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 e_a
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
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) = ↑↑(sign (Equiv.sumCongr σ₁ σ₂)) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rw [sign_sumCongr, Units.val_mul, Int.cast_mul] | /-- 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
⊢ ∀ (a₁ a₂ : Perm m × Perm n) (ha₁ : a₁ ∈ univ) (ha₂ : a₂ ∈ univ),
(fun σ x => Equiv.sumCongr σ.1 σ.2) a₁ ha₁ = (fun σ x => E... | /-
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 h₁ h₂ | /-- 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 σ₂ : Perm m × Perm n
h₁ : σ₁ ∈ univ
h₂ : σ₂ ∈ univ
⊢ (fun σ x => Equiv.sumCongr σ.1 σ.2) σ₁ h₁ = (fun σ x => Equiv.sumCongr σ.... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | dsimp 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
σ₁ σ₂ : Perm m × Perm n
h₁ : σ₁ ∈ univ
h₂ : σ₂ ∈ univ
⊢ Equiv.sumCongr σ₁.1 σ₁.2 = Equiv.sumCongr σ₂.1 σ₂.2 → σ₁ = σ₂ | /-
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 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
σ₁ σ₂ : Perm m × Perm n
h₁ : σ₁ ∈ univ
h₂ : σ₂ ∈ univ
h : Equiv.sumCongr σ₁.1 σ₁.2 = Equiv.sumCongr σ₂.1 σ₂.2
⊢ σ₁ = σ₂ | /-
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 h2 : ∀ x, Perm.sumCongr σ₁.fst σ₁.snd x = Perm.sumCongr σ₂.fst σ₂.snd x :=
FunLike.congr_fun 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
σ₁ σ₂ : Perm m × Perm n
h₁ : σ₁ ∈ univ
h₂ : σ₂ ∈ univ
h : Equiv.sumCongr σ₁.1 σ₁.2 = Equiv.sumCongr σ₂.1 σ₂.2
h2 : ∀ (x : 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... | simp only [Sum.map_inr, Sum.map_inl, Perm.sumCongr_apply, Sum.forall, Sum.inl.injEq,
Sum.inr.injEq] at h2 | /-- 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 σ₂ : Perm m × Perm n
h₁ : σ₁ ∈ univ
h₂ : σ₂ ∈ univ
h : Equiv.sumCongr σ₁.1 σ₁.2 = Equiv.sumCongr σ₂.1 σ₂.2
h2 : (∀ (a : m), σ₁... | /-
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 | /-- 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 a.H
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A : Matrix m m R
B : Matrix m n R
D : Matrix n n R
σ₁ σ₂ : Perm m × Perm n
h₁ : σ₁ ∈ univ
h₂ : σ₂ ∈ univ
h : Equiv.sumCongr σ₁.1 σ₁.2 = Equiv.sumCongr σ₂.1 σ₂.2
h2 : (∀ (a... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | exact h2.left x | /-- 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 a.H
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A : Matrix m m R
B : Matrix m n R
D : Matrix n n R
σ₁ σ₂ : Perm m × Perm n
h₁ : σ₁ ∈ univ
h₂ : σ₂ ∈ univ
h : Equiv.sumCongr σ₁.1 σ₁.2 = Equiv.sumCongr σ₂.1 σ₂.2
h2 : (∀ (a... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | exact h2.right x | /-- 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
⊢ ∀ b ∈ Set.toFinset ↑(MonoidHom.range (sumCongrHom m n)),
∃ a, ∃ (ha : a ∈ univ), b = (fun σ x => Equiv.sumCongr σ.1 σ.2) a ... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | intro σ hσ | /-- 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 : m ⊕ n ≃ m ⊕ n
hσ : σ ∈ Set.toFinset ↑(MonoidHom.range (sumCongrHom m n))
⊢ ∃ a, ∃ (ha : a ∈ univ), σ = (fun σ x => Equiv.sumC... | /-
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] at hσ | /-- 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 : m ⊕ n ≃ m ⊕ n
hσ : ∃ x, (sumCongrHom m n) x = σ
⊢ ∃ a, ∃ (ha : a ∈ univ), σ = (fun σ x => Equiv.sumCongr σ.1 σ.2) a ha | /-
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 ⟨σ₁₂, hσ₁₂⟩ := hσ | /-- 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 intro
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 : m ⊕ n ≃ m ⊕ n
σ₁₂ : Perm m × Perm n
hσ₁₂ : (sumCongrHom m n) σ₁₂ = σ
⊢ ∃ a, ∃ (ha : a ∈ univ), σ = (fun σ x => Equ... | /-
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
σ : m ⊕ n ≃ m ⊕ n
σ₁₂ : Perm m × Perm n
hσ₁₂ : (sumCongrHom m n) σ₁₂ = σ
⊢ ∃ (ha : σ₁₂ ∈ univ), σ = (fun σ x => Equiv.sumC... | /-
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 [← hσ₁₂] | /-- 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
σ : m ⊕ n ≃ m ⊕ n
σ₁₂ : Perm m × Perm n
hσ₁₂ : (sumCongrHom m n) σ₁₂ = σ
⊢ ∃ (ha : σ₁₂ ∈ univ), (sumCongrHom m n) σ₁₂ = (f... | /-
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 | /-- 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 convert_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
A : Matrix m m R
B : Matrix m n R
D : Matrix n n R
⊢ ∀ x ∈ univ,
x ∉ Set.toFinset ↑(MonoidHom.range (sumCongrHom m n)) → ↑↑(sign x) * ∏ i : m ⊕ n, fromBlocks A B... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rintro σ - hσn | /-- 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 convert_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
A : Matrix m m R
B : Matrix m n R
D : Matrix n n R
σ : Perm (m ⊕ n)
hσn : σ ∉ Set.toFinset ↑(MonoidHom.range (sumCongrHom m n))
⊢ ↑↑(sign σ) * ∏ i : m ⊕ n, fromBlock... | /-
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 h1 : ¬∀ x, ∃ y, Sum.inl y = σ (Sum.inl x) := by
rw [Set.mem_toFinset] at hσn
-- Porting note: golfed
simpa only [Set.MapsTo, Set.mem_range, forall_exists_index, forall_apply_eq_imp_iff] using
mt mem_sumCongrHom_range_of_perm_mapsTo_inl hσn | /-- 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)
hσn : σ ∉ Set.toFinset ↑(MonoidHom.range (sumCongrHom m n))
⊢ ¬∀ (x : m), ∃ y, Sum.inl y = σ (Sum.inl x) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rw [Set.mem_toFinset] at hσn | /-- 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)
hσn : σ ∉ ↑(MonoidHom.range (sumCongrHom m n))
⊢ ¬∀ (x : m), ∃ y, Sum.inl y = σ (Sum.inl x) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | simpa only [Set.MapsTo, Set.mem_range, forall_exists_index, forall_apply_eq_imp_iff] using
mt mem_sumCongrHom_range_of_perm_mapsTo_inl hσn | /-- 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 convert_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
A : Matrix m m R
B : Matrix m n R
D : Matrix n n R
σ : Perm (m ⊕ n)
hσn : σ ∉ Set.toFinset ↑(MonoidHom.range (sumCongrHom m n))
h1 : ¬∀ (x : m), ∃ y, Sum.inl y = σ (... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | obtain ⟨a, ha⟩ := not_forall.mp h1 | /-- 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 convert_2.intro
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)
hσn : σ ∉ Set.toFinset ↑(MonoidHom.range (sumCongrHom m n))
h1 : ¬∀ (x : m), ∃ y, Sum.inl y... | /-
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... | cases' hx : σ (Sum.inl a) with a2 b | /-- 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 convert_2.intro.inl
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)
hσn : σ ∉ Set.toFinset ↑(MonoidHom.range (sumCongrHom m n))
h1 : ¬∀ (x : m), ∃ y, Sum.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... | have hn := (not_exists.mp ha) a2 | /-- 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 convert_2.intro.inl
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)
hσn : σ ∉ Set.toFinset ↑(MonoidHom.range (sumCongrHom m n))
h1 : ¬∀ (x : m), ∃ y, Sum.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... | exact absurd hx.symm hn | /-- 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 convert_2.intro.inr
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)
hσn : σ ∉ Set.toFinset ↑(MonoidHom.range (sumCongrHom m n))
h1 : ¬∀ (x : m), ∃ y, Sum.i... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rw [Finset.prod_eq_zero (Finset.mem_univ (Sum.inl a)), mul_zero] | /-- 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 convert_2.intro.inr
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)
hσn : σ ∉ Set.toFinset ↑(MonoidHom.range (sumCongrHom m n))
h1 : ¬∀ (x : m), ∃ y, Sum.i... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rw [hx, fromBlocks_apply₂₁, zero_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
C : Matrix n m R
D : Matrix n n R
⊢ det (fromBlocks A 0 C 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... | rw [← det_transpose, fromBlocks_transpose, transpose_zero, det_fromBlocks_zero₂₁, det_transpose,
det_transpose] | /-- The determinant of a 2×2 block matrix with the upper-right 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_lower_triangular`. -/
@[simp]
theorem det_fromBlocks_zero₁₂ (A : Matrix m m R) (C : Matrix n m R) (D : Matrix n... | Mathlib.LinearAlgebra.Matrix.Determinant.699_0.U1f6HO8zRbnvZ95 | /-- The determinant of a 2×2 block matrix with the upper-right 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_lower_triangular`. -/
@[simp]
theorem det_fromBlocks_zero₁₂ (A : Matrix m m R) (C : Matrix n m 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
n : ℕ
A : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R
⊢ det A = ∑ i : Fin (Nat.succ n), (-1) ^ ↑i * A i 0 * det (submatrix A (Fin.succAbove i) Fin.succ) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rw [Matrix.det_apply, Finset.univ_perm_fin_succ, ← Finset.univ_product_univ] | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along column 0. -/
theorem det_succ_column_zero {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) :
det A = ∑ i : Fin n.succ, (-1) ^ (i : ℕ) * A i 0 * det (A.submatrix i.succAbove Fin.succ) := by
| Mathlib.LinearAlgebra.Matrix.Determinant.709_0.U1f6HO8zRbnvZ95 | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along column 0. -/
theorem det_succ_column_zero {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) :
det A = ∑ i : Fin n.succ, (-1) ^ (i : ℕ) * A i 0 * det (A.submatrix i.succAbove Fin.succ) | 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 : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R
