state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
M N : Matrix n n R
x✝³ x✝² : n → n
x✝¹ : x✝³ ∈ filter Bijective univ
x✝ : x✝² ∈ filter Bijective univ
h : (fun p h => ofBijective p (_ : Bijective p)) x✝³ x✝¹ = (fun p h => ofBijec... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | injection h | @[simp]
theorem det_mul (M N : Matrix n n R) : det (M * N) = det M * det N :=
calc
det (M * N) = ∑ p : n → n, ∑ σ : Perm n, ε σ * ∏ i, M (σ i) (p i) * N (p i) i := by
simp only [det_apply', mul_apply, prod_univ_sum, mul_sum, Fintype.piFinset_univ]
rw [Finset.sum_comm]
_ =
∑ p in (@univ (n ... | Mathlib.LinearAlgebra.Matrix.Determinant.150_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_mul (M N : Matrix n n R) : det (M * N) = det M * det N | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
M N : Matrix n n R
⊢ ∑ τ : Perm n, ∑ σ : Perm n, ↑↑(sign σ) * ∏ i : n, M (σ i) (τ i) * N (τ i) i =
∑ σ : Perm n, ∑ τ : Perm n, (∏ i : n, N (σ i) i) * ↑↑(sign τ) * ∏ j : n, M (τ... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | simp only [mul_comm, mul_left_comm, prod_mul_distrib, mul_assoc] | @[simp]
theorem det_mul (M N : Matrix n n R) : det (M * N) = det M * det N :=
calc
det (M * N) = ∑ p : n → n, ∑ σ : Perm n, ε σ * ∏ i, M (σ i) (p i) * N (p i) i := by
simp only [det_apply', mul_apply, prod_univ_sum, mul_sum, Fintype.piFinset_univ]
rw [Finset.sum_comm]
_ =
∑ p in (@univ (n ... | Mathlib.LinearAlgebra.Matrix.Determinant.150_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_mul (M N : Matrix n n R) : det (M * N) = det M * det N | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
M N : Matrix n n R
σ : Perm n
x✝ : σ ∈ univ
τ : Perm n
⊢ (∏ i : n, N (σ i) i) * ↑↑(sign τ) * ∏ j : n, M (τ j) (σ j) =
(∏ i : n, N (σ i) i) * (↑↑(sign σ) * ↑↑(sign ((Equiv.mulRi... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | have : (∏ j, M (τ j) (σ j)) = ∏ j, M ((τ * σ⁻¹) j) j := by
rw [← (σ⁻¹ : _ ≃ _).prod_comp]
simp only [Equiv.Perm.coe_mul, apply_inv_self, Function.comp_apply] | @[simp]
theorem det_mul (M N : Matrix n n R) : det (M * N) = det M * det N :=
calc
det (M * N) = ∑ p : n → n, ∑ σ : Perm n, ε σ * ∏ i, M (σ i) (p i) * N (p i) i := by
simp only [det_apply', mul_apply, prod_univ_sum, mul_sum, Fintype.piFinset_univ]
rw [Finset.sum_comm]
_ =
∑ p in (@univ (n ... | Mathlib.LinearAlgebra.Matrix.Determinant.150_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_mul (M N : Matrix n n R) : det (M * N) = det M * det N | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
M N : Matrix n n R
σ : Perm n
x✝ : σ ∈ univ
τ : Perm n
⊢ ∏ j : n, M (τ j) (σ j) = ∏ j : n, M ((τ * σ⁻¹) j) j | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rw [← (σ⁻¹ : _ ≃ _).prod_comp] | @[simp]
theorem det_mul (M N : Matrix n n R) : det (M * N) = det M * det N :=
calc
det (M * N) = ∑ p : n → n, ∑ σ : Perm n, ε σ * ∏ i, M (σ i) (p i) * N (p i) i := by
simp only [det_apply', mul_apply, prod_univ_sum, mul_sum, Fintype.piFinset_univ]
rw [Finset.sum_comm]
_ =
∑ p in (@univ (n ... | Mathlib.LinearAlgebra.Matrix.Determinant.150_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_mul (M N : Matrix n n R) : det (M * N) = det M * det N | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
M N : Matrix n n R
σ : Perm n
x✝ : σ ∈ univ
τ : Perm n
⊢ ∏ i : n, M (τ (σ⁻¹ i)) (σ (σ⁻¹ i)) = ∏ j : n, M ((τ * σ⁻¹) j) j | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | simp only [Equiv.Perm.coe_mul, apply_inv_self, Function.comp_apply] | @[simp]
theorem det_mul (M N : Matrix n n R) : det (M * N) = det M * det N :=
calc
det (M * N) = ∑ p : n → n, ∑ σ : Perm n, ε σ * ∏ i, M (σ i) (p i) * N (p i) i := by
simp only [det_apply', mul_apply, prod_univ_sum, mul_sum, Fintype.piFinset_univ]
rw [Finset.sum_comm]
_ =
∑ p in (@univ (n ... | Mathlib.LinearAlgebra.Matrix.Determinant.150_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_mul (M N : Matrix n n R) : det (M * N) = det M * det N | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
M N : Matrix n n R
σ : Perm n
x✝ : σ ∈ univ
τ : Perm n
this : ∏ j : n, M (τ j) (σ j) = ∏ j : n, M ((τ * σ⁻¹) j) j
⊢ (∏ i : n, N (σ i) i) * ↑↑(sign τ) * ∏ j : n, M (τ j) (σ j) =
... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | have h : ε σ * ε (τ * σ⁻¹) = ε τ :=
calc
ε σ * ε (τ * σ⁻¹) = ε (τ * σ⁻¹ * σ) := by
rw [mul_comm, sign_mul (τ * σ⁻¹)]
simp only [Int.cast_mul, Units.val_mul]
_ = ε τ := by simp only [inv_mul_cancel_right] | @[simp]
theorem det_mul (M N : Matrix n n R) : det (M * N) = det M * det N :=
calc
det (M * N) = ∑ p : n → n, ∑ σ : Perm n, ε σ * ∏ i, M (σ i) (p i) * N (p i) i := by
simp only [det_apply', mul_apply, prod_univ_sum, mul_sum, Fintype.piFinset_univ]
rw [Finset.sum_comm]
_ =
∑ p in (@univ (n ... | Mathlib.LinearAlgebra.Matrix.Determinant.150_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_mul (M N : Matrix n n R) : det (M * N) = det M * det N | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
M N : Matrix n n R
σ : Perm n
x✝ : σ ∈ univ
τ : Perm n
this : ∏ j : n, M (τ j) (σ j) = ∏ j : n, M ((τ * σ⁻¹) j) j
⊢ ↑↑(sign σ) * ↑↑(sign (τ * σ⁻¹)) = ↑↑(sign (τ * σ⁻¹ * σ)) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rw [mul_comm, sign_mul (τ * σ⁻¹)] | @[simp]
theorem det_mul (M N : Matrix n n R) : det (M * N) = det M * det N :=
calc
det (M * N) = ∑ p : n → n, ∑ σ : Perm n, ε σ * ∏ i, M (σ i) (p i) * N (p i) i := by
simp only [det_apply', mul_apply, prod_univ_sum, mul_sum, Fintype.piFinset_univ]
rw [Finset.sum_comm]
_ =
∑ p in (@univ (n ... | Mathlib.LinearAlgebra.Matrix.Determinant.150_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_mul (M N : Matrix n n R) : det (M * N) = det M * det N | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
M N : Matrix n n R
σ : Perm n
x✝ : σ ∈ univ
τ : Perm n
this : ∏ j : n, M (τ j) (σ j) = ∏ j : n, M ((τ * σ⁻¹) j) j
⊢ ↑↑(sign (τ * σ⁻¹)) * ↑↑(sign σ) = ↑↑(sign (τ * σ⁻¹) * sign σ) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | simp only [Int.cast_mul, Units.val_mul] | @[simp]
theorem det_mul (M N : Matrix n n R) : det (M * N) = det M * det N :=
calc
det (M * N) = ∑ p : n → n, ∑ σ : Perm n, ε σ * ∏ i, M (σ i) (p i) * N (p i) i := by
simp only [det_apply', mul_apply, prod_univ_sum, mul_sum, Fintype.piFinset_univ]
rw [Finset.sum_comm]
_ =
∑ p in (@univ (n ... | Mathlib.LinearAlgebra.Matrix.Determinant.150_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_mul (M N : Matrix n n R) : det (M * N) = det M * det N | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
M N : Matrix n n R
σ : Perm n
x✝ : σ ∈ univ
τ : Perm n
this : ∏ j : n, M (τ j) (σ j) = ∏ j : n, M ((τ * σ⁻¹) j) j
⊢ ↑↑(sign (τ * σ⁻¹ * σ)) = ↑↑(sign τ) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | simp only [inv_mul_cancel_right] | @[simp]
theorem det_mul (M N : Matrix n n R) : det (M * N) = det M * det N :=
calc
det (M * N) = ∑ p : n → n, ∑ σ : Perm n, ε σ * ∏ i, M (σ i) (p i) * N (p i) i := by
simp only [det_apply', mul_apply, prod_univ_sum, mul_sum, Fintype.piFinset_univ]
rw [Finset.sum_comm]
_ =
∑ p in (@univ (n ... | Mathlib.LinearAlgebra.Matrix.Determinant.150_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_mul (M N : Matrix n n R) : det (M * N) = det M * det N | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
M N : Matrix n n R
σ : Perm n
x✝ : σ ∈ univ
τ : Perm n
this : ∏ j : n, M (τ j) (σ j) = ∏ j : n, M ((τ * σ⁻¹) j) j
h : ↑↑(sign σ) * ↑↑(sign (τ * σ⁻¹)) = ↑↑(sign τ)
⊢ (∏ i : n, N (σ ... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | simp_rw [Equiv.coe_mulRight, h] | @[simp]
theorem det_mul (M N : Matrix n n R) : det (M * N) = det M * det N :=
calc
det (M * N) = ∑ p : n → n, ∑ σ : Perm n, ε σ * ∏ i, M (σ i) (p i) * N (p i) i := by
simp only [det_apply', mul_apply, prod_univ_sum, mul_sum, Fintype.piFinset_univ]
rw [Finset.sum_comm]
_ =
∑ p in (@univ (n ... | Mathlib.LinearAlgebra.Matrix.Determinant.150_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_mul (M N : Matrix n n R) : det (M * N) = det M * det N | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
M N : Matrix n n R
σ : Perm n
x✝ : σ ∈ univ
τ : Perm n
this : ∏ j : n, M (τ j) (σ j) = ∏ j : n, M ((τ * σ⁻¹) j) j
h : ↑↑(sign σ) * ↑↑(sign (τ * σ⁻¹)) = ↑↑(sign τ)
⊢ (∏ i : n, N (σ ... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | simp only [this] | @[simp]
theorem det_mul (M N : Matrix n n R) : det (M * N) = det M * det N :=
calc
det (M * N) = ∑ p : n → n, ∑ σ : Perm n, ε σ * ∏ i, M (σ i) (p i) * N (p i) i := by
simp only [det_apply', mul_apply, prod_univ_sum, mul_sum, Fintype.piFinset_univ]
rw [Finset.sum_comm]
_ =
∑ p in (@univ (n ... | Mathlib.LinearAlgebra.Matrix.Determinant.150_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_mul (M N : Matrix n n R) : det (M * N) = det M * det N | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
M N : Matrix n n R
⊢ ∑ σ : Perm n, ∑ τ : Perm n, (∏ i : n, N (σ i) i) * (↑↑(sign σ) * ↑↑(sign τ)) * ∏ i : n, M (τ i) i = det M * det N | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | simp only [det_apply', Finset.mul_sum, mul_comm, mul_left_comm, mul_assoc] | @[simp]
