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Theorem List for Metamath Proof Explorer - 20401-20500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmavmulcl 20401 Multiplication of an NxN matrix with an N-dimensional vector results in an N-dimensional vector. (Contributed by AV, 6-Dec-2018.) (Revised by AV, 23-Feb-2019.) (Proof shortened by AV, 23-Jul-2019.)
𝐴 = (𝑁 Mat 𝑅)    &    × = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑋 ∈ (Base‘𝐴))    &   (𝜑𝑌 ∈ (𝐵𝑚 𝑁))       (𝜑 → (𝑋 × 𝑌) ∈ (𝐵𝑚 𝑁))

Theorem1mavmul 20402 Multiplication of the identity NxN matrix with an N-dimensional vector results in the vector itself. (Contributed by AV, 9-Feb-2019.) (Revised by AV, 23-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝑅)    &    · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑌 ∈ (𝐵𝑚 𝑁))       (𝜑 → ((1r𝐴) · 𝑌) = 𝑌)

Theoremmavmulass 20403 Associativity of the multiplication of two NxN matrices with an N-dimensional vector. (Contributed by AV, 9-Feb-2019.) (Revised by AV, 25-Feb-2019.) (Proof shortened by AV, 22-Jul-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝑅)    &    · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑌 ∈ (𝐵𝑚 𝑁))    &    × = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)    &   (𝜑𝑋 ∈ (Base‘𝐴))    &   (𝜑𝑍 ∈ (Base‘𝐴))       (𝜑 → ((𝑋 × 𝑍) · 𝑌) = (𝑋 · (𝑍 · 𝑌)))

Theoremmavmuldm 20404 The domain of the matrix vector multiplication function. (Contributed by AV, 27-Feb-2019.)
𝐵 = (Base‘𝑅)    &   𝐶 = (𝐵𝑚 (𝑀 × 𝑁))    &   𝐷 = (𝐵𝑚 𝑁)    &    · = (𝑅 maVecMul ⟨𝑀, 𝑁⟩)       ((𝑅𝑉𝑀 ∈ Fin ∧ 𝑁 ∈ Fin) → dom · = (𝐶 × 𝐷))

Theoremmavmulsolcl 20405 Every solution of the equation 𝐴𝑋 = 𝑌 for a matrix 𝐴 and a vector 𝐵 is a vector. (Contributed by AV, 27-Feb-2019.)
𝐵 = (Base‘𝑅)    &   𝐶 = (𝐵𝑚 (𝑀 × 𝑁))    &   𝐷 = (𝐵𝑚 𝑁)    &    · = (𝑅 maVecMul ⟨𝑀, 𝑁⟩)    &   𝐸 = (𝐵𝑚 𝑀)       (((𝑀 ∈ Fin ∧ 𝑁 ∈ Fin ∧ 𝑀 ≠ ∅) ∧ (𝑅𝑉𝑌𝐸)) → ((𝐴 · 𝑋) = 𝑌𝑋𝐷))

Theoremmavmul0 20406 Multiplication of a 0-dimensional matrix with a 0-dimensional vector. (Contributed by AV, 28-Feb-2019.)
· = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)       ((𝑁 = ∅ ∧ 𝑅𝑉) → (∅ · ∅) = ∅)

Theoremmavmul0g 20407 The result of the 0-dimensional multiplication of a matrix with a vector is always the empty set. (Contributed by AV, 1-Mar-2019.)
· = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)       ((𝑁 = ∅ ∧ 𝑅𝑉) → (𝑋 · 𝑌) = ∅)

Theoremmvmumamul1 20408* The multiplication of an MxN matrix with an N-dimensional vector corresponds to the matrix multiplication of an MxN matrix with an Nx1 matrix. (Contributed by AV, 14-Mar-2019.)
× = (𝑅 maMul ⟨𝑀, 𝑁, {∅}⟩)    &    · = (𝑅 maVecMul ⟨𝑀, 𝑁⟩)    &   𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝐴 ∈ (𝐵𝑚 (𝑀 × 𝑁)))    &   (𝜑𝑌 ∈ (𝐵𝑚 𝑁))    &   (𝜑𝑍 ∈ (𝐵𝑚 (𝑁 × {∅})))       (𝜑 → (∀𝑗𝑁 (𝑌𝑗) = (𝑗𝑍∅) → ∀𝑖𝑀 ((𝐴 · 𝑌)‘𝑖) = (𝑖(𝐴 × 𝑍)∅)))