⊢ ∑ σ in Finset.map (Equiv.toEmbedding decomposeFin.symm) (univ ×ˢ univ), sign σ • ∏ i : Fin (Nat.succ n), 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... | simp only [Finset.sum_map, Equiv.toEmbedding_apply, Finset.sum_product, Matrix.submatrix] | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along column 0. -/
theorem det_succ_column_zero {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) :
det A = ∑ i : Fin n.succ, (-1) ^ (i : ℕ) * A i 0 * det (A.submatrix i.succAbove Fin.succ) := by
rw [Matrix.det_apply, Finset.univ_perm_fin_succ, ← ... | Mathlib.LinearAlgebra.Matrix.Determinant.709_0.U1f6HO8zRbnvZ95 | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along column 0. -/
theorem det_succ_column_zero {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) :
det A = ∑ i : Fin n.succ, (-1) ^ (i : ℕ) * A i 0 * det (A.submatrix i.succAbove Fin.succ) | 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 : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R
⊢ ∑ x : Fin (Nat.succ n),
∑ y : Perm (Fin n),
sign (decomposeFin.symm (x, y)) • ∏ x_1 : 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... | refine' Finset.sum_congr rfl fun i _ => Fin.cases _ (fun i => _) i | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along column 0. -/
theorem det_succ_column_zero {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) :
det A = ∑ i : Fin n.succ, (-1) ^ (i : ℕ) * A i 0 * det (A.submatrix i.succAbove Fin.succ) := by
rw [Matrix.det_apply, Finset.univ_perm_fin_succ, ← ... | Mathlib.LinearAlgebra.Matrix.Determinant.709_0.U1f6HO8zRbnvZ95 | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along column 0. -/
theorem det_succ_column_zero {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) :
det A = ∑ i : Fin n.succ, (-1) ^ (i : ℕ) * A i 0 * det (A.submatrix i.succAbove Fin.succ) | Mathlib_LinearAlgebra_Matrix_Determinant |
case refine'_1
m : Type u_1
n✝ : Type u_2
inst✝⁴ : DecidableEq n✝
inst✝³ : Fintype n✝
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
n : ℕ
A : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R
i : Fin (Nat.succ n)
x✝ : i ∈ univ
⊢ ∑ y : Perm (Fin n), sign (decomposeFin.symm (0, y)) • ∏ x : Fin (Nat... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | simp only [Fin.prod_univ_succ, Matrix.det_apply, Finset.mul_sum,
Equiv.Perm.decomposeFin_symm_apply_zero, Fin.val_zero, one_mul,
Equiv.Perm.decomposeFin.symm_sign, Equiv.swap_self, if_true, id.def, eq_self_iff_true,
Equiv.Perm.decomposeFin_symm_apply_succ, Fin.succAbove_zero, Equiv.coe_refl, pow_zero,... | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along column 0. -/
theorem det_succ_column_zero {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) :
det A = ∑ i : Fin n.succ, (-1) ^ (i : ℕ) * A i 0 * det (A.submatrix i.succAbove Fin.succ) := by
rw [Matrix.det_apply, Finset.univ_perm_fin_succ, ← ... | Mathlib.LinearAlgebra.Matrix.Determinant.709_0.U1f6HO8zRbnvZ95 | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along column 0. -/
theorem det_succ_column_zero {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) :
det A = ∑ i : Fin n.succ, (-1) ^ (i : ℕ) * A i 0 * det (A.submatrix i.succAbove Fin.succ) | 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
A : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R
i✝ : Fin (Nat.succ n)
x✝ : i✝ ∈ univ
i : Fin n
⊢ ∑ y : Perm (Fin n),
sign (decomposeFin.symm (Fin.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... | have : (-1 : R) ^ (i : ℕ) = (Perm.sign i.cycleRange) := by simp [Fin.sign_cycleRange] | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along column 0. -/
theorem det_succ_column_zero {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) :
det A = ∑ i : Fin n.succ, (-1) ^ (i : ℕ) * A i 0 * det (A.submatrix i.succAbove Fin.succ) := by
rw [Matrix.det_apply, Finset.univ_perm_fin_succ, ← ... | Mathlib.LinearAlgebra.Matrix.Determinant.709_0.U1f6HO8zRbnvZ95 | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along column 0. -/
theorem det_succ_column_zero {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) :
det A = ∑ i : Fin n.succ, (-1) ^ (i : ℕ) * A i 0 * det (A.submatrix i.succAbove Fin.succ) | 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 : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R
i✝ : Fin (Nat.succ n)
x✝ : i✝ ∈ univ
i : Fin n
⊢ (-1) ^ ↑i = ↑↑(sign (Fin.cycleRange 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 [Fin.sign_cycleRange] | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along column 0. -/
theorem det_succ_column_zero {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) :
det A = ∑ i : Fin n.succ, (-1) ^ (i : ℕ) * A i 0 * det (A.submatrix i.succAbove Fin.succ) := by
rw [Matrix.det_apply, Finset.univ_perm_fin_succ, ← ... | Mathlib.LinearAlgebra.Matrix.Determinant.709_0.U1f6HO8zRbnvZ95 | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along column 0. -/
theorem det_succ_column_zero {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) :
det A = ∑ i : Fin n.succ, (-1) ^ (i : ℕ) * A i 0 * det (A.submatrix i.succAbove Fin.succ) | 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
A : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R
i✝ : Fin (Nat.succ n)
x✝ : i✝ ∈ univ
i : Fin n
this : (-1) ^ ↑i = ↑↑(sign (Fin.cycleRange i))
⊢ ∑ y : Per... | /-
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.val_succ, pow_succ, this, mul_assoc, mul_assoc, mul_left_comm (ε _), ←
det_permute, Matrix.det_apply, Finset.mul_sum, Finset.mul_sum] | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along column 0. -/
theorem det_succ_column_zero {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) :
det A = ∑ i : Fin n.succ, (-1) ^ (i : ℕ) * A i 0 * det (A.submatrix i.succAbove Fin.succ) := by
rw [Matrix.det_apply, Finset.univ_perm_fin_succ, ← ... | Mathlib.LinearAlgebra.Matrix.Determinant.709_0.U1f6HO8zRbnvZ95 | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along column 0. -/
theorem det_succ_column_zero {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) :
det A = ∑ i : Fin n.succ, (-1) ^ (i : ℕ) * A i 0 * det (A.submatrix i.succAbove Fin.succ) | 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
A : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R
i✝ : Fin (Nat.succ n)
x✝ : i✝ ∈ univ
i : Fin n
this : (-1) ^ ↑i = ↑↑(sign (Fin.cycleRange i))
⊢ ∑ y : Per... | /-
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' Finset.sum_congr rfl fun σ _ => _ | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along column 0. -/
theorem det_succ_column_zero {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) :
det A = ∑ i : Fin n.succ, (-1) ^ (i : ℕ) * A i 0 * det (A.submatrix i.succAbove Fin.succ) := by
rw [Matrix.det_apply, Finset.univ_perm_fin_succ, ← ... | Mathlib.LinearAlgebra.Matrix.Determinant.709_0.U1f6HO8zRbnvZ95 | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along column 0. -/
theorem det_succ_column_zero {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) :
det A = ∑ i : Fin n.succ, (-1) ^ (i : ℕ) * A i 0 * det (A.submatrix i.succAbove Fin.succ) | 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
A : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R
i✝ : Fin (Nat.succ n)
x✝¹ : i✝ ∈ univ
i : Fin n
this : (-1) ^ ↑i = ↑↑(sign (Fin.cycleRange i))
σ : Perm (... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rw [Equiv.Perm.decomposeFin.symm_sign, if_neg (Fin.succ_ne_zero i)] | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along column 0. -/
theorem det_succ_column_zero {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) :
det A = ∑ i : Fin n.succ, (-1) ^ (i : ℕ) * A i 0 * det (A.submatrix i.succAbove Fin.succ) := by
rw [Matrix.det_apply, Finset.univ_perm_fin_succ, ← ... | Mathlib.LinearAlgebra.Matrix.Determinant.709_0.U1f6HO8zRbnvZ95 | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along column 0. -/
theorem det_succ_column_zero {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) :
det A = ∑ i : Fin n.succ, (-1) ^ (i : ℕ) * A i 0 * det (A.submatrix i.succAbove Fin.succ) | 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
A : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R
i✝ : Fin (Nat.succ n)
x✝¹ : i✝ ∈ univ
i : Fin n
this : (-1) ^ ↑i = ↑↑(sign (Fin.cycleRange i))
σ : Perm (... | /-
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... | calc
((-1 * Perm.sign σ : ℤ) • ∏ i', A (Perm.decomposeFin.symm (Fin.succ i, σ) i') i') =
(-1 * Perm.sign σ : ℤ) • (A (Fin.succ i) 0 *
∏ i', A ((Fin.succ i).succAbove (Fin.cycleRange i (σ i'))) i'.succ) := by
simp only [Fin.prod_univ_succ, Fin.succAbove_cycleRange,
Equiv.Perm.decompos... | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along column 0. -/
theorem det_succ_column_zero {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) :
det A = ∑ i : Fin n.succ, (-1) ^ (i : ℕ) * A i 0 * det (A.submatrix i.succAbove Fin.succ) := by
rw [Matrix.det_apply, Finset.univ_perm_fin_succ, ← ... | Mathlib.LinearAlgebra.Matrix.Determinant.709_0.U1f6HO8zRbnvZ95 | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along column 0. -/
theorem det_succ_column_zero {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) :
det A = ∑ i : Fin n.succ, (-1) ^ (i : ℕ) * A i 0 * det (A.submatrix i.succAbove Fin.succ) | 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 : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R
i✝ : Fin (Nat.succ n)
x✝¹ : i✝ ∈ univ
i : Fin n
this : (-1) ^ ↑i = ↑↑(sign (Fin.cycleRange i))