theorem det_mul (M N : Matrix n n R) : det (M * N) = det M * det N :=
calc
det (M * N) = ∑ p : n → n, ∑ σ : Perm n, ε σ * ∏ i, M (σ i) (p i) * N (p i) i := by
simp only [det_apply', mul_apply, prod_univ_sum, mul_sum, Fintype.piFinset_univ]
rw [Finset.sum_comm]
_ =
∑ p in (@univ (n ... | Mathlib.LinearAlgebra.Matrix.Determinant.150_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_mul (M N : Matrix n n R) : det (M * N) = det M * det N | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
M N : Matrix m m R
⊢ det (M * N) = det (N * M) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rw [det_mul, det_mul, mul_comm] | /-- On square matrices, `mul_comm` applies under `det`. -/
theorem det_mul_comm (M N : Matrix m m R) : det (M * N) = det (N * M) := by
| Mathlib.LinearAlgebra.Matrix.Determinant.199_0.U1f6HO8zRbnvZ95 | /-- On square matrices, `mul_comm` applies under `det`. -/
theorem det_mul_comm (M N : Matrix m m R) : det (M * N) = det (N * M) | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
M N P : Matrix m m R
⊢ det (M * (N * P)) = det (N * (M * P)) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rw [← Matrix.mul_assoc, ← Matrix.mul_assoc, det_mul, det_mul_comm M N, ← det_mul] | /-- On square matrices, `mul_left_comm` applies under `det`. -/
theorem det_mul_left_comm (M N P : Matrix m m R) : det (M * (N * P)) = det (N * (M * P)) := by
| Mathlib.LinearAlgebra.Matrix.Determinant.204_0.U1f6HO8zRbnvZ95 | /-- On square matrices, `mul_left_comm` applies under `det`. -/
theorem det_mul_left_comm (M N P : Matrix m m R) : det (M * (N * P)) = det (N * (M * P)) | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
M N P : Matrix m m R
⊢ det (M * N * P) = det (M * P * N) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rw [Matrix.mul_assoc, Matrix.mul_assoc, det_mul, det_mul_comm N P, ← det_mul] | /-- On square matrices, `mul_right_comm` applies under `det`. -/
theorem det_mul_right_comm (M N P : Matrix m m R) : det (M * N * P) = det (M * P * N) := by
| Mathlib.LinearAlgebra.Matrix.Determinant.209_0.U1f6HO8zRbnvZ95 | /-- On square matrices, `mul_right_comm` applies under `det`. -/
theorem det_mul_right_comm (M N P : Matrix m m R) : det (M * N * P) = det (M * P * N) | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
M : (Matrix m m R)ˣ
N : Matrix m m R
⊢ det (↑M * N * ↑M⁻¹) = det N | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rw [det_mul_right_comm, Units.mul_inv, one_mul] | theorem det_units_conj (M : (Matrix m m R)ˣ) (N : Matrix m m R) :
det ((M : Matrix _ _ _) * N * (↑M⁻¹ : Matrix _ _ _)) = det N := by
| Mathlib.LinearAlgebra.Matrix.Determinant.215_0.U1f6HO8zRbnvZ95 | theorem det_units_conj (M : (Matrix m m R)ˣ) (N : Matrix m m R) :
det ((M : Matrix _ _ _) * N * (↑M⁻¹ : Matrix _ _ _)) = det N | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
M : Matrix n n R
⊢ det Mᵀ = det M | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rw [det_apply', det_apply'] | /-- Transposing a matrix preserves the determinant. -/
@[simp]
theorem det_transpose (M : Matrix n n R) : Mᵀ.det = M.det := by
| Mathlib.LinearAlgebra.Matrix.Determinant.226_0.U1f6HO8zRbnvZ95 | /-- Transposing a matrix preserves the determinant. -/
@[simp]
theorem det_transpose (M : Matrix n n R) : Mᵀ.det = M.det | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
M : Matrix n n R
⊢ ∑ σ : Perm n, ↑↑(sign σ) * ∏ i : n, Mᵀ (σ i) i = ∑ σ : Perm n, ↑↑(sign σ) * ∏ i : n, M (σ i) i | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | refine' Fintype.sum_bijective _ inv_involutive.bijective _ _ _ | /-- Transposing a matrix preserves the determinant. -/
@[simp]
theorem det_transpose (M : Matrix n n R) : Mᵀ.det = M.det := by
rw [det_apply', det_apply']
| Mathlib.LinearAlgebra.Matrix.Determinant.226_0.U1f6HO8zRbnvZ95 | /-- Transposing a matrix preserves the determinant. -/
@[simp]
theorem det_transpose (M : Matrix n n R) : Mᵀ.det = M.det | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
M : Matrix n n R
⊢ ∀ (x : Perm n), ↑↑(sign x) * ∏ i : n, Mᵀ (x i) i = ↑↑(sign x⁻¹) * ∏ i : n, M (x⁻¹ i) i | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | intro σ | /-- Transposing a matrix preserves the determinant. -/
@[simp]
theorem det_transpose (M : Matrix n n R) : Mᵀ.det = M.det := by
rw [det_apply', det_apply']
refine' Fintype.sum_bijective _ inv_involutive.bijective _ _ _
| Mathlib.LinearAlgebra.Matrix.Determinant.226_0.U1f6HO8zRbnvZ95 | /-- Transposing a matrix preserves the determinant. -/
@[simp]
theorem det_transpose (M : Matrix n n R) : Mᵀ.det = M.det | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
M : Matrix n n R
σ : Perm n
⊢ ↑↑(sign σ) * ∏ i : n, Mᵀ (σ i) i = ↑↑(sign σ⁻¹) * ∏ i : n, M (σ⁻¹ i) i | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rw [sign_inv] | /-- Transposing a matrix preserves the determinant. -/
@[simp]
theorem det_transpose (M : Matrix n n R) : Mᵀ.det = M.det := by
rw [det_apply', det_apply']
refine' Fintype.sum_bijective _ inv_involutive.bijective _ _ _
intro σ
| Mathlib.LinearAlgebra.Matrix.Determinant.226_0.U1f6HO8zRbnvZ95 | /-- Transposing a matrix preserves the determinant. -/
@[simp]
theorem det_transpose (M : Matrix n n R) : Mᵀ.det = M.det | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
M : Matrix n n R
σ : Perm n
⊢ ↑↑(sign σ) * ∏ i : n, Mᵀ (σ i) i = ↑↑(sign σ) * ∏ i : n, M (σ⁻¹ i) i | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | congr 1 | /-- Transposing a matrix preserves the determinant. -/
@[simp]
theorem det_transpose (M : Matrix n n R) : Mᵀ.det = M.det := by
rw [det_apply', det_apply']
refine' Fintype.sum_bijective _ inv_involutive.bijective _ _ _
intro σ
rw [sign_inv]
| Mathlib.LinearAlgebra.Matrix.Determinant.226_0.U1f6HO8zRbnvZ95 | /-- Transposing a matrix preserves the determinant. -/
@[simp]
theorem det_transpose (M : Matrix n n R) : Mᵀ.det = M.det | Mathlib_LinearAlgebra_Matrix_Determinant |
case e_a
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
M : Matrix n n R
σ : Perm n
⊢ ∏ i : n, Mᵀ (σ i) i = ∏ i : n, M (σ⁻¹ i) i | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | apply Fintype.prod_equiv σ | /-- Transposing a matrix preserves the determinant. -/
@[simp]
theorem det_transpose (M : Matrix n n R) : Mᵀ.det = M.det := by
rw [det_apply', det_apply']
refine' Fintype.sum_bijective _ inv_involutive.bijective _ _ _
intro σ
rw [sign_inv]
congr 1
| Mathlib.LinearAlgebra.Matrix.Determinant.226_0.U1f6HO8zRbnvZ95 | /-- Transposing a matrix preserves the determinant. -/
@[simp]
theorem det_transpose (M : Matrix n n R) : Mᵀ.det = M.det | Mathlib_LinearAlgebra_Matrix_Determinant |
case e_a.h
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
M : Matrix n n R
σ : Perm n
⊢ ∀ (x : n), Mᵀ (σ x) x = M (σ⁻¹ (σ x)) (σ x) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | intros | /-- Transposing a matrix preserves the determinant. -/
@[simp]
theorem det_transpose (M : Matrix n n R) : Mᵀ.det = M.det := by
rw [det_apply', det_apply']
refine' Fintype.sum_bijective _ inv_involutive.bijective _ _ _
intro σ
rw [sign_inv]
congr 1
apply Fintype.prod_equiv σ
| Mathlib.LinearAlgebra.Matrix.Determinant.226_0.U1f6HO8zRbnvZ95 | /-- Transposing a matrix preserves the determinant. -/
@[simp]
theorem det_transpose (M : Matrix n n R) : Mᵀ.det = M.det | Mathlib_LinearAlgebra_Matrix_Determinant |
case e_a.h
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
M : Matrix n n R
σ : Perm n
x✝ : n
⊢ Mᵀ (σ x✝) x✝ = M (σ⁻¹ (σ x✝)) (σ x✝) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | simp | /-- Transposing a matrix preserves the determinant. -/
@[simp]
theorem det_transpose (M : Matrix n n R) : Mᵀ.det = M.det := by
rw [det_apply', det_apply']
refine' Fintype.sum_bijective _ inv_involutive.bijective _ _ _
intro σ
rw [sign_inv]
congr 1
apply Fintype.prod_equiv σ
intros
| Mathlib.LinearAlgebra.Matrix.Determinant.226_0.U1f6HO8zRbnvZ95 | /-- Transposing a matrix preserves the determinant. -/
@[simp]
theorem det_transpose (M : Matrix n n R) : Mᵀ.det = M.det | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
σ : Perm n
M : Matrix n n R
⊢ sign σ • detRowAlternating M = ↑↑(sign σ) * det M | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | simp [Units.smul_def] | /-- Permuting the columns changes the sign of the determinant. -/
theorem det_permute (σ : Perm n) (M : Matrix n n R) :
(Matrix.det fun i => M (σ i)) = Perm.sign σ * M.det :=
((detRowAlternating : (n → R) [Λ^n]→ₗ[R] R).map_perm M σ).trans (by | Mathlib.LinearAlgebra.Matrix.Determinant.239_0.U1f6HO8zRbnvZ95 | /-- Permuting the columns changes the sign of the determinant. -/
theorem det_permute (σ : Perm n) (M : Matrix n n R) :
(Matrix.det fun i => M (σ i)) = Perm.sign σ * M.det | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
e : n ≃ m
A : Matrix m m R
⊢ det (submatrix A ⇑e ⇑e) = det A | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rw [det_apply', det_apply'] | /-- Permuting rows and columns with the same equivalence has no effect. -/
@[simp]
theorem det_submatrix_equiv_self (e : n ≃ m) (A : Matrix m m R) :
det (A.submatrix e e) = det A := by
