[GOAL]
R : Type u_1
inst✝⁴ : StrictOrderedCommSemiring R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
ι : Type u_4
⊢ map ι (LinearEquiv.refl R M) = Equiv.refl (Orientation R M ι)
[PROOFSTEP]
rw [Orientation.map, AlternatingMap.domLCongr_refl, Module.Ray.map_refl]
[GOAL]
R : Type u_1
inst✝⁶ : StrictOrderedCommSemiring R
M : Type u_2
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
N : Type u_3
inst✝³ : AddCommMonoid N
inst✝² : Module R N
ι : Type u_4
inst✝¹ : Nontrivial R
inst✝ : IsEmpty ι
⊢ 1 ≠ AddMonoid.toZero.1
[PROOFSTEP]
exact one_ne_zero
[GOAL]
R : Type u_1
inst✝⁵ : StrictOrderedCommSemiring R
M : Type u_2
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
N : Type u_3
inst✝² : AddCommMonoid N
inst✝¹ : Module R N
ι : Type u_4
inst✝ : IsEmpty ι
x : Orientation R M ι
f : M ≃ₗ[R] M
⊢ ↑(map ι f) x = x
[PROOFSTEP]
induction' x using Module.Ray.ind with g hg
[GOAL]
case h
R : Type u_1
inst✝⁵ : StrictOrderedCommSemiring R
M : Type u_2
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
N : Type u_3
inst✝² : AddCommMonoid N
inst✝¹ : Module R N
ι : Type u_4
inst✝ : IsEmpty ι
f : M ≃ₗ[R] M
g : AlternatingMap R M R ι
hg : g ≠ 0
⊢ ↑(map ι f) (rayOfNeZero R g hg) = rayOfNeZero R g hg
[PROOFSTEP]
rw [Orientation.map_apply]
[GOAL]
case h
R : Type u_1
inst✝⁵ : StrictOrderedCommSemiring R
M : Type u_2
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
N : Type u_3
inst✝² : AddCommMonoid N
inst✝¹ : Module R N
ι : Type u_4
inst✝ : IsEmpty ι
f : M ≃ₗ[R] M
g : AlternatingMap R M R ι
hg : g ≠ 0
⊢ rayOfNeZero R (AlternatingMap.compLinearMap g ↑(LinearEquiv.symm f))
      (_ : ¬AlternatingMap.compLinearMap g ↑(LinearEquiv.symm f) = 0) =
    rayOfNeZero R g hg
[PROOFSTEP]
congr
[GOAL]
case h.e_v
R : Type u_1
inst✝⁵ : StrictOrderedCommSemiring R
M : Type u_2
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
N : Type u_3
inst✝² : AddCommMonoid N
inst✝¹ : Module R N
ι : Type u_4
inst✝ : IsEmpty ι
f : M ≃ₗ[R] M
g : AlternatingMap R M R ι
hg : g ≠ 0
⊢ AlternatingMap.compLinearMap g ↑(LinearEquiv.symm f) = g
[PROOFSTEP]
ext i
[GOAL]
case h.e_v.H
R : Type u_1
inst✝⁵ : StrictOrderedCommSemiring R
M : Type u_2
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
N : Type u_3
inst✝² : AddCommMonoid N
inst✝¹ : Module R N
ι : Type u_4
inst✝ : IsEmpty ι
f : M ≃ₗ[R] M
g : AlternatingMap R M R ι
hg : g ≠ 0
i : ι → M
⊢ ↑(AlternatingMap.compLinearMap g ↑(LinearEquiv.symm f)) i = ↑g i
[PROOFSTEP]
rw [AlternatingMap.compLinearMap_apply]
[GOAL]
case h.e_v.H
R : Type u_1
inst✝⁵ : StrictOrderedCommSemiring R
M : Type u_2
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
N : Type u_3
inst✝² : AddCommMonoid N
inst✝¹ : Module R N
ι : Type u_4
inst✝ : IsEmpty ι
f : M ≃ₗ[R] M
g : AlternatingMap R M R ι
hg : g ≠ 0
i : ι → M
⊢ (↑g fun i_1 => ↑↑(LinearEquiv.symm f) (i i_1)) = ↑g i
[PROOFSTEP]
congr
[GOAL]
case h.e_v.H.h.e_6.h
R : Type u_1
inst✝⁵ : StrictOrderedCommSemiring R
M : Type u_2
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
N : Type u_3
inst✝² : AddCommMonoid N
inst✝¹ : Module R N
ι : Type u_4
inst✝ : IsEmpty ι
f : M ≃ₗ[R] M
g : AlternatingMap R M R ι
hg : g ≠ 0
i : ι → M
⊢ (fun i_1 => ↑↑(LinearEquiv.symm f) (i i_1)) = i
[PROOFSTEP]
simp only [LinearEquiv.coe_coe, eq_iff_true_of_subsingleton]
[GOAL]
R : Type u_1
inst✝⁵ : StrictOrderedCommRing R
M : Type u_2
N : Type u_3
inst✝⁴ : AddCommGroup M
inst✝³ : AddCommGroup N
inst✝² : Module R M
inst✝¹ : Module R N
ι : Type u_4
inst✝ : Finite ι
e : Basis ι R M
x : Orientation R M ι
f : M ≃ₗ[R] M
⊢ ↑(Orientation.map ι f) x = (↑LinearEquiv.det f)⁻¹ • x
[PROOFSTEP]
cases nonempty_fintype ι
[GOAL]
case intro
R : Type u_1
inst✝⁵ : StrictOrderedCommRing R
M : Type u_2
N : Type u_3
inst✝⁴ : AddCommGroup M
inst✝³ : AddCommGroup N
inst✝² : Module R M
inst✝¹ : Module R N
ι : Type u_4
inst✝ : Finite ι
e : Basis ι R M
x : Orientation R M ι
f : M ≃ₗ[R] M
val✝ : Fintype ι
⊢ ↑(Orientation.map ι f) x = (↑LinearEquiv.det f)⁻¹ • x
[PROOFSTEP]
letI := Classical.decEq ι
[GOAL]
case intro
R : Type u_1
inst✝⁵ : StrictOrderedCommRing R
M : Type u_2
N : Type u_3
inst✝⁴ : AddCommGroup M
inst✝³ : AddCommGroup N
inst✝² : Module R M
inst✝¹ : Module R N
ι : Type u_4
inst✝ : Finite ι
e : Basis ι R M
x : Orientation R M ι
f : M ≃ₗ[R] M
val✝ : Fintype ι
this : DecidableEq ι := Classical.decEq ι
⊢ ↑(Orientation.map ι f) x = (↑LinearEquiv.det f)⁻¹ • x
[PROOFSTEP]
induction' x using Module.Ray.ind with g hg
[GOAL]
case intro.h
R : Type u_1
inst✝⁵ : StrictOrderedCommRing R
M : Type u_2
N : Type u_3
inst✝⁴ : AddCommGroup M
inst✝³ : AddCommGroup N
inst✝² : Module R M
inst✝¹ : Module R N
ι : Type u_4
inst✝ : Finite ι
e : Basis ι R M
f : M ≃ₗ[R] M
val✝ : Fintype ι
this : DecidableEq ι := Classical.decEq ι
g : AlternatingMap R M R ι
