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Theorem List for Metamath Proof Explorer - 20101-20200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremma1repvcl 20101 Closure of the column replacement function for identity matrices. (Contributed by AV, 15-Feb-2019.) (Revised by AV, 26-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑉 = ((Base‘𝑅) ↑𝑚 𝑁)    &    1 = (1r𝐴)       (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐶𝑉𝐾𝑁)) → (( 1 (𝑁 matRepV 𝑅)𝐶)‘𝐾) ∈ 𝐵)

Theoremma1repveval 20102 An entry of an identity matrix with a replaced column. (Contributed by AV, 16-Feb-2019.) (Revised by AV, 26-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑉 = ((Base‘𝑅) ↑𝑚 𝑁)    &    1 = (1r𝐴)    &    0 = (0g𝑅)    &   𝐸 = (( 1 (𝑁 matRepV 𝑅)𝐶)‘𝐾)       ((𝑅 ∈ Ring ∧ (𝑀𝐵𝐶𝑉𝐾𝑁) ∧ (𝐼𝑁𝐽𝑁)) → (𝐼𝐸𝐽) = if(𝐽 = 𝐾, (𝐶𝐼), if(𝐽 = 𝐼, (1r𝑅), 0 )))

Theoremmulmarep1el 20103 Element by element multiplication of a matrix with an identity matrix with a column replaced by a vector. (Contributed by AV, 16-Feb-2019.) (Revised by AV, 26-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑉 = ((Base‘𝑅) ↑𝑚 𝑁)    &    1 = (1r𝐴)    &    0 = (0g𝑅)    &   𝐸 = (( 1 (𝑁 matRepV 𝑅)𝐶)‘𝐾)       ((𝑅 ∈ Ring ∧ (𝑋𝐵𝐶𝑉𝐾𝑁) ∧ (𝐼𝑁𝐽𝑁𝐿𝑁)) → ((𝐼𝑋𝐿)(.r𝑅)(𝐿𝐸𝐽)) = if(𝐽 = 𝐾, ((𝐼𝑋𝐿)(.r𝑅)(𝐶𝐿)), if(𝐽 = 𝐿, (𝐼𝑋𝐿), 0 )))

Theoremmulmarep1gsum1 20104* The sum of element by element multiplications of a matrix with an identity matrix with a column replaced by a vector. (Contributed by AV, 16-Feb-2019.) (Revised by AV, 26-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑉 = ((Base‘𝑅) ↑𝑚 𝑁)    &    1 = (1r𝐴)    &    0 = (0g𝑅)    &   𝐸 = (( 1 (𝑁 matRepV 𝑅)𝐶)‘𝐾)       ((𝑅 ∈ Ring ∧ (𝑋𝐵𝐶𝑉𝐾𝑁) ∧ (𝐼𝑁𝐽𝑁𝐽𝐾)) → (𝑅 Σg (𝑙𝑁 ↦ ((𝐼𝑋𝑙)(.r𝑅)(𝑙𝐸𝐽)))) = (𝐼𝑋𝐽))

Theoremmulmarep1gsum2 20105* The sum of element by element multiplications of a matrix with an identity matrix with a column replaced by a vector. (Contributed by AV, 18-Feb-2019.) (Revised by AV, 26-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑉 = ((Base‘𝑅) ↑𝑚 𝑁)    &    1 = (1r𝐴)    &    0 = (0g𝑅)    &   𝐸 = (( 1 (𝑁 matRepV 𝑅)𝐶)‘𝐾)    &    × = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)       ((𝑅 ∈ Ring ∧ (𝑋𝐵𝐶𝑉𝐾𝑁) ∧ (𝐼𝑁𝐽𝑁 ∧ (𝑋 × 𝐶) = 𝑍)) → (𝑅 Σg (𝑙𝑁 ↦ ((𝐼𝑋𝑙)(.r𝑅)(𝑙𝐸𝐽)))) = if(𝐽 = 𝐾, (𝑍𝐼), (𝐼𝑋𝐽)))

Theorem1marepvmarrepid 20106 Replacing the ith row by 0's and the ith component of a (column) vector at the diagonal position for the identity matrix with the ith column replaced by the vector results in the matrix itself. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 27-Feb-2019.)
𝑉 = ((Base‘𝑅) ↑𝑚 𝑁)    &    1 = (1r‘(𝑁 Mat 𝑅))    &   𝑋 = (( 1 (𝑁 matRepV 𝑅)𝑍)‘𝐼)       (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → (𝐼(𝑋(𝑁 matRRep 𝑅)(𝑍𝐼))𝐼) = 𝑋)

11.2.8  Submatrices

Syntaxcsubma 20107 Syntax for submatrices of a square matrix.
class subMat

Definitiondf-subma 20108* Define the submatrices of a square matrix. A submatrix is obtained by deleting a row and a column of the original matrix. Since the indices of a matrix need not to be sequential integers, it does not matter that there may be gaps in the numbering of the indices for the submatrix. The determinants of such submatrices are called the "minors" of the original matrix. (Contributed by AV, 27-Dec-2018.)
subMat = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑘𝑛, 𝑙𝑛 ↦ (𝑖 ∈ (𝑛 ∖ {𝑘}), 𝑗 ∈ (𝑛 ∖ {𝑙}) ↦ (𝑖𝑚𝑗)))))

Theoremsubmabas 20109* Any subset of the index set of a square matrix defines a submatrix of the matrix. (Contributed by AV, 1-Jan-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)       ((𝑀𝐵𝐷𝑁) → (𝑖𝐷, 𝑗𝐷 ↦ (𝑖𝑀𝑗)) ∈ (Base‘(𝐷 Mat 𝑅)))

