁 ∧ (𝐸 ↾ ({𝐻} × 𝑁)) = ((({𝐻} × 𝑁) × {𝐹}) ∘f · (𝐺 ↾ ({𝐻} × 𝑁))) ∧ (𝐸 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)) = (𝐺 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)))) → (𝐷‘𝐸) = (𝐹 · (𝐷‘𝐺)))
Theorem	mdetunilem5 22932*	Lemma for mdetuni 22938. (Contributed by SO, 15-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐷:𝐵⟶𝐾)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑁 ∀𝑧 ∈ 𝑁 ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷‘𝑥) = 0 ))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧))))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐾 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧))))    &   ⊢ (𝜓 → 𝜑)    &   ⊢ (𝜓 → 𝐸 ∈ 𝑁)    &   ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (𝐹 ∈ 𝐾 ∧ 𝐺 ∈ 𝐾 ∧ 𝐻 ∈ 𝐾))    ⇒   ⊢ (𝜓 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, (𝐹 + 𝐺), 𝐻))) = ((𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐹, 𝐻))) + (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, 𝐻)))))
Theorem	mdetunilem6 22933*	Lemma for mdetuni 22938. (Contributed by SO, 15-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐷:𝐵⟶𝐾)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑁 ∀𝑧 ∈ 𝑁 ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷‘𝑥) = 0 ))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧))))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐾 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧))))    &   ⊢ (𝜓 → 𝜑)    &   ⊢ (𝜓 → (𝐸 ∈ 𝑁 ∧ 𝐹 ∈ 𝑁 ∧ 𝐸 ≠ 𝐹))    &   ⊢ ((𝜓 ∧ 𝑏 ∈ 𝑁) → (𝐺 ∈ 𝐾 ∧ 𝐻 ∈ 𝐾))    &   ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝐼 ∈ 𝐾)    ⇒   ⊢ (𝜓 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, if(𝑎 = 𝐹, 𝐻, 𝐼)))) = ((invg‘𝑅)‘(𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐻, if(𝑎 = 𝐹, 𝐺, 𝐼))))))
Theorem	mdetunilem7 22934*	Lemma for mdetuni 22938. (Contributed by SO, 15-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐷:𝐵⟶𝐾)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑁 ∀𝑧 ∈ 𝑁 ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷‘𝑥) = 0 ))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧))))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐾 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧))))    ⇒   ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝐸‘𝑎)𝐹𝑏))) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝐸) · (𝐷‘𝐹)))
Theorem	mdetunilem8 22935*	Lemma for mdetuni 22938. (Contributed by SO, 15-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐷:𝐵⟶𝐾)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑁 ∀𝑧 ∈ 𝑁 ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷‘𝑥) = 0 ))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧))))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐾 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧))))    &   ⊢ (𝜑 → (𝐷‘(1r‘𝐴)) = 0 )    ⇒   ⊢ ((𝜑 ∧ 𝐸:𝑁⟶𝑁) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if((𝐸‘𝑎) = 𝑏, 1 , 0 ))) = 0 )
Theorem	mdetunilem9 22936*	Lemma for mdetuni 22938. (Contributed by SO, 15-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐷:𝐵⟶𝐾)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑁 ∀𝑧 ∈ 𝑁 ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷‘𝑥) = 0 ))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧))))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐾 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧))))    &   ⊢ (𝜑 → (𝐷‘(1r‘𝐴)) = 0 )    &   ⊢ 𝑌 = {𝑥 ∣ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑m 𝑁)(∀𝑤 ∈ 𝑥 (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 )}    ⇒   ⊢ (𝜑 → 𝐷 = (𝐵 × { 0 }))
Theorem	mdetuni0 22937*	Lemma for mdetuni 22938. (Contributed by SO, 15-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐷:𝐵⟶𝐾)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑁 ∀𝑧 ∈ 𝑁 ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷‘𝑥) = 0 ))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧))))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐾 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧))))    &   ⊢ 𝐸 = (𝑁 maDet 𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐷‘𝐹) = ((𝐷‘(1r‘𝐴)) · (𝐸‘𝐹)))
Theorem	mdetuni 22938*	According to the definition in [Weierstrass] p. 272, the determinant function is the unique multilinear, alternating and normalized function from the algebra of square matrices of the same dimension over a commutative ring to this ring. So for any multilinear (mdetuni.li and mdetuni.sc), alternating (mdetuni.al) and normalized (mdetuni.no) function D (mdetuni.ff) from the algebra of square matrices (mdetuni.a) to their underlying commutative ring (mdetuni.cr), the function value of this function D for a matrix F (mdetuni.f) is the determinant of this matrix. (Contributed by Stefan O'Rear, 15-Jul-2018.) (Revised by Alexander van der Vekens, 8-Feb-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐷:𝐵⟶𝐾)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑁 ∀𝑧 ∈ 𝑁 ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷‘𝑥) = 0 ))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧))))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐾 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧))))    &   ⊢ 𝐸 = (𝑁 maDet 𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐷‘(1r‘𝐴)) = 1 )    ⇒   ⊢ (𝜑 → (𝐷‘𝐹) = (𝐸‘𝐹))
Theorem	mdetmul 22939	Multiplicativity of the determinant function: the determinant of a matrix product of square matrices equals the product of their determinants. Proposition 4.15 in [Lang] p. 517. (Contributed by Stefan O'Rear, 16-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ ∙ = (.r‘𝐴)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐷‘(𝐹 ∙ 𝐺)) = ((𝐷‘𝐹) · (𝐷‘𝐺)))
11.5.2  Determinants of 2 x 2 -matrices
Theorem	m2detleiblem1 22940	Lemma 1 for m2detleib 22947. (Contributed by AV, 12-Dec-2018.)