hg : g ≠ 0
⊢ ↑(Orientation.map ι f) (rayOfNeZero R g hg) = (↑LinearEquiv.det f)⁻¹ • rayOfNeZero R g hg
[PROOFSTEP]
rw [Orientation.map_apply, smul_rayOfNeZero, ray_eq_iff, Units.smul_def, (g.compLinearMap f.symm).eq_smul_basis_det e,
  g.eq_smul_basis_det e, AlternatingMap.compLinearMap_apply, AlternatingMap.smul_apply,
  show (fun i ↦ (LinearEquiv.symm f).toLinearMap (e i)) = (LinearEquiv.symm f).toLinearMap ∘ e by rfl, Basis.det_comp,
  Basis.det_self, mul_one, smul_eq_mul, mul_comm, mul_smul, LinearEquiv.coe_inv_det]
[GOAL]
R : Type u_1
inst✝⁵ : StrictOrderedCommRing R
M : Type u_2
N : Type u_3
inst✝⁴ : AddCommGroup M
inst✝³ : AddCommGroup N
inst✝² : Module R M
inst✝¹ : Module R N
ι : Type u_4
inst✝ : Finite ι
e : Basis ι R M
f : M ≃ₗ[R] M
val✝ : Fintype ι
this : DecidableEq ι := Classical.decEq ι
g : AlternatingMap R M R ι
hg : g ≠ 0
⊢ (fun i => ↑↑(LinearEquiv.symm f) (↑e i)) = ↑↑(LinearEquiv.symm f) ∘ ↑e
[PROOFSTEP]
rfl
[GOAL]
R : Type u_1
inst✝⁷ : StrictOrderedCommRing R
M : Type u_2
N : Type u_3
inst✝⁶ : AddCommGroup M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R M
inst✝³ : Module R N
ι : Type u_4
inst✝² : Fintype ι
inst✝¹ : DecidableEq ι
inst✝ : Nontrivial R
e : Basis ι R M
f : M ≃ₗ[R] N
⊢ Basis.orientation (Basis.map e f) = ↑(Orientation.map ι f) (Basis.orientation e)
[PROOFSTEP]
simp_rw [Basis.orientation, Orientation.map_apply, Basis.det_map']
[GOAL]
R : Type u_1
inst✝⁷ : StrictOrderedCommRing R
M : Type u_2
N : Type u_3
inst✝⁶ : AddCommGroup M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R M
inst✝³ : Module R N
ι : Type u_4
inst✝² : Fintype ι
inst✝¹ : DecidableEq ι
inst✝ : Nontrivial R
e : Basis ι R M
w : ι → Rˣ
⊢ Basis.orientation (unitsSMul e w) = (∏ i : ι, w i)⁻¹ • Basis.orientation e
[PROOFSTEP]
rw [Basis.orientation, Basis.orientation, smul_rayOfNeZero, ray_eq_iff, e.det.eq_smul_basis_det (e.unitsSMul w),
  det_unitsSMul_self, Units.smul_def, smul_smul]
[GOAL]
R : Type u_1
inst✝⁷ : StrictOrderedCommRing R
M : Type u_2
N : Type u_3
inst✝⁶ : AddCommGroup M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R M
inst✝³ : Module R N
ι : Type u_4
inst✝² : Fintype ι
inst✝¹ : DecidableEq ι
inst✝ : Nontrivial R
e : Basis ι R M
w : ι → Rˣ
⊢ SameRay R (det (unitsSMul e w)) ((↑(∏ i : ι, w i)⁻¹ * ∏ i : ι, ↑(w i)) • det (unitsSMul e w))
[PROOFSTEP]
norm_cast
[GOAL]
R : Type u_1
inst✝⁷ : StrictOrderedCommRing R
M : Type u_2
N : Type u_3
inst✝⁶ : AddCommGroup M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R M
inst✝³ : Module R N
ι : Type u_4
inst✝² : Fintype ι
inst✝¹ : DecidableEq ι
inst✝ : Nontrivial R
e : Basis ι R M
w : ι → Rˣ
⊢ SameRay R (det (unitsSMul e w)) (↑((∏ i : ι, w i)⁻¹ * ∏ i : ι, w i) • det (unitsSMul e w))
[PROOFSTEP]
simp
[GOAL]
R : Type u_1
inst✝⁷ : StrictOrderedCommRing R
M : Type u_2
N : Type u_3
inst✝⁶ : AddCommGroup M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R M
inst✝³ : Module R N
ι : Type u_4
inst✝² : Fintype ι
inst✝¹ : DecidableEq ι
inst✝ : Nontrivial R
e : Basis ι R M
w : ι → Rˣ
⊢ SameRay R (det (unitsSMul e w)) (det (unitsSMul e w))
[PROOFSTEP]
exact SameRay.rfl
[GOAL]
R : Type u_1
inst✝⁸ : StrictOrderedCommRing R
M : Type u_2
N : Type u_3
inst✝⁷ : AddCommGroup M
inst✝⁶ : AddCommGroup N
inst✝⁵ : Module R M
inst✝⁴ : Module R N
ι : Type u_4
inst✝³ : Fintype ι
inst✝² : DecidableEq ι
inst✝¹ : Nontrivial R
inst✝ : IsEmpty ι
b : Basis ι R M
⊢ Basis.orientation b = positiveOrientation
[PROOFSTEP]
rw [Basis.orientation]
[GOAL]
R : Type u_1
inst✝⁸ : StrictOrderedCommRing R
M : Type u_2
N : Type u_3
inst✝⁷ : AddCommGroup M
inst✝⁶ : AddCommGroup N
inst✝⁵ : Module R M
inst✝⁴ : Module R N
ι : Type u_4
inst✝³ : Fintype ι
inst✝² : DecidableEq ι
inst✝¹ : Nontrivial R
inst✝ : IsEmpty ι
b : Basis ι R M
⊢ rayOfNeZero R (det b) (_ : det b ≠ 0) = positiveOrientation
[PROOFSTEP]
congr
[GOAL]
case e_v
R : Type u_1
inst✝⁸ : StrictOrderedCommRing R
M : Type u_2
N : Type u_3
inst✝⁷ : AddCommGroup M
inst✝⁶ : AddCommGroup N
inst✝⁵ : Module R M
inst✝⁴ : Module R N
ι : Type u_4
inst✝³ : Fintype ι
inst✝² : DecidableEq ι
inst✝¹ : Nontrivial R
inst✝ : IsEmpty ι
b : Basis ι R M
⊢ det b = ↑AlternatingMap.constLinearEquivOfIsEmpty 1
[PROOFSTEP]
exact b.det_isEmpty
[GOAL]
R : Type u_1
inst✝⁴ : LinearOrderedCommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_3
inst✝¹ : Nontrivial R
inst✝ : IsEmpty ι
o : Orientation R M ι
⊢ o = positiveOrientation ∨ o = -positiveOrientation
[PROOFSTEP]
induction' o using Module.Ray.ind with x hx
[GOAL]
case h
R : Type u_1
inst✝⁴ : LinearOrderedCommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_3
inst✝¹ : Nontrivial R
inst✝ : IsEmpty ι
x : AlternatingMap R M R ι
hx : x ≠ 0
⊢ rayOfNeZero R x hx = positiveOrientation ∨ rayOfNeZero R x hx = -positiveOrientation
