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 |
|---|---|---|---|---|---|---|
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
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 (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... | by_cases hi : i = k.succ | 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 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
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 (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 [hi, hM', hsucc, updateRow_self] | 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 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
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 (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 [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 |
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 k_ne_succ : (Fin.castSucc k) ≠ k.succ := (Fin.castSucc_lt_succ k).ne | 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 M_k : M (Fin.castSucc k) = M' (Fin.castSucc k) := (updateRow_ne k_ne_succ).symm | 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... | rw [hM, M_k, det_updateRow_add_smul_self M' k_ne_succ.symm, ih (Function.update c k 0)] | 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._hc
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 i 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 |
case refine'_2._hc
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... | rw [Fin.lt_iff_val_lt_val, Fin.coe_castSucc, Fin.val_succ, Nat.lt_succ_iff] at 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 |
case refine'_2._hc
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... | rw [Function.update_apply] | 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._hc
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... | split_ifs with hik | 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 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
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 (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... | rfl | 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 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
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 (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... | exact hc _ (Fin.succ_lt_succ_iff.mpr (lt_of_le_of_ne hi (Ne.symm hik))) | 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._h0
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... | rwa [hM', updateRow_ne (Fin.succ_ne_zero _).symm] | 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._hsucc
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.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... | intro 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'_2._hsucc
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.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... | rw [Function.update_apply] | 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._hsucc
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.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... | split_ifs with hik | 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 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
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 (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 [zero_mul, add_zero, hM', hik, updateRow_self] | 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 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
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 (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 [hM', updateRow_ne ((Fin.succ_injective _).ne hik), 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 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
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 (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... | by_cases hik2 : k < 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 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
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 (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 [hc i (Fin.succ_lt_succ_iff.mpr hik2)] | 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 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
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 (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 [updateRow_ne] | 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 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