⊢ 𝑁 = {1, 2}    &   ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝑌 = (ℤRHom‘𝑅)    &   ⊢ 𝑆 = (pmSgn‘𝑁)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃) → (𝑌‘(𝑆‘𝑄)) = (((pmSgn‘𝑁)‘𝑄)(.g‘𝑅) 1 ))
Theorem	m2detleiblem5 22941	Lemma 5 for m2detleib 22947. (Contributed by AV, 20-Dec-2018.)
⊢ 𝑁 = {1, 2}    &   ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝑌 = (ℤRHom‘𝑅)    &   ⊢ 𝑆 = (pmSgn‘𝑁)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩}) → (𝑌‘(𝑆‘𝑄)) = 1 )
Theorem	m2detleiblem6 22942	Lemma 6 for m2detleib 22947. (Contributed by AV, 20-Dec-2018.)
⊢ 𝑁 = {1, 2}    &   ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝑌 = (ℤRHom‘𝑅)    &   ⊢ 𝑆 = (pmSgn‘𝑁)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝐼 = (invg‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 2⟩, ⟨2, 1⟩}) → (𝑌‘(𝑆‘𝑄)) = (𝐼‘ 1 ))
Theorem	m2detleiblem7 22943	Lemma 7 for m2detleib 22947. (Contributed by AV, 20-Dec-2018.)
⊢ 𝑁 = {1, 2}    &   ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝑌 = (ℤRHom‘𝑅)    &   ⊢ 𝑆 = (pmSgn‘𝑁)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝐼 = (invg‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ − = (-g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ (Base‘𝑅) ∧ 𝑍 ∈ (Base‘𝑅)) → (𝑋(+g‘𝑅)((𝐼‘ 1 ) · 𝑍)) = (𝑋 − 𝑍))
Theorem	m2detleiblem2 22944*	Lemma 2 for m2detleib 22947. (Contributed by AV, 16-Dec-2018.) (Proof shortened by AV, 1-Jan-2019.)
⊢ 𝑁 = {1, 2}    &   ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐺 = (mulGrp‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃 ∧ 𝑀 ∈ 𝐵) → (𝐺 Σg (𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))) ∈ (Base‘𝑅))
Theorem	m2detleiblem3 22945*	Lemma 3 for m2detleib 22947. (Contributed by AV, 16-Dec-2018.) (Proof shortened by AV, 2-Jan-2019.)
⊢ 𝑁 = {1, 2}    &   ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ · = (+g‘𝐺)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → (𝐺 Σg (𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))) = ((1𝑀1) · (2𝑀2)))
Theorem	m2detleiblem4 22946*	Lemma 4 for m2detleib 22947. (Contributed by AV, 20-Dec-2018.) (Proof shortened by AV, 2-Jan-2019.)
⊢ 𝑁 = {1, 2}    &   ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ · = (+g‘𝐺)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 2⟩, ⟨2, 1⟩} ∧ 𝑀 ∈ 𝐵) → (𝐺 Σg (𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))) = ((2𝑀1) · (1𝑀2)))
Theorem	m2detleib 22947	Leibniz' Formula for 2x2-matrices. (Contributed by AV, 21-Dec-2018.) (Revised by AV, 26-Dec-2018.) (Proof shortened by AV, 23-Jul-2019.)
⊢ 𝑁 = {1, 2}    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ − = (-g‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝐷‘𝑀) = (((1𝑀1) · (2𝑀2)) − ((2𝑀1) · (1𝑀2))))
11.5.3  The matrix adjugate/adjunct
Syntax	cmadu 22948	Syntax for the matrix adjugate/adjunct function.
class maAdju
Syntax	cminmar1 22949	Syntax for the minor matrices of a square matrix.
class minMatR1
Definition	df-madu 22950*	Define the adjugate or adjunct (matrix of cofactors) of a square matrix. This definition gives the standard cofactors, however the internal minors are not the standard minors, see definition in [Lang] p. 518. (Contributed by Stefan O'Rear, 7-Sep-2015.) (Revised by SO, 10-Jul-2018.)