Theoremsubmafval 20110* First substitution for a submatrix. (Contributed by AV, 28-Dec-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (𝑁 subMat 𝑅)    &   𝐵 = (Base‘𝐴)       𝑄 = (𝑚𝐵 ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑚𝑗))))

Theoremsubmaval0 20111* Second substitution for a submatrix. (Contributed by AV, 28-Dec-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (𝑁 subMat 𝑅)    &   𝐵 = (Base‘𝐴)       (𝑀𝐵 → (𝑄𝑀) = (𝑘𝑁, 𝑙𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑀𝑗))))

Theoremsubmaval 20112* Third substitution for a submatrix. (Contributed by AV, 28-Dec-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (𝑁 subMat 𝑅)    &   𝐵 = (Base‘𝐴)       ((𝑀𝐵𝐾𝑁𝐿𝑁) → (𝐾(𝑄𝑀)𝐿) = (𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐿}) ↦ (𝑖𝑀𝑗)))

Theoremsubmaeval 20113 An entry of a submatrix of a square matrix. (Contributed by AV, 28-Dec-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (𝑁 subMat 𝑅)    &   𝐵 = (Base‘𝐴)       ((𝑀𝐵 ∧ (𝐾𝑁𝐿𝑁) ∧ (𝐼 ∈ (𝑁 ∖ {𝐾}) ∧ 𝐽 ∈ (𝑁 ∖ {𝐿}))) → (𝐼(𝐾(𝑄𝑀)𝐿)𝐽) = (𝐼𝑀𝐽))

Theorem1marepvsma1 20114 The submatrix of the identity matrix with the ith column replaced by the vector obtained by removing the ith row and the ith column is an identity matrix. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 27-Feb-2019.)
𝑉 = ((Base‘𝑅) ↑𝑚 𝑁)    &    1 = (1r‘(𝑁 Mat 𝑅))    &   𝑋 = (( 1 (𝑁 matRepV 𝑅)𝑍)‘𝐼)       (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → (𝐼((𝑁 subMat 𝑅)‘𝑋)𝐼) = (1r‘((𝑁 ∖ {𝐼}) Mat 𝑅)))

11.3  The determinant

11.3.1  Definition and basic properties

Syntaxcmdat 20115 Syntax for the matrix determinant function.

Definitiondf-mdet 20116* Determinant of a square matrix. This definition is based on Leibniz' Formula (see mdetleib 20118). 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". The functionality is shown by mdetf 20126. Multilineary means "linear for each row" - the additivity is shown by mdetrlin 20133, the homogeneity by mdetrsca 20134. Furthermore, it is shown that the determinant function is alternating (see mdetralt 20139) and normalized (see mdet1 20132). Finally, the uniqueness is shown by mdetuni 20153. As a consequence, the "determinant of a square matrix" is the function value of the determinant function for this square matrix, see mdetleib 20118. (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)(.r𝑟)((mulGrp‘𝑟) Σg (𝑥𝑛 ↦ ((𝑝𝑥)𝑚𝑥))))))))

Theoremmdetfval 20117* 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)    &    · = (.r𝑅)    &   𝑈 = (mulGrp‘𝑅)       𝐷 = (𝑚𝐵 ↦ (𝑅 Σg (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥)))))))

Theoremmdetleib 20118* 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)    &    · = (.r𝑅)    &   𝑈 = (mulGrp‘𝑅)       (𝑀𝐵 → (𝐷𝑀) = (𝑅 Σg (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑀𝑥)))))))

Theoremmdetleib2 20119* 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)    &    · = (.r𝑅)    &   𝑈 = (mulGrp‘𝑅)       ((𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝐷𝑀) = (𝑅 Σg (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ (𝑥𝑀(𝑝𝑥))))))))

Theoremnfimdetndef 20120 The determinant is not defined for an infinite matrix. (Contributed by AV, 27-Dec-2018.)
𝐷 = (𝑁 maDet 𝑅)       (𝑁 ∉ Fin → 𝐷 = ∅)

Theoremmdetfval1 20121* 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)    &    · = (.r𝑅)    &   𝑈 = (mulGrp‘𝑅)       𝐷 = (𝑚𝐵 ↦ (𝑅 Σg (𝑝𝑃 ↦ ((𝑌‘(𝑆𝑝)) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥)))))))

Theoremmdetleib1 20122* 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)    &    · = (.r𝑅)    &   𝑈 = (mulGrp‘𝑅)       (𝑀𝐵 → (𝐷𝑀) = (𝑅 Σg (𝑝𝑃 ↦ ((𝑌‘(𝑆𝑝)) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑀𝑥)))))))

Theoremmdet0pr 20123 The determinant for 0-dimensional matrices is a singleton containing an ordered pair with the singleton containing the empty set as first component, and the singleton containing the 1 element of the underlying ring as second component. (Contributed by AV, 28-Feb-2019.)
(𝑅 ∈ Ring → (∅ maDet 𝑅) = {⟨∅, (1r𝑅)⟩})

Theoremmdet0f1o 20124 The determinant for 0-dimensional matrices is a one-to-one function from the singleton containing the empty set onto the singleton containing the 1 element of the underlying ring.function x is . (Contributed by AV, 28-Feb-2019.)
(𝑅 ∈ Ring → (∅ maDet 𝑅):{∅}–1-1-onto→{(1r𝑅)})

Theoremmdet0fv0 20125 The determinant of a 0-dimensional matrix is the 1 element of the underlying ring . (Contributed by AV, 28-Feb-2019.)
(𝑅 ∈ Ring → ((∅ maDet 𝑅)‘∅) = (1r𝑅))