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 (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... | apply ne_of_lt | 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 neg.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)
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 ... | /-
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... | rwa [Fin.lt_iff_val_lt_val, Fin.coe_castSucc, Fin.val_succ, Nat.lt_succ_iff, ← not_lt] | 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
A B : Matrix (Fin (n + 1)) (Fin (n + 1)) R
c : Fin n → R
A_zero : ∀ (i : Fin (n + 1)), A i 0 = B i 0
A_succ : ∀ (i : Fin (n + 1)) (j : Fin n), A i (Fin.succ j) = B i (Fin.... | /-
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 A, ← det_transpose B] | /-- If you add multiples of previous columns to the next columns, the determinant doesn't change. -/
theorem det_eq_of_forall_col_eq_smul_add_pred {n : ℕ} {A B : Matrix (Fin (n + 1)) (Fin (n + 1)) R}
(c : Fin n → R) (A_zero : ∀ i, A i 0 = B i 0)
(A_succ : ∀ (i) (j : Fin n), A i j.succ = B i j.succ + c j * A i (... | Mathlib.LinearAlgebra.Matrix.Determinant.560_0.U1f6HO8zRbnvZ95 | /-- If you add multiples of previous columns to the next columns, the determinant doesn't change. -/
theorem det_eq_of_forall_col_eq_smul_add_pred {n : ℕ} {A B : Matrix (Fin (n + 1)) (Fin (n + 1)) R}
(c : Fin n → R) (A_zero : ∀ i, A i 0 = B i 0)
(A_succ : ∀ (i) (j : Fin n), A i j.succ = B i j.succ + c j * A i (... | 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
A B : Matrix (Fin (n + 1)) (Fin (n + 1)) R
c : Fin n → R
A_zero : ∀ (i : Fin (n + 1)), A i 0 = B i 0
A_succ : ∀ (i : Fin (n + 1)) (j : Fin n), A i (Fin.succ j) = B i (Fin.... | /-
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_of_forall_row_eq_smul_add_pred c A_zero fun i j => A_succ j i | /-- If you add multiples of previous columns to the next columns, the determinant doesn't change. -/
theorem det_eq_of_forall_col_eq_smul_add_pred {n : ℕ} {A B : Matrix (Fin (n + 1)) (Fin (n + 1)) R}
(c : Fin n → R) (A_zero : ∀ i, A i 0 = B i 0)
(A_succ : ∀ (i) (j : Fin n), A i j.succ = B i j.succ + c j * A i (... | Mathlib.LinearAlgebra.Matrix.Determinant.560_0.U1f6HO8zRbnvZ95 | /-- If you add multiples of previous columns to the next columns, the determinant doesn't change. -/
theorem det_eq_of_forall_col_eq_smul_add_pred {n : ℕ} {A B : Matrix (Fin (n + 1)) (Fin (n + 1)) R}
(c : Fin n → R) (A_zero : ∀ i, A i 0 = B i 0)
(A_succ : ∀ (i) (j : Fin n), A i j.succ = B i j.succ + c j * A i (... | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
⊢ det (blockDiagonal M) = ∏ k : o, det (M k) | /-
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 [det_apply'] | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
| Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
⊢ ∑ σ : Perm (n × o), ↑↑(sign σ) * ∏ i : n × o, blockDiagonal M (σ i) i =
∏ x : o, ∑ σ : Perm 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 [Finset.prod_sum] | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
| Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
⊢ ∑ σ : Perm (n × o), ↑↑(sign σ) * ∏ i : n × o, blockDiagonal M (σ i) i =
∑ p in pi univ fun x => un... | /-
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 [Finset.prod_attach_univ, Finset.univ_pi_univ] | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
⊢ ∑ σ : Perm (n × o), ↑↑(sign σ) * ∏ i : n × o, blockDiagonal M (σ i) i =
∑ x : (a : o) → a ∈ univ →... | /-
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... | let preserving_snd : Finset (Equiv.Perm (n × o)) :=
Finset.univ.filter fun σ => ∀ x, (σ x).snd = x.snd | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) univ
⊢ ∑ σ : Pe... | /-
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 mem_preserving_snd :
∀ {σ : Equiv.Perm (n × o)}, σ ∈ preserving_snd ↔ ∀ x, (σ x).snd = x.snd := fun {σ} =>
Finset.mem_filter.trans ⟨fun h => h.2, fun h => ⟨Finset.mem_univ _, h⟩⟩ | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) univ
mem_preser... | /-
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 [← Finset.sum_subset (Finset.subset_univ preserving_snd) _] | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) univ
mem_preser... | /-
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 [(Finset.sum_bij
(fun (σ : ∀ k : o, k ∈ Finset.univ → Equiv.Perm n) _ =>
prodCongrLeft fun k => σ k (Finset.mem_univ k))
_ _ _ _).symm] | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) univ
mem_preser... | /-
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 _ | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) univ
mem_preser... | /-