⊢ maAdju = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ ((𝑛 maDet 𝑟)‘(𝑘 ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r‘𝑟), (0g‘𝑟)), (𝑘𝑚𝑙)))))))
Definition	df-minmar1 22951*	Define the matrices whose determinants are the minors of a square matrix. In contrast to the standard definition of minors, a row is replaced by 0's and one 1 instead of deleting the column and row (e.g., definition in [Lang] p. 515). By this, the determinant of such a matrix is equal to the minor determined in the standard way (as determinant of a submatrix, see df-subma 22893- note that the matrix is transposed compared with the submatrix defined in df-subma 22893, but this does not matter because the determinants are the same, see mdettpos 22927). Such matrices are used in the definition of an adjunct of a square matrix, see df-madu 22950. (Contributed by AV, 27-Dec-2018.)
⊢ minMatR1 = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑘 ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, (1r‘𝑟), (0g‘𝑟)), (𝑖𝑚𝑗))))))
Theorem	mndifsplit 22952	Lemma for maducoeval2 22956. (Contributed by SO, 16-Jul-2018.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 0 = (0g‘𝑀)    &   ⊢ + = (+g‘𝑀)    ⇒   ⊢ ((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) → if((𝜑 ∨ 𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )))
Theorem	madufval 22953*	First substitution for the adjunct (cofactor) matrix. (Contributed by SO, 11-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐽 = (𝑁 maAdju 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ 𝐽 = (𝑚 ∈ 𝐵 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙))))))
Theorem	maduval 22954*	Second substitution for the adjunct (cofactor) matrix. (Contributed by SO, 11-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐽 = (𝑁 maAdju 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑀 ∈ 𝐵 → (𝐽‘𝑀) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙))))))
Theorem	maducoeval 22955*	An entry of the adjunct (cofactor) matrix. (Contributed by SO, 11-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐽 = (𝑁 maAdju 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → (𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙)))))
Theorem	maducoeval2 22956*	An entry of the adjunct (cofactor) matrix. (Contributed by SO, 17-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐽 = (𝑁 maAdju 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → (𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if((𝑘 = 𝐻 ∨ 𝑙 = 𝐼), if((𝑙 = 𝐼 ∧ 𝑘 = 𝐻), 1 , 0 ), (𝑘𝑀𝑙)))))
Theorem	maduf 22957	Creating the adjunct of matrices is a function from the set of matrices into the set of matrices. (Contributed by Stefan O'Rear, 11-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐽 = (𝑁 maAdju 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    ⇒   ⊢ (𝑅 ∈ CRing → 𝐽:𝐵⟶𝐵)
Theorem	madutpos 22958	The adjuct of a transposed matrix is the transposition of the adjunct of the matrix. (Contributed by Stefan O'Rear, 17-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐽 = (𝑁 maAdju 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐽‘tpos 𝑀) = tpos (𝐽‘𝑀))
Theorem	madugsum 22959*	The determinant of a matrix with a row 𝐿 consisting of the same element 𝑋 is the sum of the elements of the 𝐿-th column of the adjunct of the matrix multiplied with 𝑋. (Contributed by Stefan O'Rear, 16-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐽 = (𝑁 maAdju 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑀 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → 𝑋 ∈ 𝐾)    &   ⊢ (𝜑 → 𝐿 ∈ 𝑁)    ⇒   ⊢ (𝜑 → (𝑅 Σg (𝑖 ∈ 𝑁 ↦ (𝑋 · (𝑖(𝐽‘𝑀)𝐿)))) = (𝐷‘(𝑗 ∈ 𝑁, 𝑖 ∈ 𝑁 ↦ if(𝑗 = 𝐿, 𝑋, (𝑗𝑀𝑖)))))
Theorem	madurid 22960	Multiplying a matrix with its adjunct results in the identity matrix multiplied with the determinant of the matrix. See Proposition 4.16 in [Lang] p. 518. (Contributed by Stefan O'Rear, 16-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐽 = (𝑁 maAdju 𝑅)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 1 = (1r‘𝐴)    &   ⊢ · = (.r‘𝐴)    &   ⊢ ∙ = ( ·𝑠 ‘𝐴)    ⇒   ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → (𝑀 · (𝐽‘𝑀)) = ((𝐷‘𝑀) ∙ 1 ))
Theorem	madulid 22961	Multiplying the adjunct of a matrix with the matrix results in the identity matrix multiplied with the determinant of the matrix. See Proposition 4.16 in [Lang] p. 518. (Contributed by Stefan O'Rear, 17-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐽 = (𝑁 maAdju 𝑅)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 1 = (1r‘𝐴)    &   ⊢ · = (.r‘𝐴)    &   ⊢ ∙ = ( ·𝑠 ‘𝐴)    ⇒   ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → ((𝐽‘𝑀) · 𝑀) = ((𝐷‘𝑀) ∙ 1 ))