| Mathlib.LinearAlgebra.Matrix.Determinant.245_0.U1f6HO8zRbnvZ95 | /-- Permuting rows and columns with the same equivalence has no effect. -/
@[simp]
theorem det_submatrix_equiv_self (e : n ≃ m) (A : Matrix m m R) :
det (A.submatrix e e) = det A | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
e : n ≃ m
A : Matrix m m R
⊢ ∑ σ : Perm n, ↑↑(sign σ) * ∏ i : n, submatrix A (⇑e) (⇑e) (σ i) i = ∑ σ : Perm m, ↑↑(sign σ) * ∏ i : m, A (σ i) i | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | apply Fintype.sum_equiv (Equiv.permCongr e) | /-- Permuting rows and columns with the same equivalence has no effect. -/
@[simp]
theorem det_submatrix_equiv_self (e : n ≃ m) (A : Matrix m m R) :
det (A.submatrix e e) = det A := by
rw [det_apply', det_apply']
| Mathlib.LinearAlgebra.Matrix.Determinant.245_0.U1f6HO8zRbnvZ95 | /-- Permuting rows and columns with the same equivalence has no effect. -/
@[simp]
theorem det_submatrix_equiv_self (e : n ≃ m) (A : Matrix m m R) :
det (A.submatrix e e) = det A | Mathlib_LinearAlgebra_Matrix_Determinant |
case h
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
e : n ≃ m
A : Matrix m m R
⊢ ∀ (x : Perm n),
↑↑(sign x) * ∏ i : n, submatrix A (⇑e) (⇑e) (x i) i =
↑↑(sign ((permCongr e) x)) * ∏ i : m, A (((permCongr e) x) i) i | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | intro σ | /-- Permuting rows and columns with the same equivalence has no effect. -/
@[simp]
theorem det_submatrix_equiv_self (e : n ≃ m) (A : Matrix m m R) :
det (A.submatrix e e) = det A := by
rw [det_apply', det_apply']
apply Fintype.sum_equiv (Equiv.permCongr e)
| Mathlib.LinearAlgebra.Matrix.Determinant.245_0.U1f6HO8zRbnvZ95 | /-- Permuting rows and columns with the same equivalence has no effect. -/
@[simp]
theorem det_submatrix_equiv_self (e : n ≃ m) (A : Matrix m m R) :
det (A.submatrix e e) = det A | Mathlib_LinearAlgebra_Matrix_Determinant |
case h
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
e : n ≃ m
A : Matrix m m R
σ : Perm n
⊢ ↑↑(sign σ) * ∏ i : n, submatrix A (⇑e) (⇑e) (σ i) i = ↑↑(sign ((permCongr e) σ)) * ∏ i : m, A (((permCongr e) σ) i) i | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rw [Equiv.Perm.sign_permCongr e σ] | /-- Permuting rows and columns with the same equivalence has no effect. -/
@[simp]
theorem det_submatrix_equiv_self (e : n ≃ m) (A : Matrix m m R) :
det (A.submatrix e e) = det A := by
rw [det_apply', det_apply']
apply Fintype.sum_equiv (Equiv.permCongr e)
intro σ
| Mathlib.LinearAlgebra.Matrix.Determinant.245_0.U1f6HO8zRbnvZ95 | /-- Permuting rows and columns with the same equivalence has no effect. -/
@[simp]
theorem det_submatrix_equiv_self (e : n ≃ m) (A : Matrix m m R) :
det (A.submatrix e e) = det A | Mathlib_LinearAlgebra_Matrix_Determinant |
case h
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
e : n ≃ m
A : Matrix m m R
σ : Perm n
⊢ ↑↑(sign σ) * ∏ i : n, submatrix A (⇑e) (⇑e) (σ i) i = ↑↑(sign σ) * ∏ i : m, A (((permCongr e) σ) i) i | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | congr 1 | /-- Permuting rows and columns with the same equivalence has no effect. -/
@[simp]
theorem det_submatrix_equiv_self (e : n ≃ m) (A : Matrix m m R) :
det (A.submatrix e e) = det A := by
rw [det_apply', det_apply']
apply Fintype.sum_equiv (Equiv.permCongr e)
intro σ
rw [Equiv.Perm.sign_permCongr e σ]
| Mathlib.LinearAlgebra.Matrix.Determinant.245_0.U1f6HO8zRbnvZ95 | /-- Permuting rows and columns with the same equivalence has no effect. -/
@[simp]
theorem det_submatrix_equiv_self (e : n ≃ m) (A : Matrix m m R) :
det (A.submatrix e e) = det A | Mathlib_LinearAlgebra_Matrix_Determinant |
case h.e_a
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
e : n ≃ m
A : Matrix m m R
σ : Perm n
⊢ ∏ i : n, submatrix A (⇑e) (⇑e) (σ i) i = ∏ i : m, A (((permCongr e) σ) i) i | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | apply Fintype.prod_equiv e | /-- Permuting rows and columns with the same equivalence has no effect. -/
@[simp]
theorem det_submatrix_equiv_self (e : n ≃ m) (A : Matrix m m R) :
det (A.submatrix e e) = det A := by
rw [det_apply', det_apply']
apply Fintype.sum_equiv (Equiv.permCongr e)
intro σ
rw [Equiv.Perm.sign_permCongr e σ]
congr ... | Mathlib.LinearAlgebra.Matrix.Determinant.245_0.U1f6HO8zRbnvZ95 | /-- Permuting rows and columns with the same equivalence has no effect. -/
@[simp]
theorem det_submatrix_equiv_self (e : n ≃ m) (A : Matrix m m R) :
det (A.submatrix e e) = det A | Mathlib_LinearAlgebra_Matrix_Determinant |
case h.e_a.h
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
e : n ≃ m
A : Matrix m m R
σ : Perm n
⊢ ∀ (x : n), submatrix A (⇑e) (⇑e) (σ x) x = A (((permCongr e) σ) (e x)) (e x) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | intro i | /-- Permuting rows and columns with the same equivalence has no effect. -/
@[simp]
theorem det_submatrix_equiv_self (e : n ≃ m) (A : Matrix m m R) :
det (A.submatrix e e) = det A := by
rw [det_apply', det_apply']
apply Fintype.sum_equiv (Equiv.permCongr e)
intro σ
rw [Equiv.Perm.sign_permCongr e σ]
congr ... | Mathlib.LinearAlgebra.Matrix.Determinant.245_0.U1f6HO8zRbnvZ95 | /-- Permuting rows and columns with the same equivalence has no effect. -/
@[simp]
theorem det_submatrix_equiv_self (e : n ≃ m) (A : Matrix m m R) :
det (A.submatrix e e) = det A | Mathlib_LinearAlgebra_Matrix_Determinant |
case h.e_a.h
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
e : n ≃ m
A : Matrix m m R
σ : Perm n
i : n
⊢ submatrix A (⇑e) (⇑e) (σ i) i = A (((permCongr e) σ) (e i)) (e i) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rw [Equiv.permCongr_apply, Equiv.symm_apply_apply, submatrix_apply] | /-- Permuting rows and columns with the same equivalence has no effect. -/
@[simp]
theorem det_submatrix_equiv_self (e : n ≃ m) (A : Matrix m m R) :
det (A.submatrix e e) = det A := by
rw [det_apply', det_apply']
apply Fintype.sum_equiv (Equiv.permCongr e)
intro σ
rw [Equiv.Perm.sign_permCongr e σ]
congr ... | Mathlib.LinearAlgebra.Matrix.Determinant.245_0.U1f6HO8zRbnvZ95 | /-- Permuting rows and columns with the same equivalence has no effect. -/
@[simp]
theorem det_submatrix_equiv_self (e : n ≃ m) (A : Matrix m m R) :
det (A.submatrix e e) = det A | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
σ : Perm n
⊢ det (PEquiv.toMatrix (toPEquiv σ)) = ↑↑(sign σ) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rw [← Matrix.mul_one (σ.toPEquiv.toMatrix : Matrix n n R), PEquiv.toPEquiv_mul_matrix,
det_permute, det_one, mul_one] | /-- The determinant of a permutation matrix equals its sign. -/
@[simp]
theorem det_permutation (σ : Perm n) :
Matrix.det (σ.toPEquiv.toMatrix : Matrix n n R) = Perm.sign σ := by
| Mathlib.LinearAlgebra.Matrix.Determinant.268_0.U1f6HO8zRbnvZ95 | /-- The determinant of a permutation matrix equals its sign. -/
@[simp]
theorem det_permutation (σ : Perm n) :
Matrix.det (σ.toPEquiv.toMatrix : Matrix n n R) = Perm.sign σ | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A : Matrix n n R
c : R
⊢ det (c • A) = det ((diagonal fun x => c) * A) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rw [smul_eq_diagonal_mul] | theorem det_smul (A : Matrix n n R) (c : R) : det (c • A) = c ^ Fintype.card n * det A :=
calc
det (c • A) = det ((diagonal fun _ => c) * A) := by | Mathlib.LinearAlgebra.Matrix.Determinant.276_0.U1f6HO8zRbnvZ95 | theorem det_smul (A : Matrix n n R) (c : R) : det (c • A) = c ^ Fintype.card n * det A | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A : Matrix n n R
c : R
⊢ det (diagonal fun x => c) * det A = c ^ Fintype.card n * det A | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | simp [card_univ] | theorem det_smul (A : Matrix n n R) (c : R) : det (c • A) = c ^ Fintype.card n * det A :=
calc
det (c • A) = det ((diagonal fun _ => c) * A) := by rw [smul_eq_diagonal_mul]
_ = det (diagonal fun _ => c) * det A := (det_mul _ _)
_ = c ^ Fintype.card n * det A := by | Mathlib.LinearAlgebra.Matrix.Determinant.276_0.U1f6HO8zRbnvZ95 | theorem det_smul (A : Matrix n n R) (c : R) : det (c • A) = c ^ Fintype.card n * det A | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁸ : DecidableEq n
inst✝⁷ : Fintype n
inst✝⁶ : DecidableEq m
inst✝⁵ : Fintype m
R : Type v
inst✝⁴ : CommRing R
α : Type u_3
inst✝³ : Monoid α
inst✝² : DistribMulAction α R
inst✝¹ : IsScalarTower α R R
inst✝ : SMulCommClass α R R
c : α
A : Matrix n n R
⊢ det (c • A) = c ^ Fintype.card n • d... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rw [← smul_one_smul R c A, det_smul, smul_pow, one_pow, smul_mul_assoc, one_mul] | @[simp]
theorem det_smul_of_tower {α} [Monoid α] [DistribMulAction α R] [IsScalarTower α R R]
[SMulCommClass α R R] (c : α) (A : Matrix n n R) : det (c • A) = c ^ Fintype.card n • det A :=
by | Mathlib.LinearAlgebra.Matrix.Determinant.283_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_smul_of_tower {α} [Monoid α] [DistribMulAction α R] [IsScalarTower α R R]
[SMulCommClass α R R] (c : α) (A : Matrix n n R) : det (c • A) = c ^ Fintype.card n • det A | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A : Matrix n n R