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 [mem_preserving_snd] | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) univ
mem_preser... | /-
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 ⟨-, x⟩ | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
case mk
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) univ
me... | /-
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 [prodCongrLeft_apply] | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) univ
mem_preser... | /-
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 _ | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) univ
mem_preser... | /-
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 [Finset.prod_mul_distrib, ← Finset.univ_product_univ, Finset.prod_product_right] | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) univ
mem_preser... | /-
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 [sign_prodCongrLeft, Units.coe_prod, Int.cast_prod, blockDiagonal_apply_eq,
prodCongrLeft_apply] | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) univ
mem_preser... | /-
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 σ' _ _ eq | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) univ
mem_preser... | /-
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 x hx k | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
case h.h.H
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) univ... | /-
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 at eq | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
case h.h.H
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) univ... | /-
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 :
∀ k x,
prodCongrLeft (fun k => σ k (Finset.mem_univ _)) (k, x) =
prodCongrLeft (fun k => σ' k (Finset.mem_univ _)) (k, x) :=
fun k x => by rw [eq] | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) univ
mem_preser... | /-
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 [eq] | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
case h.h.H
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) univ... | /-
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 [prodCongrLeft_apply, Prod.mk.inj_iff] at this | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
case h.h.H
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) univ... | /-
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 (this k x).1 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) univ
mem_preser... | /-
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 hσ | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) univ
mem_preser... | /-
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 [mem_preserving_snd] at hσ | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) univ
mem_preser... | /-
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 hσ' : ∀ x, (σ⁻¹ x).snd = x.snd := by
intro x
conv_rhs => rw [← Perm.apply_inv_self σ x, hσ] | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) univ
mem_preser... | /-
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 x | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) univ
mem_preser... | /-
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... | conv_rhs => rw [← Perm.apply_inv_self σ x, hσ] | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) univ
mem_preser... | /-
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 [← Perm.apply_inv_self σ x, hσ] | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) univ
mem_preser... | /-
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 [← Perm.apply_inv_self σ x, hσ] | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) univ
mem_preser... | /-
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 [← Perm.apply_inv_self σ x, hσ] | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) univ
mem_preser... | /-
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 mk_apply_eq : ∀ k x, ((σ (x, k)).fst, k) = σ (x, k) := by
intro k x
ext
· simp only
· simp only [hσ] | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) univ
mem_preser... | /-
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 k x | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) univ
mem_preser... | /-
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 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
case a
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) univ
mem... | /-
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 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
case a
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) univ
mem... | /-