Theoremmavmumamul1 20409* The multiplication of an NxN matrix with an N-dimensional vector corresponds to the matrix multiplication of an NxN matrix with an Nx1 matrix. (Contributed by AV, 14-Mar-2019.)
𝐴 = (𝑁 Mat 𝑅)    &    × = (𝑅 maMul ⟨𝑁, 𝑁, {∅}⟩)    &    · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)    &   𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑋 ∈ (Base‘𝐴))    &   (𝜑𝑌 ∈ (𝐵𝑚 𝑁))    &   (𝜑𝑍 ∈ (𝐵𝑚 (𝑁 × {∅})))       (𝜑 → (∀𝑗𝑁 (𝑌𝑗) = (𝑗𝑍∅) → ∀𝑖𝑁 ((𝑋 · 𝑌)‘𝑖) = (𝑖(𝑋 × 𝑍)∅)))

11.2.7  Replacement functions for a square matrix

Syntaxcmarrep 20410 Syntax for the row replacing function for a square matrix.
class matRRep

SyntaxcmatrepV 20411 Syntax for the function replacing a column of a matrix by a vector.
class matRepV

Definitiondf-marrep 20412* Define the matrices whose k-th row is replaced by 0's and an arbitrary element of the underlying ring at the l-th column. (Contributed by AV, 12-Feb-2019.)
matRRep = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)), 𝑠 ∈ (Base‘𝑟) ↦ (𝑘𝑛, 𝑙𝑛 ↦ (𝑖𝑛, 𝑗𝑛 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, (0g𝑟)), (𝑖𝑚𝑗))))))

Definitiondf-marepv 20413* Function replacing a column of a matrix by a vector. (Contributed by AV, 9-Feb-2019.) (Revised by AV, 26-Feb-2019.)
matRepV = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)), 𝑣 ∈ ((Base‘𝑟) ↑𝑚 𝑛) ↦ (𝑘𝑛 ↦ (𝑖𝑛, 𝑗𝑛 ↦ if(𝑗 = 𝑘, (𝑣𝑖), (𝑖𝑚𝑗))))))

Theoremmarrepfval 20414* First substitution for the definition of the matrix row replacement function. (Contributed by AV, 12-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑄 = (𝑁 matRRep 𝑅)    &    0 = (0g𝑅)       𝑄 = (𝑚𝐵, 𝑠 ∈ (Base‘𝑅) ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗)))))

Theoremmarrepval0 20415* Second substitution for the definition of the matrix row replacement function. (Contributed by AV, 12-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑄 = (𝑁 matRRep 𝑅)    &    0 = (0g𝑅)       ((𝑀𝐵𝑆 ∈ (Base‘𝑅)) → (𝑀𝑄𝑆) = (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗)))))

Theoremmarrepval 20416* Third substitution for the definition of the matrix row replacement function. (Contributed by AV, 12-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑄 = (𝑁 matRRep 𝑅)    &    0 = (0g𝑅)       (((𝑀𝐵𝑆 ∈ (Base‘𝑅)) ∧ (𝐾𝑁𝐿𝑁)) → (𝐾(𝑀𝑄𝑆)𝐿) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, 0 ), (𝑖𝑀𝑗))))

Theoremmarrepeval 20417 An entry of a matrix with a replaced row. (Contributed by AV, 12-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑄 = (𝑁 matRRep 𝑅)    &    0 = (0g𝑅)       (((𝑀𝐵𝑆 ∈ (Base‘𝑅)) ∧ (𝐾𝑁𝐿𝑁) ∧ (𝐼𝑁𝐽𝑁)) → (𝐼(𝐾(𝑀𝑄𝑆)𝐿)𝐽) = if(𝐼 = 𝐾, if(𝐽 = 𝐿, 𝑆, 0 ), (𝐼𝑀𝐽)))

Theoremmarrepcl 20418 Closure of the row replacement function for square matrices. (Contributed by AV, 13-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)       (((𝑅 ∈ Ring ∧ 𝑀𝐵𝑆 ∈ (Base‘𝑅)) ∧ (𝐾𝑁𝐿𝑁)) → (𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐿) ∈ 𝐵)

Theoremmarepvfval 20419* First substitution for the definition of the function replacing a column of a matrix by a vector. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 26-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑄 = (𝑁 matRepV 𝑅)    &   𝑉 = ((Base‘𝑅) ↑𝑚 𝑁)       𝑄 = (𝑚𝐵, 𝑣𝑉 ↦ (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝑣𝑖), (𝑖𝑚𝑗)))))

Theoremmarepvval0 20420* Second substitution for the definition of the function replacing a column of a matrix by a vector. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 26-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑄 = (𝑁 matRepV 𝑅)    &   𝑉 = ((Base‘𝑅) ↑𝑚 𝑁)       ((𝑀𝐵𝐶𝑉) → (𝑀𝑄𝐶) = (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗)))))