⊢ det (-A) = (-1) ^ Fintype.card n * det A | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rw [← det_smul, neg_one_smul] | theorem det_neg (A : Matrix n n R) : det (-A) = (-1) ^ Fintype.card n * det A := by
| Mathlib.LinearAlgebra.Matrix.Determinant.289_0.U1f6HO8zRbnvZ95 | theorem det_neg (A : Matrix n n R) : det (-A) = (-1) ^ Fintype.card n * det A | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A : Matrix n n R
⊢ det (-A) = (-1) ^ Fintype.card n • det A | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rw [← det_smul_of_tower, Units.neg_smul, one_smul] | /-- A variant of `Matrix.det_neg` with scalar multiplication by `Units ℤ` instead of multiplication
by `R`. -/
theorem det_neg_eq_smul (A : Matrix n n R) : det (-A) = (-1 : Units ℤ) ^ Fintype.card n • det A :=
by | Mathlib.LinearAlgebra.Matrix.Determinant.293_0.U1f6HO8zRbnvZ95 | /-- A variant of `Matrix.det_neg` with scalar multiplication by `Units ℤ` instead of multiplication
by `R`. -/
theorem det_neg_eq_smul (A : Matrix n n R) : det (-A) = (-1 : Units ℤ) ^ Fintype.card n • det A | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
v : n → R
A : Matrix n n R
⊢ (of fun i j => v j * A i j) = A * diagonal v | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | ext | /-- Multiplying each row by a fixed `v i` multiplies the determinant by
the product of the `v`s. -/
theorem det_mul_row (v : n → R) (A : Matrix n n R) :
det (of fun i j => v j * A i j) = (∏ i, v i) * det A :=
calc
det (of fun i j => v j * A i j) = det (A * diagonal v) :=
congr_arg det <| by
| Mathlib.LinearAlgebra.Matrix.Determinant.299_0.U1f6HO8zRbnvZ95 | /-- Multiplying each row by a fixed `v i` multiplies the determinant by
the product of the `v`s. -/
theorem det_mul_row (v : n → R) (A : Matrix n n R) :
det (of fun i j => v j * A i j) = (∏ i, v i) * det A | Mathlib_LinearAlgebra_Matrix_Determinant |
case a.h
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
v : n → R
A : Matrix n n R
i✝ x✝ : n
⊢ of (fun i j => v j * A i j) i✝ x✝ = (A * diagonal v) i✝ x✝ | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | simp [mul_comm] | /-- Multiplying each row by a fixed `v i` multiplies the determinant by
the product of the `v`s. -/
theorem det_mul_row (v : n → R) (A : Matrix n n R) :
det (of fun i j => v j * A i j) = (∏ i, v i) * det A :=
calc
det (of fun i j => v j * A i j) = det (A * diagonal v) :=
congr_arg det <| by
ext
... | Mathlib.LinearAlgebra.Matrix.Determinant.299_0.U1f6HO8zRbnvZ95 | /-- Multiplying each row by a fixed `v i` multiplies the determinant by
the product of the `v`s. -/
theorem det_mul_row (v : n → R) (A : Matrix n n R) :
det (of fun i j => v j * A i j) = (∏ i, v i) * det A | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
v : n → R
A : Matrix n n R
⊢ det (A * diagonal v) = (∏ i : n, v i) * det A | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rw [det_mul, det_diagonal, mul_comm] | /-- Multiplying each row by a fixed `v i` multiplies the determinant by
the product of the `v`s. -/
theorem det_mul_row (v : n → R) (A : Matrix n n R) :
det (of fun i j => v j * A i j) = (∏ i, v i) * det A :=
calc
det (of fun i j => v j * A i j) = det (A * diagonal v) :=
congr_arg det <| by
ext
... | Mathlib.LinearAlgebra.Matrix.Determinant.299_0.U1f6HO8zRbnvZ95 | /-- Multiplying each row by a fixed `v i` multiplies the determinant by
the product of the `v`s. -/
theorem det_mul_row (v : n → R) (A : Matrix n n R) :
det (of fun i j => v j * A i j) = (∏ i, v i) * det A | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁵ : DecidableEq n
inst✝⁴ : Fintype n
inst✝³ : DecidableEq m
inst✝² : Fintype m
R : Type v
inst✝¹ : CommRing R
S : Type w
inst✝ : CommRing S
f : R →+* S
M : Matrix n n R
⊢ f (det M) = det ((RingHom.mapMatrix f) M) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | simp [Matrix.det_apply', f.map_sum, f.map_prod] | theorem _root_.RingHom.map_det (f : R →+* S) (M : Matrix n n R) :
f M.det = Matrix.det (f.mapMatrix M) :=
by | Mathlib.LinearAlgebra.Matrix.Determinant.327_0.U1f6HO8zRbnvZ95 | theorem _root_.RingHom.map_det (f : R →+* S) (M : Matrix n n R) :
f M.det = Matrix.det (f.mapMatrix M) | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A : Matrix n n R
j : n
h : ∀ (i : n), A i j = 0
⊢ det A = 0 | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rw [← det_transpose] | theorem det_eq_zero_of_column_eq_zero {A : Matrix n n R} (j : n) (h : ∀ i, A i j = 0) :
det A = 0 := by
| Mathlib.LinearAlgebra.Matrix.Determinant.367_0.U1f6HO8zRbnvZ95 | theorem det_eq_zero_of_column_eq_zero {A : Matrix n n R} (j : n) (h : ∀ i, A i j = 0) :
det A = 0 | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A : Matrix n n R
j : n
h : ∀ (i : n), A i j = 0
⊢ det Aᵀ = 0 | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | exact det_eq_zero_of_row_eq_zero j h | theorem det_eq_zero_of_column_eq_zero {A : Matrix n n R} (j : n) (h : ∀ i, A i j = 0) :
det A = 0 := by
rw [← det_transpose]
| Mathlib.LinearAlgebra.Matrix.Determinant.367_0.U1f6HO8zRbnvZ95 | theorem det_eq_zero_of_column_eq_zero {A : Matrix n n R} (j : n) (h : ∀ i, A i j = 0) :
det A = 0 | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
M : Matrix n n R
i j : n
i_ne_j : i ≠ j
hij : ∀ (k : n), M k i = M k j
⊢ det M = 0 | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rw [← det_transpose, det_zero_of_row_eq i_ne_j] | /-- If a matrix has a repeated column, the determinant will be zero. -/
theorem det_zero_of_column_eq (i_ne_j : i ≠ j) (hij : ∀ k, M k i = M k j) : M.det = 0 := by
| Mathlib.LinearAlgebra.Matrix.Determinant.380_0.U1f6HO8zRbnvZ95 | /-- If a matrix has a repeated column, the determinant will be zero. -/
theorem det_zero_of_column_eq (i_ne_j : i ≠ j) (hij : ∀ k, M k i = M k j) : M.det = 0 | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
M : Matrix n n R
i j : n
i_ne_j : i ≠ j
hij : ∀ (k : n), M k i = M k j
⊢ Mᵀ i = Mᵀ j | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | exact funext hij | /-- If a matrix has a repeated column, the determinant will be zero. -/
theorem det_zero_of_column_eq (i_ne_j : i ≠ j) (hij : ∀ k, M k i = M k j) : M.det = 0 := by
rw [← det_transpose, det_zero_of_row_eq i_ne_j]
| Mathlib.LinearAlgebra.Matrix.Determinant.380_0.U1f6HO8zRbnvZ95 | /-- If a matrix has a repeated column, the determinant will be zero. -/
theorem det_zero_of_column_eq (i_ne_j : i ≠ j) (hij : ∀ k, M k i = M k j) : M.det = 0 | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
M : Matrix n n R
j : n
u v : n → R
⊢ det (updateColumn M j (u + v)) = det (updateColumn M j u) + det (updateColumn M j v) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rw [← det_transpose, ← updateRow_transpose, det_updateRow_add] | theorem det_updateColumn_add (M : Matrix n n R) (j : n) (u v : n → R) :
det (updateColumn M j <| u + v) = det (updateColumn M j u) + det (updateColumn M j v) := by
| Mathlib.LinearAlgebra.Matrix.Determinant.393_0.U1f6HO8zRbnvZ95 | theorem det_updateColumn_add (M : Matrix n n R) (j : n) (u v : n → R) :
det (updateColumn M j <| u + v) = det (updateColumn M j u) + det (updateColumn M j v) | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
M : Matrix n n R
j : n
u v : n → R
⊢ det (updateRow Mᵀ j u) + det (updateRow Mᵀ j v) = det (updateColumn M j u) + det (updateColumn M j v) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | simp [updateRow_transpose, det_transpose] | theorem det_updateColumn_add (M : Matrix n n R) (j : n) (u v : n → R) :
det (updateColumn M j <| u + v) = det (updateColumn M j u) + det (updateColumn M j v) := by
rw [← det_transpose, ← updateRow_transpose, det_updateRow_add]
| Mathlib.LinearAlgebra.Matrix.Determinant.393_0.U1f6HO8zRbnvZ95 | theorem det_updateColumn_add (M : Matrix n n R) (j : n) (u v : n → R) :
det (updateColumn M j <| u + v) = det (updateColumn M j u) + det (updateColumn M j v) | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
M : Matrix n n R
j : n
s : R
u : n → R
⊢ det (updateColumn M j (s • u)) = s * det (updateColumn M j u) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rw [← det_transpose, ← updateRow_transpose, det_updateRow_smul] | theorem det_updateColumn_smul (M : Matrix n n R) (j : n) (s : R) (u : n → R) :
det (updateColumn M j <| s • u) = s * det (updateColumn M j u) := by
| Mathlib.LinearAlgebra.Matrix.Determinant.404_0.U1f6HO8zRbnvZ95 | theorem det_updateColumn_smul (M : Matrix n n R) (j : n) (s : R) (u : n → R) :
det (updateColumn M j <| s • u) = s * det (updateColumn M j u) | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
M : Matrix n n R
j : n
s : R
u : n → R
⊢ s * det (updateRow Mᵀ j u) = s * det (updateColumn M j u) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | simp [updateRow_transpose, det_transpose] | theorem det_updateColumn_smul (M : Matrix n n R) (j : n) (s : R) (u : n → R) :
det (updateColumn M j <| s • u) = s * det (updateColumn M j u) := by
rw [← det_transpose, ← updateRow_transpose, det_updateRow_smul]
| Mathlib.LinearAlgebra.Matrix.Determinant.404_0.U1f6HO8zRbnvZ95 | theorem det_updateColumn_smul (M : Matrix n n R) (j : n) (s : R) (u : n → R) :