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 [hσ] | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) univ
mem_preser... | /-
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 mk_inv_apply_eq : ∀ k x, ((σ⁻¹ (x, k)).fst, k) = σ⁻¹ (x, k) := by
intro k x
conv_lhs => rw [← Perm.apply_inv_self σ (x, k)]
ext
· simp only [apply_inv_self]
· simp only [hσ'] | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) univ
mem_preser... | /-
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 k x | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) univ
mem_preser... | /-
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... | conv_lhs => rw [← Perm.apply_inv_self σ (x, k)] | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) univ
mem_preser... | /-
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 [← Perm.apply_inv_self σ (x, k)] | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) univ
mem_preser... | /-
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 [← Perm.apply_inv_self σ (x, k)] | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) univ
mem_preser... | /-
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 [← Perm.apply_inv_self σ (x, k)] | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) univ
mem_preser... | /-
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 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
case a
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) univ
mem... | /-
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 [apply_inv_self] | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
case a
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) univ
mem... | /-
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 [hσ'] | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) univ
mem_preser... | /-
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' ⟨fun k _ => ⟨fun x => (σ (x, k)).fst, fun x => (σ⁻¹ (x, k)).fst, _, _⟩, _, _⟩ | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
case refine'_1
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) ... | /-
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 x | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
case refine'_1
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) ... | /-
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 [mk_apply_eq, inv_apply_self] | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
case refine'_2
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) ... | /-
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 x | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
case refine'_2
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) ... | /-
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 [mk_inv_apply_eq, apply_inv_self] | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
case refine'_3
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) ... | /-
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 Finset.mem_univ | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
case refine'_4
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) ... | /-
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 ⟨k, x⟩ | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
case refine'_4.H.mk.a
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 ... | /-
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 [coe_fn_mk, prodCongrLeft_apply] | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
case refine'_4.H.mk.a
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 ... | /-
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 [prodCongrLeft_apply, hσ] | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) univ
mem_preser... | /-
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 _ hσ | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) univ
mem_preser... | /-
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 [mem_preserving_snd] at hσ | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) univ
mem_preser... | /-
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... | obtain ⟨⟨k, x⟩, hkx⟩ := not_forall.mp hσ | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
case intro.mk
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) 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 [Finset.prod_eq_zero (Finset.mem_univ (k, x)), mul_zero] | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
case intro.mk
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) 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 [blockDiagonal_apply_ne] | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | Mathlib_LinearAlgebra_Matrix_Determinant |