σ : Perm (Fin n)
x✝ : σ ∈... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | simp only [Fin.prod_univ_succ, Fin.succAbove_cycleRange,
Equiv.Perm.decomposeFin_symm_apply_zero, Equiv.Perm.decomposeFin_symm_apply_succ] | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along column 0. -/
theorem det_succ_column_zero {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) :
det A = ∑ i : Fin n.succ, (-1) ^ (i : ℕ) * A i 0 * det (A.submatrix i.succAbove Fin.succ) := by
rw [Matrix.det_apply, Finset.univ_perm_fin_succ, ← ... | Mathlib.LinearAlgebra.Matrix.Determinant.709_0.U1f6HO8zRbnvZ95 | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along column 0. -/
theorem det_succ_column_zero {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) :
det A = ∑ i : Fin n.succ, (-1) ^ (i : ℕ) * A i 0 * det (A.submatrix i.succAbove Fin.succ) | 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 : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R
i✝ : Fin (Nat.succ n)
x✝¹ : i✝ ∈ univ
i : Fin n
this : (-1) ^ ↑i = ↑↑(sign (Fin.cycleRange i))
σ : Perm (Fin n)
x✝ : σ ∈... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | simp [mul_assoc, mul_comm, _root_.neg_mul, one_mul, zsmul_eq_mul, neg_inj, neg_smul,
Fin.succAbove_cycleRange, mul_left_comm] | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along column 0. -/
theorem det_succ_column_zero {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) :
det A = ∑ i : Fin n.succ, (-1) ^ (i : ℕ) * A i 0 * det (A.submatrix i.succAbove Fin.succ) := by
rw [Matrix.det_apply, Finset.univ_perm_fin_succ, ← ... | Mathlib.LinearAlgebra.Matrix.Determinant.709_0.U1f6HO8zRbnvZ95 | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along column 0. -/
theorem det_succ_column_zero {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) :
det A = ∑ i : Fin n.succ, (-1) ^ (i : ℕ) * A i 0 * det (A.submatrix i.succAbove Fin.succ) | 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 : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R
⊢ det A = ∑ j : Fin (Nat.succ n), (-1) ^ ↑j * A 0 j * det (submatrix A Fin.succ (Fin.succAbove j)) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rw [← det_transpose A, det_succ_column_zero] | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along row 0. -/
theorem det_succ_row_zero {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) :
det A = ∑ j : Fin n.succ, (-1) ^ (j : ℕ) * A 0 j * det (A.submatrix Fin.succ j.succAbove) := by
| Mathlib.LinearAlgebra.Matrix.Determinant.741_0.U1f6HO8zRbnvZ95 | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along row 0. -/
theorem det_succ_row_zero {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) :
det A = ∑ j : Fin n.succ, (-1) ^ (j : ℕ) * A 0 j * det (A.submatrix Fin.succ j.succAbove) | 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 : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R
⊢ ∑ i : Fin (Nat.succ n), (-1) ^ ↑i * Aᵀ i 0 * det (submatrix Aᵀ (Fin.succAbove i) Fin.succ) =
∑ j : Fin (Nat.succ n... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | refine' Finset.sum_congr rfl fun i _ => _ | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along row 0. -/
theorem det_succ_row_zero {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) :
det A = ∑ j : Fin n.succ, (-1) ^ (j : ℕ) * A 0 j * det (A.submatrix Fin.succ j.succAbove) := by
rw [← det_transpose A, det_succ_column_zero]
| Mathlib.LinearAlgebra.Matrix.Determinant.741_0.U1f6HO8zRbnvZ95 | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along row 0. -/
theorem det_succ_row_zero {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) :
det A = ∑ j : Fin n.succ, (-1) ^ (j : ℕ) * A 0 j * det (A.submatrix Fin.succ j.succAbove) | 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 : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R
i : Fin (Nat.succ n)
x✝ : i ∈ univ
⊢ (-1) ^ ↑i * Aᵀ i 0 * det (submatrix Aᵀ (Fin.succAbove i) Fin.succ) =
(-1) ^ ↑i ... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rw [← det_transpose] | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along row 0. -/
theorem det_succ_row_zero {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) :
det A = ∑ j : Fin n.succ, (-1) ^ (j : ℕ) * A 0 j * det (A.submatrix Fin.succ j.succAbove) := by
rw [← det_transpose A, det_succ_column_zero]
refine' Fi... | Mathlib.LinearAlgebra.Matrix.Determinant.741_0.U1f6HO8zRbnvZ95 | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along row 0. -/
theorem det_succ_row_zero {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) :
det A = ∑ j : Fin n.succ, (-1) ^ (j : ℕ) * A 0 j * det (A.submatrix Fin.succ j.succAbove) | 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 : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R
i : Fin (Nat.succ n)
x✝ : i ∈ univ
⊢ (-1) ^ ↑i * Aᵀ i 0 * det (submatrix Aᵀ (Fin.succAbove i) Fin.succ)ᵀ =
(-1) ^ ↑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 [transpose_apply, transpose_submatrix, transpose_transpose] | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along row 0. -/
theorem det_succ_row_zero {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) :
det A = ∑ j : Fin n.succ, (-1) ^ (j : ℕ) * A 0 j * det (A.submatrix Fin.succ j.succAbove) := by
rw [← det_transpose A, det_succ_column_zero]
refine' Fi... | Mathlib.LinearAlgebra.Matrix.Determinant.741_0.U1f6HO8zRbnvZ95 | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along row 0. -/
theorem det_succ_row_zero {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) :
det A = ∑ j : Fin n.succ, (-1) ^ (j : ℕ) * A 0 j * det (A.submatrix Fin.succ j.succAbove) | 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 : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R
i : Fin (Nat.succ n)
⊢ det A = ∑ j : Fin (Nat.succ n), (-1) ^ (↑i + ↑j) * A i j * det (submatrix A (Fin.succAbove i) (Fi... | /-
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 [pow_add, mul_assoc, ← mul_sum] | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along row `i`. -/
theorem det_succ_row {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) (i : Fin n.succ) :
det A =
∑ j : Fin n.succ, (-1) ^ (i + j : ℕ) * A i j * det (A.submatrix i.succAbove j.succAbove) := by
| Mathlib.LinearAlgebra.Matrix.Determinant.750_0.U1f6HO8zRbnvZ95 | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along row `i`. -/
theorem det_succ_row {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) (i : Fin n.succ) :
det A =
∑ j : Fin n.succ, (-1) ^ (i + j : ℕ) * A i j * det (A.submatrix i.succAbove j.succAbove) | 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 : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R
i : Fin (Nat.succ n)
⊢ det A =
(-1) ^ ↑i * ∑ x : Fin (Nat.succ n), (-1) ^ ↑x * (A i x * det (submatrix A (Fin.succAb... | /-
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 : det A = (-1 : R) ^ (i : ℕ) * (Perm.sign i.cycleRange⁻¹) * det A := by
calc
det A = ↑((-1 : ℤˣ) ^ (i : ℕ) * (-1 : ℤˣ) ^ (i : ℕ) : ℤˣ) * det A := by simp
_ = (-1 : R) ^ (i : ℕ) * (Perm.sign i.cycleRange⁻¹) * det A := by simp [-Int.units_mul_self] | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along row `i`. -/
theorem det_succ_row {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) (i : Fin n.succ) :
det A =
∑ j : Fin n.succ, (-1) ^ (i + j : ℕ) * A i j * det (A.submatrix i.succAbove j.succAbove) := by
simp_rw [pow_add, mul_assoc, ←... | Mathlib.LinearAlgebra.Matrix.Determinant.750_0.U1f6HO8zRbnvZ95 | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along row `i`. -/
theorem det_succ_row {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) (i : Fin n.succ) :
det A =
∑ j : Fin n.succ, (-1) ^ (i + j : ℕ) * A i j * det (A.submatrix i.succAbove j.succAbove) | 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 : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R
i : Fin (Nat.succ n)
⊢ det A = (-1) ^ ↑i * ↑↑(sign (Fin.cycleRange i)⁻¹) * det A | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | calc
det A = ↑((-1 : ℤˣ) ^ (i : ℕ) * (-1 : ℤˣ) ^ (i : ℕ) : ℤˣ) * det A := by simp
_ = (-1 : R) ^ (i : ℕ) * (Perm.sign i.cycleRange⁻¹) * det A := by simp [-Int.units_mul_self] | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along row `i`. -/
theorem det_succ_row {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) (i : Fin n.succ) :
det A =
∑ j : Fin n.succ, (-1) ^ (i + j : ℕ) * A i j * det (A.submatrix i.succAbove j.succAbove) := by
simp_rw [pow_add, mul_assoc, ←... | Mathlib.LinearAlgebra.Matrix.Determinant.750_0.U1f6HO8zRbnvZ95 | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along row `i`. -/
theorem det_succ_row {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) (i : Fin n.succ) :
det A =
∑ j : Fin n.succ, (-1) ^ (i + j : ℕ) * A i j * det (A.submatrix i.succAbove j.succAbove) | 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 : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R
i : Fin (Nat.succ n)
⊢ det A = ↑↑((-1) ^ ↑i * (-1) ^ ↑i) * det A | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | simp | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along row `i`. -/
theorem det_succ_row {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) (i : Fin n.succ) :
det A =
∑ j : Fin n.succ, (-1) ^ (i + j : ℕ) * A i j * det (A.submatrix i.succAbove j.succAbove) := by
simp_rw [pow_add, mul_assoc, ←... | Mathlib.LinearAlgebra.Matrix.Determinant.750_0.U1f6HO8zRbnvZ95 | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along row `i`. -/
theorem det_succ_row {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) (i : Fin n.succ) :