[PROOFSTEP]
dsimp [positiveOrientation]
[GOAL]
case h
R : Type u_1
inst✝⁴ : LinearOrderedCommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_3
inst✝¹ : Nontrivial R
inst✝ : IsEmpty ι
x : AlternatingMap R M R ι
hx : x ≠ 0
⊢ rayOfNeZero R x hx =
      rayOfNeZero R (AlternatingMap.constOfIsEmpty R M ι 1)
        (_ :
          ↑AlternatingMap.constLinearEquivOfIsEmpty 1 ≠ ↑AlternatingMap.constLinearEquivOfIsEmpty AddMonoid.toZero.1) ∨
    rayOfNeZero R x hx =
      rayOfNeZero R (-AlternatingMap.constOfIsEmpty R M ι 1) (_ : ¬-AlternatingMap.constOfIsEmpty R M ι 1 = 0)
[PROOFSTEP]
simp only [ray_eq_iff, sameRay_neg_swap]
[GOAL]
case h
R : Type u_1
inst✝⁴ : LinearOrderedCommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_3
inst✝¹ : Nontrivial R
inst✝ : IsEmpty ι
x : AlternatingMap R M R ι
hx : x ≠ 0
⊢ SameRay R x (AlternatingMap.constOfIsEmpty R M ι 1) ∨ SameRay R x (-AlternatingMap.constOfIsEmpty R M ι 1)
[PROOFSTEP]
rw [sameRay_or_sameRay_neg_iff_not_linearIndependent]
[GOAL]
case h
R : Type u_1
inst✝⁴ : LinearOrderedCommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_3
inst✝¹ : Nontrivial R
inst✝ : IsEmpty ι
x : AlternatingMap R M R ι
hx : x ≠ 0
⊢ ¬LinearIndependent R ![x, AlternatingMap.constOfIsEmpty R M ι 1]
[PROOFSTEP]
intro h
[GOAL]
case h
R : Type u_1
inst✝⁴ : LinearOrderedCommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_3
inst✝¹ : Nontrivial R
inst✝ : IsEmpty ι
x : AlternatingMap R M R ι
hx : x ≠ 0
h : LinearIndependent R ![x, AlternatingMap.constOfIsEmpty R M ι 1]
⊢ False
[PROOFSTEP]
set f : AlternatingMap R M R ι ≃ₗ[R] R := AlternatingMap.constLinearEquivOfIsEmpty.symm
[GOAL]
case h
R : Type u_1
inst✝⁴ : LinearOrderedCommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_3
inst✝¹ : Nontrivial R
inst✝ : IsEmpty ι
x : AlternatingMap R M R ι
hx : x ≠ 0
h : LinearIndependent R ![x, AlternatingMap.constOfIsEmpty R M ι 1]
f : AlternatingMap R M R ι ≃ₗ[R] R := LinearEquiv.symm AlternatingMap.constLinearEquivOfIsEmpty
⊢ False
[PROOFSTEP]
have H : LinearIndependent R ![f x, 1] := by
  convert h.map' f.toLinearMap f.ker
  ext i
  fin_cases i <;> simp
[GOAL]
R : Type u_1
inst✝⁴ : LinearOrderedCommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_3
inst✝¹ : Nontrivial R
inst✝ : IsEmpty ι
x : AlternatingMap R M R ι
hx : x ≠ 0
h : LinearIndependent R ![x, AlternatingMap.constOfIsEmpty R M ι 1]
f : AlternatingMap R M R ι ≃ₗ[R] R := LinearEquiv.symm AlternatingMap.constLinearEquivOfIsEmpty
⊢ LinearIndependent R ![↑f x, 1]
[PROOFSTEP]
convert h.map' f.toLinearMap f.ker
[GOAL]
case h.e'_4
R : Type u_1
inst✝⁴ : LinearOrderedCommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_3
inst✝¹ : Nontrivial R
inst✝ : IsEmpty ι
x : AlternatingMap R M R ι
hx : x ≠ 0
h : LinearIndependent R ![x, AlternatingMap.constOfIsEmpty R M ι 1]
f : AlternatingMap R M R ι ≃ₗ[R] R := LinearEquiv.symm AlternatingMap.constLinearEquivOfIsEmpty
⊢ ![↑f x, 1] = ↑↑f ∘ ![x, AlternatingMap.constOfIsEmpty R M ι 1]
[PROOFSTEP]
ext i
[GOAL]
case h.e'_4.h
R : Type u_1
inst✝⁴ : LinearOrderedCommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_3
inst✝¹ : Nontrivial R
inst✝ : IsEmpty ι
x : AlternatingMap R M R ι
hx : x ≠ 0
h : LinearIndependent R ![x, AlternatingMap.constOfIsEmpty R M ι 1]
f : AlternatingMap R M R ι ≃ₗ[R] R := LinearEquiv.symm AlternatingMap.constLinearEquivOfIsEmpty
i : Fin (Nat.succ (Nat.succ 0))
⊢ Matrix.vecCons (↑f x) ![1] i = (↑↑f ∘ ![x, AlternatingMap.constOfIsEmpty R M ι 1]) i
[PROOFSTEP]
fin_cases i
[GOAL]
case h.e'_4.h.head
R : Type u_1
inst✝⁴ : LinearOrderedCommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_3
inst✝¹ : Nontrivial R
inst✝ : IsEmpty ι
x : AlternatingMap R M R ι
hx : x ≠ 0
h : LinearIndependent R ![x, AlternatingMap.constOfIsEmpty R M ι 1]
f : AlternatingMap R M R ι ≃ₗ[R] R := LinearEquiv.symm AlternatingMap.constLinearEquivOfIsEmpty
⊢ Matrix.vecCons (↑f x) ![1] { val := 0, isLt := (_ : 0 < Nat.succ (Nat.succ 0)) } =
    (↑↑f ∘ ![x, AlternatingMap.constOfIsEmpty R M ι 1]) { val := 0, isLt := (_ : 0 < Nat.succ (Nat.succ 0)) }
[PROOFSTEP]
simp
[GOAL]
case h.e'_4.h.tail.head
R : Type u_1
inst✝⁴ : LinearOrderedCommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_3
inst✝¹ : Nontrivial R
inst✝ : IsEmpty ι
x : AlternatingMap R M R ι
hx : x ≠ 0
h : LinearIndependent R ![x, AlternatingMap.constOfIsEmpty R M ι 1]
f : AlternatingMap R M R ι ≃ₗ[R] R := LinearEquiv.symm AlternatingMap.constLinearEquivOfIsEmpty
⊢ Matrix.vecCons (↑f x) ![1] { val := 1, isLt := (_ : (fun a => a < Nat.succ (Nat.succ 0)) 1) } =
    (↑↑f ∘ ![x, AlternatingMap.constOfIsEmpty R M ι 1])
      { val := 1, isLt := (_ : (fun a => a < Nat.succ (Nat.succ 0)) 1) }
[PROOFSTEP]
simp
[GOAL]
case h
R : Type u_1