case intro.mk.h
m : Type u_1
n : Type u_2
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype m
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2)... | /-
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 hkx | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finse... | Mathlib.LinearAlgebra.Matrix.Determinant.571_0.U1f6HO8zRbnvZ95 | @[simp]
theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).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
A : Matrix m m R
B : Matrix m n R
D : Matrix n n R
⊢ det (fromBlocks A B 0 D) = det A * det 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... | classical
simp_rw [det_apply']
convert Eq.symm <|
sum_subset (β := R) (subset_univ ((sumCongrHom m n).range : Set (Perm (Sum m n))).toFinset) ?_
rw [sum_mul_sum]
simp_rw [univ_product_univ]
rw [(sum_bij (fun (σ : Perm m × Perm n) _ => Equiv.sumCongr σ.fst σ.snd) _ _ _ _).symm]
· intro σ₁₂ ... | /-- The determinant of a 2×2 block matrix with the lower-left block equal to zero is the product of
the determinants of the diagonal blocks. For the generalization to any number of blocks, see
`Matrix.det_of_upper_triangular`. -/
@[simp]
theorem det_fromBlocks_zero₂₁ (A : Matrix m m R) (B : Matrix m n R) (D : Matrix n ... | Mathlib.LinearAlgebra.Matrix.Determinant.643_0.U1f6HO8zRbnvZ95 | /-- The determinant of a 2×2 block matrix with the lower-left block equal to zero is the product of
the determinants of the diagonal blocks. For the generalization to any number of blocks, see
`Matrix.det_of_upper_triangular`. -/
@[simp]
theorem det_fromBlocks_zero₂₁ (A : Matrix m m R) (B : Matrix m n R) (D : Matrix 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
A : Matrix m m R
B : Matrix m n R
D : Matrix n n R
⊢ det (fromBlocks A B 0 D) = det A * det 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... | simp_rw [det_apply'] | /-- The determinant of a 2×2 block matrix with the lower-left block equal to zero is the product of
the determinants of the diagonal blocks. For the generalization to any number of blocks, see
`Matrix.det_of_upper_triangular`. -/
@[simp]
theorem det_fromBlocks_zero₂₁ (A : Matrix m m R) (B : Matrix m n R) (D : Matrix n ... | Mathlib.LinearAlgebra.Matrix.Determinant.643_0.U1f6HO8zRbnvZ95 | /-- The determinant of a 2×2 block matrix with the lower-left block equal to zero is the product of
the determinants of the diagonal blocks. For the generalization to any number of blocks, see
`Matrix.det_of_upper_triangular`. -/
@[simp]
theorem det_fromBlocks_zero₂₁ (A : Matrix m m R) (B : Matrix m n R) (D : Matrix 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
A : Matrix m m R
B : Matrix m n R
D : Matrix n n R
⊢ ∑ σ : Perm (m ⊕ n), ↑↑(sign σ) * ∏ i : m ⊕ n, fromBlocks A B 0 D (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... | convert Eq.symm <|
sum_subset (β := R) (subset_univ ((sumCongrHom m n).range : Set (Perm (Sum m n))).toFinset) ?_ | /-- The determinant of a 2×2 block matrix with the lower-left block equal to zero is the product of
the determinants of the diagonal blocks. For the generalization to any number of blocks, see
`Matrix.det_of_upper_triangular`. -/
@[simp]
theorem det_fromBlocks_zero₂₁ (A : Matrix m m R) (B : Matrix m n R) (D : Matrix n ... | Mathlib.LinearAlgebra.Matrix.Determinant.643_0.U1f6HO8zRbnvZ95 | /-- The determinant of a 2×2 block matrix with the lower-left block equal to zero is the product of
the determinants of the diagonal blocks. For the generalization to any number of blocks, see
`Matrix.det_of_upper_triangular`. -/
@[simp]
theorem det_fromBlocks_zero₂₁ (A : Matrix m m R) (B : Matrix m n R) (D : Matrix n ... | Mathlib_LinearAlgebra_Matrix_Determinant |
case h.e'_3
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 m m R
B : Matrix m n R
D : Matrix n n R
⊢ (∑ σ : Perm m, ↑↑(sign σ) * ∏ i : m, A (σ i) i) * ∑ σ : Perm n, ↑↑(sign σ) * ∏ i : n, D (σ i) i =
∑ x in Set.to... | /-
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 [sum_mul_sum] | /-- The determinant of a 2×2 block matrix with the lower-left block equal to zero is the product of
the determinants of the diagonal blocks. For the generalization to any number of blocks, see
`Matrix.det_of_upper_triangular`. -/
@[simp]
theorem det_fromBlocks_zero₂₁ (A : Matrix m m R) (B : Matrix m n R) (D : Matrix n ... | Mathlib.LinearAlgebra.Matrix.Determinant.643_0.U1f6HO8zRbnvZ95 | /-- The determinant of a 2×2 block matrix with the lower-left block equal to zero is the product of