det A =
∑ j : Fin n.succ, (-1) ^ (i + j : ℕ) * A i j * det (A.submatrix i.succAbove j.succAbove) | 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 : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R
i : Fin (Nat.succ n)
⊢ ↑↑((-1) ^ ↑i * (-1) ^ ↑i) * det A = (-1) ^ ↑i * ↑↑(sign (Fin.cycleRange i)⁻¹) * det A | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | simp [-Int.units_mul_self] | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along row `i`. -/
theorem det_succ_row {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) (i : Fin n.succ) :
det A =
∑ j : Fin n.succ, (-1) ^ (i + j : ℕ) * A i j * det (A.submatrix i.succAbove j.succAbove) := by
simp_rw [pow_add, mul_assoc, ←... | Mathlib.LinearAlgebra.Matrix.Determinant.750_0.U1f6HO8zRbnvZ95 | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along row `i`. -/
theorem det_succ_row {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) (i : Fin n.succ) :
det A =
∑ j : Fin n.succ, (-1) ^ (i + j : ℕ) * A i j * det (A.submatrix i.succAbove j.succAbove) | 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 : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R
i : Fin (Nat.succ n)
this : det A = (-1) ^ ↑i * ↑↑(sign (Fin.cycleRange i)⁻¹) * det A
⊢ det A =
(-1) ^ ↑i * ∑ x : Fi... | /-
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 [this, mul_assoc] | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along row `i`. -/
theorem det_succ_row {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) (i : Fin n.succ) :
det A =
∑ j : Fin n.succ, (-1) ^ (i + j : ℕ) * A i j * det (A.submatrix i.succAbove j.succAbove) := by
simp_rw [pow_add, mul_assoc, ←... | Mathlib.LinearAlgebra.Matrix.Determinant.750_0.U1f6HO8zRbnvZ95 | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along row `i`. -/
theorem det_succ_row {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) (i : Fin n.succ) :
det A =
∑ j : Fin n.succ, (-1) ^ (i + j : ℕ) * A i j * det (A.submatrix i.succAbove j.succAbove) | 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 : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R
i : Fin (Nat.succ n)
this : det A = (-1) ^ ↑i * ↑↑(sign (Fin.cycleRange i)⁻¹) * det A
⊢ (-1) ^ ↑i * (↑↑(sign (Fin.cycleR... | /-
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 | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along row `i`. -/
theorem det_succ_row {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) (i : Fin n.succ) :
det A =
∑ j : Fin n.succ, (-1) ^ (i + j : ℕ) * A i j * det (A.submatrix i.succAbove j.succAbove) := by
simp_rw [pow_add, mul_assoc, ←... | Mathlib.LinearAlgebra.Matrix.Determinant.750_0.U1f6HO8zRbnvZ95 | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along row `i`. -/
theorem det_succ_row {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) (i : Fin n.succ) :
det A =
∑ j : Fin n.succ, (-1) ^ (i + j : ℕ) * A i j * det (A.submatrix i.succAbove j.succAbove) | Mathlib_LinearAlgebra_Matrix_Determinant |
case e_a
m : Type u_1
n✝ : Type u_2
inst✝⁴ : DecidableEq n✝
inst✝³ : Fintype n✝
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
n : ℕ
A : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R
i : Fin (Nat.succ n)
this : det A = (-1) ^ ↑i * ↑↑(sign (Fin.cycleRange i)⁻¹) * det A
⊢ ↑↑(sign (Fin.cycleRange... | /-
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_permute, det_succ_row_zero] | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along row `i`. -/
theorem det_succ_row {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) (i : Fin n.succ) :
det A =
∑ j : Fin n.succ, (-1) ^ (i + j : ℕ) * A i j * det (A.submatrix i.succAbove j.succAbove) := by
simp_rw [pow_add, mul_assoc, ←... | Mathlib.LinearAlgebra.Matrix.Determinant.750_0.U1f6HO8zRbnvZ95 | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along row `i`. -/
theorem det_succ_row {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) (i : Fin n.succ) :
det A =
∑ j : Fin n.succ, (-1) ^ (i + j : ℕ) * A i j * det (A.submatrix i.succAbove j.succAbove) | Mathlib_LinearAlgebra_Matrix_Determinant |
case e_a
m : Type u_1
n✝ : Type u_2
inst✝⁴ : DecidableEq n✝
inst✝³ : Fintype n✝
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
n : ℕ
A : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R
i : Fin (Nat.succ n)
this : det A = (-1) ^ ↑i * ↑↑(sign (Fin.cycleRange i)⁻¹) * det A
⊢ ∑ j : Fin (Nat.succ n),... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | refine' Finset.sum_congr rfl fun j _ => _ | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along row `i`. -/
theorem det_succ_row {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) (i : Fin n.succ) :
det A =
∑ j : Fin n.succ, (-1) ^ (i + j : ℕ) * A i j * det (A.submatrix i.succAbove j.succAbove) := by
simp_rw [pow_add, mul_assoc, ←... | Mathlib.LinearAlgebra.Matrix.Determinant.750_0.U1f6HO8zRbnvZ95 | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along row `i`. -/
theorem det_succ_row {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) (i : Fin n.succ) :
det A =
∑ j : Fin n.succ, (-1) ^ (i + j : ℕ) * A i j * det (A.submatrix i.succAbove j.succAbove) | Mathlib_LinearAlgebra_Matrix_Determinant |
case e_a
m : Type u_1
n✝ : Type u_2
inst✝⁴ : DecidableEq n✝
inst✝³ : Fintype n✝
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
n : ℕ
A : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R
i : Fin (Nat.succ n)
this : det A = (-1) ^ ↑i * ↑↑(sign (Fin.cycleRange i)⁻¹) * det A
j : Fin (Nat.succ n)
x✝ :... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rw [mul_assoc, Matrix.submatrix, Matrix.submatrix] | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along row `i`. -/
theorem det_succ_row {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) (i : Fin n.succ) :
det A =
∑ j : Fin n.succ, (-1) ^ (i + j : ℕ) * A i j * det (A.submatrix i.succAbove j.succAbove) := by
simp_rw [pow_add, mul_assoc, ←... | Mathlib.LinearAlgebra.Matrix.Determinant.750_0.U1f6HO8zRbnvZ95 | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along row `i`. -/
theorem det_succ_row {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) (i : Fin n.succ) :
det A =
∑ j : Fin n.succ, (-1) ^ (i + j : ℕ) * A i j * det (A.submatrix i.succAbove j.succAbove) | Mathlib_LinearAlgebra_Matrix_Determinant |
case e_a
m : Type u_1
n✝ : Type u_2
inst✝⁴ : DecidableEq n✝
inst✝³ : Fintype n✝
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
n : ℕ
A : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R
i : Fin (Nat.succ n)
this : det A = (-1) ^ ↑i * ↑↑(sign (Fin.cycleRange i)⁻¹) * det A
j : Fin (Nat.succ n)
x✝ :... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | congr | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along row `i`. -/
theorem det_succ_row {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) (i : Fin n.succ) :
det A =
∑ j : Fin n.succ, (-1) ^ (i + j : ℕ) * A i j * det (A.submatrix i.succAbove j.succAbove) := by
simp_rw [pow_add, mul_assoc, ←... | Mathlib.LinearAlgebra.Matrix.Determinant.750_0.U1f6HO8zRbnvZ95 | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along row `i`. -/
theorem det_succ_row {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) (i : Fin n.succ) :
det A =
∑ j : Fin n.succ, (-1) ^ (i + j : ℕ) * A i j * det (A.submatrix i.succAbove j.succAbove) | Mathlib_LinearAlgebra_Matrix_Determinant |
case e_a.e_a.e_a.e_a
m : Type u_1
n✝ : Type u_2
inst✝⁴ : DecidableEq n✝
inst✝³ : Fintype n✝
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
n : ℕ
A : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R
i : Fin (Nat.succ n)
this : det A = (-1) ^ ↑i * ↑↑(sign (Fin.cycleRange i)⁻¹) * det A
j : Fin (Nat.... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rw [Equiv.Perm.inv_def, Fin.cycleRange_symm_zero] | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along row `i`. -/
theorem det_succ_row {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) (i : Fin n.succ) :
det A =
∑ j : Fin n.succ, (-1) ^ (i + j : ℕ) * A i j * det (A.submatrix i.succAbove j.succAbove) := by
simp_rw [pow_add, mul_assoc, ←... | Mathlib.LinearAlgebra.Matrix.Determinant.750_0.U1f6HO8zRbnvZ95 | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along row `i`. -/
theorem det_succ_row {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) (i : Fin n.succ) :
det A =
∑ j : Fin n.succ, (-1) ^ (i + j : ℕ) * A i j * det (A.submatrix i.succAbove j.succAbove) | Mathlib_LinearAlgebra_Matrix_Determinant |
case e_a.e_a.e_a.e_M.h.e_6.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
A : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R
i : Fin (Nat.succ n)
this : det A = (-1) ^ ↑i * ↑↑(sign (Fin.cycleRange i)⁻¹) * det A
j : F... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | ext i' j' | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along row `i`. -/
theorem det_succ_row {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) (i : Fin n.succ) :
det A =
∑ j : Fin n.succ, (-1) ^ (i + j : ℕ) * A i j * det (A.submatrix i.succAbove j.succAbove) := by
simp_rw [pow_add, mul_assoc, ←... | Mathlib.LinearAlgebra.Matrix.Determinant.750_0.U1f6HO8zRbnvZ95 | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along row `i`. -/
theorem det_succ_row {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) (i : Fin n.succ) :
det A =
∑ j : Fin n.succ, (-1) ^ (i + j : ℕ) * A i j * det (A.submatrix i.succAbove j.succAbove) | Mathlib_LinearAlgebra_Matrix_Determinant |
case e_a.e_a.e_a.e_M.h.e_6.h.h.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
A : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R
i : Fin (Nat.succ n)
this : det A = (-1) ^ ↑i * ↑↑(sign (Fin.cycleRange i)⁻¹) * det A
j... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rw [Equiv.Perm.inv_def, Fin.cycleRange_symm_succ] | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along row `i`. -/
theorem det_succ_row {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) (i : Fin n.succ) :
det A =
∑ j : Fin n.succ, (-1) ^ (i + j : ℕ) * A i j * det (A.submatrix i.succAbove j.succAbove) := by
simp_rw [pow_add, mul_assoc, ←... | Mathlib.LinearAlgebra.Matrix.Determinant.750_0.U1f6HO8zRbnvZ95 | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along row `i`. -/
theorem det_succ_row {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) (i : Fin n.succ) :
det A =
∑ j : Fin n.succ, (-1) ^ (i + j : ℕ) * A i j * det (A.submatrix i.succAbove j.succAbove) | 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 : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R
j : Fin (Nat.succ n)
⊢ det A = ∑ i : Fin (Nat.succ n), (-1) ^ (↑i + ↑j) * A i j * det (submatrix A (Fin.succAbove i) (Fi... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rw [← det_transpose, det_succ_row _ j] | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along column `j`. -/
theorem det_succ_column {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) (j : Fin n.succ) :
det A =
∑ i : Fin n.succ, (-1) ^ (i + j : ℕ) * A i j * det (A.submatrix i.succAbove j.succAbove) := by
| Mathlib.LinearAlgebra.Matrix.Determinant.770_0.U1f6HO8zRbnvZ95 | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along column `j`. -/
theorem det_succ_column {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) (j : Fin n.succ) :
det A =
∑ i : Fin n.succ, (-1) ^ (i + j : ℕ) * A i j * det (A.submatrix i.succAbove j.succAbove) | 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 : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R
j : Fin (Nat.succ n)
⊢ ∑ j_1 : Fin (Nat.succ n), (-1) ^ (↑j + ↑j_1) * Aᵀ j j_1 * det (submatrix Aᵀ (Fin.succAbove j) (Fi... | /-
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' Finset.sum_congr rfl fun i _ => _ | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along column `j`. -/
theorem det_succ_column {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) (j : Fin n.succ) :
det A =
∑ i : Fin n.succ, (-1) ^ (i + j : ℕ) * A i j * det (A.submatrix i.succAbove j.succAbove) := by
rw [← det_transpose, det... | Mathlib.LinearAlgebra.Matrix.Determinant.770_0.U1f6HO8zRbnvZ95 | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along column `j`. -/
theorem det_succ_column {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) (j : Fin n.succ) :
det A =
∑ i : Fin n.succ, (-1) ^ (i + j : ℕ) * A i j * det (A.submatrix i.succAbove j.succAbove) | 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 : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R
j i : Fin (Nat.succ n)
x✝ : i ∈ univ
⊢ (-1) ^ (↑j + ↑i) * Aᵀ j i * det (submatrix Aᵀ (Fin.succAbove j) (Fin.succAbove i)... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | rw [add_comm, ← det_transpose, transpose_apply, transpose_submatrix, transpose_transpose] | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along column `j`. -/
theorem det_succ_column {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) (j : Fin n.succ) :
det A =
∑ i : Fin n.succ, (-1) ^ (i + j : ℕ) * A i j * det (A.submatrix i.succAbove j.succAbove) := by
rw [← det_transpose, det... | Mathlib.LinearAlgebra.Matrix.Determinant.770_0.U1f6HO8zRbnvZ95 | /-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along column `j`. -/
theorem det_succ_column {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) (j : Fin n.succ) :
det A =
∑ i : Fin n.succ, (-1) ^ (i + j : ℕ) * A i j * det (A.submatrix i.succAbove j.succAbove) | 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 (Fin 2) (Fin 2) R
⊢ det A = A 0 0 * A 1 1 - A 0 1 * A 1 0 | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | simp only [det_succ_row_zero, det_unique, Fin.default_eq_zero, submatrix_apply,
Fin.succ_zero_eq_one, Fin.sum_univ_succ, Fin.val_zero, Fin.zero_succAbove, univ_unique,
Fin.val_succ, Fin.coe_fin_one, Fin.succ_succAbove_zero, sum_singleton] | /-- Determinant of 2x2 matrix -/
theorem det_fin_two (A : Matrix (Fin 2) (Fin 2) R) : det A = A 0 0 * A 1 1 - A 0 1 * A 1 0 := by
| Mathlib.LinearAlgebra.Matrix.Determinant.794_0.U1f6HO8zRbnvZ95 | /-- Determinant of 2x2 matrix -/
theorem det_fin_two (A : Matrix (Fin 2) (Fin 2) R) : det A = A 0 0 * A 1 1 - A 0 1 * A 1 0 | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A : Matrix (Fin 2) (Fin 2) R
⊢ (-1) ^ 0 * A 0 0 * A 1 1 + (-1) ^ (0 + 1) * A 0 1 * A 1 0 = A 0 0 * A 1 1 - A 0 1 * A 1 0 | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | ring | /-- Determinant of 2x2 matrix -/
theorem det_fin_two (A : Matrix (Fin 2) (Fin 2) R) : det A = A 0 0 * A 1 1 - A 0 1 * A 1 0 := by
simp only [det_succ_row_zero, det_unique, Fin.default_eq_zero, submatrix_apply,
Fin.succ_zero_eq_one, Fin.sum_univ_succ, Fin.val_zero, Fin.zero_succAbove, univ_unique,
Fin.val_succ... | Mathlib.LinearAlgebra.Matrix.Determinant.794_0.U1f6HO8zRbnvZ95 | /-- Determinant of 2x2 matrix -/
theorem det_fin_two (A : Matrix (Fin 2) (Fin 2) R) : det A = A 0 0 * A 1 1 - A 0 1 * A 1 0 | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A : Matrix (Fin 3) (Fin 3) R
⊢ det A =
A 0 0 * A 1 1 * A 2 2 - A 0 0 * A 1 2 * A 2 1 - A 0 1 * A 1 0 * A 2 2 + A 0 1 * A 1 2 * A 2 0 +
A 0 2 * A 1 0 * A 2 1 -
A 0... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fintype.BigOperators
impor... | simp only [det_succ_row_zero, Nat.odd_iff_not_even, submatrix_apply, Fin.succ_zero_eq_one,
submatrix_submatrix, det_unique, Fin.default_eq_zero, comp_apply, Fin.succ_one_eq_two,
Fin.sum_univ_succ, Fin.val_zero, Fin.zero_succAbove, univ_unique, Fin.val_succ,
Fin.coe_fin_one, Fin.succ_succAbove_zero, sum_sing... | /-- Determinant of 3x3 matrix -/
theorem det_fin_three (A : Matrix (Fin 3) (Fin 3) R) :
det A =
A 0 0 * A 1 1 * A 2 2 - A 0 0 * A 1 2 * A 2 1
- A 0 1 * A 1 0 * A 2 2 + A 0 1 * A 1 2 * A 2 0
+ A 0 2 * A 1 0 * A 2 1 - A 0 2 * A 1 1 * A 2 0 := by
| Mathlib.LinearAlgebra.Matrix.Determinant.807_0.U1f6HO8zRbnvZ95 | /-- Determinant of 3x3 matrix -/
theorem det_fin_three (A : Matrix (Fin 3) (Fin 3) R) :
det A =
A 0 0 * A 1 1 * A 2 2 - A 0 0 * A 1 2 * A 2 1
- A 0 1 * A 1 0 * A 2 2 + A 0 1 * A 1 2 * A 2 0
+ A 0 2 * A 1 0 * A 2 1 - A 0 2 * A 1 1 * A 2 0 | Mathlib_LinearAlgebra_Matrix_Determinant |
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A : Matrix (Fin 3) (Fin 3) R
⊢ (-1) ^ 0 * A 0 0 * ((-1) ^ 0 * A 1 1 * A 2 2 + (-1) ^ (0 + 1) * A 1 2 * A 2 1) +
((-1) ^ (0 + 1) * A 0 1 * ((-1) ^ 0 * A 1 0 * A 2 2 + (-1) ^ (... | /-
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... | ring | /-- Determinant of 3x3 matrix -/
theorem det_fin_three (A : Matrix (Fin 3) (Fin 3) R) :
det A =
A 0 0 * A 1 1 * A 2 2 - A 0 0 * A 1 2 * A 2 1
- A 0 1 * A 1 0 * A 2 2 + A 0 1 * A 1 2 * A 2 0
+ A 0 2 * A 1 0 * A 2 1 - A 0 2 * A 1 1 * A 2 0 := by
simp only [det_succ_row_zero, Nat.odd_iff_not_even, ... | Mathlib.LinearAlgebra.Matrix.Determinant.807_0.U1f6HO8zRbnvZ95 | /-- Determinant of 3x3 matrix -/
theorem det_fin_three (A : Matrix (Fin 3) (Fin 3) R) :
det A =
A 0 0 * A 1 1 * A 2 2 - A 0 0 * A 1 2 * A 2 1
- A 0 1 * A 1 0 * A 2 2 + A 0 1 * A 1 2 * A 2 0
+ A 0 2 * A 1 0 * A 2 1 - A 0 2 * A 1 1 * A 2 0 | Mathlib_LinearAlgebra_Matrix_Determinant |
A : Type v
inst✝ : Ring A
x✝² x✝¹ x✝ : A
⊢ ⁅x✝² + x✝¹, x✝⁆ = ⁅x✝², x✝⁆ + ⁅x✝¹, x✝⁆ | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | simp only [Ring.lie_def, right_distrib, left_distrib] | /-- An associative ring gives rise to a Lie ring by taking the bracket to be the ring commutator. -/
instance (priority := 100) ofAssociativeRing : LieRing A where
add_lie _ _ _ := by | Mathlib.Algebra.Lie.OfAssociative.67_0.ll51mLev4p7Z1wP | /-- An associative ring gives rise to a Lie ring by taking the bracket to be the ring commutator. -/
instance (priority | Mathlib_Algebra_Lie_OfAssociative |
A : Type v
inst✝ : Ring A
x✝² x✝¹ x✝ : A
⊢ x✝² * x✝ + x✝¹ * x✝ - (x✝ * x✝² + x✝ * x✝¹) = x✝² * x✝ - x✝ * x✝² + (x✝¹ * x✝ - x✝ * x✝¹) | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | abel | /-- An associative ring gives rise to a Lie ring by taking the bracket to be the ring commutator. -/
instance (priority := 100) ofAssociativeRing : LieRing A where
add_lie _ _ _ := by simp only [Ring.lie_def, right_distrib, left_distrib]; | Mathlib.Algebra.Lie.OfAssociative.67_0.ll51mLev4p7Z1wP | /-- An associative ring gives rise to a Lie ring by taking the bracket to be the ring commutator. -/
instance (priority | Mathlib_Algebra_Lie_OfAssociative |
A : Type v
inst✝ : Ring A
x✝² x✝¹ x✝ : A
⊢ x✝² * x✝ + x✝¹ * x✝ - (x✝ * x✝² + x✝ * x✝¹) = x✝² * x✝ - x✝ * x✝² + (x✝¹ * x✝ - x✝ * x✝¹) | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | abel | /-- An associative ring gives rise to a Lie ring by taking the bracket to be the ring commutator. -/
instance (priority := 100) ofAssociativeRing : LieRing A where
add_lie _ _ _ := by simp only [Ring.lie_def, right_distrib, left_distrib]; | Mathlib.Algebra.Lie.OfAssociative.67_0.ll51mLev4p7Z1wP | /-- An associative ring gives rise to a Lie ring by taking the bracket to be the ring commutator. -/
instance (priority | Mathlib_Algebra_Lie_OfAssociative |
A : Type v
inst✝ : Ring A
x✝² x✝¹ x✝ : A