det (updateColumn M j <| s • u) = s * det (updateColumn M j u) | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
M : Matrix n n R
j : n
s : R
u : n → R
⊢ det (updateColumn (s • M) j u) = s ^ (Fintype.card n - 1) * det (updateColumn M j u) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rw [← det_transpose, ← updateRow_transpose, transpose_smul, det_updateRow_smul'] | theorem det_updateColumn_smul' (M : Matrix n n R) (j : n) (s : R) (u : n → R) :
det (updateColumn (s • M) j u) = s ^ (Fintype.card n - 1) * det (updateColumn M j u) := by
| Mathlib.LinearAlgebra.Matrix.Determinant.415_0.U1f6HO8zRbnvZ95 | theorem det_updateColumn_smul' (M : Matrix n n R) (j : n) (s : R) (u : n → R) :
det (updateColumn (s • M) j u) = s ^ (Fintype.card n - 1) * det (updateColumn M j u) | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
M : Matrix n n R
j : n
s : R
u : n → R
⊢ s ^ (Fintype.card n - 1) * det (updateRow Mᵀ j u) = s ^ (Fintype.card n - 1) * det (updateColumn M j u) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | simp [updateRow_transpose, det_transpose] | theorem det_updateColumn_smul' (M : Matrix n n R) (j : n) (s : R) (u : n → R) :
det (updateColumn (s • M) j u) = s ^ (Fintype.card n - 1) * det (updateColumn M j u) := by
rw [← det_transpose, ← updateRow_transpose, transpose_smul, det_updateRow_smul']
| Mathlib.LinearAlgebra.Matrix.Determinant.415_0.U1f6HO8zRbnvZ95 | theorem det_updateColumn_smul' (M : Matrix n n R) (j : n) (s : R) (u : n → R) :
det (updateColumn (s • M) j u) = s ^ (Fintype.card n - 1) * det (updateColumn M j u) | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A B C : Matrix n n R
hC : det C = 1
hA : A = B * C
⊢ det B * det C = det B | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rw [hC, mul_one] | theorem det_eq_of_eq_mul_det_one {A B : Matrix n n R} (C : Matrix n n R) (hC : det C = 1)
(hA : A = B * C) : det A = det B :=
calc
det A = det (B * C) := congr_arg _ hA
_ = det B * det C := (det_mul _ _)
_ = det B := by | Mathlib.LinearAlgebra.Matrix.Determinant.429_0.U1f6HO8zRbnvZ95 | theorem det_eq_of_eq_mul_det_one {A B : Matrix n n R} (C : Matrix n n R) (hC : det C = 1)
(hA : A = B * C) : det A = det B | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A B C : Matrix n n R
hC : det C = 1
hA : A = C * B
⊢ det C * det B = det B | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rw [hC, one_mul] | theorem det_eq_of_eq_det_one_mul {A B : Matrix n n R} (C : Matrix n n R) (hC : det C = 1)
(hA : A = C * B) : det A = det B :=
calc
det A = det (C * B) := congr_arg _ hA
_ = det C * det B := (det_mul _ _)
_ = det B := by | Mathlib.LinearAlgebra.Matrix.Determinant.437_0.U1f6HO8zRbnvZ95 | theorem det_eq_of_eq_det_one_mul {A B : Matrix n n R} (C : Matrix n n R) (hC : det C = 1)
(hA : A = C * B) : det A = det B | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A : Matrix n n R
i j : n
hij : i ≠ j
⊢ det (updateRow A i (A i + A j)) = det A | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | simp [det_updateRow_add,
det_zero_of_row_eq hij (updateRow_self.trans (updateRow_ne hij.symm).symm)] | theorem det_updateRow_add_self (A : Matrix n n R) {i j : n} (hij : i ≠ j) :
det (updateRow A i (A i + A j)) = det A := by
| Mathlib.LinearAlgebra.Matrix.Determinant.445_0.U1f6HO8zRbnvZ95 | theorem det_updateRow_add_self (A : Matrix n n R) {i j : n} (hij : i ≠ j) :
det (updateRow A i (A i + A j)) = det A | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A : Matrix n n R
i j : n
hij : i ≠ j
⊢ det (updateColumn A i fun k => A k i + A k j) = det A | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rw [← det_transpose, ← updateRow_transpose, ← det_transpose A] | theorem det_updateColumn_add_self (A : Matrix n n R) {i j : n} (hij : i ≠ j) :
det (updateColumn A i fun k => A k i + A k j) = det A := by
| Mathlib.LinearAlgebra.Matrix.Determinant.451_0.U1f6HO8zRbnvZ95 | theorem det_updateColumn_add_self (A : Matrix n n R) {i j : n} (hij : i ≠ j) :
det (updateColumn A i fun k => A k i + A k j) = det A | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A : Matrix n n R
i j : n
hij : i ≠ j
⊢ det (updateRow Aᵀ i fun k => A k i + A k j) = det Aᵀ | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | exact det_updateRow_add_self Aᵀ hij | theorem det_updateColumn_add_self (A : Matrix n n R) {i j : n} (hij : i ≠ j) :
det (updateColumn A i fun k => A k i + A k j) = det A := by
rw [← det_transpose, ← updateRow_transpose, ← det_transpose A]
| Mathlib.LinearAlgebra.Matrix.Determinant.451_0.U1f6HO8zRbnvZ95 | theorem det_updateColumn_add_self (A : Matrix n n R) {i j : n} (hij : i ≠ j) :
det (updateColumn A i fun k => A k i + A k j) = det A | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A : Matrix n n R
i j : n
hij : i ≠ j
c : R
⊢ det (updateRow A i (A i + c • A j)) = det A | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | simp [det_updateRow_add, det_updateRow_smul,
det_zero_of_row_eq hij (updateRow_self.trans (updateRow_ne hij.symm).symm)] | theorem det_updateRow_add_smul_self (A : Matrix n n R) {i j : n} (hij : i ≠ j) (c : R) :
det (updateRow A i (A i + c • A j)) = det A := by
| Mathlib.LinearAlgebra.Matrix.Determinant.457_0.U1f6HO8zRbnvZ95 | theorem det_updateRow_add_smul_self (A : Matrix n n R) {i j : n} (hij : i ≠ j) (c : R) :
det (updateRow A i (A i + c • A j)) = det A | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A : Matrix n n R
i j : n
hij : i ≠ j
c : R
⊢ det (updateColumn A i fun k => A k i + c • A k j) = det A | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rw [← det_transpose, ← updateRow_transpose, ← det_transpose A] | theorem det_updateColumn_add_smul_self (A : Matrix n n R) {i j : n} (hij : i ≠ j) (c : R) :
det (updateColumn A i fun k => A k i + c • A k j) = det A := by
| Mathlib.LinearAlgebra.Matrix.Determinant.463_0.U1f6HO8zRbnvZ95 | theorem det_updateColumn_add_smul_self (A : Matrix n n R) {i j : n} (hij : i ≠ j) (c : R) :
det (updateColumn A i fun k => A k i + c • A k j) = det A | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A : Matrix n n R
i j : n
hij : i ≠ j
c : R
⊢ det (updateRow Aᵀ i fun k => A k i + c • A k j) = det Aᵀ | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | exact det_updateRow_add_smul_self Aᵀ hij c | theorem det_updateColumn_add_smul_self (A : Matrix n n R) {i j : n} (hij : i ≠ j) (c : R) :
det (updateColumn A i fun k => A k i + c • A k j) = det A := by
rw [← det_transpose, ← updateRow_transpose, ← det_transpose A]
| Mathlib.LinearAlgebra.Matrix.Determinant.463_0.U1f6HO8zRbnvZ95 | theorem det_updateColumn_add_smul_self (A : Matrix n n R) {i j : n} (hij : i ≠ j) (c : R) :
det (updateColumn A i fun k => A k i + c • A k j) = det A | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A B : Matrix n n R
s : Finset n
⊢ ∀ (c : n → R), (∀ i ∉ s, c i = 0) → ∀ k ∉ s, (∀ (i j : n), A i j = B i j + c i * B k j) → det A = det B | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | induction s using Finset.induction_on generalizing B with
| empty =>
rintro c hs k - A_eq
have : ∀ i, c i = 0 := by
intro i
specialize hs i
contrapose! hs
simp [hs]
congr
ext i j
rw [A_eq, this, zero_mul, add_zero]
| @insert i s _hi ih =>
intro c hs k hk A_eq
have... | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B := by
| Mathlib.LinearAlgebra.Matrix.Determinant.469_0.U1f6HO8zRbnvZ95 | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A B : Matrix n n R
s : Finset n
⊢ ∀ (c : n → R), (∀ i ∉ s, c i = 0) → ∀ k ∉ s, (∀ (i j : n), A i j = B i j + c i * B k j) → det A = det B | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | induction s using Finset.induction_on generalizing B with
| empty =>
rintro c hs k - A_eq
have : ∀ i, c i = 0 := by
intro i
specialize hs i
contrapose! hs
simp [hs]
congr
ext i j
rw [A_eq, this, zero_mul, add_zero]
| @insert i s _hi ih =>
intro c hs k hk A_eq
have... | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B := by
| Mathlib.LinearAlgebra.Matrix.Determinant.469_0.U1f6HO8zRbnvZ95 | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B | Mathlib_LinearAlgebra_Matrix_Determinant |
case empty
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A B : Matrix n n R
⊢ ∀ (c : n → R), (∀ i ∉ ∅, c i = 0) → ∀ k ∉ ∅, (∀ (i j : n), A i j = B i j + c i * B k j) → det A = det B | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | | empty =>
rintro c hs k - A_eq
have : ∀ i, c i = 0 := by
intro i
specialize hs i
contrapose! hs
simp [hs]
congr
ext i j
rw [A_eq, this, zero_mul, add_zero] | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B := by
induction s using Finset.induction_on generalizing B with
| Mathlib.LinearAlgebra.Matrix.Determinant.469_0.U1f6HO8zRbnvZ95 | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B | Mathlib_LinearAlgebra_Matrix_Determinant |
case empty
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A B : Matrix n n R
⊢ ∀ (c : n → R), (∀ i ∉ ∅, c i = 0) → ∀ k ∉ ∅, (∀ (i j : n), A i j = B i j + c i * B k j) → det A = det B | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rintro c hs k - A_eq | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B := by