the determinants of the diagonal blocks. For the generalization to any number of blocks, see
`Matrix.det_of_upper_triangular`. -/
@[simp]
theorem det_fromBlocks_zero₂₁ (A : Matrix m m R) (B : Matrix m n R) (D : Matrix n ... | Mathlib_LinearAlgebra_Matrix_Determinant |
case h.e'_3
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 m m R
B : Matrix m n R
D : Matrix n n R
⊢ ∑ p in univ ×ˢ univ, (↑↑(sign p.1) * ∏ i : m, A (p.1 i) i) * (↑↑(sign p.2) * ∏ i : n, D (p.2 i) i) =
∑ x in Set... | /-
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 [univ_product_univ] | /-- The determinant of a 2×2 block matrix with the lower-left block equal to zero is the product of
the determinants of the diagonal blocks. For the generalization to any number of blocks, see
`Matrix.det_of_upper_triangular`. -/
@[simp]
theorem det_fromBlocks_zero₂₁ (A : Matrix m m R) (B : Matrix m n R) (D : Matrix n ... | Mathlib.LinearAlgebra.Matrix.Determinant.643_0.U1f6HO8zRbnvZ95 | /-- The determinant of a 2×2 block matrix with the lower-left block equal to zero is the product of
the determinants of the diagonal blocks. For the generalization to any number of blocks, see
`Matrix.det_of_upper_triangular`. -/
@[simp]
theorem det_fromBlocks_zero₂₁ (A : Matrix m m R) (B : Matrix m n R) (D : Matrix n ... | Mathlib_LinearAlgebra_Matrix_Determinant |
case h.e'_3
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 m m R
B : Matrix m n R
D : Matrix n n R
⊢ ∑ p : Perm m × Perm n, (↑↑(sign p.1) * ∏ i : m, A (p.1 i) i) * (↑↑(sign p.2) * ∏ i : n, D (p.2 i) i) =
∑ x in S... | /-
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 [(sum_bij (fun (σ : Perm m × Perm n) _ => Equiv.sumCongr σ.fst σ.snd) _ _ _ _).symm] | /-- The determinant of a 2×2 block matrix with the lower-left block equal to zero is the product of
the determinants of the diagonal blocks. For the generalization to any number of blocks, see
`Matrix.det_of_upper_triangular`. -/
@[simp]
theorem det_fromBlocks_zero₂₁ (A : Matrix m m R) (B : Matrix m n R) (D : Matrix n ... | Mathlib.LinearAlgebra.Matrix.Determinant.643_0.U1f6HO8zRbnvZ95 | /-- The determinant of a 2×2 block matrix with the lower-left block equal to zero is the product of
the determinants of the diagonal blocks. For the generalization to any number of blocks, see
`Matrix.det_of_upper_triangular`. -/
@[simp]
theorem det_fromBlocks_zero₂₁ (A : Matrix m m R) (B : Matrix m n R) (D : Matrix 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
A : Matrix m m R
B : Matrix m n R
D : Matrix n n R
⊢ ∀ (a : Perm m × Perm n) (ha : a ∈ univ),
(fun σ x => Equiv.sumCongr σ.1 σ.2) a ha ∈ Set.toFinset ↑(MonoidHom.range (sumCong... | /-
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 σ₁₂ h | /-- The determinant of a 2×2 block matrix with the lower-left block equal to zero is the product of
the determinants of the diagonal blocks. For the generalization to any number of blocks, see
`Matrix.det_of_upper_triangular`. -/
@[simp]
theorem det_fromBlocks_zero₂₁ (A : Matrix m m R) (B : Matrix m n R) (D : Matrix n ... | Mathlib.LinearAlgebra.Matrix.Determinant.643_0.U1f6HO8zRbnvZ95 | /-- The determinant of a 2×2 block matrix with the lower-left block equal to zero is the product of
the determinants of the diagonal blocks. For the generalization to any number of blocks, see
`Matrix.det_of_upper_triangular`. -/
@[simp]
theorem det_fromBlocks_zero₂₁ (A : Matrix m m R) (B : Matrix m n R) (D : Matrix 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
A : Matrix m m R
B : Matrix m n R
D : Matrix n n R
σ₁₂ : Perm m × Perm n
h : σ₁₂ ∈ univ
⊢ (fun σ x => Equiv.sumCongr σ.1 σ.2) σ₁₂ h ∈ Set.toFinset ↑(MonoidHom.range (sumCongrHom 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 only | /-- The determinant of a 2×2 block matrix with the lower-left block equal to zero is the product of
the determinants of the diagonal blocks. For the generalization to any number of blocks, see
`Matrix.det_of_upper_triangular`. -/
@[simp]
theorem det_fromBlocks_zero₂₁ (A : Matrix m m R) (B : Matrix m n R) (D : Matrix n ... | Mathlib.LinearAlgebra.Matrix.Determinant.643_0.U1f6HO8zRbnvZ95 | /-- The determinant of a 2×2 block matrix with the lower-left block equal to zero is the product of
the determinants of the diagonal blocks. For the generalization to any number of blocks, see
`Matrix.det_of_upper_triangular`. -/
@[simp]
theorem det_fromBlocks_zero₂₁ (A : Matrix m m R) (B : Matrix m n R) (D : Matrix 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
A : Matrix m m R
B : Matrix m n R
D : Matrix n n R
σ₁₂ : Perm m × Perm n
h : σ₁₂ ∈ univ