⊢ ⁅x✝², x✝¹ + x✝⁆ = ⁅x✝², x✝¹⁆ + ⁅x✝², x✝⁆ | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | simp only [Ring.lie_def, right_distrib, left_distrib] | /-- An associative ring gives rise to a Lie ring by taking the bracket to be the ring commutator. -/
instance (priority := 100) ofAssociativeRing : LieRing A where
add_lie _ _ _ := by simp only [Ring.lie_def, right_distrib, left_distrib]; abel
lie_add _ _ _ := by | Mathlib.Algebra.Lie.OfAssociative.67_0.ll51mLev4p7Z1wP | /-- An associative ring gives rise to a Lie ring by taking the bracket to be the ring commutator. -/
instance (priority | Mathlib_Algebra_Lie_OfAssociative |
A : Type v
inst✝ : Ring A
x✝² x✝¹ x✝ : A
⊢ x✝² * x✝¹ + x✝² * x✝ - (x✝¹ * x✝² + x✝ * x✝²) = x✝² * x✝¹ - x✝¹ * x✝² + (x✝² * x✝ - x✝ * x✝²) | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | abel | /-- An associative ring gives rise to a Lie ring by taking the bracket to be the ring commutator. -/
instance (priority := 100) ofAssociativeRing : LieRing A where
add_lie _ _ _ := by simp only [Ring.lie_def, right_distrib, left_distrib]; abel
lie_add _ _ _ := by simp only [Ring.lie_def, right_distrib, left_distrib... | Mathlib.Algebra.Lie.OfAssociative.67_0.ll51mLev4p7Z1wP | /-- An associative ring gives rise to a Lie ring by taking the bracket to be the ring commutator. -/
instance (priority | Mathlib_Algebra_Lie_OfAssociative |
A : Type v
inst✝ : Ring A
x✝² x✝¹ x✝ : A
⊢ x✝² * x✝¹ + x✝² * x✝ - (x✝¹ * x✝² + x✝ * x✝²) = x✝² * x✝¹ - x✝¹ * x✝² + (x✝² * x✝ - x✝ * x✝²) | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | abel | /-- An associative ring gives rise to a Lie ring by taking the bracket to be the ring commutator. -/
instance (priority := 100) ofAssociativeRing : LieRing A where
add_lie _ _ _ := by simp only [Ring.lie_def, right_distrib, left_distrib]; abel
lie_add _ _ _ := by simp only [Ring.lie_def, right_distrib, left_distrib... | Mathlib.Algebra.Lie.OfAssociative.67_0.ll51mLev4p7Z1wP | /-- An associative ring gives rise to a Lie ring by taking the bracket to be the ring commutator. -/
instance (priority | Mathlib_Algebra_Lie_OfAssociative |
A : Type v
inst✝ : Ring A
⊢ ∀ (x : A), ⁅x, x⁆ = 0 | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | simp only [Ring.lie_def, forall_const, sub_self] | /-- An associative ring gives rise to a Lie ring by taking the bracket to be the ring commutator. -/
instance (priority := 100) ofAssociativeRing : LieRing A where
add_lie _ _ _ := by simp only [Ring.lie_def, right_distrib, left_distrib]; abel
lie_add _ _ _ := by simp only [Ring.lie_def, right_distrib, left_distrib... | Mathlib.Algebra.Lie.OfAssociative.67_0.ll51mLev4p7Z1wP | /-- An associative ring gives rise to a Lie ring by taking the bracket to be the ring commutator. -/
instance (priority | Mathlib_Algebra_Lie_OfAssociative |
A : Type v
inst✝ : Ring A
x✝² x✝¹ x✝ : A
⊢ ⁅x✝², ⁅x✝¹, x✝⁆⁆ = ⁅⁅x✝², x✝¹⁆, x✝⁆ + ⁅x✝¹, ⁅x✝², x✝⁆⁆ | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | simp only [Ring.lie_def, mul_sub_left_distrib, mul_sub_right_distrib, mul_assoc] | /-- An associative ring gives rise to a Lie ring by taking the bracket to be the ring commutator. -/
instance (priority := 100) ofAssociativeRing : LieRing A where
add_lie _ _ _ := by simp only [Ring.lie_def, right_distrib, left_distrib]; abel
lie_add _ _ _ := by simp only [Ring.lie_def, right_distrib, left_distrib... | Mathlib.Algebra.Lie.OfAssociative.67_0.ll51mLev4p7Z1wP | /-- An associative ring gives rise to a Lie ring by taking the bracket to be the ring commutator. -/
instance (priority | Mathlib_Algebra_Lie_OfAssociative |
A : Type v
inst✝ : Ring A
x✝² x✝¹ x✝ : A
⊢ x✝² * (x✝¹ * x✝) - x✝² * (x✝ * x✝¹) - (x✝¹ * (x✝ * x✝²) - x✝ * (x✝¹ * x✝²)) =
x✝² * (x✝¹ * x✝) - x✝¹ * (x✝² * x✝) - (x✝ * (x✝² * x✝¹) - x✝ * (x✝¹ * x✝²)) +
(x✝¹ * (x✝² * x✝) - x✝¹ * (x✝ * x✝²) - (x✝² * (x✝ * x✝¹) - x✝ * (x✝² * x✝¹))) | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | abel | /-- An associative ring gives rise to a Lie ring by taking the bracket to be the ring commutator. -/
instance (priority := 100) ofAssociativeRing : LieRing A where
add_lie _ _ _ := by simp only [Ring.lie_def, right_distrib, left_distrib]; abel
lie_add _ _ _ := by simp only [Ring.lie_def, right_distrib, left_distrib... | Mathlib.Algebra.Lie.OfAssociative.67_0.ll51mLev4p7Z1wP | /-- An associative ring gives rise to a Lie ring by taking the bracket to be the ring commutator. -/
instance (priority | Mathlib_Algebra_Lie_OfAssociative |
A : Type v
inst✝ : Ring A
x✝² x✝¹ x✝ : A
⊢ x✝² * (x✝¹ * x✝) - x✝² * (x✝ * x✝¹) - (x✝¹ * (x✝ * x✝²) - x✝ * (x✝¹ * x✝²)) =
x✝² * (x✝¹ * x✝) - x✝¹ * (x✝² * x✝) - (x✝ * (x✝² * x✝¹) - x✝ * (x✝¹ * x✝²)) +
(x✝¹ * (x✝² * x✝) - x✝¹ * (x✝ * x✝²) - (x✝² * (x✝ * x✝¹) - x✝ * (x✝² * x✝¹))) | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | abel | /-- An associative ring gives rise to a Lie ring by taking the bracket to be the ring commutator. -/
instance (priority := 100) ofAssociativeRing : LieRing A where
add_lie _ _ _ := by simp only [Ring.lie_def, right_distrib, left_distrib]; abel
lie_add _ _ _ := by simp only [Ring.lie_def, right_distrib, left_distrib... | Mathlib.Algebra.Lie.OfAssociative.67_0.ll51mLev4p7Z1wP | /-- An associative ring gives rise to a Lie ring by taking the bracket to be the ring commutator. -/
instance (priority | Mathlib_Algebra_Lie_OfAssociative |
A : Type v
inst✝² : Ring A
M : Type w
inst✝¹ : AddCommGroup M
inst✝ : Module A M
⊢ ∀ (x y : A) (m : M), ⁅x, ⁅y, m⁆⁆ = ⁅⁅x, y⁆, m⁆ + ⁅y, ⁅x, m⁆⁆ | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | simp [LieRing.of_associative_ring_bracket, sub_smul, mul_smul, sub_add_cancel] | /-- We can regard a module over an associative ring `A` as a Lie ring module over `A` with Lie
bracket equal to its ring commutator.
Note that this cannot be a global instance because it would create a diamond when `M = A`,
specifically we can build two mathematically-different `bracket A A`s:
1. `@Ring.bracket A _` ... | Mathlib.Algebra.Lie.OfAssociative.91_0.ll51mLev4p7Z1wP | /-- We can regard a module over an associative ring `A` as a Lie ring module over `A` with Lie
bracket equal to its ring commutator.
Note that this cannot be a global instance because it would create a diamond when `M = A`,
specifically we can build two mathematically-different `bracket A A`s:
1. `@Ring.bracket A _` ... | Mathlib_Algebra_Lie_OfAssociative |
A : Type v
inst✝² : Ring A
R : Type u
inst✝¹ : CommRing R
inst✝ : Algebra R A
t : R
x y : A
⊢ ⁅x, t • y⁆ = t • ⁅x, y⁆ | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | rw [LieRing.of_associative_ring_bracket, LieRing.of_associative_ring_bracket,
Algebra.mul_smul_comm, Algebra.smul_mul_assoc, smul_sub] | /-- An associative algebra gives rise to a Lie algebra by taking the bracket to be the ring
commutator. -/
instance (priority := 100) LieAlgebra.ofAssociativeAlgebra : LieAlgebra R A where
lie_smul t x y := by
| Mathlib.Algebra.Lie.OfAssociative.121_0.ll51mLev4p7Z1wP | /-- An associative algebra gives rise to a Lie algebra by taking the bracket to be the ring
commutator. -/
instance (priority | Mathlib_Algebra_Lie_OfAssociative |
A : Type v
inst✝⁶ : Ring A
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : Algebra R A
B : Type w
C : Type w₁
inst✝³ : Ring B
inst✝² : Ring C
inst✝¹ : Algebra R B
inst✝ : Algebra R C
f : A →ₐ[R] B
g : B →ₐ[R] C
src✝ : A →ₗ[R] B := toLinearMap f
x✝¹ x✝ : A
⊢ AddHom.toFun
{ toAddHom := src✝.toAddHom,
map_smul' :=... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | simp [LieRing.of_associative_ring_bracket] | /-- The map `ofAssociativeAlgebra` associating a Lie algebra to an associative algebra is
functorial. -/
def toLieHom : A →ₗ⁅R⁆ B :=
{ f.toLinearMap with
map_lie' := fun {_ _} => by | Mathlib.Algebra.Lie.OfAssociative.161_0.ll51mLev4p7Z1wP | /-- The map `ofAssociativeAlgebra` associating a Lie algebra to an associative algebra is
functorial. -/
def toLieHom : A →ₗ⁅R⁆ B | Mathlib_Algebra_Lie_OfAssociative |
A : Type v
inst✝⁶ : Ring A
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : Algebra R A
B : Type w
C : Type w₁
inst✝³ : Ring B
inst✝² : Ring C
inst✝¹ : Algebra R B
inst✝ : Algebra R C
f✝ : A →ₐ[R] B
g✝ : B →ₐ[R] C
f g : A →ₐ[R] B
h : toLieHom f = toLieHom g
⊢ f = g | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | ext a | theorem toLieHom_injective {f g : A →ₐ[R] B} (h : (f : A →ₗ⁅R⁆ B) = (g : A →ₗ⁅R⁆ B)) : f = g := by
| Mathlib.Algebra.Lie.OfAssociative.197_0.ll51mLev4p7Z1wP | theorem toLieHom_injective {f g : A →ₐ[R] B} (h : (f : A →ₗ⁅R⁆ B) = (g : A →ₗ⁅R⁆ B)) : f = g | Mathlib_Algebra_Lie_OfAssociative |
case H
A : Type v
inst✝⁶ : Ring A
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : Algebra R A
B : Type w
C : Type w₁
inst✝³ : Ring B
inst✝² : Ring C
inst✝¹ : Algebra R B
inst✝ : Algebra R C
f✝ : A →ₐ[R] B
g✝ : B →ₐ[R] C
f g : A →ₐ[R] B
h : toLieHom f = toLieHom g
a : A
⊢ f a = g a | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | exact LieHom.congr_fun h a | theorem toLieHom_injective {f g : A →ₐ[R] B} (h : (f : A →ₗ⁅R⁆ B) = (g : A →ₗ⁅R⁆ B)) : f = g := by
ext a; | Mathlib.Algebra.Lie.OfAssociative.197_0.ll51mLev4p7Z1wP | theorem toLieHom_injective {f g : A →ₐ[R] B} (h : (f : A →ₗ⁅R⁆ B) = (g : A →ₗ⁅R⁆ B)) : f = g | Mathlib_Algebra_Lie_OfAssociative |
R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
x y : L
⊢ (fun x =>
{ toAddHom := { toFun := fun m => ⁅x, m⁆, map_add' := (_ : ∀ (m n : M), ⁅x, m + n⁆ = ⁅x, m⁆ + ⁅x, n⁆)... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | ext m | /-- A Lie module yields a Lie algebra morphism into the linear endomorphisms of the module.