Theorem	minmar1fval 22962*	First substitution for the definition of a matrix for a minor. (Contributed by AV, 31-Dec-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑄 = (𝑁 minMatR1 𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ 𝑄 = (𝑚 ∈ 𝐵 ↦ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗)))))
Theorem	minmar1val0 22963*	Second substitution for the definition of a matrix for a minor. (Contributed by AV, 31-Dec-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑄 = (𝑁 minMatR1 𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑀 ∈ 𝐵 → (𝑄‘𝑀) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑀𝑗)))))
Theorem	minmar1val 22964*	Third substitution for the definition of a matrix for a minor. (Contributed by AV, 31-Dec-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑄 = (𝑁 minMatR1 𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → (𝐾(𝑄‘𝑀)𝐿) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗))))
Theorem	minmar1eval 22965	An entry of a matrix for a minor. (Contributed by AV, 31-Dec-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑄 = (𝑁 minMatR1 𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑀 ∈ 𝐵 ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝐾(𝑄‘𝑀)𝐿)𝐽) = if(𝐼 = 𝐾, if(𝐽 = 𝐿, 1 , 0 ), (𝐼𝑀𝐽)))
Theorem	minmar1marrep 22966	The minor matrix is a special case of a matrix with a replaced row. (Contributed by AV, 12-Feb-2019.) (Revised by AV, 4-Jul-2022.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((𝑁 minMatR1 𝑅)‘𝑀) = (𝑀(𝑁 matRRep 𝑅) 1 ))
Theorem	minmar1cl 22967	Closure of the row replacement function for square matrices: The matrix for a minor is a matrix. (Contributed by AV, 13-Feb-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    ⇒   ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁)) → (𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐿) ∈ 𝐵)
Theorem	maducoevalmin1 22968	The coefficients of an adjunct (matrix of cofactors) expressed as determinants of the minor matrices (alternative definition) of the original matrix. (Contributed by AV, 31-Dec-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐽 = (𝑁 maAdju 𝑅)    ⇒   ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → (𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝐻((𝑁 minMatR1 𝑅)‘𝑀)𝐼)))
11.5.4  Laplace expansion of determinants (special case) According to Wikipedia ("Laplace expansion", 08-Mar-2019, https://en.wikipedia.org/wiki/Laplace_expansion) "In linear algebra, the Laplace expansion, named after Pierre-Simon Laplace, also called cofactor expansion, is an expression for the determinant det(B) of an n x n -matrix B that is a weighted sum of the determinants of n sub-matrices of B, each of size (n-1) x (n-1)". The expansion is usually performed for a row of matrix B (alternately for a column of matrix B). The mentioned "sub-matrices" are the matrices resultung from deleting the i-th row and the j-th column of matrix B. The mentioned "weights" (factors/coefficients) are the elements at position i and j in matrix B. If the expansion is performed for a row, the coefficients are the elements of the selected row. In the following, only the case where the row for the expansion contains only the zero element of the underlying ring except at the diagonal position. By this, the sum for the Laplace expansion is reduced to one summand, consisting of the element at the diagonal position multiplied with the determinant of the corresponding submatrix, see smadiadetg 22989 or smadiadetr 22991.
Theorem	symgmatr01lem 22969*	Lemma for symgmatr01 22970. (Contributed by AV, 3-Jan-2019.)
⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    ⇒   ⊢ ((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → (𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) → ∃𝑘 ∈ 𝑁 if(𝑘 = 𝐾, if((𝑄‘𝑘) = 𝐿, 𝐴, 𝐵), (𝑘𝑀(𝑄‘𝑘))) = 𝐵))
Theorem	symgmatr01 22970*	Applying a permutation that does not fix a certain element of a set to a second element to an index of a matrix a row with 0's and a 1. (Contributed by AV, 3-Jan-2019.)
⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ ((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → (𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) → ∃𝑘 ∈ 𝑁 (𝑘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑄‘𝑘)) = 0 ))
Theorem	gsummatr01lem1 22971*	Lemma A for gsummatr01 22975. (Contributed by AV, 8-Jan-2019.)
⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝑅 = {𝑟 ∈ 𝑃 ∣ (𝑟‘𝐾) = 𝐿}    ⇒   ⊢ ((𝑄 ∈ 𝑅 ∧ 𝑋 ∈ 𝑁) → (𝑄‘𝑋) ∈ 𝑁)
Theorem	gsummatr01lem2 22972*	Lemma B for gsummatr01 22975. (Contributed by AV, 8-Jan-2019.)
⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝑅 = {𝑟 ∈ 𝑃 ∣ (𝑟‘𝐾) = 𝐿}    ⇒   ⊢ ((𝑄 ∈ 𝑅 ∧ 𝑋 ∈ 𝑁) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝐴𝑗) ∈ (Base‘𝐺) → (𝑋𝐴(𝑄‘𝑋)) ∈ (Base‘𝐺)))
Theorem	gsummatr01lem3 22973*	Lemma 1 for gsummatr01 22975. (Contributed by AV, 8-Jan-2019.)
⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝑅 = {𝑟 ∈ 𝑃 ∣ (𝑟‘𝐾) = 𝐿}    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑆 = (Base‘𝐺)    ⇒   ⊢ (((𝐺 ∈ CMnd ∧ 𝑁 ∈ Fin) ∧ (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝐴𝑗) ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝑄 ∈ 𝑅)) → (𝐺 Σg (𝑛 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾}) ↦ (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 0 , 𝐵), (𝑖𝐴𝑗)))(𝑄‘𝑛)))) = ((𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 0 , 𝐵), (𝑖𝐴𝑗)))(𝑄‘𝑛))))(+g‘𝐺)(𝐾(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 0 , 𝐵), (𝑖𝐴𝑗)))(𝑄‘𝐾))))
Theorem	gsummatr01lem4 22974*	Lemma 2 for gsummatr01 22975. (Contributed by AV, 8-Jan-2019.)
⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝑅 = {𝑟 ∈ 𝑃 ∣ (𝑟‘𝐾) = 𝐿}    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑆 = (Base‘𝐺)    ⇒   ⊢ ((((𝐺 ∈ CMnd ∧ 𝑁 ∈ Fin) ∧ (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝐴𝑗) ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝑄 ∈ 𝑅)) ∧ 𝑛 ∈ (𝑁 ∖ {𝐾})) → (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 0 , 𝐵), (𝑖𝐴𝑗)))(𝑄‘𝑛)) = (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐿}) ↦ (𝑖𝐴𝑗))(𝑄‘𝑛)))
Theorem	gsummatr01 22975*	Lemma 1 for smadiadetlem4 22985. (Contributed by AV, 8-Jan-2019.)
⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝑅 = {𝑟 ∈ 𝑃 ∣ (𝑟‘𝐾) = 𝐿}    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑆 = (Base‘𝐺)    ⇒   ⊢ (((𝐺 ∈ CMnd ∧ 𝑁 ∈ Fin) ∧ (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝐴𝑗) ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝑄 ∈ 𝑅)) → (𝐺 Σg (𝑛 ∈ 𝑁 ↦ (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 0 , 𝐵), (𝑖𝐴𝑗)))(𝑄‘𝑛)))) = (𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐿}) ↦ (𝑖𝐴𝑗))(𝑄‘𝑛)))))
Theorem	marep01ma 22976*	Replacing a row of a square matrix by a row with 0's and a 1 results in a square matrix of the same dimension. (Contributed by AV, 30-Dec-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑅 ∈ CRing    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝑀 ∈ 𝐵 → (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙))) ∈ 𝐵)
Theorem	smadiadetlem0 22977*	Lemma 0 for smadiadet 22986: The products of the Leibniz' formula vanish for all permutations fixing the index of the row containing the 0's and the 1 to the column with the 1. (Contributed by AV, 3-Jan-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑅 ∈ CRing    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝐺 = (mulGrp‘𝑅)    ⇒   ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → (𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) → (𝐺 Σg (𝑛 ∈ 𝑁 ↦ (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑄‘𝑛)))) = 0 ))
Theorem	smadiadetlem1 22978*	Lemma 1 for smadiadet 22986: A summand of the determinant of a matrix belongs to the underlying ring. (Contributed by AV, 1-Jan-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑅 ∈ CRing    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ 𝑌 = (ℤRHom‘𝑅)    &   ⊢ 𝑆 = (pmSgn‘𝑁)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) ∧ 𝑝 ∈ 𝑃) → (((𝑌 ∘ 𝑆)‘𝑝)(.r‘𝑅)(𝐺 Σg (𝑛 ∈ 𝑁 ↦ (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 1 , 0 ), (𝑖𝑀𝑗)))(𝑝‘𝑛))))) ∈ (Base‘𝑅))
Theorem	smadiadetlem1a 22979*	Lemma 1a for smadiadet 22986: The summands of the Leibniz' formula vanish for all permutations fixing the index of the row containing the 0's and the 1 to the column with the 1. (Contributed by AV, 3-Jan-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑅 ∈ CRing    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ 𝑌 = (ℤRHom‘𝑅)    &   ⊢ 𝑆 = (pmSgn‘𝑁)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → (𝑅 Σg (𝑝 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝐺 Σg (𝑛 ∈ 𝑁 ↦ (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑝‘𝑛))))))) = 0 )
Theorem	smadiadetlem2 22980*	Lemma 2 for smadiadet 22986: The summands of the Leibniz' formula vanish for all permutations fixing the index of the row containing the 0's and the 1 to itself. (Contributed by AV, 31-Dec-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑅 ∈ CRing    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ 𝑌 = (ℤRHom‘𝑅)    &   ⊢ 𝑆 = (pmSgn‘𝑁)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) → (𝑅 Σg (𝑝 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝐺 Σg (𝑛 ∈ 𝑁 ↦ (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 1 , 0 ), (𝑖𝑀𝑗)))(𝑝‘𝑛))))))) = 0 )