⊢ Equiv.sumCongr σ₁₂.1 σ₁₂.2 ∈ Set.toFinset ↑(MonoidHom.range (sumCongrHom m 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... | erw [Set.mem_toFinset, MonoidHom.mem_range] | /-- The determinant of a 2×2 block matrix with the lower-left block equal to zero is the product of
the determinants of the diagonal blocks. For the generalization to any number of blocks, see
`Matrix.det_of_upper_triangular`. -/
@[simp]
theorem det_fromBlocks_zero₂₁ (A : Matrix m m R) (B : Matrix m n R) (D : Matrix n ... | Mathlib.LinearAlgebra.Matrix.Determinant.643_0.U1f6HO8zRbnvZ95 | /-- The determinant of a 2×2 block matrix with the lower-left block equal to zero is the product of
the determinants of the diagonal blocks. For the generalization to any number of blocks, see
`Matrix.det_of_upper_triangular`. -/
@[simp]
theorem det_fromBlocks_zero₂₁ (A : Matrix m m R) (B : Matrix m n R) (D : Matrix 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
A : Matrix m m R
B : Matrix m n R
D : Matrix n n R
σ₁₂ : Perm m × Perm n
h : σ₁₂ ∈ univ
⊢ ∃ x, (sumCongrHom m n) x = Equiv.sumCongr σ₁₂.1 σ₁₂.2 | /-
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... | use σ₁₂ | /-- The determinant of a 2×2 block matrix with the lower-left block equal to zero is the product of
the determinants of the diagonal blocks. For the generalization to any number of blocks, see
`Matrix.det_of_upper_triangular`. -/
@[simp]
theorem det_fromBlocks_zero₂₁ (A : Matrix m m R) (B : Matrix m n R) (D : Matrix n ... | Mathlib.LinearAlgebra.Matrix.Determinant.643_0.U1f6HO8zRbnvZ95 | /-- The determinant of a 2×2 block matrix with the lower-left block equal to zero is the product of
the determinants of the diagonal blocks. For the generalization to any number of blocks, see
`Matrix.det_of_upper_triangular`. -/
@[simp]
theorem det_fromBlocks_zero₂₁ (A : Matrix m m R) (B : Matrix m n R) (D : Matrix n ... | 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
A : Matrix m m R
B : Matrix m n R
D : Matrix n n R
σ₁₂ : Perm m × Perm n
h : σ₁₂ ∈ univ
⊢ (sumCongrHom m n) σ₁₂ = Equiv.sumCongr σ₁₂.1 σ₁₂.2 | /-
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 [sumCongrHom_apply] | /-- The determinant of a 2×2 block matrix with the lower-left block equal to zero is the product of
the determinants of the diagonal blocks. For the generalization to any number of blocks, see
`Matrix.det_of_upper_triangular`. -/
@[simp]
theorem det_fromBlocks_zero₂₁ (A : Matrix m m R) (B : Matrix m n R) (D : Matrix n ... | Mathlib.LinearAlgebra.Matrix.Determinant.643_0.U1f6HO8zRbnvZ95 | /-- The determinant of a 2×2 block matrix with the lower-left block equal to zero is the product of
the determinants of the diagonal blocks. For the generalization to any number of blocks, see
`Matrix.det_of_upper_triangular`. -/
@[simp]
theorem det_fromBlocks_zero₂₁ (A : Matrix m m R) (B : Matrix m n R) (D : Matrix 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
A : Matrix m m R
B : Matrix m n R
D : Matrix n n R
⊢ ∀ (a : Perm m × Perm n) (ha : a ∈ univ),
(↑↑(sign a.1) * ∏ i : m, A (a.1 i) i) * (↑↑(sign a.2) * ∏ i : n, D (a.2 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... | simp only [forall_prop_of_true, Prod.forall, mem_univ] | /-- The determinant of a 2×2 block matrix with the lower-left block equal to zero is the product of
the determinants of the diagonal blocks. For the generalization to any number of blocks, see
`Matrix.det_of_upper_triangular`. -/
@[simp]
theorem det_fromBlocks_zero₂₁ (A : Matrix m m R) (B : Matrix m n R) (D : Matrix n ... | Mathlib.LinearAlgebra.Matrix.Determinant.643_0.U1f6HO8zRbnvZ95 | /-- The determinant of a 2×2 block matrix with the lower-left block equal to zero is the product of
the determinants of the diagonal blocks. For the generalization to any number of blocks, see
`Matrix.det_of_upper_triangular`. -/
@[simp]
theorem det_fromBlocks_zero₂₁ (A : Matrix m m R) (B : Matrix m n R) (D : Matrix 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
A : Matrix m m R
B : Matrix m n R
D : Matrix n n R
⊢ ∀ (a : Perm m) (b : Perm n),
(↑↑(sign a) * ∏ x : m, A (a x) x) * (↑↑(sign b) * ∏ x : n, D (b x) x) =
↑↑(sign (Equiv.s... | /-
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 σ₁ σ₂ | /-- The determinant of a 2×2 block matrix with the lower-left block equal to zero is the product of
the determinants of the diagonal blocks. For the generalization to any number of blocks, see
`Matrix.det_of_upper_triangular`. -/
@[simp]
theorem det_fromBlocks_zero₂₁ (A : Matrix m m R) (B : Matrix m n R) (D : Matrix n ... | Mathlib.LinearAlgebra.Matrix.Determinant.643_0.U1f6HO8zRbnvZ95 | /-- The determinant of a 2×2 block matrix with the lower-left block equal to zero is the product of