Theoremmdetf 20126 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:𝐵𝐾)

Theoremmdetcl 20127 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 ∧ 𝑀𝐵) → (𝐷𝑀) ∈ 𝐾)

Theoremm1detdiag 20128 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} ∧ 𝐼𝑉) ∧ 𝑀𝐵) → (𝐷𝑀) = (𝐼𝑀𝐼))

Theoremmdetdiaglem 20129* Lemma for mdetdiag 20130. Previously part of proof for mdet1 20132. (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 = (0g𝑅)    &   𝐻 = (Base‘(SymGrp‘𝑁))    &   𝑍 = (ℤRHom‘𝑅)    &   𝑆 = (pmSgn‘𝑁)    &    · = (.r𝑅)       (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀𝐵) ∧ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑀𝑗) = 0 ) ∧ (𝑃𝐻𝑃 ≠ ( I ↾ 𝑁))) → (((𝑍𝑆)‘𝑃) · (𝐺 Σg (𝑘𝑁 ↦ ((𝑃𝑘)𝑀𝑘)))) = 0 )

Theoremmdetdiag 20130* 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 = (0g𝑅)       ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀𝐵) → (∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑀𝑗) = 0 ) → (𝐷𝑀) = (𝐺 Σg (𝑘𝑁 ↦ (𝑘𝑀𝑘)))))

Theoremmdetdiagid 20131* 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 = (0g𝑅)    &   𝐶 = (Base‘𝑅)    &    · = (.g𝐺)       (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀𝐵𝑋𝐶)) → (∀𝑖𝑁𝑗𝑁 (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑋, 0 ) → (𝐷𝑀) = ((#‘𝑁) · 𝑋)))

Theoremmdet1 20132 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 = (1r𝐴)    &    1 = (1r𝑅)       ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → (𝐷𝐼) = 1 )

Theoremmdetrlin 20133 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)    &    + = (+g𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝐼𝑁)    &   (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 → (𝐷𝑋) = ((𝐷𝑌) + (𝐷𝑍)))

Theoremmdetrsca 20134 The determinant function is homogeneous for each row: The matrices X and Z are identical except for the I's row, and the I's row of the matrix X is the componentwise product of the I's row of the matrix Z and the scalar Y. In this case 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)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐾)    &   (𝜑𝑍𝐵)    &   (𝜑𝐼𝑁)    &   (𝜑 → (𝑋 ↾ ({𝐼} × 𝑁)) = ((({𝐼} × 𝑁) × {𝑌}) ∘𝑓 · (𝑍 ↾ ({𝐼} × 𝑁))))    &   (𝜑 → (𝑋 ↾ ((𝑁 ∖ {𝐼}) × 𝑁)) = (𝑍 ↾ ((𝑁 ∖ {𝐼}) × 𝑁)))       (𝜑 → (𝐷𝑋) = (𝑌 · (𝐷𝑍)))

Theoremmdetrsca2 20135* 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)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑁 ∈ Fin)    &   ((𝜑𝑖𝑁𝑗𝑁) → 𝑋𝐾)    &   ((𝜑𝑖𝑁𝑗𝑁) → 𝑌𝐾)    &   (𝜑𝐹𝐾)    &   (𝜑𝐼𝑁)       (𝜑 → (𝐷‘(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐼, (𝐹 · 𝑋), 𝑌))) = (𝐹 · (𝐷‘(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐼, 𝑋, 𝑌)))))

Theoremmdetr0 20136* 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 = (0g𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑁 ∈ Fin)    &   ((𝜑𝑖𝑁𝑗𝑁) → 𝑋𝐾)    &   (𝜑𝐼𝑁)       (𝜑 → (𝐷‘(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐼, 0 , 𝑋))) = 0 )

Theoremmdet0 20137 The determinant of the zero matrix (of dimension greater 0!) is 0. (Contributed by AV, 17-Aug-2019.)
𝐷 = (𝑁 maDet 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑍 = (0g𝐴)    &    0 = (0g𝑅)       ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑁 ≠ ∅) → (𝐷𝑍) = 0 )

Theoremmdetrlin2 20138* 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)    &    + = (+g𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑁 ∈ Fin)    &   ((𝜑𝑖𝑁𝑗𝑁) → 𝑋𝐾)    &   ((𝜑𝑖𝑁𝑗𝑁) → 𝑌𝐾)    &   ((𝜑𝑖𝑁𝑗𝑁) → 𝑍𝐾)    &   (𝜑𝐼𝑁)       (𝜑 → (𝐷‘(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐼, (𝑋 + 𝑌), 𝑍))) = ((𝐷‘(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐼, 𝑋, 𝑍))) + (𝐷‘(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐼, 𝑌, 𝑍)))))

Theoremmdetralt 20139* 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 = (0g𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑋𝐵)    &   (𝜑𝐼𝑁)    &   (𝜑𝐽𝑁)    &   (𝜑𝐼𝐽)    &   (𝜑 → ∀𝑎𝑁 (𝐼𝑋𝑎) = (𝐽𝑋𝑎))       (𝜑 → (𝐷𝑋) = 0 )

Theoremmdetralt2 20140* 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 = (0g𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑁 ∈ Fin)    &   ((𝜑𝑗𝑁) → 𝑋𝐾)    &   ((𝜑𝑖𝑁𝑗𝑁) → 𝑌𝐾)    &   (𝜑𝐼𝑁)    &   (𝜑𝐽𝑁)    &   (𝜑𝐼𝐽)       (𝜑 → (𝐷‘(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)))) = 0 )