induction s using Finset.induction_on generalizing B with
| empty =>
| Mathlib.LinearAlgebra.Matrix.Determinant.469_0.U1f6HO8zRbnvZ95 | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B | Mathlib_LinearAlgebra_Matrix_Determinant |
case empty
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A B : Matrix n n R
c : n → R
hs : ∀ i ∉ ∅, c i = 0
k : n
A_eq : ∀ (i j : n), A i j = B i j + c i * B k j
⊢ det A = det B | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | have : ∀ i, c i = 0 := by
intro i
specialize hs i
contrapose! hs
simp [hs] | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B := by
induction s using Finset.induction_on generalizing B with
| empty =>
rintro c hs k - A_eq
... | Mathlib.LinearAlgebra.Matrix.Determinant.469_0.U1f6HO8zRbnvZ95 | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A B : Matrix n n R
c : n → R
hs : ∀ i ∉ ∅, c i = 0
k : n
A_eq : ∀ (i j : n), A i j = B i j + c i * B k j
⊢ ∀ (i : n), c i = 0 | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | intro i | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B := by
induction s using Finset.induction_on generalizing B with
| empty =>
rintro c hs k - A_eq
... | Mathlib.LinearAlgebra.Matrix.Determinant.469_0.U1f6HO8zRbnvZ95 | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A B : Matrix n n R
c : n → R
hs : ∀ i ∉ ∅, c i = 0
k : n
A_eq : ∀ (i j : n), A i j = B i j + c i * B k j
i : n
⊢ c i = 0 | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | specialize hs i | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B := by
induction s using Finset.induction_on generalizing B with
| empty =>
rintro c hs k - A_eq
... | Mathlib.LinearAlgebra.Matrix.Determinant.469_0.U1f6HO8zRbnvZ95 | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A B : Matrix n n R
c : n → R
k : n
A_eq : ∀ (i j : n), A i j = B i j + c i * B k j
i : n
hs : i ∉ ∅ → c i = 0
⊢ c i = 0 | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | contrapose! hs | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B := by
induction s using Finset.induction_on generalizing B with
| empty =>
rintro c hs k - A_eq
... | Mathlib.LinearAlgebra.Matrix.Determinant.469_0.U1f6HO8zRbnvZ95 | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A B : Matrix n n R
c : n → R
k : n
A_eq : ∀ (i j : n), A i j = B i j + c i * B k j
i : n
hs : c i ≠ 0
⊢ i ∉ ∅ ∧ c i ≠ 0 | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | simp [hs] | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B := by
induction s using Finset.induction_on generalizing B with
| empty =>
rintro c hs k - A_eq
... | Mathlib.LinearAlgebra.Matrix.Determinant.469_0.U1f6HO8zRbnvZ95 | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B | Mathlib_LinearAlgebra_Matrix_Determinant |
case empty
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A B : Matrix n n R
c : n → R
hs : ∀ i ∉ ∅, c i = 0
k : n
A_eq : ∀ (i j : n), A i j = B i j + c i * B k j
this : ∀ (i : n), c i = 0
⊢ det A = det B | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | congr | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B := by
induction s using Finset.induction_on generalizing B with
| empty =>
rintro c hs k - A_eq
... | Mathlib.LinearAlgebra.Matrix.Determinant.469_0.U1f6HO8zRbnvZ95 | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B | Mathlib_LinearAlgebra_Matrix_Determinant |
case empty.e_M
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A B : Matrix n n R
c : n → R
hs : ∀ i ∉ ∅, c i = 0
k : n
A_eq : ∀ (i j : n), A i j = B i j + c i * B k j
this : ∀ (i : n), c i = 0
⊢ A = B | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | ext i j | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B := by
induction s using Finset.induction_on generalizing B with
| empty =>
rintro c hs k - A_eq
... | Mathlib.LinearAlgebra.Matrix.Determinant.469_0.U1f6HO8zRbnvZ95 | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B | Mathlib_LinearAlgebra_Matrix_Determinant |
case empty.e_M.a.h
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A B : Matrix n n R
c : n → R
hs : ∀ i ∉ ∅, c i = 0
k : n
A_eq : ∀ (i j : n), A i j = B i j + c i * B k j
this : ∀ (i : n), c i = 0
i j : n
⊢ A i j = B i j | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rw [A_eq, this, zero_mul, add_zero] | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B := by
induction s using Finset.induction_on generalizing B with
| empty =>
rintro c hs k - A_eq
... | Mathlib.LinearAlgebra.Matrix.Determinant.469_0.U1f6HO8zRbnvZ95 | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B | Mathlib_LinearAlgebra_Matrix_Determinant |
case insert
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A : Matrix n n R
i : n
s : Finset n
_hi : i ∉ s
ih :
∀ {B : Matrix n n R} (c : n → R),
(∀ i ∉ s, c i = 0) → ∀ k ∉ s, (∀ (i j : n), A i j = B i j + c i * B k j) → ... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | | @insert i s _hi ih =>
intro c hs k hk A_eq
have hAi : A i = B i + c i • B k := funext (A_eq i)
rw [@ih (updateRow B i (A i)) (Function.update c i 0), hAi, det_updateRow_add_smul_self]
· exact mt (fun h => show k ∈ insert i s from h ▸ Finset.mem_insert_self _ _) hk
· intro i' hi'
rw [Function... | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B := by
induction s using Finset.induction_on generalizing B with
| empty =>
rintro c hs k - A_eq
... | Mathlib.LinearAlgebra.Matrix.Determinant.469_0.U1f6HO8zRbnvZ95 | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B | Mathlib_LinearAlgebra_Matrix_Determinant |
case insert
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A : Matrix n n R
i : n
s : Finset n
_hi : i ∉ s
ih :
∀ {B : Matrix n n R} (c : n → R),
(∀ i ∉ s, c i = 0) → ∀ k ∉ s, (∀ (i j : n), A i j = B i j + c i * B k j) → ... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | intro c hs k hk A_eq | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B := by
induction s using Finset.induction_on generalizing B with
| empty =>
rintro c hs k - A_eq
... | Mathlib.LinearAlgebra.Matrix.Determinant.469_0.U1f6HO8zRbnvZ95 | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B | Mathlib_LinearAlgebra_Matrix_Determinant |
case insert
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A : Matrix n n R
i : n
s : Finset n
_hi : i ∉ s
ih :
∀ {B : Matrix n n R} (c : n → R),
(∀ i ∉ s, c i = 0) → ∀ k ∉ s, (∀ (i j : n), A i j = B i j + c i * B k j) → ... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | have hAi : A i = B i + c i • B k := funext (A_eq i) | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B := by
induction s using Finset.induction_on generalizing B with
| empty =>
rintro c hs k - A_eq
... | Mathlib.LinearAlgebra.Matrix.Determinant.469_0.U1f6HO8zRbnvZ95 | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B | Mathlib_LinearAlgebra_Matrix_Determinant |
case insert
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A : Matrix n n R
i : n
s : Finset n
_hi : i ∉ s
ih :
∀ {B : Matrix n n R} (c : n → R),
(∀ i ∉ s, c i = 0) → ∀ k ∉ s, (∀ (i j : n), A i j = B i j + c i * B k j) → ... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rw [@ih (updateRow B i (A i)) (Function.update c i 0), hAi, det_updateRow_add_smul_self] | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B := by
induction s using Finset.induction_on generalizing B with
| empty =>
rintro c hs k - A_eq
... | Mathlib.LinearAlgebra.Matrix.Determinant.469_0.U1f6HO8zRbnvZ95 | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B | Mathlib_LinearAlgebra_Matrix_Determinant |
case insert.hij
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A : Matrix n n R
i : n
s : Finset n
_hi : i ∉ s
ih :
∀ {B : Matrix n n R} (c : n → R),
(∀ i ∉ s, c i = 0) → ∀ k ∉ s, (∀ (i j : n), A i j = B i j + c i * B k j... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | exact mt (fun h => show k ∈ insert i s from h ▸ Finset.mem_insert_self _ _) hk | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B := by
induction s using Finset.induction_on generalizing B with
| empty =>
rintro c hs k - A_eq
... | Mathlib.LinearAlgebra.Matrix.Determinant.469_0.U1f6HO8zRbnvZ95 | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B | Mathlib_LinearAlgebra_Matrix_Determinant |
case insert.x
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A : Matrix n n R
i : n
s : Finset n
_hi : i ∉ s
ih :
∀ {B : Matrix n n R} (c : n → R),
(∀ i ∉ s, c i = 0) → ∀ k ∉ s, (∀ (i j : n), A i j = B i j + c i * B k j) ... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | intro i' hi' | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B := by
induction s using Finset.induction_on generalizing B with
| empty =>
rintro c hs k - A_eq
... | Mathlib.LinearAlgebra.Matrix.Determinant.469_0.U1f6HO8zRbnvZ95 | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B | Mathlib_LinearAlgebra_Matrix_Determinant |
case insert.x
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A : Matrix n n R
i : n
s : Finset n
_hi : i ∉ s
ih :
∀ {B : Matrix n n R} (c : n → R),