the determinants of the diagonal blocks. For the generalization to any number of blocks, see
`Matrix.det_of_upper_triangular`. -/
@[simp]
theorem det_fromBlocks_zero₂₁ (A : Matrix m m R) (B : Matrix m n R) (D : Matrix 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
A : Matrix m m R
B : Matrix m n R
D : Matrix n n R
σ₁ : Perm m
σ₂ : Perm n
⊢ (↑↑(sign σ₁) * ∏ x : m, A (σ₁ x) x) * (↑↑(sign σ₂) * ∏ x : n, D (σ₂ x) x) =
↑↑(sign (Equiv.sumCongr... | /-
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 [Fintype.prod_sum_type] | /-- The determinant of a 2×2 block matrix with the lower-left block equal to zero is the product of
the determinants of the diagonal blocks. For the generalization to any number of blocks, see
`Matrix.det_of_upper_triangular`. -/
@[simp]
theorem det_fromBlocks_zero₂₁ (A : Matrix m m R) (B : Matrix m n R) (D : Matrix n ... | Mathlib.LinearAlgebra.Matrix.Determinant.643_0.U1f6HO8zRbnvZ95 | /-- The determinant of a 2×2 block matrix with the lower-left block equal to zero is the product of
the determinants of the diagonal blocks. For the generalization to any number of blocks, see
`Matrix.det_of_upper_triangular`. -/
@[simp]
theorem det_fromBlocks_zero₂₁ (A : Matrix m m R) (B : Matrix m n R) (D : Matrix 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
A : Matrix m m R
B : Matrix m n R
D : Matrix n n R
σ₁ : Perm m
σ₂ : Perm n
⊢ (↑↑(sign σ₁) * ∏ x : m, A (σ₁ x) x) * (↑↑(sign σ₂) * ∏ x : n, D (σ₂ x) x) =
↑↑(sign (Equiv.sumCongr... | /-
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.sumCongr_apply, Sum.map_inr, Sum.map_inl, fromBlocks_apply₁₁,
fromBlocks_apply₂₂] | /-- The determinant of a 2×2 block matrix with the lower-left block equal to zero is the product of
the determinants of the diagonal blocks. For the generalization to any number of blocks, see
`Matrix.det_of_upper_triangular`. -/
@[simp]
theorem det_fromBlocks_zero₂₁ (A : Matrix m m R) (B : Matrix m n R) (D : Matrix n ... | Mathlib.LinearAlgebra.Matrix.Determinant.643_0.U1f6HO8zRbnvZ95 | /-- The determinant of a 2×2 block matrix with the lower-left block equal to zero is the product of
the determinants of the diagonal blocks. For the generalization to any number of blocks, see
`Matrix.det_of_upper_triangular`. -/
@[simp]
theorem det_fromBlocks_zero₂₁ (A : Matrix m m R) (B : Matrix m n R) (D : Matrix 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
A : Matrix m m R
B : Matrix m n R
D : Matrix n n R
σ₁ : Perm m
σ₂ : Perm n
⊢ (↑↑(sign σ₁) * ∏ x : m, A (σ₁ x) x) * (↑↑(sign σ₂) * ∏ x : n, D (σ₂ x) x) =
↑↑(sign (Equiv.sumCongr... | /-
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_mul_mul_comm] | /-- The determinant of a 2×2 block matrix with the lower-left block equal to zero is the product of
the determinants of the diagonal blocks. For the generalization to any number of blocks, see
`Matrix.det_of_upper_triangular`. -/
@[simp]
theorem det_fromBlocks_zero₂₁ (A : Matrix m m R) (B : Matrix m n R) (D : Matrix n ... | Mathlib.LinearAlgebra.Matrix.Determinant.643_0.U1f6HO8zRbnvZ95 | /-- The determinant of a 2×2 block matrix with the lower-left block equal to zero is the product of
the determinants of the diagonal blocks. For the generalization to any number of blocks, see
`Matrix.det_of_upper_triangular`. -/
@[simp]
theorem det_fromBlocks_zero₂₁ (A : Matrix m m R) (B : Matrix m n R) (D : Matrix 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
A : Matrix m m R
B : Matrix m n R
D : Matrix n n R
σ₁ : Perm m
σ₂ : Perm n
⊢ ↑↑(sign σ₁) * ↑↑(sign σ₂) * ((∏ x : m, A (σ₁ x) x) * ∏ x : n, D (σ₂ x) x) =
↑↑(sign (Equiv.sumCongr... | /-
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 | /-- The determinant of a 2×2 block matrix with the lower-left block equal to zero is the product of
the determinants of the diagonal blocks. For the generalization to any number of blocks, see
`Matrix.det_of_upper_triangular`. -/
@[simp]
theorem det_fromBlocks_zero₂₁ (A : Matrix m m R) (B : Matrix m n R) (D : Matrix n ... | Mathlib.LinearAlgebra.Matrix.Determinant.643_0.U1f6HO8zRbnvZ95 | /-- The determinant of a 2×2 block matrix with the lower-left block equal to zero is the product of
the determinants of the diagonal blocks. For the generalization to any number of blocks, see
`Matrix.det_of_upper_triangular`. -/
@[simp]
theorem det_fromBlocks_zero₂₁ (A : Matrix m m R) (B : Matrix m n R) (D : Matrix n ... | Mathlib_LinearAlgebra_Matrix_Determinant |
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