inst✝⁴ : LinearOrderedCommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_3
inst✝¹ : Nontrivial R
inst✝ : IsEmpty ι
x : AlternatingMap R M R ι
hx : x ≠ 0
h : LinearIndependent R ![x, AlternatingMap.constOfIsEmpty R M ι 1]
f : AlternatingMap R M R ι ≃ₗ[R] R := LinearEquiv.symm AlternatingMap.constLinearEquivOfIsEmpty
H : LinearIndependent R ![↑f x, 1]
⊢ False
[PROOFSTEP]
rw [linearIndependent_iff'] at H 
[GOAL]
case h
R : Type u_1
inst✝⁴ : LinearOrderedCommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_3
inst✝¹ : Nontrivial R
inst✝ : IsEmpty ι
x : AlternatingMap R M R ι
hx : x ≠ 0
h : LinearIndependent R ![x, AlternatingMap.constOfIsEmpty R M ι 1]
f : AlternatingMap R M R ι ≃ₗ[R] R := LinearEquiv.symm AlternatingMap.constLinearEquivOfIsEmpty
H :
  ∀ (s : Finset (Fin (Nat.succ (Nat.succ 0)))) (g : Fin (Nat.succ (Nat.succ 0)) → R),
    ∑ i in s, g i • Matrix.vecCons (↑f x) ![1] i = 0 → ∀ (i : Fin (Nat.succ (Nat.succ 0))), i ∈ s → g i = 0
⊢ False
[PROOFSTEP]
simpa using H Finset.univ ![1, -f x] (by simp [Fin.sum_univ_succ]) 0 (by simp)
[GOAL]
R : Type u_1
inst✝⁴ : LinearOrderedCommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_3
inst✝¹ : Nontrivial R
inst✝ : IsEmpty ι
x : AlternatingMap R M R ι
hx : x ≠ 0
h : LinearIndependent R ![x, AlternatingMap.constOfIsEmpty R M ι 1]
f : AlternatingMap R M R ι ≃ₗ[R] R := LinearEquiv.symm AlternatingMap.constLinearEquivOfIsEmpty
H :
  ∀ (s : Finset (Fin (Nat.succ (Nat.succ 0)))) (g : Fin (Nat.succ (Nat.succ 0)) → R),
    ∑ i in s, g i • Matrix.vecCons (↑f x) ![1] i = 0 → ∀ (i : Fin (Nat.succ (Nat.succ 0))), i ∈ s → g i = 0
⊢ ∑ i : Fin (Nat.succ (Nat.succ 0)), Matrix.vecCons 1 ![-↑f x] i • Matrix.vecCons (↑f x) ![1] i = 0
[PROOFSTEP]
simp [Fin.sum_univ_succ]
[GOAL]
R : Type u_1
inst✝⁴ : LinearOrderedCommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_3
inst✝¹ : Nontrivial R
inst✝ : IsEmpty ι
x : AlternatingMap R M R ι
hx : x ≠ 0
h : LinearIndependent R ![x, AlternatingMap.constOfIsEmpty R M ι 1]
f : AlternatingMap R M R ι ≃ₗ[R] R := LinearEquiv.symm AlternatingMap.constLinearEquivOfIsEmpty
H :
  ∀ (s : Finset (Fin (Nat.succ (Nat.succ 0)))) (g : Fin (Nat.succ (Nat.succ 0)) → R),
    ∑ i in s, g i • Matrix.vecCons (↑f x) ![1] i = 0 → ∀ (i : Fin (Nat.succ (Nat.succ 0))), i ∈ s → g i = 0
⊢ 0 ∈ Finset.univ
[PROOFSTEP]
simp
[GOAL]
R : Type u_1
inst✝⁴ : LinearOrderedCommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
e₁ e₂ : Basis ι R M
⊢ SameRay R (det e₁) (det e₂) ↔ SameRay R (↑(det e₁) ↑e₂ • det e₂) (det e₂)
[PROOFSTEP]
rw [← e₁.det.eq_smul_basis_det e₂]
[GOAL]
R : Type u_1
inst✝⁴ : LinearOrderedCommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
e : Basis ι R M
x : Orientation R M ι
⊢ x = Basis.orientation e ∨ x = -Basis.orientation e
[PROOFSTEP]
induction' x using Module.Ray.ind with x hx
[GOAL]
case h
R : Type u_1
inst✝⁴ : LinearOrderedCommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
e : Basis ι R M
x : AlternatingMap R M R ι
hx : x ≠ 0
⊢ rayOfNeZero R x hx = Basis.orientation e ∨ rayOfNeZero R x hx = -Basis.orientation e
[PROOFSTEP]
rw [← x.map_basis_ne_zero_iff e] at hx 
[GOAL]
case h
R : Type u_1
inst✝⁴ : LinearOrderedCommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
e : Basis ι R M
x : AlternatingMap R M R ι
hx✝ : x ≠ 0
hx : ↑x ↑e ≠ 0
⊢ rayOfNeZero R x hx✝ = Basis.orientation e ∨ rayOfNeZero R x hx✝ = -Basis.orientation e
[PROOFSTEP]
rwa [Basis.orientation, ray_eq_iff, neg_rayOfNeZero, ray_eq_iff, x.eq_smul_basis_det e,
  sameRay_neg_smul_left_iff_of_ne e.det_ne_zero hx, sameRay_smul_left_iff_of_ne e.det_ne_zero hx, lt_or_lt_iff_ne,
  ne_comm]
[GOAL]
R : Type u_1
inst✝⁴ : LinearOrderedCommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
e : Basis ι R M
f : M ≃ₗ[R] M
⊢ Basis.orientation (Basis.map e f) = Basis.orientation e ↔ 0 < ↑LinearMap.det ↑f
[PROOFSTEP]
rw [orientation_map, e.map_orientation_eq_det_inv_smul, units_inv_smul, units_smul_eq_self_iff, LinearEquiv.coe_det]
[GOAL]
R : Type u_1
inst✝⁴ : LinearOrderedCommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
e : Basis ι R M
f : M ≃ₗ[R] M
⊢ Basis.orientation (Basis.map e f) = -Basis.orientation e ↔ ↑LinearMap.det ↑f < 0
[PROOFSTEP]
rw [orientation_map, e.map_orientation_eq_det_inv_smul, units_inv_smul, units_smul_eq_neg_iff, LinearEquiv.coe_det]
[GOAL]
R : Type u_1
inst✝⁵ : LinearOrderedCommRing R
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
ι : Type u_3
inst✝² : Fintype ι
inst✝¹ : DecidableEq ι
inst✝ : Nontrivial R
e : Basis ι R M
i : ι
⊢ Basis.orientation (unitsSMul e (Function.update 1 i (-1))) = -Basis.orientation e
[PROOFSTEP]
rw [orientation_unitsSMul, Finset.prod_update_of_mem (Finset.mem_univ _)]
[GOAL]
R : Type u_1
inst✝⁵ : LinearOrderedCommRing R