(∀ i ∉ s, c i = 0) → ∀ k ∉ s, (∀ (i j : n), A i j = B i j + c i * B k j) ... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rw [Function.update_apply] | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B := by
induction s using Finset.induction_on generalizing B with
| empty =>
rintro c hs k - A_eq
... | Mathlib.LinearAlgebra.Matrix.Determinant.469_0.U1f6HO8zRbnvZ95 | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B | Mathlib_LinearAlgebra_Matrix_Determinant |
case insert.x
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A : Matrix n n R
i : n
s : Finset n
_hi : i ∉ s
ih :
∀ {B : Matrix n n R} (c : n → R),
(∀ i ∉ s, c i = 0) → ∀ k ∉ s, (∀ (i j : n), A i j = B i j + c i * B k j) ... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | split_ifs with hi'i | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B := by
induction s using Finset.induction_on generalizing B with
| empty =>
rintro c hs k - A_eq
... | Mathlib.LinearAlgebra.Matrix.Determinant.469_0.U1f6HO8zRbnvZ95 | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B | Mathlib_LinearAlgebra_Matrix_Determinant |
case pos
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A : Matrix n n R
i : n
s : Finset n
_hi : i ∉ s
ih :
∀ {B : Matrix n n R} (c : n → R),
(∀ i ∉ s, c i = 0) → ∀ k ∉ s, (∀ (i j : n), A i j = B i j + c i * B k j) → det... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rfl | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B := by
induction s using Finset.induction_on generalizing B with
| empty =>
rintro c hs k - A_eq
... | Mathlib.LinearAlgebra.Matrix.Determinant.469_0.U1f6HO8zRbnvZ95 | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B | Mathlib_LinearAlgebra_Matrix_Determinant |
case neg
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A : Matrix n n R
i : n
s : Finset n
_hi : i ∉ s
ih :
∀ {B : Matrix n n R} (c : n → R),
(∀ i ∉ s, c i = 0) → ∀ k ∉ s, (∀ (i j : n), A i j = B i j + c i * B k j) → det... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | exact hs i' fun h => hi' ((Finset.mem_insert.mp h).resolve_left hi'i) | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B := by
induction s using Finset.induction_on generalizing B with
| empty =>
rintro c hs k - A_eq
... | Mathlib.LinearAlgebra.Matrix.Determinant.469_0.U1f6HO8zRbnvZ95 | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B | Mathlib_LinearAlgebra_Matrix_Determinant |
case insert.k
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A : Matrix n n R
i : n
s : Finset n
_hi : i ∉ s
ih :
∀ {B : Matrix n n R} (c : n → R),
(∀ i ∉ s, c i = 0) → ∀ k ∉ s, (∀ (i j : n), A i j = B i j + c i * B k j) ... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | exact k | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B := by
induction s using Finset.induction_on generalizing B with
| empty =>
rintro c hs k - A_eq
... | Mathlib.LinearAlgebra.Matrix.Determinant.469_0.U1f6HO8zRbnvZ95 | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B | Mathlib_LinearAlgebra_Matrix_Determinant |
case insert.x
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A : Matrix n n R
i : n
s : Finset n
_hi : i ∉ s
ih :
∀ {B : Matrix n n R} (c : n → R),
(∀ i ∉ s, c i = 0) → ∀ k ∉ s, (∀ (i j : n), A i j = B i j + c i * B k j) ... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | exact fun h => hk (Finset.mem_insert_of_mem h) | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B := by
induction s using Finset.induction_on generalizing B with
| empty =>
rintro c hs k - A_eq
... | Mathlib.LinearAlgebra.Matrix.Determinant.469_0.U1f6HO8zRbnvZ95 | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B | Mathlib_LinearAlgebra_Matrix_Determinant |
case insert.x
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A : Matrix n n R
i : n
s : Finset n
_hi : i ∉ s
ih :
∀ {B : Matrix n n R} (c : n → R),
(∀ i ∉ s, c i = 0) → ∀ k ∉ s, (∀ (i j : n), A i j = B i j + c i * B k j) ... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | intro i' j' | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B := by
induction s using Finset.induction_on generalizing B with
| empty =>
rintro c hs k - A_eq
... | Mathlib.LinearAlgebra.Matrix.Determinant.469_0.U1f6HO8zRbnvZ95 | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B | Mathlib_LinearAlgebra_Matrix_Determinant |
case insert.x
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A : Matrix n n R
i : n
s : Finset n
_hi : i ∉ s
ih :
∀ {B : Matrix n n R} (c : n → R),
(∀ i ∉ s, c i = 0) → ∀ k ∉ s, (∀ (i j : n), A i j = B i j + c i * B k j) ... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rw [updateRow_apply, Function.update_apply] | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B := by
induction s using Finset.induction_on generalizing B with
| empty =>
rintro c hs k - A_eq
... | Mathlib.LinearAlgebra.Matrix.Determinant.469_0.U1f6HO8zRbnvZ95 | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B | Mathlib_LinearAlgebra_Matrix_Determinant |
case insert.x
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A : Matrix n n R
i : n
s : Finset n
_hi : i ∉ s
ih :
∀ {B : Matrix n n R} (c : n → R),
(∀ i ∉ s, c i = 0) → ∀ k ∉ s, (∀ (i j : n), A i j = B i j + c i * B k j) ... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | split_ifs with hi'i | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B := by
induction s using Finset.induction_on generalizing B with
| empty =>
rintro c hs k - A_eq
... | Mathlib.LinearAlgebra.Matrix.Determinant.469_0.U1f6HO8zRbnvZ95 | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B | Mathlib_LinearAlgebra_Matrix_Determinant |
case pos
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A : Matrix n n R
i : n
s : Finset n
_hi : i ∉ s
ih :
∀ {B : Matrix n n R} (c : n → R),
(∀ i ∉ s, c i = 0) → ∀ k ∉ s, (∀ (i j : n), A i j = B i j + c i * B k j) → det... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | simp [hi'i] | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B := by
induction s using Finset.induction_on generalizing B with
| empty =>
rintro c hs k - A_eq
... | Mathlib.LinearAlgebra.Matrix.Determinant.469_0.U1f6HO8zRbnvZ95 | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B | Mathlib_LinearAlgebra_Matrix_Determinant |
case neg
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A : Matrix n n R
i : n
s : Finset n
_hi : i ∉ s
ih :
∀ {B : Matrix n n R} (c : n → R),
(∀ i ∉ s, c i = 0) → ∀ k ∉ s, (∀ (i j : n), A i j = B i j + c i * B k j) → det... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rw [A_eq, updateRow_ne fun h : k = i => hk <| h ▸ Finset.mem_insert_self k s] | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B := by
induction s using Finset.induction_on generalizing B with
| empty =>
rintro c hs k - A_eq
... | Mathlib.LinearAlgebra.Matrix.Determinant.469_0.U1f6HO8zRbnvZ95 | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_: ∀ i j, A i j = B i j + c i * B k j), det A = det B | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n✝ : Type u_2
inst✝⁴ : DecidableEq n✝
inst✝³ : Fintype n✝
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
n : ℕ
k : Fin (n + 1)
⊢ ∀ (c : Fin n → R),
(∀ (i : Fin n), k < Fin.succ i → c i = 0) →
∀ {M N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R},
(∀ (j : Fin (N... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | refine' Fin.induction _ (fun k ih => _) k | theorem det_eq_of_forall_row_eq_smul_add_pred_aux {n : ℕ} (k : Fin (n + 1)) :
∀ (c : Fin n → R) (_hc : ∀ i : Fin n, k < i.succ → c i = 0)
{M N : Matrix (Fin n.succ) (Fin n.succ) R} (_h0 : ∀ j, M 0 j = N 0 j)
(_hsucc : ∀ (i : Fin n) (j), M i.succ j = N i.succ j + c i * M (Fin.castSucc i) j),
det M ... | Mathlib.LinearAlgebra.Matrix.Determinant.513_0.U1f6HO8zRbnvZ95 | theorem det_eq_of_forall_row_eq_smul_add_pred_aux {n : ℕ} (k : Fin (n + 1)) :
∀ (c : Fin n → R) (_hc : ∀ i : Fin n, k < i.succ → c i = 0)
{M N : Matrix (Fin n.succ) (Fin n.succ) R} (_h0 : ∀ j, M 0 j = N 0 j)
(_hsucc : ∀ (i : Fin n) (j), M i.succ j = N i.succ j + c i * M (Fin.castSucc i) j),
det M ... | Mathlib_LinearAlgebra_Matrix_Determinant |
case refine'_1
m : Type u_1
n✝ : Type u_2
inst✝⁴ : DecidableEq n✝
inst✝³ : Fintype n✝
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
n : ℕ
k : Fin (n + 1)
⊢ ∀ (c : Fin n → R),
(∀ (i : Fin n), 0 < Fin.succ i → c i = 0) →
∀ {M N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R},
... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | intro c hc M N h0 hsucc | theorem det_eq_of_forall_row_eq_smul_add_pred_aux {n : ℕ} (k : Fin (n + 1)) :
∀ (c : Fin n → R) (_hc : ∀ i : Fin n, k < i.succ → c i = 0)
{M N : Matrix (Fin n.succ) (Fin n.succ) R} (_h0 : ∀ j, M 0 j = N 0 j)
(_hsucc : ∀ (i : Fin n) (j), M i.succ j = N i.succ j + c i * M (Fin.castSucc i) j),
det M ... | Mathlib.LinearAlgebra.Matrix.Determinant.513_0.U1f6HO8zRbnvZ95 | theorem det_eq_of_forall_row_eq_smul_add_pred_aux {n : ℕ} (k : Fin (n + 1)) :
∀ (c : Fin n → R) (_hc : ∀ i : Fin n, k < i.succ → c i = 0)
{M N : Matrix (Fin n.succ) (Fin n.succ) R} (_h0 : ∀ j, M 0 j = N 0 j)
(_hsucc : ∀ (i : Fin n) (j), M i.succ j = N i.succ j + c i * M (Fin.castSucc i) j),