Theoremmarepvval 20421* Third substitution for the definition of the function replacing a column of a matrix by a vector. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 26-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑄 = (𝑁 matRepV 𝑅)    &   𝑉 = ((Base‘𝑅) ↑𝑚 𝑁)       ((𝑀𝐵𝐶𝑉𝐾𝑁) → ((𝑀𝑄𝐶)‘𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝐾, (𝐶𝑖), (𝑖𝑀𝑗))))

Theoremmarepveval 20422 An entry of a matrix with a replaced column. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 26-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑄 = (𝑁 matRepV 𝑅)    &   𝑉 = ((Base‘𝑅) ↑𝑚 𝑁)       (((𝑀𝐵𝐶𝑉𝐾𝑁) ∧ (𝐼𝑁𝐽𝑁)) → (𝐼((𝑀𝑄𝐶)‘𝐾)𝐽) = if(𝐽 = 𝐾, (𝐶𝐼), (𝐼𝑀𝐽)))

Theoremmarepvcl 20423 Closure of the column replacement function for square matrices. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 26-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑉 = ((Base‘𝑅) ↑𝑚 𝑁)       ((𝑅 ∈ Ring ∧ (𝑀𝐵𝐶𝑉𝐾𝑁)) → ((𝑀(𝑁 matRepV 𝑅)𝐶)‘𝐾) ∈ 𝐵)

Theoremma1repvcl 20424 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 20425 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 20426 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 20427* 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 20428* 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 20429 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 20430 Syntax for submatrices of a square matrix.
class subMat

Definitiondf-subma 20431* 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 20432* 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 20433* 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 20434* 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 20435* 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 20436 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 20437 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 20438 Syntax for the matrix determinant function.

Definitiondf-mdet 20439* Determinant of a square matrix. This definition is based on Leibniz' Formula (see mdetleib 20441). 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 20449. Multilineary means "linear for each row" - the additivity is shown by mdetrlin 20456, the homogeneity by mdetrsca 20457. Furthermore, it is shown that the determinant function is alternating (see mdetralt 20462) and normalized (see mdet1 20455). Finally, the uniqueness is shown by mdetuni 20476. As a consequence, the "determinant of a square matrix" is the function value of the determinant function for this square matrix, see mdetleib 20441. (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 20440* 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 20441* 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 20442* 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 20443 The determinant is not defined for an infinite matrix. (Contributed by AV, 27-Dec-2018.)
𝐷 = (𝑁 maDet 𝑅)       (𝑁 ∉ Fin → 𝐷 = ∅)

Theoremmdetfval1 20444* 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 20445* 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 20446 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 20447 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 20448 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 20449 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 20450 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 20451 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 20452* Lemma for mdetdiag 20453. Previously part of proof for mdet1 20455. (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 20453* 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 20454* 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 20455 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 20456 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 20457 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 20458* 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 20459* 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 20460 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 20461* 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 20462* 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 20463* 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 20464* 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 20465 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 20466* Lemma for mdetuni 20476. (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 20467* Lemma for mdetuni 20476. (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 20468* Lemma for mdetuni 20476. (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 20469* Lemma for mdetuni 20476. (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 20470* Lemma for mdetuni 20476. (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 20471* Lemma for mdetuni 20476. (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 20472* Lemma for mdetuni 20476. (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 20473* Lemma for mdetuni 20476. (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 20474* Lemma for mdetuni 20476. (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 20475* Lemma for mdetuni 20476. (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 20476* 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 20477 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 20478 Lemma 1 for m2detleib 20485. (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 20479 Lemma 5 for m2detleib 20485. (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 20480 Lemma 6 for m2detleib 20485. (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 20481 Lemma 7 for m2detleib 20485. (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 20482* Lemma 2 for m2detleib 20485. (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 20483* Lemma 3 for m2detleib 20485. (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 20484* Lemma 4 for m2detleib 20485. (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 20485 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 20487 Syntax for the minor matrices of a square matrix.
class minMatR1

Definitiondf-madu 20488* 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 20489* 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 20431- note that the matrix is transposed compared with the submatrix defined in df-subma 20431, but this does not matter because the determinants are the same, see mdettpos 20465). Such matrices are used in the definition of an adjunct of a square matrix, see df-madu 20488. (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 20490 Lemma for maducoeval2 20494. (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 20491* 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 20492* 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 20493* 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 20494* 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 20495 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 20496 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 20497* 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 20498 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 20499 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 20500* 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)))))

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