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
ι : Type u_3
inst✝² : Fintype ι
inst✝¹ : DecidableEq ι
inst✝ : Nontrivial R
e : Basis ι R M
i : ι
⊢ (-1 * ∏ x in Finset.univ \ {i}, OfNat.ofNat 1 x)⁻¹ • Basis.orientation e = -Basis.orientation e
[PROOFSTEP]
simp
[GOAL]
R : Type u_1
inst✝⁶ : LinearOrderedCommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
ι : Type u_3
inst✝³ : Fintype ι
inst✝² : DecidableEq ι
inst✝¹ : Nontrivial R
inst✝ : Nonempty ι
e : Basis ι R M
x : Orientation R M ι
⊢ Basis.orientation (adjustToOrientation e x) = x
[PROOFSTEP]
rw [adjustToOrientation]
[GOAL]
R : Type u_1
inst✝⁶ : LinearOrderedCommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
ι : Type u_3
inst✝³ : Fintype ι
inst✝² : DecidableEq ι
inst✝¹ : Nontrivial R
inst✝ : Nonempty ι
e : Basis ι R M
x : Orientation R M ι
⊢ Basis.orientation
      (if Basis.orientation e = x then e else unitsSMul e (Function.update 1 (Classical.arbitrary ι) (-1))) =
    x
[PROOFSTEP]
split_ifs with h
[GOAL]
case pos
R : Type u_1
inst✝⁶ : LinearOrderedCommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
ι : Type u_3
inst✝³ : Fintype ι
inst✝² : DecidableEq ι
inst✝¹ : Nontrivial R
inst✝ : Nonempty ι
e : Basis ι R M
x : Orientation R M ι
h : Basis.orientation e = x
⊢ Basis.orientation e = x
[PROOFSTEP]
exact h
[GOAL]
case neg
R : Type u_1
inst✝⁶ : LinearOrderedCommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
ι : Type u_3
inst✝³ : Fintype ι
inst✝² : DecidableEq ι
inst✝¹ : Nontrivial R
inst✝ : Nonempty ι
e : Basis ι R M
x : Orientation R M ι
h : ¬Basis.orientation e = x
⊢ Basis.orientation (unitsSMul e (Function.update 1 (Classical.arbitrary ι) (-1))) = x
[PROOFSTEP]
rw [orientation_neg_single, eq_comm, ← orientation_ne_iff_eq_neg, ne_comm]
[GOAL]
case neg
R : Type u_1
inst✝⁶ : LinearOrderedCommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
ι : Type u_3
inst✝³ : Fintype ι
inst✝² : DecidableEq ι
inst✝¹ : Nontrivial R
inst✝ : Nonempty ι
e : Basis ι R M
x : Orientation R M ι
h : ¬Basis.orientation e = x
⊢ Basis.orientation e ≠ x
[PROOFSTEP]
exact h
[GOAL]
R : Type u_1
inst✝⁶ : LinearOrderedCommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
ι : Type u_3
inst✝³ : Fintype ι
inst✝² : DecidableEq ι
inst✝¹ : Nontrivial R
inst✝ : Nonempty ι
e : Basis ι R M
x : Orientation R M ι
i : ι
⊢ ↑(adjustToOrientation e x) i = ↑e i ∨ ↑(adjustToOrientation e x) i = -↑e i
[PROOFSTEP]
rw [adjustToOrientation]
[GOAL]
R : Type u_1
inst✝⁶ : LinearOrderedCommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
ι : Type u_3
inst✝³ : Fintype ι
inst✝² : DecidableEq ι
inst✝¹ : Nontrivial R
inst✝ : Nonempty ι
e : Basis ι R M
x : Orientation R M ι
i : ι
⊢ ↑(if Basis.orientation e = x then e else unitsSMul e (Function.update 1 (Classical.arbitrary ι) (-1))) i = ↑e i ∨
    ↑(if Basis.orientation e = x then e else unitsSMul e (Function.update 1 (Classical.arbitrary ι) (-1))) i = -↑e i
[PROOFSTEP]
split_ifs with h
[GOAL]
case pos
R : Type u_1
inst✝⁶ : LinearOrderedCommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
ι : Type u_3
inst✝³ : Fintype ι
inst✝² : DecidableEq ι
inst✝¹ : Nontrivial R
inst✝ : Nonempty ι
e : Basis ι R M
x : Orientation R M ι
i : ι
h : Basis.orientation e = x
⊢ ↑e i = ↑e i ∨ ↑e i = -↑e i
[PROOFSTEP]
simp
[GOAL]
case neg
R : Type u_1
inst✝⁶ : LinearOrderedCommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
ι : Type u_3
inst✝³ : Fintype ι
inst✝² : DecidableEq ι
inst✝¹ : Nontrivial R
inst✝ : Nonempty ι
e : Basis ι R M
x : Orientation R M ι
i : ι
h : ¬Basis.orientation e = x
⊢ ↑(unitsSMul e (Function.update 1 (Classical.arbitrary ι) (-1))) i = ↑e i ∨
    ↑(unitsSMul e (Function.update 1 (Classical.arbitrary ι) (-1))) i = -↑e i
[PROOFSTEP]
by_cases hi : i = Classical.arbitrary ι
[GOAL]
case pos
R : Type u_1
inst✝⁶ : LinearOrderedCommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
ι : Type u_3
inst✝³ : Fintype ι
inst✝² : DecidableEq ι
inst✝¹ : Nontrivial R
inst✝ : Nonempty ι
e : Basis ι R M
x : Orientation R M ι
i : ι
h : ¬Basis.orientation e = x
hi : i = Classical.arbitrary ι
⊢ ↑(unitsSMul e (Function.update 1 (Classical.arbitrary ι) (-1))) i = ↑e i ∨
    ↑(unitsSMul e (Function.update 1 (Classical.arbitrary ι) (-1))) i = -↑e i
[PROOFSTEP]
simp [unitsSMul_apply, hi]
[GOAL]
case neg
R : Type u_1
inst✝⁶ : LinearOrderedCommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
ι : Type u_3
inst✝³ : Fintype ι
inst✝² : DecidableEq ι
inst✝¹ : Nontrivial R
inst✝ : Nonempty ι
e : Basis ι R M
x : Orientation R M ι
i : ι
h : ¬Basis.orientation e = x
hi : ¬i = Classical.arbitrary ι
⊢ ↑(unitsSMul e (Function.update 1 (Classical.arbitrary ι) (-1))) i = ↑e i ∨
    ↑(unitsSMul e (Function.update 1 (Classical.arbitrary ι) (-1))) i = -↑e i