Theoremmdetero 20141* 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)    &    + = (+g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑁 ∈ Fin)    &   ((𝜑𝑗𝑁) → 𝑋𝐾)    &   ((𝜑𝑗𝑁) → 𝑌𝐾)    &   ((𝜑𝑖𝑁𝑗𝑁) → 𝑍𝐾)    &   (𝜑𝑊𝐾)    &   (𝜑𝐼𝑁)    &   (𝜑𝐽𝑁)    &   (𝜑𝐼𝐽)       (𝜑 → (𝐷‘(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐼, (𝑋 + (𝑊 · 𝑌)), if(𝑖 = 𝐽, 𝑌, 𝑍)))) = (𝐷‘(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑌, 𝑍)))))

Theoremmdettpos 20142 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 ∧ 𝑀𝐵) → (𝐷‘tpos 𝑀) = (𝐷𝑀))

Theoremmdetunilem1 20143* Lemma for mdetuni 20153. (Contributed by SO, 14-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐾 = (Base‘𝑅)    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐷:𝐵𝐾)    &   (𝜑 → ∀𝑥𝐵𝑦𝑁𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷𝑥) = 0 ))    &   (𝜑 → ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑤𝑁 (((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))) → (𝐷𝑥) = ((𝐷𝑦) + (𝐷𝑧))))    &   (𝜑 → ∀𝑥𝐵𝑦𝐾𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = (𝑦 · (𝐷𝑧))))       (((𝜑𝐸𝐵 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹𝑁𝐺𝑁𝐹𝐺)) → (𝐷𝐸) = 0 )

Theoremmdetunilem2 20144* Lemma for mdetuni 20153. (Contributed by SO, 15-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐾 = (Base‘𝑅)    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐷:𝐵𝐾)    &   (𝜑 → ∀𝑥𝐵𝑦𝑁𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷𝑥) = 0 ))    &   (𝜑 → ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑤𝑁 (((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))) → (𝐷𝑥) = ((𝐷𝑦) + (𝐷𝑧))))    &   (𝜑 → ∀𝑥𝐵𝑦𝐾𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = (𝑦 · (𝐷𝑧))))    &   (𝜓𝜑)    &   (𝜓 → (𝐸𝑁𝐺𝑁𝐸𝐺))    &   ((𝜓𝑏𝑁) → 𝐹𝐾)    &   ((𝜓𝑎𝑁𝑏𝑁) → 𝐻𝐾)       (𝜓 → (𝐷‘(𝑎𝑁, 𝑏𝑁 ↦ if(𝑎 = 𝐸, 𝐹, if(𝑎 = 𝐺, 𝐹, 𝐻)))) = 0 )

Theoremmdetunilem3 20145* Lemma for mdetuni 20153. (Contributed by SO, 15-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐾 = (Base‘𝑅)    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐷:𝐵𝐾)    &   (𝜑 → ∀𝑥𝐵𝑦𝑁𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷𝑥) = 0 ))    &   (𝜑 → ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑤𝑁 (((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))) → (𝐷𝑥) = ((𝐷𝑦) + (𝐷𝑧))))    &   (𝜑 → ∀𝑥𝐵𝑦𝐾𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = (𝑦 · (𝐷𝑧))))       (((𝜑𝐸𝐵𝐹𝐵) ∧ (𝐺𝐵𝐻𝑁 ∧ (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)))) → (𝐷𝐸) = ((𝐷𝐹) + (𝐷𝐺)))

Theoremmdetunilem4 20146* Lemma for mdetuni 20153. (Contributed by SO, 15-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐾 = (Base‘𝑅)    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐷:𝐵𝐾)    &   (𝜑 → ∀𝑥𝐵𝑦𝑁𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷𝑥) = 0 ))    &   (𝜑 → ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑤𝑁 (((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))) → (𝐷𝑥) = ((𝐷𝑦) + (𝐷𝑧))))    &   (𝜑 → ∀𝑥𝐵𝑦𝐾𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = (𝑦 · (𝐷𝑧))))       ((𝜑 ∧ (𝐸𝐵𝐹𝐾𝐺𝐵) ∧ (𝐻𝑁 ∧ (𝐸 ↾ ({𝐻} × 𝑁)) = ((({𝐻} × 𝑁) × {𝐹}) ∘𝑓 · (𝐺 ↾ ({𝐻} × 𝑁))) ∧ (𝐸 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)) = (𝐺 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)))) → (𝐷𝐸) = (𝐹 · (𝐷𝐺)))

Theoremmdetunilem5 20147* Lemma for mdetuni 20153. (Contributed by SO, 15-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐾 = (Base‘𝑅)    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐷:𝐵𝐾)    &   (𝜑 → ∀𝑥𝐵𝑦𝑁𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷𝑥) = 0 ))    &   (𝜑 → ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑤𝑁 (((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))) → (𝐷𝑥) = ((𝐷𝑦) + (𝐷𝑧))))    &   (𝜑 → ∀𝑥𝐵𝑦𝐾𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = (𝑦 · (𝐷𝑧))))    &   (𝜓𝜑)    &   (𝜓𝐸𝑁)    &   ((𝜓𝑎𝑁𝑏𝑁) → (𝐹𝐾𝐺𝐾𝐻𝐾))       (𝜓 → (𝐷‘(𝑎𝑁, 𝑏𝑁 ↦ if(𝑎 = 𝐸, (𝐹 + 𝐺), 𝐻))) = ((𝐷‘(𝑎𝑁, 𝑏𝑁 ↦ if(𝑎 = 𝐸, 𝐹, 𝐻))) + (𝐷‘(𝑎𝑁, 𝑏𝑁 ↦ if(𝑎 = 𝐸, 𝐺, 𝐻)))))