See also `LieModule.toModuleHom`. -/
@[simps]
def LieModule.toEndomorphism : L →ₗ⁅R⁆ Module.End R M where
toFun x :=
{ toFun := fun m => ⁅x, m⁆
map_add' := lie_add x
map_smul' := fun t => lie_smul t x }
map_... | Mathlib.Algebra.Lie.OfAssociative.215_0.ll51mLev4p7Z1wP | /-- A Lie module yields a Lie algebra morphism into the linear endomorphisms of the module.
See also `LieModule.toModuleHom`. -/
@[simps]
def LieModule.toEndomorphism : L →ₗ⁅R⁆ Module.End R M where
toFun x | Mathlib_Algebra_Lie_OfAssociative |
case h
R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
x y : L
m : M
⊢ ((fun x =>
{ toAddHom := { toFun := fun m => ⁅x, m⁆, map_add' := (_ : ∀ (m n : M), ⁅x, m + n⁆ = ... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | apply add_lie | /-- A Lie module yields a Lie algebra morphism into the linear endomorphisms of the module.
See also `LieModule.toModuleHom`. -/
@[simps]
def LieModule.toEndomorphism : L →ₗ⁅R⁆ Module.End R M where
toFun x :=
{ toFun := fun m => ⁅x, m⁆
map_add' := lie_add x
map_smul' := fun t => lie_smul t x }
map_... | Mathlib.Algebra.Lie.OfAssociative.215_0.ll51mLev4p7Z1wP | /-- A Lie module yields a Lie algebra morphism into the linear endomorphisms of the module.
See also `LieModule.toModuleHom`. -/
@[simps]
def LieModule.toEndomorphism : L →ₗ⁅R⁆ Module.End R M where
toFun x | Mathlib_Algebra_Lie_OfAssociative |
R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
t : R
x : L
⊢ AddHom.toFun
{
toFun := fun x =>
{ toAddHom := { toFun := fun m => ⁅x, m⁆, map_add' := (_ :... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | ext m | /-- A Lie module yields a Lie algebra morphism into the linear endomorphisms of the module.
See also `LieModule.toModuleHom`. -/
@[simps]
def LieModule.toEndomorphism : L →ₗ⁅R⁆ Module.End R M where
toFun x :=
{ toFun := fun m => ⁅x, m⁆
map_add' := lie_add x
map_smul' := fun t => lie_smul t x }
map_... | Mathlib.Algebra.Lie.OfAssociative.215_0.ll51mLev4p7Z1wP | /-- A Lie module yields a Lie algebra morphism into the linear endomorphisms of the module.
See also `LieModule.toModuleHom`. -/
@[simps]
def LieModule.toEndomorphism : L →ₗ⁅R⁆ Module.End R M where
toFun x | Mathlib_Algebra_Lie_OfAssociative |
case h
R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
t : R
x : L
m : M
⊢ (AddHom.toFun
{
toFun := fun x =>
{ toAddHom := { toFun := fun m => ⁅x, ... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | apply smul_lie | /-- A Lie module yields a Lie algebra morphism into the linear endomorphisms of the module.
See also `LieModule.toModuleHom`. -/
@[simps]
def LieModule.toEndomorphism : L →ₗ⁅R⁆ Module.End R M where
toFun x :=
{ toFun := fun m => ⁅x, m⁆
map_add' := lie_add x
map_smul' := fun t => lie_smul t x }
map_... | Mathlib.Algebra.Lie.OfAssociative.215_0.ll51mLev4p7Z1wP | /-- A Lie module yields a Lie algebra morphism into the linear endomorphisms of the module.
See also `LieModule.toModuleHom`. -/
@[simps]
def LieModule.toEndomorphism : L →ₗ⁅R⁆ Module.End R M where
toFun x | Mathlib_Algebra_Lie_OfAssociative |
R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
x y : L
⊢ AddHom.toFun
{
toAddHom :=
{
toFun := fun x =>
{ toAddHom := ... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | ext m | /-- A Lie module yields a Lie algebra morphism into the linear endomorphisms of the module.
See also `LieModule.toModuleHom`. -/
@[simps]
def LieModule.toEndomorphism : L →ₗ⁅R⁆ Module.End R M where
toFun x :=
{ toFun := fun m => ⁅x, m⁆
map_add' := lie_add x
map_smul' := fun t => lie_smul t x }
map_... | Mathlib.Algebra.Lie.OfAssociative.215_0.ll51mLev4p7Z1wP | /-- A Lie module yields a Lie algebra morphism into the linear endomorphisms of the module.
See also `LieModule.toModuleHom`. -/
@[simps]
def LieModule.toEndomorphism : L →ₗ⁅R⁆ Module.End R M where
toFun x | Mathlib_Algebra_Lie_OfAssociative |
case h
R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
x y : L
m : M
⊢ (AddHom.toFun
{
toAddHom :=
{
toFun := fun x =>
... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | apply lie_lie | /-- A Lie module yields a Lie algebra morphism into the linear endomorphisms of the module.
See also `LieModule.toModuleHom`. -/
@[simps]
def LieModule.toEndomorphism : L →ₗ⁅R⁆ Module.End R M where
toFun x :=
{ toFun := fun m => ⁅x, m⁆
map_add' := lie_add x
map_smul' := fun t => lie_smul t x }
map_... | Mathlib.Algebra.Lie.OfAssociative.215_0.ll51mLev4p7Z1wP | /-- A Lie module yields a Lie algebra morphism into the linear endomorphisms of the module.
See also `LieModule.toModuleHom`. -/
@[simps]
def LieModule.toEndomorphism : L →ₗ⁅R⁆ Module.End R M where
toFun x | Mathlib_Algebra_Lie_OfAssociative |
R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
⊢ toEndomorphism R (Module.End R M) M = LieHom.id | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | ext g m | @[simp]
theorem LieModule.toEndomorphism_module_end :
LieModule.toEndomorphism R (Module.End R M) M = LieHom.id := by | Mathlib.Algebra.Lie.OfAssociative.239_0.ll51mLev4p7Z1wP | @[simp]
theorem LieModule.toEndomorphism_module_end :
LieModule.toEndomorphism R (Module.End R M) M = LieHom.id | Mathlib_Algebra_Lie_OfAssociative |
case h.h
R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
g : Module.End R M
m : M
⊢ ((toEndomorphism R (Module.End R M) M) g) m = (LieHom.id g) m | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | simp [lie_eq_smul] | @[simp]
theorem LieModule.toEndomorphism_module_end :
LieModule.toEndomorphism R (Module.End R M) M = LieHom.id := by ext g m; | Mathlib.Algebra.Lie.OfAssociative.239_0.ll51mLev4p7Z1wP | @[simp]
theorem LieModule.toEndomorphism_module_end :
LieModule.toEndomorphism R (Module.End R M) M = LieHom.id | Mathlib_Algebra_Lie_OfAssociative |
R : Type u
L : Type v
M : Type w
inst✝¹⁰ : CommRing R
inst✝⁹ : LieRing L
inst✝⁸ : LieAlgebra R L
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : LieRingModule L M
inst✝⁴ : LieModule R L M
M₂ : Type w₁
inst✝³ : AddCommGroup M₂
inst✝² : Module R M₂
inst✝¹ : LieRingModule L M₂
inst✝ : LieModule R L M₂
f : M →ₗ⁅R,L⁆ M... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | apply LinearMap.commute_pow_left_of_commute | lemma toEndomorphism_pow_comp_lieHom :
(toEndomorphism R L M₂ x ^ k) ∘ₗ f = f ∘ₗ toEndomorphism R L M x ^ k := by
| Mathlib.Algebra.Lie.OfAssociative.262_0.ll51mLev4p7Z1wP | lemma toEndomorphism_pow_comp_lieHom :
(toEndomorphism R L M₂ x ^ k) ∘ₗ f = f ∘ₗ toEndomorphism R L M x ^ k | Mathlib_Algebra_Lie_OfAssociative |
case h
R : Type u
L : Type v
M : Type w
inst✝¹⁰ : CommRing R
inst✝⁹ : LieRing L
inst✝⁸ : LieAlgebra R L
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : LieRingModule L M
inst✝⁴ : LieModule R L M
M₂ : Type w₁
inst✝³ : AddCommGroup M₂
inst✝² : Module R M₂
inst✝¹ : LieRingModule L M₂
inst✝ : LieModule R L M₂
f : M →ₗ... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | ext | lemma toEndomorphism_pow_comp_lieHom :
(toEndomorphism R L M₂ x ^ k) ∘ₗ f = f ∘ₗ toEndomorphism R L M x ^ k := by
apply LinearMap.commute_pow_left_of_commute
| Mathlib.Algebra.Lie.OfAssociative.262_0.ll51mLev4p7Z1wP | lemma toEndomorphism_pow_comp_lieHom :
(toEndomorphism R L M₂ x ^ k) ∘ₗ f = f ∘ₗ toEndomorphism R L M x ^ k | Mathlib_Algebra_Lie_OfAssociative |
case h.h
R : Type u
L : Type v
M : Type w
inst✝¹⁰ : CommRing R
inst✝⁹ : LieRing L
inst✝⁸ : LieAlgebra R L
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : LieRingModule L M
inst✝⁴ : LieModule R L M
M₂ : Type w₁
inst✝³ : AddCommGroup M₂
inst✝² : Module R M₂
inst✝¹ : LieRingModule L M₂
inst✝ : LieModule R L M₂
f : M ... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | simp | lemma toEndomorphism_pow_comp_lieHom :