[PROOFSTEP]
simp [unitsSMul_apply, hi]
[GOAL]
R : Type u_1
inst✝⁶ : LinearOrderedCommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
ι : Type u_3
inst✝³ : Fintype ι
inst✝² : DecidableEq ι
inst✝¹ : Nontrivial R
inst✝ : Nonempty ι
e : Basis ι R M
x : Orientation R M ι
⊢ det (adjustToOrientation e x) = det e ∨ det (adjustToOrientation e x) = -det e
[PROOFSTEP]
dsimp [Basis.adjustToOrientation]
[GOAL]
R : Type u_1
inst✝⁶ : LinearOrderedCommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
ι : Type u_3
inst✝³ : Fintype ι
inst✝² : DecidableEq ι
inst✝¹ : Nontrivial R
inst✝ : Nonempty ι
e : Basis ι R M
x : Orientation R M ι
⊢ det (if Basis.orientation e = x then e else unitsSMul e (Function.update 1 (Classical.arbitrary ι) (-1))) = det e ∨
    det (if Basis.orientation e = x then e else unitsSMul e (Function.update 1 (Classical.arbitrary ι) (-1))) = -det e
[PROOFSTEP]
split_ifs
[GOAL]
case pos
R : Type u_1
inst✝⁶ : LinearOrderedCommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
ι : Type u_3
inst✝³ : Fintype ι
inst✝² : DecidableEq ι
inst✝¹ : Nontrivial R
inst✝ : Nonempty ι
e : Basis ι R M
x : Orientation R M ι
h✝ : Basis.orientation e = x
⊢ det e = det e ∨ det e = -det e
[PROOFSTEP]
left
[GOAL]
case pos.h
R : Type u_1
inst✝⁶ : LinearOrderedCommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
ι : Type u_3
inst✝³ : Fintype ι
inst✝² : DecidableEq ι
inst✝¹ : Nontrivial R
inst✝ : Nonempty ι
e : Basis ι R M
x : Orientation R M ι
h✝ : Basis.orientation e = x
⊢ det e = det e
[PROOFSTEP]
rfl
[GOAL]
case neg
R : Type u_1
inst✝⁶ : LinearOrderedCommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
ι : Type u_3
inst✝³ : Fintype ι
inst✝² : DecidableEq ι
inst✝¹ : Nontrivial R
inst✝ : Nonempty ι
e : Basis ι R M
x : Orientation R M ι
h✝ : ¬Basis.orientation e = x
⊢ det (unitsSMul e (Function.update 1 (Classical.arbitrary ι) (-1))) = det e ∨
    det (unitsSMul e (Function.update 1 (Classical.arbitrary ι) (-1))) = -det e
[PROOFSTEP]
right
[GOAL]
case neg.h
R : Type u_1
inst✝⁶ : LinearOrderedCommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
ι : Type u_3
inst✝³ : Fintype ι
inst✝² : DecidableEq ι
inst✝¹ : Nontrivial R
inst✝ : Nonempty ι
e : Basis ι R M
x : Orientation R M ι
h✝ : ¬Basis.orientation e = x
⊢ det (unitsSMul e (Function.update 1 (Classical.arbitrary ι) (-1))) = -det e
[PROOFSTEP]
simp [e.det_unitsSMul, ← Units.coe_prod, Finset.prod_update_of_mem]
[GOAL]
case neg.h
R : Type u_1
inst✝⁶ : LinearOrderedCommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
ι : Type u_3
inst✝³ : Fintype ι
inst✝² : DecidableEq ι
inst✝¹ : Nontrivial R
inst✝ : Nonempty ι
e : Basis ι R M
x : Orientation R M ι
h✝ : ¬Basis.orientation e = x
⊢ -1 • det e = -det e
[PROOFSTEP]
ext
[GOAL]
case neg.h.H
R : Type u_1
inst✝⁶ : LinearOrderedCommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
ι : Type u_3
inst✝³ : Fintype ι
inst✝² : DecidableEq ι
inst✝¹ : Nontrivial R
inst✝ : Nonempty ι
e : Basis ι R M
x : Orientation R M ι
h✝ : ¬Basis.orientation e = x
x✝ : ι → M
⊢ ↑(-1 • det e) x✝ = ↑(-det e) x✝
[PROOFSTEP]
simp
[GOAL]
R : Type u_1
inst✝⁶ : LinearOrderedCommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
ι : Type u_3
inst✝³ : Fintype ι
inst✝² : DecidableEq ι
inst✝¹ : Nontrivial R
inst✝ : Nonempty ι
e : Basis ι R M
x : Orientation R M ι
v : ι → M
⊢ |↑(det (adjustToOrientation e x)) v| = |↑(det e) v|
[PROOFSTEP]
cases' e.det_adjustToOrientation x with h h
[GOAL]
case inl
R : Type u_1
inst✝⁶ : LinearOrderedCommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
ι : Type u_3
inst✝³ : Fintype ι
inst✝² : DecidableEq ι
inst✝¹ : Nontrivial R
inst✝ : Nonempty ι
e : Basis ι R M
x : Orientation R M ι
v : ι → M
h : det (adjustToOrientation e x) = det e
⊢ |↑(det (adjustToOrientation e x)) v| = |↑(det e) v|
[PROOFSTEP]
simp [h]
[GOAL]
case inr
R : Type u_1
inst✝⁶ : LinearOrderedCommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
ι : Type u_3
inst✝³ : Fintype ι
inst✝² : DecidableEq ι
inst✝¹ : Nontrivial R
inst✝ : Nonempty ι
e : Basis ι R M
x : Orientation R M ι
v : ι → M
h : det (adjustToOrientation e x) = -det e
⊢ |↑(det (adjustToOrientation e x)) v| = |↑(det e) v|
[PROOFSTEP]
simp [h]
[GOAL]
R : Type u_1
inst✝³ : LinearOrderedField R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : Fintype ι
_i : FiniteDimensional R M
x₁ x₂ : Orientation R M ι
h : Fintype.card ι = finrank R M
⊢ x₁ = x₂ ∨ x₁ = -x₂
[PROOFSTEP]
have e := (finBasis R M).reindex (Fintype.equivFinOfCardEq h).symm
[GOAL]
R : Type u_1
inst✝³ : LinearOrderedField R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : Fintype ι
_i : FiniteDimensional R M
x₁ x₂ : Orientation R M ι
h : Fintype.card ι = finrank R M
e : Basis ι R M
⊢ x₁ = x₂ ∨ x₁ = -x₂
[PROOFSTEP]