Theorem	smadiadetlem3lem0 22981*	Lemma 0 for smadiadetlem3 22984. (Contributed by AV, 12-Jan-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑅 ∈ CRing    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ 𝑌 = (ℤRHom‘𝑅)    &   ⊢ 𝑆 = (pmSgn‘𝑁)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑊 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   ⊢ 𝑍 = (pmSgn‘(𝑁 ∖ {𝐾}))    ⇒   ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ 𝑊) → (((𝑌 ∘ 𝑍)‘𝑄)(.r‘𝑅)(𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑖𝑀𝑗))(𝑄‘𝑛))))) ∈ (Base‘𝑅))
Theorem	smadiadetlem3lem1 22982*	Lemma 1 for smadiadetlem3 22984. (Contributed by AV, 12-Jan-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑅 ∈ CRing    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ 𝑌 = (ℤRHom‘𝑅)    &   ⊢ 𝑆 = (pmSgn‘𝑁)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑊 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   ⊢ 𝑍 = (pmSgn‘(𝑁 ∖ {𝐾}))    ⇒   ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) → (𝑝 ∈ 𝑊 ↦ (((𝑌 ∘ 𝑍)‘𝑝)(.r‘𝑅)(𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑖𝑀𝑗))(𝑝‘𝑛)))))):𝑊⟶(Base‘𝑅))
Theorem	smadiadetlem3lem2 22983*	Lemma 2 for smadiadetlem3 22984. (Contributed by AV, 12-Jan-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑅 ∈ CRing    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ 𝑌 = (ℤRHom‘𝑅)    &   ⊢ 𝑆 = (pmSgn‘𝑁)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑊 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   ⊢ 𝑍 = (pmSgn‘(𝑁 ∖ {𝐾}))    ⇒   ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) → ran (𝑝 ∈ 𝑊 ↦ (((𝑌 ∘ 𝑍)‘𝑝)(.r‘𝑅)(𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑖𝑀𝑗))(𝑝‘𝑛)))))) ⊆ ((Cntz‘𝑅)‘ran (𝑝 ∈ 𝑊 ↦ (((𝑌 ∘ 𝑍)‘𝑝)(.r‘𝑅)(𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑖𝑀𝑗))(𝑝‘𝑛))))))))
Theorem	smadiadetlem3 22984*	Lemma 3 for smadiadet 22986. (Contributed by AV, 31-Jan-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑅 ∈ CRing    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ 𝑌 = (ℤRHom‘𝑅)    &   ⊢ 𝑆 = (pmSgn‘𝑁)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑊 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   ⊢ 𝑍 = (pmSgn‘(𝑁 ∖ {𝐾}))    ⇒   ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) → (𝑅 Σg (𝑝 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} ↦ (((𝑌 ∘ 𝑆)‘𝑝)(.r‘𝑅)(𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑖𝑀𝑗))(𝑝‘𝑛))))))) = (𝑅 Σg (𝑝 ∈ 𝑊 ↦ (((𝑌 ∘ 𝑍)‘𝑝)(.r‘𝑅)(𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑖𝑀𝑗))(𝑝‘𝑛))))))))
Theorem	smadiadetlem4 22985*	Lemma 4 for smadiadet 22986. (Contributed by AV, 31-Jan-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑅 ∈ CRing    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ 𝑌 = (ℤRHom‘𝑅)    &   ⊢ 𝑆 = (pmSgn‘𝑁)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑊 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   ⊢ 𝑍 = (pmSgn‘(𝑁 ∖ {𝐾}))    ⇒   ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) → (𝑅 Σg (𝑝 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} ↦ (((𝑌 ∘ 𝑆)‘𝑝)(.r‘𝑅)(𝐺 Σg (𝑛 ∈ 𝑁 ↦ (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 1 , 0 ), (𝑖𝑀𝑗)))(𝑝‘𝑛))))))) = (𝑅 Σg (𝑝 ∈ 𝑊 ↦ (((𝑌 ∘ 𝑍)‘𝑝)(.r‘𝑅)(𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑖𝑀𝑗))(𝑝‘𝑛))))))))
Theorem	smadiadet 22986	The determinant of a submatrix of a square matrix obtained by removing a row and a column at the same index equals the determinant of the original matrix with the row replaced with 0's and a 1 at the diagonal position. (Contributed by AV, 31-Jan-2019.) (Proof shortened by AV, 24-Jul-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑅 ∈ CRing    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐸 = ((𝑁 ∖ {𝐾}) maDet 𝑅)    ⇒   ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) → (𝐸‘(𝐾((𝑁 subMat 𝑅)‘𝑀)𝐾)) = (𝐷‘(𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾)))
