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Type | Label | Description |
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Statement | ||
Theorem | nfimdetndef 22401 | The determinant is not defined for an infinite matrix. (Contributed by AV, 27-Dec-2018.) |
⊢ 𝐷 = (𝑁 maDet 𝑅) ⇒ ⊢ (𝑁 ∉ Fin → 𝐷 = ∅) | ||
Theorem | mdetfval1 22402* | 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 22403* | 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 22404 | 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 22405 | 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 22406 | 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 22407 | 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 22408 | 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 22409 | 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 22410* | Lemma for mdetdiag 22411. Previously part of proof for mdet1 22413. (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 22411* | 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 22412* | 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 22413 | 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 22414 | 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 22415 | 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 22416* | 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 22417* | 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 22418 | 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 22419* | 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 22420* | 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 22421* | 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 22422* | 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 22423 | 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 22424* | Lemma for mdetuni 22434. (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 22425* | Lemma for mdetuni 22434. (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 22426* | Lemma for mdetuni 22434. (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 22427* | Lemma for mdetuni 22434. (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 22428* | Lemma for mdetuni 22434. (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 22429* | Lemma for mdetuni 22434. (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 22430* | Lemma for mdetuni 22434. (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 22431* | Lemma for mdetuni 22434. (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 22432* | Lemma for mdetuni 22434. (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 22433* | Lemma for mdetuni 22434. (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 22434* | 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 22435 | 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))) | ||
Theorem | m2detleiblem1 22436 | Lemma 1 for m2detleib 22443. (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 22437 | Lemma 5 for m2detleib 22443. (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 22438 | Lemma 6 for m2detleib 22443. (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 22439 | Lemma 7 for m2detleib 22443. (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 22440* | Lemma 2 for m2detleib 22443. (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 22441* | Lemma 3 for m2detleib 22443. (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 22442* | Lemma 4 for m2detleib 22443. (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 22443 | 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)))) | ||
Syntax | cmadu 22444 | Syntax for the matrix adjugate/adjunct function. |
class maAdju | ||
Syntax | cminmar1 22445 | Syntax for the minor matrices of a square matrix. |
class minMatR1 | ||
Definition | df-madu 22446* | 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 22447* | 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 22389- note that the matrix is transposed compared with the submatrix defined in df-subma 22389, but this does not matter because the determinants are the same, see mdettpos 22423). Such matrices are used in the definition of an adjunct of a square matrix, see df-madu 22446. (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 22448 | Lemma for maducoeval2 22452. (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 22449* | 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 22450* | 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 22451* | 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 22452* | 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 22453 | 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 22454 | 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 22455* | 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 22456 | 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 22457 | 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 22458* | 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 22459* | 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 22460* | 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 22461 | 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 22462 | 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 22463 | 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 22464 | 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))) | ||
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 22485 or smadiadetr 22487. | ||
Theorem | symgmatr01lem 22465* | Lemma for symgmatr01 22466. (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 22466* | 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 22467* | Lemma A for gsummatr01 22471. (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 22468* | Lemma B for gsummatr01 22471. (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 22469* | Lemma 1 for gsummatr01 22471. (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 22470* | Lemma 2 for gsummatr01 22471. (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 22471* | Lemma 1 for smadiadetlem4 22481. (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 22472* | 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 22473* | Lemma 0 for smadiadet 22482: 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 22474* | Lemma 1 for smadiadet 22482: 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 22475* | Lemma 1a for smadiadet 22482: 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 22476* | Lemma 2 for smadiadet 22482: 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 22477* | Lemma 0 for smadiadetlem3 22480. (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 22478* | Lemma 1 for smadiadetlem3 22480. (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 22479* | Lemma 2 for smadiadetlem3 22480. (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 22480* | Lemma 3 for smadiadet 22482. (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 22481* | Lemma 4 for smadiadet 22482. (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 22482 | 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 22483 | Lemma 1 for smadiadetg 22485. (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 22484 | Lemma 2 for smadiadetg 22485. (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 22485 | 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 22486 | Lemma for smadiadetr 22487: version of smadiadetg 22485 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 22487 | 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 22485. 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)))) | ||