letI := Classical.decEq ι
[GOAL]
R : Type u_1
inst✝³ : LinearOrderedField R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : Fintype ι
_i : FiniteDimensional R M
x₁ x₂ : Orientation R M ι
h : Fintype.card ι = finrank R M
e : Basis ι R M
this : DecidableEq ι := Classical.decEq ι
⊢ x₁ = x₂ ∨ x₁ = -x₂
[PROOFSTEP]
have orientation_neg_neg : ∀ f : Basis ι R M, - -Basis.orientation f = Basis.orientation f := by simp
[GOAL]
R : Type u_1
inst✝³ : LinearOrderedField R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : Fintype ι
_i : FiniteDimensional R M
x₁ x₂ : Orientation R M ι
h : Fintype.card ι = finrank R M
e : Basis ι R M
this : DecidableEq ι := Classical.decEq ι
⊢ ∀ (f : Basis ι R M), - -Basis.orientation f = Basis.orientation f
[PROOFSTEP]
simp
[GOAL]
R : Type u_1
inst✝³ : LinearOrderedField R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : Fintype ι
_i : FiniteDimensional R M
x₁ x₂ : Orientation R M ι
h : Fintype.card ι = finrank R M
e : Basis ι R M
this : DecidableEq ι := Classical.decEq ι
orientation_neg_neg : ∀ (f : Basis ι R M), - -Basis.orientation f = Basis.orientation f
⊢ x₁ = x₂ ∨ x₁ = -x₂
[PROOFSTEP]
rcases e.orientation_eq_or_eq_neg x₁ with (h₁ | h₁)
[GOAL]
case inl
R : Type u_1
inst✝³ : LinearOrderedField R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : Fintype ι
_i : FiniteDimensional R M
x₁ x₂ : Orientation R M ι
h : Fintype.card ι = finrank R M
e : Basis ι R M
this : DecidableEq ι := Classical.decEq ι
orientation_neg_neg : ∀ (f : Basis ι R M), - -Basis.orientation f = Basis.orientation f
h₁ : x₁ = Basis.orientation e
⊢ x₁ = x₂ ∨ x₁ = -x₂
[PROOFSTEP]
rcases e.orientation_eq_or_eq_neg x₂ with (h₂ | h₂)
[GOAL]
case inr
R : Type u_1
inst✝³ : LinearOrderedField R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : Fintype ι
_i : FiniteDimensional R M
x₁ x₂ : Orientation R M ι
h : Fintype.card ι = finrank R M
e : Basis ι R M
this : DecidableEq ι := Classical.decEq ι
orientation_neg_neg : ∀ (f : Basis ι R M), - -Basis.orientation f = Basis.orientation f
h₁ : x₁ = -Basis.orientation e
⊢ x₁ = x₂ ∨ x₁ = -x₂
[PROOFSTEP]
rcases e.orientation_eq_or_eq_neg x₂ with (h₂ | h₂)
[GOAL]
case inl.inl
R : Type u_1
inst✝³ : LinearOrderedField R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : Fintype ι
_i : FiniteDimensional R M
x₁ x₂ : Orientation R M ι
h : Fintype.card ι = finrank R M
e : Basis ι R M
this : DecidableEq ι := Classical.decEq ι
orientation_neg_neg : ∀ (f : Basis ι R M), - -Basis.orientation f = Basis.orientation f
h₁ : x₁ = Basis.orientation e
h₂ : x₂ = Basis.orientation e
⊢ x₁ = x₂ ∨ x₁ = -x₂
[PROOFSTEP]
simp [h₁, h₂, orientation_neg_neg]
[GOAL]
case inl.inr
R : Type u_1
inst✝³ : LinearOrderedField R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : Fintype ι
_i : FiniteDimensional R M
x₁ x₂ : Orientation R M ι
h : Fintype.card ι = finrank R M
e : Basis ι R M
this : DecidableEq ι := Classical.decEq ι
orientation_neg_neg : ∀ (f : Basis ι R M), - -Basis.orientation f = Basis.orientation f
h₁ : x₁ = Basis.orientation e
h₂ : x₂ = -Basis.orientation e
⊢ x₁ = x₂ ∨ x₁ = -x₂
[PROOFSTEP]
simp [h₁, h₂, orientation_neg_neg]
[GOAL]
case inr.inl
R : Type u_1
inst✝³ : LinearOrderedField R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : Fintype ι
_i : FiniteDimensional R M
x₁ x₂ : Orientation R M ι
h : Fintype.card ι = finrank R M
e : Basis ι R M
this : DecidableEq ι := Classical.decEq ι
orientation_neg_neg : ∀ (f : Basis ι R M), - -Basis.orientation f = Basis.orientation f
h₁ : x₁ = -Basis.orientation e
h₂ : x₂ = Basis.orientation e
⊢ x₁ = x₂ ∨ x₁ = -x₂
[PROOFSTEP]
simp [h₁, h₂, orientation_neg_neg]
[GOAL]
case inr.inr
R : Type u_1
inst✝³ : LinearOrderedField R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : Fintype ι
_i : FiniteDimensional R M
x₁ x₂ : Orientation R M ι
h : Fintype.card ι = finrank R M
e : Basis ι R M
this : DecidableEq ι := Classical.decEq ι
orientation_neg_neg : ∀ (f : Basis ι R M), - -Basis.orientation f = Basis.orientation f
h₁ : x₁ = -Basis.orientation e
h₂ : x₂ = -Basis.orientation e
⊢ x₁ = x₂ ∨ x₁ = -x₂
[PROOFSTEP]
simp [h₁, h₂, orientation_neg_neg]
[GOAL]
R : Type u_1
inst✝³ : LinearOrderedField R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : Fintype ι
_i : FiniteDimensional R M
x : Orientation R M ι
f : M ≃ₗ[R] M
h : Fintype.card ι = finrank R M
⊢ ↑(map ι f) x = x ↔ 0 < ↑LinearMap.det ↑f
[PROOFSTEP]
cases isEmpty_or_nonempty ι
[GOAL]
case inl
R : Type u_1
inst✝³ : LinearOrderedField R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : Fintype ι
_i : FiniteDimensional R M
x : Orientation R M ι
f : M ≃ₗ[R] M
h : Fintype.card ι = finrank R M
h✝ : IsEmpty ι
⊢ ↑(map ι f) x = x ↔ 0 < ↑LinearMap.det ↑f
[PROOFSTEP]
have H : finrank R M = 0 := by
  refine' h.symm.trans _