Theorem	smadiadetglem1 22987	Lemma 1 for smadiadetg 22989. (Contributed by AV, 13-Feb-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑅 ∈ CRing    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐸 = ((𝑁 ∖ {𝐾}) maDet 𝑅)    ⇒   ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → ((𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾) ↾ ((𝑁 ∖ {𝐾}) × 𝑁)) = ((𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾) ↾ ((𝑁 ∖ {𝐾}) × 𝑁)))
Theorem	smadiadetglem2 22988	Lemma 2 for smadiadetg 22989. (Contributed by AV, 14-Feb-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑅 ∈ CRing    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐸 = ((𝑁 ∖ {𝐾}) maDet 𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → ((𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾) ↾ ({𝐾} × 𝑁)) = ((({𝐾} × 𝑁) × {𝑆}) ∘f · ((𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾) ↾ ({𝐾} × 𝑁))))
Theorem	smadiadetg 22989	The determinant of a square matrix with one row replaced with 0's and an arbitrary element of the underlying ring at the diagonal position equals the ring element multiplied with the determinant of a submatrix of the square matrix obtained by removing the row and the column at the same index. (Contributed by AV, 14-Feb-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑅 ∈ CRing    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐸 = ((𝑁 ∖ {𝐾}) maDet 𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → (𝐷‘(𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾)) = (𝑆 · (𝐸‘(𝐾((𝑁 subMat 𝑅)‘𝑀)𝐾))))
Theorem	smadiadetg0 22990	Lemma for smadiadetr 22991: version of smadiadetg 22989 with all hypotheses defining class variables removed, i.e. all class variables defined in the hypotheses replaced in the theorem by their definition. (Contributed by AV, 15-Feb-2019.)
⊢ 𝑅 ∈ CRing    ⇒   ⊢ ((𝑀 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → ((𝑁 maDet 𝑅)‘(𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾)) = (𝑆(.r‘𝑅)(((𝑁 ∖ {𝐾}) maDet 𝑅)‘(𝐾((𝑁 subMat 𝑅)‘𝑀)𝐾))))
Theorem	smadiadetr 22991	The determinant of a square matrix with one row replaced with 0's and an arbitrary element of the underlying ring at the diagonal position equals the ring element multiplied with the determinant of a submatrix of the square matrix obtained by removing the row and the column at the same index. Closed form of smadiadetg 22989. Special case of the "Laplace expansion", see definition in [Lang] p. 515. (Contributed by AV, 15-Feb-2019.)
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝑁 Mat 𝑅))) ∧ (𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅))) → ((𝑁 maDet 𝑅)‘(𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾)) = (𝑆(.r‘𝑅)(((𝑁 ∖ {𝐾}) maDet 𝑅)‘(𝐾((𝑁 subMat 𝑅)‘𝑀)𝐾))))
11.5.5  Inverse matrix
Theorem	invrvald 22992	If a matrix multiplied with a given matrix (from the left as well as from the right) results in the identity matrix, this matrix is the inverse (matrix) of the given matrix. (Contributed by Stefan O'Rear, 17-Jul-2018.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 𝐼 = (invr‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → (𝑋 · 𝑌) = 1 )    &   ⊢ (𝜑 → (𝑌 · 𝑋) = 1 )    ⇒   ⊢ (𝜑 → (𝑋 ∈ 𝑈 ∧ (𝐼‘𝑋) = 𝑌))
Theorem	matinv 22993	The inverse of a matrix is the adjunct of the matrix multiplied with the inverse of the determinant of the matrix if the determinant is a unit in the underlying ring. Proposition 4.16 in [Lang] p. 518. (Contributed by Stefan O'Rear, 17-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐽 = (𝑁 maAdju 𝑅)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑈 = (Unit‘𝐴)    &   ⊢ 𝑉 = (Unit‘𝑅)    &   ⊢ 𝐻 = (invr‘𝑅)    &   ⊢ 𝐼 = (invr‘𝐴)    &   ⊢ ∙ = ( ·𝑠 ‘𝐴)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ (𝐷‘𝑀) ∈ 𝑉) → (𝑀 ∈ 𝑈 ∧ (𝐼‘𝑀) = ((𝐻‘(𝐷‘𝑀)) ∙ (𝐽‘𝑀))))
Theorem	matunit 22994	A matrix is a unit in the ring of matrices iff its determinant is a unit in the underlying ring. (Contributed by Stefan O'Rear, 17-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑈 = (Unit‘𝐴)    &   ⊢ 𝑉 = (Unit‘𝑅)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑀 ∈ 𝑈 ↔ (𝐷‘𝑀) ∈ 𝑉))
Theorem	matunitlindflem1 22995	One direction of matunitlindf 22997. (Contributed by Brendan Leahy, 2-Jun-2021.)