Theorem | invrvald 22488 | 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 22489 | 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 22490 | 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)) | ||
In the following, Cramer's rule cramer 22503 is proven. According to Wikipedia "Cramer's rule", 21-Feb-2019, https://en.wikipedia.org/wiki/Cramer%27s_rule 22503: "[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), 22503 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 22496), using matrix multiplication (mamuval 22198) and multiplication of a vector with a matrix (mulmarep1gsum2 22386). By using the multiplicativity of the determinant (mdetmul 22435) it follows that det (𝐴‘𝑖) = det (𝐴 × (𝑋‘𝑖)) = det 𝐴 · det (𝑋‘𝑖) (cramerimplem3 22497). Furthermore, it follows that det (𝑋‘𝑖) = (𝑥‘𝑖) (cramerimplem1 22495). To show this, a special case of the Laplace expansion is used (smadiadetg 22485). From these equations and the cancellation law for division in a ring (dvrcan3 20297) it follows that (𝑥‘𝑖) = det (𝑋‘𝑖) = det (𝐴‘𝑖) / det 𝐴. This is the right to left implication (cramerimp 22498, cramerlem1 22499, cramerlem2 22500) of Cramer's rule (cramer 22503). The left to right implication is shown by cramerlem3 22501, using the fact that a solution of the system of linear equations exists (slesolex 22494). Notice that for the special case of 0-dimensional matrices/vectors only the left to right implication is valid (see cramer0 22502), because assuming the right-hand side of the implication ((𝑋 · 𝑍) = 𝑌), 𝑍 could be anything (see mavmul0g 22365). | ||
Theorem | slesolvec 22491 | 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 22492 | 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 22493 | 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))) | ||
Theorem | slesolex 22494* | Every system of linear equations represented by a matrix with a unit as determinant has a solution. (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) ⇒ ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → ∃𝑧 ∈ 𝑉 (𝑋 · 𝑧) = 𝑌) | ||
Theorem | cramerimplem1 22495 | Lemma 1 for cramerimp 22498: The determinant of the identity matrix with the ith column replaced by a (column) vector equals the ith component of the vector. (Contributed by AV, 15-Feb-2019.) (Revised by AV, 5-Jul-2022.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑉 = ((Base‘𝑅) ↑m 𝑁) & ⊢ 𝐸 = (((1r‘𝐴)(𝑁 matRepV 𝑅)𝑍)‘𝐼) & ⊢ 𝐷 = (𝑁 maDet 𝑅) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ 𝑍 ∈ 𝑉) → (𝐷‘𝐸) = (𝑍‘𝐼)) | ||
Theorem | cramerimplem2 22496 | Lemma 2 for cramerimp 22498: The matrix of a system of linear equations multiplied with the identity matrix with the ith column replaced by the solution vector of the system of linear equations equals the matrix of the system of linear equations with the ith column replaced by the right-hand side vector of the system of linear equations. (Contributed by AV, 19-Feb-2019.) (Revised by AV, 1-Mar-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑉 = ((Base‘𝑅) ↑m 𝑁) & ⊢ 𝐸 = (((1r‘𝐴)(𝑁 matRepV 𝑅)𝑍)‘𝐼) & ⊢ 𝐻 = ((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝐼) & ⊢ · = (𝑅 maVecMul 〈𝑁, 𝑁〉) & ⊢ × = (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉) ⇒ ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝑋 · 𝑍) = 𝑌) → (𝑋 × 𝐸) = 𝐻) | ||
Theorem | cramerimplem3 22497 | Lemma 3 for cramerimp 22498: The determinant of the matrix of a system of linear equations multiplied with the determinant of the identity matrix with the ith column replaced by the solution vector of the system of linear equations equals the determinant of the matrix of the system of linear equations with the ith column replaced by the right-hand side vector of the system of linear equations. (Contributed by AV, 19-Feb-2019.) (Revised by AV, 1-Mar-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑉 = ((Base‘𝑅) ↑m 𝑁) & ⊢ 𝐸 = (((1r‘𝐴)(𝑁 matRepV 𝑅)𝑍)‘𝐼) & ⊢ 𝐻 = ((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝐼) & ⊢ · = (𝑅 maVecMul 〈𝑁, 𝑁〉) & ⊢ 𝐷 = (𝑁 maDet 𝑅) & ⊢ ⊗ = (.r‘𝑅) ⇒ ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝑋 · 𝑍) = 𝑌) → ((𝐷‘𝑋) ⊗ (𝐷‘𝐸)) = (𝐷‘𝐻)) | ||
Theorem | cramerimp 22498 | One direction of Cramer's rule (according to Wikipedia "Cramer's rule", 21-Feb-2019, https://en.wikipedia.org/wiki/Cramer%27s_rule: "[Cramer's rule] ... expresses the 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 ith component of the solution vector of a system of linear equations equals the determinant of the matrix of the system of linear equations with the ith column replaced by the righthand side vector of the system of linear equations divided by the determinant of the matrix of the system of linear equations. (Contributed by AV, 19-Feb-2019.) (Revised by AV, 1-Mar-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑉 = ((Base‘𝑅) ↑m 𝑁) & ⊢ 𝐸 = (((1r‘𝐴)(𝑁 matRepV 𝑅)𝑍)‘𝐼) & ⊢ 𝐻 = ((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝐼) & ⊢ · = (𝑅 maVecMul 〈𝑁, 𝑁〉) & ⊢ 𝐷 = (𝑁 maDet 𝑅) & ⊢ / = (/r‘𝑅) ⇒ ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅))) → (𝑍‘𝐼) = ((𝐷‘𝐻) / (𝐷‘𝑋))) | ||
Theorem | cramerlem1 22499* | Lemma 1 for cramer 22503. (Contributed by AV, 21-Feb-2019.) (Revised by AV, 1-Mar-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑉 = ((Base‘𝑅) ↑m 𝑁) & ⊢ 𝐷 = (𝑁 maDet 𝑅) & ⊢ · = (𝑅 maVecMul 〈𝑁, 𝑁〉) & ⊢ / = (/r‘𝑅) ⇒ ⊢ ((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝐷‘𝑋) ∈ (Unit‘𝑅) ∧ 𝑍 ∈ 𝑉 ∧ (𝑋 · 𝑍) = 𝑌)) → 𝑍 = (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋)))) | ||
Theorem | cramerlem2 22500* | Lemma 2 for cramer 22503. (Contributed by AV, 21-Feb-2019.) (Revised by AV, 1-Mar-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑉 = ((Base‘𝑅) ↑m 𝑁) & ⊢ 𝐷 = (𝑁 maDet 𝑅) & ⊢ · = (𝑅 maVecMul 〈𝑁, 𝑁〉) & ⊢ / = (/r‘𝑅) ⇒ ⊢ ((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → ∀𝑧 ∈ 𝑉 ((𝑋 · 𝑧) = 𝑌 → 𝑧 = (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋))))) |
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