det M ... | Mathlib_LinearAlgebra_Matrix_Determinant |
case refine'_2
m : Type u_1
n✝ : Type u_2
inst✝⁴ : DecidableEq n✝
inst✝³ : Fintype n✝
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
n : ℕ
k✝ : Fin (n + 1)
k : Fin n
ih :
∀ (c : Fin n → R),
(∀ (i : Fin n), Fin.castSucc k < Fin.succ i → c i = 0) →
∀ {M N : Matrix (Fin (Nat.succ n)) (... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | intro c hc M N h0 hsucc | theorem det_eq_of_forall_row_eq_smul_add_pred_aux {n : ℕ} (k : Fin (n + 1)) :
∀ (c : Fin n → R) (_hc : ∀ i : Fin n, k < i.succ → c i = 0)
{M N : Matrix (Fin n.succ) (Fin n.succ) R} (_h0 : ∀ j, M 0 j = N 0 j)
(_hsucc : ∀ (i : Fin n) (j), M i.succ j = N i.succ j + c i * M (Fin.castSucc i) j),
det M ... | Mathlib.LinearAlgebra.Matrix.Determinant.513_0.U1f6HO8zRbnvZ95 | theorem det_eq_of_forall_row_eq_smul_add_pred_aux {n : ℕ} (k : Fin (n + 1)) :
∀ (c : Fin n → R) (_hc : ∀ i : Fin n, k < i.succ → c i = 0)
{M N : Matrix (Fin n.succ) (Fin n.succ) R} (_h0 : ∀ j, M 0 j = N 0 j)
(_hsucc : ∀ (i : Fin n) (j), M i.succ j = N i.succ j + c i * M (Fin.castSucc i) j),
det M ... | Mathlib_LinearAlgebra_Matrix_Determinant |
case refine'_1
m : Type u_1
n✝ : Type u_2
inst✝⁴ : DecidableEq n✝
inst✝³ : Fintype n✝
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
n : ℕ
k : Fin (n + 1)
c : Fin n → R
hc : ∀ (i : Fin n), 0 < Fin.succ i → c i = 0
M N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R
h0 : ∀ (j : Fin (Nat.succ n)... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | congr | theorem det_eq_of_forall_row_eq_smul_add_pred_aux {n : ℕ} (k : Fin (n + 1)) :
∀ (c : Fin n → R) (_hc : ∀ i : Fin n, k < i.succ → c i = 0)
{M N : Matrix (Fin n.succ) (Fin n.succ) R} (_h0 : ∀ j, M 0 j = N 0 j)
(_hsucc : ∀ (i : Fin n) (j), M i.succ j = N i.succ j + c i * M (Fin.castSucc i) j),
det M ... | Mathlib.LinearAlgebra.Matrix.Determinant.513_0.U1f6HO8zRbnvZ95 | theorem det_eq_of_forall_row_eq_smul_add_pred_aux {n : ℕ} (k : Fin (n + 1)) :
∀ (c : Fin n → R) (_hc : ∀ i : Fin n, k < i.succ → c i = 0)
{M N : Matrix (Fin n.succ) (Fin n.succ) R} (_h0 : ∀ j, M 0 j = N 0 j)
(_hsucc : ∀ (i : Fin n) (j), M i.succ j = N i.succ j + c i * M (Fin.castSucc i) j),
det M ... | Mathlib_LinearAlgebra_Matrix_Determinant |
case refine'_1.e_M
m : Type u_1
n✝ : Type u_2
inst✝⁴ : DecidableEq n✝
inst✝³ : Fintype n✝
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
n : ℕ
k : Fin (n + 1)
c : Fin n → R
hc : ∀ (i : Fin n), 0 < Fin.succ i → c i = 0
M N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R
h0 : ∀ (j : Fin (Nat.suc... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | ext i j | theorem det_eq_of_forall_row_eq_smul_add_pred_aux {n : ℕ} (k : Fin (n + 1)) :
∀ (c : Fin n → R) (_hc : ∀ i : Fin n, k < i.succ → c i = 0)
{M N : Matrix (Fin n.succ) (Fin n.succ) R} (_h0 : ∀ j, M 0 j = N 0 j)
(_hsucc : ∀ (i : Fin n) (j), M i.succ j = N i.succ j + c i * M (Fin.castSucc i) j),
det M ... | Mathlib.LinearAlgebra.Matrix.Determinant.513_0.U1f6HO8zRbnvZ95 | theorem det_eq_of_forall_row_eq_smul_add_pred_aux {n : ℕ} (k : Fin (n + 1)) :
∀ (c : Fin n → R) (_hc : ∀ i : Fin n, k < i.succ → c i = 0)
{M N : Matrix (Fin n.succ) (Fin n.succ) R} (_h0 : ∀ j, M 0 j = N 0 j)
(_hsucc : ∀ (i : Fin n) (j), M i.succ j = N i.succ j + c i * M (Fin.castSucc i) j),
det M ... | Mathlib_LinearAlgebra_Matrix_Determinant |
case refine'_1.e_M.a.h
m : Type u_1
n✝ : Type u_2
inst✝⁴ : DecidableEq n✝
inst✝³ : Fintype n✝
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
n : ℕ
k : Fin (n + 1)
c : Fin n → R
hc : ∀ (i : Fin n), 0 < Fin.succ i → c i = 0
M N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R
h0 : ∀ (j : Fin (Nat... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | refine' Fin.cases (h0 j) (fun i => _) i | theorem det_eq_of_forall_row_eq_smul_add_pred_aux {n : ℕ} (k : Fin (n + 1)) :
∀ (c : Fin n → R) (_hc : ∀ i : Fin n, k < i.succ → c i = 0)
{M N : Matrix (Fin n.succ) (Fin n.succ) R} (_h0 : ∀ j, M 0 j = N 0 j)
(_hsucc : ∀ (i : Fin n) (j), M i.succ j = N i.succ j + c i * M (Fin.castSucc i) j),
det M ... | Mathlib.LinearAlgebra.Matrix.Determinant.513_0.U1f6HO8zRbnvZ95 | theorem det_eq_of_forall_row_eq_smul_add_pred_aux {n : ℕ} (k : Fin (n + 1)) :
∀ (c : Fin n → R) (_hc : ∀ i : Fin n, k < i.succ → c i = 0)
{M N : Matrix (Fin n.succ) (Fin n.succ) R} (_h0 : ∀ j, M 0 j = N 0 j)
(_hsucc : ∀ (i : Fin n) (j), M i.succ j = N i.succ j + c i * M (Fin.castSucc i) j),
det M ... | Mathlib_LinearAlgebra_Matrix_Determinant |
case refine'_1.e_M.a.h
m : Type u_1
n✝ : Type u_2
inst✝⁴ : DecidableEq n✝
inst✝³ : Fintype n✝
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
n : ℕ
k : Fin (n + 1)
c : Fin n → R
hc : ∀ (i : Fin n), 0 < Fin.succ i → c i = 0
M N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R
h0 : ∀ (j : Fin (Nat... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rw [hsucc, hc i (Fin.succ_pos _), zero_mul, add_zero] | theorem det_eq_of_forall_row_eq_smul_add_pred_aux {n : ℕ} (k : Fin (n + 1)) :
∀ (c : Fin n → R) (_hc : ∀ i : Fin n, k < i.succ → c i = 0)
{M N : Matrix (Fin n.succ) (Fin n.succ) R} (_h0 : ∀ j, M 0 j = N 0 j)
(_hsucc : ∀ (i : Fin n) (j), M i.succ j = N i.succ j + c i * M (Fin.castSucc i) j),
det M ... | Mathlib.LinearAlgebra.Matrix.Determinant.513_0.U1f6HO8zRbnvZ95 | theorem det_eq_of_forall_row_eq_smul_add_pred_aux {n : ℕ} (k : Fin (n + 1)) :
∀ (c : Fin n → R) (_hc : ∀ i : Fin n, k < i.succ → c i = 0)
{M N : Matrix (Fin n.succ) (Fin n.succ) R} (_h0 : ∀ j, M 0 j = N 0 j)
(_hsucc : ∀ (i : Fin n) (j), M i.succ j = N i.succ j + c i * M (Fin.castSucc i) j),
det M ... | Mathlib_LinearAlgebra_Matrix_Determinant |
case refine'_2
m : Type u_1
n✝ : Type u_2
inst✝⁴ : DecidableEq n✝
inst✝³ : Fintype n✝
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
n : ℕ
k✝ : Fin (n + 1)
k : Fin n
ih :
∀ (c : Fin n → R),
(∀ (i : Fin n), Fin.castSucc k < Fin.succ i → c i = 0) →
∀ {M N : Matrix (Fin (Nat.succ n)) (... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | set M' := updateRow M k.succ (N k.succ) with hM' | theorem det_eq_of_forall_row_eq_smul_add_pred_aux {n : ℕ} (k : Fin (n + 1)) :
∀ (c : Fin n → R) (_hc : ∀ i : Fin n, k < i.succ → c i = 0)
{M N : Matrix (Fin n.succ) (Fin n.succ) R} (_h0 : ∀ j, M 0 j = N 0 j)
(_hsucc : ∀ (i : Fin n) (j), M i.succ j = N i.succ j + c i * M (Fin.castSucc i) j),
det M ... | Mathlib.LinearAlgebra.Matrix.Determinant.513_0.U1f6HO8zRbnvZ95 | theorem det_eq_of_forall_row_eq_smul_add_pred_aux {n : ℕ} (k : Fin (n + 1)) :
∀ (c : Fin n → R) (_hc : ∀ i : Fin n, k < i.succ → c i = 0)
{M N : Matrix (Fin n.succ) (Fin n.succ) R} (_h0 : ∀ j, M 0 j = N 0 j)
(_hsucc : ∀ (i : Fin n) (j), M i.succ j = N i.succ j + c i * M (Fin.castSucc i) j),
det M ... | Mathlib_LinearAlgebra_Matrix_Determinant |
case refine'_2
m : Type u_1
n✝ : Type u_2
inst✝⁴ : DecidableEq n✝
inst✝³ : Fintype n✝
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
n : ℕ
k✝ : Fin (n + 1)
k : Fin n
ih :
∀ (c : Fin n → R),
(∀ (i : Fin n), Fin.castSucc k < Fin.succ i → c i = 0) →
∀ {M N : Matrix (Fin (Nat.succ n)) (... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | have hM : M = updateRow M' k.succ (M' k.succ + c k • M (Fin.castSucc k)) := by
ext i j
by_cases hi : i = k.succ
· simp [hi, hM', hsucc, updateRow_self]
rw [updateRow_ne hi, hM', updateRow_ne hi] | theorem det_eq_of_forall_row_eq_smul_add_pred_aux {n : ℕ} (k : Fin (n + 1)) :
∀ (c : Fin n → R) (_hc : ∀ i : Fin n, k < i.succ → c i = 0)
{M N : Matrix (Fin n.succ) (Fin n.succ) R} (_h0 : ∀ j, M 0 j = N 0 j)
(_hsucc : ∀ (i : Fin n) (j), M i.succ j = N i.succ j + c i * M (Fin.castSucc i) j),
det M ... | Mathlib.LinearAlgebra.Matrix.Determinant.513_0.U1f6HO8zRbnvZ95 | theorem det_eq_of_forall_row_eq_smul_add_pred_aux {n : ℕ} (k : Fin (n + 1)) :
∀ (c : Fin n → R) (_hc : ∀ i : Fin n, k < i.succ → c i = 0)
{M N : Matrix (Fin n.succ) (Fin n.succ) R} (_h0 : ∀ j, M 0 j = N 0 j)
(_hsucc : ∀ (i : Fin n) (j), M i.succ j = N i.succ j + c i * M (Fin.castSucc i) j),
det M ... | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n✝ : Type u_2
inst✝⁴ : DecidableEq n✝
inst✝³ : Fintype n✝
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
n : ℕ
k✝ : Fin (n + 1)
k : Fin n
ih :
∀ (c : Fin n → R),
(∀ (i : Fin n), Fin.castSucc k < Fin.succ i → c i = 0) →
∀ {M N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | ext i j | theorem det_eq_of_forall_row_eq_smul_add_pred_aux {n : ℕ} (k : Fin (n + 1)) :
∀ (c : Fin n → R) (_hc : ∀ i : Fin n, k < i.succ → c i = 0)
{M N : Matrix (Fin n.succ) (Fin n.succ) R} (_h0 : ∀ j, M 0 j = N 0 j)
(_hsucc : ∀ (i : Fin n) (j), M i.succ j = N i.succ j + c i * M (Fin.castSucc i) j),
det M ... | Mathlib.LinearAlgebra.Matrix.Determinant.513_0.U1f6HO8zRbnvZ95 | theorem det_eq_of_forall_row_eq_smul_add_pred_aux {n : ℕ} (k : Fin (n + 1)) :
∀ (c : Fin n → R) (_hc : ∀ i : Fin n, k < i.succ → c i = 0)
{M N : Matrix (Fin n.succ) (Fin n.succ) R} (_h0 : ∀ j, M 0 j = N 0 j)
(_hsucc : ∀ (i : Fin n) (j), M i.succ j = N i.succ j + c i * M (Fin.castSucc i) j),
det M ... | Mathlib_LinearAlgebra_Matrix_Determinant |
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