Theoremmdetunilem6 20148* Lemma for mdetuni 20153. (Contributed by SO, 15-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐾 = (Base‘𝑅)    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐷:𝐵𝐾)    &   (𝜑 → ∀𝑥𝐵𝑦𝑁𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷𝑥) = 0 ))    &   (𝜑 → ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑤𝑁 (((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))) → (𝐷𝑥) = ((𝐷𝑦) + (𝐷𝑧))))    &   (𝜑 → ∀𝑥𝐵𝑦𝐾𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = (𝑦 · (𝐷𝑧))))    &   (𝜓𝜑)    &   (𝜓 → (𝐸𝑁𝐹𝑁𝐸𝐹))    &   ((𝜓𝑏𝑁) → (𝐺𝐾𝐻𝐾))    &   ((𝜓𝑎𝑁𝑏𝑁) → 𝐼𝐾)       (𝜓 → (𝐷‘(𝑎𝑁, 𝑏𝑁 ↦ if(𝑎 = 𝐸, 𝐺, if(𝑎 = 𝐹, 𝐻, 𝐼)))) = ((invg𝑅)‘(𝐷‘(𝑎𝑁, 𝑏𝑁 ↦ if(𝑎 = 𝐸, 𝐻, if(𝑎 = 𝐹, 𝐺, 𝐼))))))

Theoremmdetunilem7 20149* Lemma for mdetuni 20153. (Contributed by SO, 15-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐾 = (Base‘𝑅)    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐷:𝐵𝐾)    &   (𝜑 → ∀𝑥𝐵𝑦𝑁𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷𝑥) = 0 ))    &   (𝜑 → ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑤𝑁 (((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))) → (𝐷𝑥) = ((𝐷𝑦) + (𝐷𝑧))))    &   (𝜑 → ∀𝑥𝐵𝑦𝐾𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = (𝑦 · (𝐷𝑧))))       ((𝜑𝐸:𝑁1-1-onto𝑁𝐹𝐵) → (𝐷‘(𝑎𝑁, 𝑏𝑁 ↦ ((𝐸𝑎)𝐹𝑏))) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝐸) · (𝐷𝐹)))

Theoremmdetunilem8 20150* Lemma for mdetuni 20153. (Contributed by SO, 15-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐾 = (Base‘𝑅)    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐷:𝐵𝐾)    &   (𝜑 → ∀𝑥𝐵𝑦𝑁𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷𝑥) = 0 ))    &   (𝜑 → ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑤𝑁 (((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))) → (𝐷𝑥) = ((𝐷𝑦) + (𝐷𝑧))))    &   (𝜑 → ∀𝑥𝐵𝑦𝐾𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = (𝑦 · (𝐷𝑧))))    &   (𝜑 → (𝐷‘(1r𝐴)) = 0 )       ((𝜑𝐸:𝑁𝑁) → (𝐷‘(𝑎𝑁, 𝑏𝑁 ↦ if((𝐸𝑎) = 𝑏, 1 , 0 ))) = 0 )

Theoremmdetunilem9 20151* Lemma for mdetuni 20153. (Contributed by SO, 15-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐾 = (Base‘𝑅)    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐷:𝐵𝐾)    &   (𝜑 → ∀𝑥𝐵𝑦𝑁𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷𝑥) = 0 ))    &   (𝜑 → ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑤𝑁 (((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))) → (𝐷𝑥) = ((𝐷𝑦) + (𝐷𝑧))))    &   (𝜑 → ∀𝑥𝐵𝑦𝐾𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = (𝑦 · (𝐷𝑧))))    &   (𝜑 → (𝐷‘(1r𝐴)) = 0 )    &   𝑌 = {𝑥 ∣ ∀𝑦𝐵𝑧 ∈ (𝑁𝑚 𝑁)(∀𝑤𝑥 (𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 )}       (𝜑𝐷 = (𝐵 × { 0 }))

Theoremmdetuni0 20152* Lemma for mdetuni 20153. (Contributed by SO, 15-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐾 = (Base‘𝑅)    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐷:𝐵𝐾)    &   (𝜑 → ∀𝑥𝐵𝑦𝑁𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷𝑥) = 0 ))    &   (𝜑 → ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑤𝑁 (((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))) → (𝐷𝑥) = ((𝐷𝑦) + (𝐷𝑧))))    &   (𝜑 → ∀𝑥𝐵𝑦𝐾𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = (𝑦 · (𝐷𝑧))))    &   𝐸 = (𝑁 maDet 𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐹𝐵)       (𝜑 → (𝐷𝐹) = ((𝐷‘(1r𝐴)) · (𝐸𝐹)))

Theoremmdetuni 20153* 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 = (0g𝑅)    &    1 = (1r𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐷:𝐵𝐾)    &   (𝜑 → ∀𝑥𝐵𝑦𝑁𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷𝑥) = 0 ))    &   (𝜑 → ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑤𝑁 (((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))) → (𝐷𝑥) = ((𝐷𝑦) + (𝐷𝑧))))    &   (𝜑 → ∀𝑥𝐵𝑦𝐾𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = (𝑦 · (𝐷𝑧))))    &   𝐸 = (𝑁 maDet 𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐹𝐵)    &   (𝜑 → (𝐷‘(1r𝐴)) = 1 )       (𝜑 → (𝐷𝐹) = (𝐸𝐹))

Theoremmdetmul 20154 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)    &    · = (.r𝑅)    &    = (.r𝐴)       ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → (𝐷‘(𝐹 𝐺)) = ((𝐷𝐹) · (𝐷𝐺)))