⊢ (((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) → (¬ curry 𝑀 LIndF (𝑅 freeLMod 𝐼) → ((𝐼 maDet 𝑅)‘𝑀) = (0g‘𝑅)))
Theorem	matunitlindflem2 22996	One direction of matunitlindf 22997. (Contributed by Brendan Leahy, 2-Jun-2021.)
⊢ ((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ((𝐼 maDet 𝑅)‘𝑀) ∈ (Unit‘𝑅))
Theorem	matunitlindf 22997	A matrix over a field is invertible iff the rows are linearly independent. (Contributed by Brendan Leahy, 2-Jun-2021.)
⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (𝑀 ∈ (Unit‘(𝐼 Mat 𝑅)) ↔ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)))
11.5.6  Cramer's rule In the following, Cramer's rule cramer 23010 is proven. According to Wikipedia "Cramer's rule", 21-Feb-2019, https://en.wikipedia.org/wiki/Cramer%27s_rule 23010: "[Cramer's rule] ... expresses the [unique] solution [of a system of linear equations] in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the column vector of right-hand sides of the equations." The outline of the proof for systems of linear equations with coefficients from a commutative ring, according to the proof in Wikipedia (https://en.wikipedia.org/wiki/Cramer's_rule#A_short_proof), 23010 is as follows: The system of linear equations 𝐴 × 𝑋 = 𝐵 to be solved shall be given by the N x N coefficient matrix 𝐴 and the N-dimensional vector 𝐵. Let (𝐴‘𝑖) be the matrix obtained by replacing the i-th column of the coefficient matrix 𝐴 by the right-hand side vector 𝐵. Additionally, let (𝑋‘𝑖) be the matrix obtained by replacing the i-th column of the identity matrix by the solution vector 𝑋, with 𝑋 = (𝑥‘𝑖). Finally, it is assumed that det 𝐴 is a unit in the underlying ring. With these definitions, it follows that 𝐴 × (𝑋‘𝑖) = (𝐴‘𝑖) (cramerimplem2 23003), using matrix multiplication (mamuval 22709) and multiplication of a vector with a matrix (mulmarep1gsum2 22890). By using the multiplicativity of the determinant (mdetmul 22939) it follows that det (𝐴‘𝑖) = det (𝐴 × (𝑋‘𝑖)) = det 𝐴 · det (𝑋‘𝑖) (cramerimplem3 23004). Furthermore, it follows that det (𝑋‘𝑖) = (𝑥‘𝑖) (cramerimplem1 23002). To show this, a special case of the Laplace expansion is used (smadiadetg 22989). From these equations and the cancellation law for division in a ring (dvrcan3 20641) it follows that (𝑥‘𝑖) = det (𝑋‘𝑖) = det (𝐴‘𝑖) / det 𝐴. This is the right to left implication (cramerimp 23005, cramerlem1 23006, cramerlem2 23007) of Cramer's rule (cramer 23010). The left to right implication is shown by cramerlem3 23008, using the fact that a solution of the system of linear equations exists (slesolex 23001). Notice that for the special case of 0-dimensional matrices/vectors only the left to right implication is valid (see cramer0 23009), because assuming the right-hand side of the implication ((𝑋 · 𝑍) = 𝑌), 𝑍 could be anything (see mavmul0g 22869).
Theorem	slesolvec 22998	Every solution of a system of linear equations represented by a matrix and a vector is a vector. (Contributed by AV, 10-Feb-2019.) (Revised by AV, 27-Feb-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑉 = ((Base‘𝑅) ↑m 𝑁)    &   ⊢ · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)    ⇒   ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉)) → ((𝑋 · 𝑍) = 𝑌 → 𝑍 ∈ 𝑉))
Theorem	slesolinv 22999	The solution of a system of linear equations represented by a matrix with a unit as determinant is the multiplication of the inverse of the matrix with the right-hand side vector. (Contributed by AV, 10-Feb-2019.) (Revised by AV, 28-Feb-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑉 = ((Base‘𝑅) ↑m 𝑁)    &   ⊢ · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐼 = (invr‘𝐴)    ⇒   ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝐷‘𝑋) ∈ (Unit‘𝑅) ∧ (𝑋 · 𝑍) = 𝑌)) → 𝑍 = ((𝐼‘𝑋) · 𝑌))
Theorem	slesolinvbi 23000	The solution of a system of linear equations represented by a matrix with a unit as determinant is the multiplication of the inverse of the matrix with the right-hand side vector. (Contributed by AV, 11-Feb-2019.) (Revised by AV, 28-Feb-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑉 = ((Base‘𝑅) ↑m 𝑁)    &   ⊢ · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐼 = (invr‘𝐴)    ⇒   ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → ((𝑋 · 𝑍) = 𝑌 ↔ 𝑍 = ((𝐼‘𝑋) · 𝑌)))