  convert @Fintype.card_of_isEmpty ι _
[GOAL]
R : Type u_1
inst✝³ : LinearOrderedField R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : Fintype ι
_i : FiniteDimensional R M
x : Orientation R M ι
f : M ≃ₗ[R] M
h : Fintype.card ι = finrank R M
h✝ : IsEmpty ι
⊢ finrank R M = 0
[PROOFSTEP]
refine' h.symm.trans _
[GOAL]
R : Type u_1
inst✝³ : LinearOrderedField R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : Fintype ι
_i : FiniteDimensional R M
x : Orientation R M ι
f : M ≃ₗ[R] M
h : Fintype.card ι = finrank R M
h✝ : IsEmpty ι
⊢ Fintype.card ι = 0
[PROOFSTEP]
convert @Fintype.card_of_isEmpty ι _
[GOAL]
case inl
R : Type u_1
inst✝³ : LinearOrderedField R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : Fintype ι
_i : FiniteDimensional R M
x : Orientation R M ι
f : M ≃ₗ[R] M
h : Fintype.card ι = finrank R M
h✝ : IsEmpty ι
H : finrank R M = 0
⊢ ↑(map ι f) x = x ↔ 0 < ↑LinearMap.det ↑f
[PROOFSTEP]
simp [LinearMap.det_eq_one_of_finrank_eq_zero H]
[GOAL]
case inr
R : Type u_1
inst✝³ : LinearOrderedField R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : Fintype ι
_i : FiniteDimensional R M
x : Orientation R M ι
f : M ≃ₗ[R] M
h : Fintype.card ι = finrank R M
h✝ : Nonempty ι
⊢ ↑(map ι f) x = x ↔ 0 < ↑LinearMap.det ↑f
[PROOFSTEP]
rw [map_eq_det_inv_smul _ _ h, units_inv_smul, units_smul_eq_self_iff, LinearEquiv.coe_det]
[GOAL]
R : Type u_1
inst✝³ : LinearOrderedField R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : Fintype ι
_i : FiniteDimensional R M
x : Orientation R M ι
f : M ≃ₗ[R] M
h : Fintype.card ι = finrank R M
⊢ ↑(map ι f) x = -x ↔ ↑LinearMap.det ↑f < 0
[PROOFSTEP]
cases isEmpty_or_nonempty ι
[GOAL]
case inl
R : Type u_1
inst✝³ : LinearOrderedField R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : Fintype ι
_i : FiniteDimensional R M
x : Orientation R M ι
f : M ≃ₗ[R] M
h : Fintype.card ι = finrank R M
h✝ : IsEmpty ι
⊢ ↑(map ι f) x = -x ↔ ↑LinearMap.det ↑f < 0
[PROOFSTEP]
have H : finrank R M = 0 := by
  refine' h.symm.trans _
  convert @Fintype.card_of_isEmpty ι _
[GOAL]
R : Type u_1
inst✝³ : LinearOrderedField R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : Fintype ι
_i : FiniteDimensional R M
x : Orientation R M ι
f : M ≃ₗ[R] M
h : Fintype.card ι = finrank R M
h✝ : IsEmpty ι
⊢ finrank R M = 0
[PROOFSTEP]
refine' h.symm.trans _
[GOAL]
R : Type u_1
inst✝³ : LinearOrderedField R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : Fintype ι
_i : FiniteDimensional R M
x : Orientation R M ι
f : M ≃ₗ[R] M
h : Fintype.card ι = finrank R M
h✝ : IsEmpty ι
⊢ Fintype.card ι = 0
[PROOFSTEP]
convert @Fintype.card_of_isEmpty ι _
[GOAL]
case inl
R : Type u_1
inst✝³ : LinearOrderedField R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : Fintype ι
_i : FiniteDimensional R M
x : Orientation R M ι
f : M ≃ₗ[R] M
h : Fintype.card ι = finrank R M
h✝ : IsEmpty ι
H : finrank R M = 0
⊢ ↑(map ι f) x = -x ↔ ↑LinearMap.det ↑f < 0
[PROOFSTEP]
simp [LinearMap.det_eq_one_of_finrank_eq_zero H, Module.Ray.ne_neg_self x]
[GOAL]
case inr
R : Type u_1
inst✝³ : LinearOrderedField R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : Fintype ι
_i : FiniteDimensional R M
x : Orientation R M ι
f : M ≃ₗ[R] M
h : Fintype.card ι = finrank R M
h✝ : Nonempty ι
⊢ ↑(map ι f) x = -x ↔ ↑LinearMap.det ↑f < 0
[PROOFSTEP]
have H : 0 < finrank R M := by
  rw [← h]
  exact Fintype.card_pos
[GOAL]
R : Type u_1
inst✝³ : LinearOrderedField R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : Fintype ι
_i : FiniteDimensional R M
x : Orientation R M ι
f : M ≃ₗ[R] M
h : Fintype.card ι = finrank R M
h✝ : Nonempty ι
⊢ 0 < finrank R M
[PROOFSTEP]
rw [← h]
[GOAL]
R : Type u_1
inst✝³ : LinearOrderedField R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : Fintype ι
_i : FiniteDimensional R M
x : Orientation R M ι
f : M ≃ₗ[R] M
h : Fintype.card ι = finrank R M
h✝ : Nonempty ι
⊢ 0 < Fintype.card ι
[PROOFSTEP]
exact Fintype.card_pos
[GOAL]
case inr
R : Type u_1
inst✝³ : LinearOrderedField R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : Fintype ι
_i : FiniteDimensional R M
x : Orientation R M ι
f : M ≃ₗ[R] M
h : Fintype.card ι = finrank R M
h✝ : Nonempty ι
H : 0 < finrank R M
⊢ ↑(map ι f) x = -x ↔ ↑LinearMap.det ↑f < 0
[PROOFSTEP]
haveI : FiniteDimensional R M := finiteDimensional_of_finrank H
[GOAL]
case inr
R : Type u_1
inst✝³ : LinearOrderedField R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : Fintype ι
_i : FiniteDimensional R M
x : Orientation R M ι
f : M ≃ₗ[R] M
h : Fintype.card ι = finrank R M
h✝ : Nonempty ι
H : 0 < finrank R M
this : FiniteDimensional R M
⊢ ↑(map ι f) x = -x ↔ ↑LinearMap.det ↑f < 0
[PROOFSTEP]
rw [map_eq_det_inv_smul _ _ h, units_inv_smul, units_smul_eq_neg_iff, LinearEquiv.coe_det]