11.3.2  Determinants of 2 x 2 -matrices

Theoremm2detleiblem1 20155 Lemma 1 for m2detleib 20162. (Contributed by AV, 12-Dec-2018.)
𝑁 = {1, 2}    &   𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑌 = (ℤRHom‘𝑅)    &   𝑆 = (pmSgn‘𝑁)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 𝑄𝑃) → (𝑌‘(𝑆𝑄)) = (((pmSgn‘𝑁)‘𝑄)(.g𝑅) 1 ))

Theoremm2detleiblem5 20156 Lemma 5 for m2detleib 20162. (Contributed by AV, 20-Dec-2018.)
𝑁 = {1, 2}    &   𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑌 = (ℤRHom‘𝑅)    &   𝑆 = (pmSgn‘𝑁)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩}) → (𝑌‘(𝑆𝑄)) = 1 )

Theoremm2detleiblem6 20157 Lemma 6 for m2detleib 20162. (Contributed by AV, 20-Dec-2018.)
𝑁 = {1, 2}    &   𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑌 = (ℤRHom‘𝑅)    &   𝑆 = (pmSgn‘𝑁)    &    1 = (1r𝑅)    &   𝐼 = (invg𝑅)       ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 2⟩, ⟨2, 1⟩}) → (𝑌‘(𝑆𝑄)) = (𝐼1 ))

Theoremm2detleiblem7 20158 Lemma 7 for m2detleib 20162. (Contributed by AV, 20-Dec-2018.)
𝑁 = {1, 2}    &   𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑌 = (ℤRHom‘𝑅)    &   𝑆 = (pmSgn‘𝑁)    &    1 = (1r𝑅)    &   𝐼 = (invg𝑅)    &    · = (.r𝑅)    &    = (-g𝑅)       ((𝑅 ∈ Ring ∧ 𝑋 ∈ (Base‘𝑅) ∧ 𝑍 ∈ (Base‘𝑅)) → (𝑋(+g𝑅)((𝐼1 ) · 𝑍)) = (𝑋 𝑍))

Theoremm2detleiblem2 20159* Lemma 2 for m2detleib 20162. (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 ∧ 𝑄𝑃𝑀𝐵) → (𝐺 Σg (𝑛𝑁 ↦ ((𝑄𝑛)𝑀𝑛))) ∈ (Base‘𝑅))

Theoremm2detleiblem3 20160* Lemma 3 for m2detleib 20162. (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)    &    · = (+g𝐺)       ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀𝐵) → (𝐺 Σg (𝑛𝑁 ↦ ((𝑄𝑛)𝑀𝑛))) = ((1𝑀1) · (2𝑀2)))

Theoremm2detleiblem4 20161* Lemma 4 for m2detleib 20162. (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)    &    · = (+g𝐺)       ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 2⟩, ⟨2, 1⟩} ∧ 𝑀𝐵) → (𝐺 Σg (𝑛𝑁 ↦ ((𝑄𝑛)𝑀𝑛))) = ((2𝑀1) · (1𝑀2)))

Theoremm2detleib 20162 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)    &    = (-g𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝐷𝑀) = (((1𝑀1) · (2𝑀2)) ((2𝑀1) · (1𝑀2))))

Syntaxcminmar1 20164 Syntax for the minor matrices of a square matrix.
class minMatR1

Definitiondf-madu 20165* 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)) ↦ (𝑖𝑛, 𝑗𝑛 ↦ ((𝑛 maDet 𝑟)‘(𝑘𝑛, 𝑙𝑛 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r𝑟), (0g𝑟)), (𝑘𝑚𝑙)))))))

Definitiondf-minmar1 20166* 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 20108- note that the matrix is transposed compared with the submatrix defined in df-subma 20108, but this does not matter because the determinants are the same, see mdettpos 20142). Such matrices are used in the definition of an adjunct of a square matrix, see df-madu 20165. (Contributed by AV, 27-Dec-2018.)
minMatR1 = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑘𝑛, 𝑙𝑛 ↦ (𝑖𝑛, 𝑗𝑛 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, (1r𝑟), (0g𝑟)), (𝑖𝑚𝑗))))))

Theoremmndifsplit 20167 Lemma for maducoeval2 20171. (Contributed by SO, 16-Jul-2018.)
𝐵 = (Base‘𝑀)    &    0 = (0g𝑀)    &    + = (+g𝑀)       ((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) → if((𝜑𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )))

Theoremmadufval 20168* 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 = (1r𝑅)    &    0 = (0g𝑅)       𝐽 = (𝑚𝐵 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙))))))

Theoremmaduval 20169* 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 = (1r𝑅)    &    0 = (0g𝑅)       (𝑀𝐵 → (𝐽𝑀) = (𝑖𝑁, 𝑗𝑁 ↦ (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙))))))

Theoremmaducoeval 20170* 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 = (1r𝑅)    &    0 = (0g𝑅)       ((𝑀𝐵𝐼𝑁𝐻𝑁) → (𝐼(𝐽𝑀)𝐻) = (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙)))))

Theoremmaducoeval2 20171* 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 = (1r𝑅)    &    0 = (0g𝑅)       (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) → (𝐼(𝐽𝑀)𝐻) = (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if((𝑘 = 𝐻𝑙 = 𝐼), if((𝑙 = 𝐼𝑘 = 𝐻), 1 , 0 ), (𝑘𝑀𝑙)))))

Theoremmaduf 20172 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:𝐵𝐵)

Theoremmadutpos 20173 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 ∧ 𝑀𝐵) → (𝐽‘tpos 𝑀) = tpos (𝐽𝑀))