(toEndomorphism R L M₂ x ^ k) ∘ₗ f = f ∘ₗ toEndomorphism R L M x ^ k := by
apply LinearMap.commute_pow_left_of_commute
ext
| Mathlib.Algebra.Lie.OfAssociative.262_0.ll51mLev4p7Z1wP | lemma toEndomorphism_pow_comp_lieHom :
(toEndomorphism R L M₂ x ^ k) ∘ₗ f = f ∘ₗ toEndomorphism R L M x ^ k | Mathlib_Algebra_Lie_OfAssociative |
R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
N : LieSubmodule R L M
x : L
⊢ Submodule.map ((toEndomorphism R L M) x) ↑N ≤ ↑N | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | rintro n ⟨m, hm, rfl⟩ | theorem coe_map_toEndomorphism_le :
(N : Submodule R M).map (LieModule.toEndomorphism R L M x) ≤ N := by
| Mathlib.Algebra.Lie.OfAssociative.280_0.ll51mLev4p7Z1wP | theorem coe_map_toEndomorphism_le :
(N : Submodule R M).map (LieModule.toEndomorphism R L M x) ≤ N | Mathlib_Algebra_Lie_OfAssociative |
case intro.intro
R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
N : LieSubmodule R L M
x : L
m : M
hm : m ∈ ↑↑N
⊢ ((toEndomorphism R L M) x) m ∈ ↑N | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | exact N.lie_mem hm | theorem coe_map_toEndomorphism_le :
(N : Submodule R M).map (LieModule.toEndomorphism R L M x) ≤ N := by
rintro n ⟨m, hm, rfl⟩
| Mathlib.Algebra.Lie.OfAssociative.280_0.ll51mLev4p7Z1wP | theorem coe_map_toEndomorphism_le :
(N : Submodule R M).map (LieModule.toEndomorphism R L M x) ≤ N | Mathlib_Algebra_Lie_OfAssociative |
R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
N : LieSubmodule R L M
x : L
m : M
hm : m ∈ ↑N
⊢ ((toEndomorphism R L M) x ∘ₗ Submodule.subtype ↑N) { val := m, property := hm } ... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | simpa using N.lie_mem hm | theorem toEndomorphism_comp_subtype_mem (m : M) (hm : m ∈ (N : Submodule R M)) :
(toEndomorphism R L M x).comp (N : Submodule R M).subtype ⟨m, hm⟩ ∈ (N : Submodule R M) := by
| Mathlib.Algebra.Lie.OfAssociative.288_0.ll51mLev4p7Z1wP | theorem toEndomorphism_comp_subtype_mem (m : M) (hm : m ∈ (N : Submodule R M)) :
(toEndomorphism R L M x).comp (N : Submodule R M).subtype ⟨m, hm⟩ ∈ (N : Submodule R M) | Mathlib_Algebra_Lie_OfAssociative |
R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
N : LieSubmodule R L M
x : L
h :
optParam
(∀ (m : M) (hm : m ∈ ↑N), ((toEndomorphism R L M) x ∘ₗ Submodule.subtype ↑N) { va... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | ext | @[simp]
theorem toEndomorphism_restrict_eq_toEndomorphism (h := N.toEndomorphism_comp_subtype_mem x) :
(toEndomorphism R L M x).restrict h = toEndomorphism R L N x := by
| Mathlib.Algebra.Lie.OfAssociative.293_0.ll51mLev4p7Z1wP | @[simp]
theorem toEndomorphism_restrict_eq_toEndomorphism (h | Mathlib_Algebra_Lie_OfAssociative |
case h.a
R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
N : LieSubmodule R L M
x : L
h :
optParam
(∀ (m : M) (hm : m ∈ ↑N), ((toEndomorphism R L M) x ∘ₗ Submodule.subtype... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | simp [LinearMap.restrict_apply] | @[simp]
theorem toEndomorphism_restrict_eq_toEndomorphism (h := N.toEndomorphism_comp_subtype_mem x) :
(toEndomorphism R L M x).restrict h = toEndomorphism R L N x := by
ext; | Mathlib.Algebra.Lie.OfAssociative.293_0.ll51mLev4p7Z1wP | @[simp]
theorem toEndomorphism_restrict_eq_toEndomorphism (h | Mathlib_Algebra_Lie_OfAssociative |
R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
N : LieSubmodule R L M
x✝ : L
φ : R
k : ℕ
x : L
⊢ MapsTo ⇑(((toEndomorphism R L M) x - (algebraMap R (Module.End R M)) φ) ^ k) ↑N... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | rw [LinearMap.coe_pow] | lemma mapsTo_pow_toEndomorphism_sub_algebraMap {φ : R} {k : ℕ} {x : L} :
MapsTo ((toEndomorphism R L M x - algebraMap R (Module.End R M) φ) ^ k) N N := by
| Mathlib.Algebra.Lie.OfAssociative.299_0.ll51mLev4p7Z1wP | lemma mapsTo_pow_toEndomorphism_sub_algebraMap {φ : R} {k : ℕ} {x : L} :
MapsTo ((toEndomorphism R L M x - algebraMap R (Module.End R M) φ) ^ k) N N | Mathlib_Algebra_Lie_OfAssociative |
R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
N : LieSubmodule R L M
x✝ : L
φ : R
k : ℕ
x : L
⊢ MapsTo (⇑((toEndomorphism R L M) x - (algebraMap R (Module.End R M)) φ))^[k] ↑N... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | exact MapsTo.iterate (fun m hm ↦ N.sub_mem (N.lie_mem hm) (N.smul_mem _ hm)) k | lemma mapsTo_pow_toEndomorphism_sub_algebraMap {φ : R} {k : ℕ} {x : L} :
MapsTo ((toEndomorphism R L M x - algebraMap R (Module.End R M) φ) ^ k) N N := by
rw [LinearMap.coe_pow]
| Mathlib.Algebra.Lie.OfAssociative.299_0.ll51mLev4p7Z1wP | lemma mapsTo_pow_toEndomorphism_sub_algebraMap {φ : R} {k : ℕ} {x : L} :
MapsTo ((toEndomorphism R L M x - algebraMap R (Module.End R M) φ) ^ k) N N | Mathlib_Algebra_Lie_OfAssociative |
R : Type u
L : Type v
M : Type w
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : LieRingModule L M
inst✝² : LieModule R L M
A : Type v
inst✝¹ : Ring A
inst✝ : Algebra R A
⊢ ⇑(ad R A) = LinearMap.mulLeft R - LinearMap.mulRight R | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | ext a b | theorem LieAlgebra.ad_eq_lmul_left_sub_lmul_right (A : Type v) [Ring A] [Algebra R A] :
(ad R A : A → Module.End R A) = LinearMap.mulLeft R - LinearMap.mulRight R := by
| Mathlib.Algebra.Lie.OfAssociative.308_0.ll51mLev4p7Z1wP | theorem LieAlgebra.ad_eq_lmul_left_sub_lmul_right (A : Type v) [Ring A] [Algebra R A] :
(ad R A : A → Module.End R A) = LinearMap.mulLeft R - LinearMap.mulRight R | Mathlib_Algebra_Lie_OfAssociative |
case h.h
R : Type u
L : Type v
M : Type w
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : LieRingModule L M
inst✝² : LieModule R L M
A : Type v
inst✝¹ : Ring A
inst✝ : Algebra R A
a b : A
⊢ ((ad R A) a) b = ((LinearMap.mulLeft R - LinearMap.mulRight R)... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | simp [LieRing.of_associative_ring_bracket] | theorem LieAlgebra.ad_eq_lmul_left_sub_lmul_right (A : Type v) [Ring A] [Algebra R A] :
(ad R A : A → Module.End R A) = LinearMap.mulLeft R - LinearMap.mulRight R := by
ext a b; | Mathlib.Algebra.Lie.OfAssociative.308_0.ll51mLev4p7Z1wP | theorem LieAlgebra.ad_eq_lmul_left_sub_lmul_right (A : Type v) [Ring A] [Algebra R A] :
(ad R A : A → Module.End R A) = LinearMap.mulLeft R - LinearMap.mulRight R | Mathlib_Algebra_Lie_OfAssociative |
R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
K : LieSubalgebra R L
x : ↥K
⊢ (ad R L) ↑x ∘ₗ ↑(incl K) = ↑(incl K) ∘ₗ (ad R ↥K) x | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | ext y | theorem LieSubalgebra.ad_comp_incl_eq (K : LieSubalgebra R L) (x : K) :
(ad R L ↑x).comp (K.incl : K →ₗ[R] L) = (K.incl : K →ₗ[R] L).comp (ad R K x) := by
| Mathlib.Algebra.Lie.OfAssociative.313_0.ll51mLev4p7Z1wP | theorem LieSubalgebra.ad_comp_incl_eq (K : LieSubalgebra R L) (x : K) :
(ad R L ↑x).comp (K.incl : K →ₗ[R] L) = (K.incl : K →ₗ[R] L).comp (ad R K x) | Mathlib_Algebra_Lie_OfAssociative |
case h
R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
K : LieSubalgebra R L
x y : ↥K
⊢ ((ad R L) ↑x ∘ₗ ↑(incl K)) y = (↑(incl K) ∘ₗ (ad R ↥K) x) y | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | simp only [ad_apply, LieHom.coe_toLinearMap, LieSubalgebra.coe_incl, LinearMap.coe_comp,
LieSubalgebra.coe_bracket, Function.comp_apply] | theorem LieSubalgebra.ad_comp_incl_eq (K : LieSubalgebra R L) (x : K) :
(ad R L ↑x).comp (K.incl : K →ₗ[R] L) = (K.incl : K →ₗ[R] L).comp (ad R K x) := by
ext y
| Mathlib.Algebra.Lie.OfAssociative.313_0.ll51mLev4p7Z1wP | theorem LieSubalgebra.ad_comp_incl_eq (K : LieSubalgebra R L) (x : K) :
(ad R L ↑x).comp (K.incl : K →ₗ[R] L) = (K.incl : K →ₗ[R] L).comp (ad R K x) | Mathlib_Algebra_Lie_OfAssociative |
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