Theoremmadugsum 20174* 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)    &    · = (.r𝑅)    &   𝐾 = (Base‘𝑅)    &   (𝜑𝑀𝐵)    &   (𝜑𝑅 ∈ CRing)    &   ((𝜑𝑖𝑁) → 𝑋𝐾)    &   (𝜑𝐿𝑁)       (𝜑 → (𝑅 Σg (𝑖𝑁 ↦ (𝑋 · (𝑖(𝐽𝑀)𝐿)))) = (𝐷‘(𝑗𝑁, 𝑖𝑁 ↦ if(𝑗 = 𝐿, 𝑋, (𝑗𝑀𝑖)))))

Theoremmadurid 20175 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 = (1r𝐴)    &    · = (.r𝐴)    &    = ( ·𝑠𝐴)       ((𝑀𝐵𝑅 ∈ CRing) → (𝑀 · (𝐽𝑀)) = ((𝐷𝑀) 1 ))

Theoremmadulid 20176 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 = (1r𝐴)    &    · = (.r𝐴)    &    = ( ·𝑠𝐴)       ((𝑀𝐵𝑅 ∈ CRing) → ((𝐽𝑀) · 𝑀) = ((𝐷𝑀) 1 ))

Theoremminmar1fval 20177* 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 = (1r𝑅)    &    0 = (0g𝑅)       𝑄 = (𝑚𝐵 ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗)))))

Theoremminmar1val0 20178* 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 = (1r𝑅)    &    0 = (0g𝑅)       (𝑀𝐵 → (𝑄𝑀) = (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑀𝑗)))))

Theoremminmar1val 20179* 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 = (1r𝑅)    &    0 = (0g𝑅)       ((𝑀𝐵𝐾𝑁𝐿𝑁) → (𝐾(𝑄𝑀)𝐿) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗))))

Theoremminmar1eval 20180 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 = (1r𝑅)    &    0 = (0g𝑅)       ((𝑀𝐵 ∧ (𝐾𝑁𝐿𝑁) ∧ (𝐼𝑁𝐽𝑁)) → (𝐼(𝐾(𝑄𝑀)𝐿)𝐽) = if(𝐼 = 𝐾, if(𝐽 = 𝐿, 1 , 0 ), (𝐼𝑀𝐽)))

Theoremminmar1marrep 20181 The minor matrix is a special case of a matrix with a replaced row. (Contributed by AV, 12-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑄 = (𝑁 matRRep 𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 𝑀𝐵) → ((𝑁 minMatR1 𝑅)‘𝑀) = (𝑀(𝑁 matRRep 𝑅) 1 ))

Theoremminmar1cl 20182 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 ∧ 𝑀𝐵) ∧ (𝐾𝑁𝐿𝑁)) → (𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐿) ∈ 𝐵)

Theoremmaducoevalmin1 20183 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)       ((𝑀𝐵𝐼𝑁𝐻𝑁) → (𝐼(𝐽𝑀)𝐻) = (𝐷‘(𝐻((𝑁 minMatR1 𝑅)‘𝑀)𝐼)))

11.3.4  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 (alternatively 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 20204 or smadiadetr 20206.

Theoremsymgmatr01lem 20184* Lemma for symgmatr01 20185. (Contributed by AV, 3-Jan-2019.)
𝑃 = (Base‘(SymGrp‘𝑁))       ((𝐾𝑁𝐿𝑁) → (𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) → ∃𝑘𝑁 if(𝑘 = 𝐾, if((𝑄𝑘) = 𝐿, 𝐴, 𝐵), (𝑘𝑀(𝑄𝑘))) = 𝐵))

Theoremsymgmatr01 20185* 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 = (0g𝑅)    &    1 = (1r𝑅)       ((𝐾𝑁𝐿𝑁) → (𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) → ∃𝑘𝑁 (𝑘(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑄𝑘)) = 0 ))

Theoremgsummatr01lem1 20186* Lemma A for gsummatr01 20190. (Contributed by AV, 8-Jan-2019.)
𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑅 = {𝑟𝑃 ∣ (𝑟𝐾) = 𝐿}       ((𝑄𝑅𝑋𝑁) → (𝑄𝑋) ∈ 𝑁)

Theoremgsummatr01lem2 20187* Lemma B for gsummatr01 20190. (Contributed by AV, 8-Jan-2019.)
𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑅 = {𝑟𝑃 ∣ (𝑟𝐾) = 𝐿}       ((𝑄𝑅𝑋𝑁) → (∀𝑖𝑁𝑗𝑁 (𝑖𝐴𝑗) ∈ (Base‘𝐺) → (𝑋𝐴(𝑄𝑋)) ∈ (Base‘𝐺)))

Theoremgsummatr01lem3 20188* Lemma 1 for gsummatr01 20190. (Contributed by AV, 8-Jan-2019.)
𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑅 = {𝑟𝑃 ∣ (𝑟𝐾) = 𝐿}    &    0 = (0g𝐺)    &   𝑆 = (Base‘𝐺)       (((𝐺 ∈ CMnd ∧ 𝑁 ∈ Fin) ∧ (∀𝑖𝑁𝑗𝑁 (𝑖𝐴𝑗) ∈ 𝑆𝐵𝑆) ∧ (𝐾𝑁𝐿𝑁𝑄𝑅)) → (𝐺 Σg (𝑛 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾}) ↦ (𝑛(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 0 , 𝐵), (𝑖𝐴𝑗)))(𝑄𝑛)))) = ((𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 0 , 𝐵), (𝑖𝐴𝑗)))(𝑄𝑛))))(+g𝐺)(𝐾(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 0 , 𝐵), (𝑖𝐴𝑗)))(𝑄𝐾))))

Theoremgsummatr01lem4 20189* Lemma 2 for gsummatr01 20190. (Contributed by AV, 8-Jan-2019.)
𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑅 = {𝑟𝑃 ∣ (𝑟𝐾) = 𝐿}    &    0 = (0g𝐺)    &   𝑆 = (Base‘𝐺)       ((((𝐺 ∈ CMnd ∧ 𝑁 ∈ Fin) ∧ (∀𝑖𝑁𝑗𝑁 (𝑖𝐴𝑗) ∈ 𝑆𝐵𝑆) ∧ (𝐾𝑁𝐿𝑁𝑄𝑅)) ∧ 𝑛 ∈ (𝑁 ∖ {𝐾})) → (𝑛(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 0 , 𝐵), (𝑖𝐴𝑗)))(𝑄𝑛)) = (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐿}) ↦ (𝑖𝐴𝑗))(𝑄𝑛)))

𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑅 = {𝑟𝑃 ∣ (𝑟𝐾) = 𝐿}    &    0 = (0g𝐺)    &   𝑆 = (Base‘𝐺)       (((𝐺 ∈ CMnd ∧ 𝑁 ∈ Fin) ∧ (∀𝑖𝑁𝑗𝑁 (𝑖𝐴𝑗) ∈ 𝑆𝐵𝑆) ∧ (𝐾𝑁𝐿𝑁𝑄𝑅)) → (𝐺 Σg (𝑛𝑁 ↦ (𝑛(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 0 , 𝐵), (𝑖𝐴𝑗)))(𝑄𝑛)))) = (𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐿}) ↦ (𝑖𝐴𝑗))(𝑄𝑛)))))

Theoremmarep01ma 20191* 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 = (0g𝑅)    &    1 = (1r𝑅)       (𝑀𝐵 → (𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙))) ∈ 𝐵)

Theoremsmadiadetlem0 20192* Lemma 0 for smadiadet 20201: 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 = (0g𝑅)    &    1 = (1r𝑅)    &   𝑃 = (Base‘(SymGrp‘𝑁))    &   𝐺 = (mulGrp‘𝑅)       ((𝑀𝐵𝐾𝑁𝐿𝑁) → (𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) → (𝐺 Σg (𝑛𝑁 ↦ (𝑛(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑄𝑛)))) = 0 ))

Theoremsmadiadetlem1 20193* Lemma 1 for smadiadet 20201: 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 = (0g𝑅)    &    1 = (1r𝑅)    &   𝑃 = (Base‘(SymGrp‘𝑁))    &   𝐺 = (mulGrp‘𝑅)    &   𝑌 = (ℤRHom‘𝑅)    &   𝑆 = (pmSgn‘𝑁)    &    · = (.r𝑅)       (((𝑀𝐵𝐾𝑁) ∧ 𝑝𝑃) → (((𝑌𝑆)‘𝑝)(.r𝑅)(𝐺 Σg (𝑛𝑁 ↦ (𝑛(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 1 , 0 ), (𝑖𝑀𝑗)))(𝑝𝑛))))) ∈ (Base‘𝑅))

Theoremsmadiadetlem1a 20194* Lemma 1a for smadiadet 20201: 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 = (0g𝑅)    &    1 = (1r𝑅)    &   𝑃 = (Base‘(SymGrp‘𝑁))    &   𝐺 = (mulGrp‘𝑅)    &   𝑌 = (ℤRHom‘𝑅)    &   𝑆 = (pmSgn‘𝑁)    &    · = (.r𝑅)       ((𝑀𝐵𝐾𝑁𝐿𝑁) → (𝑅 Σg (𝑝 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) ↦ (((𝑌𝑆)‘𝑝) · (𝐺 Σg (𝑛𝑁 ↦ (𝑛(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑝𝑛))))))) = 0 )

Theoremsmadiadetlem2 20195* Lemma 2 for smadiadet 20201: 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 = (0g𝑅)    &    1 = (1r𝑅)    &   𝑃 = (Base‘(SymGrp‘𝑁))    &   𝐺 = (mulGrp‘𝑅)    &   𝑌 = (ℤRHom‘𝑅)    &   𝑆 = (pmSgn‘𝑁)    &    · = (.r𝑅)       ((𝑀𝐵𝐾𝑁) → (𝑅 Σg (𝑝 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ↦ (((𝑌𝑆)‘𝑝) · (𝐺 Σg (𝑛𝑁 ↦ (𝑛(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 1 , 0 ), (𝑖𝑀𝑗)))(𝑝𝑛))))))) = 0 )

𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑅 ∈ CRing    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   𝑃 = (Base‘(SymGrp‘𝑁))    &   𝐺 = (mulGrp‘𝑅)    &   𝑌 = (ℤRHom‘𝑅)    &   𝑆 = (pmSgn‘𝑁)    &    · = (.r𝑅)    &   𝑊 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   𝑍 = (pmSgn‘(𝑁 ∖ {𝐾}))       (((𝑀𝐵𝐾𝑁) ∧ 𝑄𝑊) → (((𝑌𝑍)‘𝑄)(.r𝑅)(𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑖𝑀𝑗))(𝑄𝑛))))) ∈ (Base‘𝑅))

𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑅 ∈ CRing    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   𝑃 = (Base‘(SymGrp‘𝑁))    &   𝐺 = (mulGrp‘𝑅)    &   𝑌 = (ℤRHom‘𝑅)    &   𝑆 = (pmSgn‘𝑁)    &    · = (.r𝑅)    &   𝑊 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   𝑍 = (pmSgn‘(𝑁 ∖ {𝐾}))       ((𝑀𝐵𝐾𝑁) → (𝑝𝑊 ↦ (((𝑌𝑍)‘𝑝)(.r𝑅)(𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑖𝑀𝑗))(𝑝𝑛)))))):𝑊⟶(Base‘𝑅))