Theorem: LinearMap.separatingLeft_dualProd
File: Mathlib/LinearAlgebra/QuadraticForm/Dual.lean

tactic: rw [separatingLeft_iff_ker_eq_bot, ker_eq_bot]
state_before: R : Type u_1
M : Type u_2
N : Type u_3
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
⊢ SeparatingLeft (dualProd R M) ↔ Function.Injective ⇑(Module.Dual.eval R M)
state_after: R : Type u_1
M : Type u_2
N : Type u_3
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
⊢ Function.Injective ⇑(dualProd R M) ↔ Function.Injective ⇑(Module.Dual.eval R M)

tactic: let e := LinearEquiv.prodComm R _ _ ≪≫ₗ Module.dualProdDualEquivDual R (Module.Dual R M) M
state_before: R : Type u_1
M : Type u_2
N : Type u_3
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
⊢ Function.Injective ⇑(dualProd R M) ↔ Function.Injective ⇑(Module.Dual.eval R M)
state_after: R : Type u_1
M : Type u_2
N : Type u_3
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
e : (Module.Dual R M × Module.Dual R (Module.Dual R M)) ≃ₗ[R] Module.Dual R (Module.Dual R M × M) :=
  LinearEquiv.prodComm R (Module.Dual R M) (Module.Dual R (Module.Dual R M)) ≪≫ₗ
    Module.dualProdDualEquivDual R (Module.Dual R M) M
⊢ Function.Injective ⇑(dualProd R M) ↔ Function.Injective ⇑(Module.Dual.eval R M)

tactic: let h_d := e.symm.toLinearMap.comp (dualProd R M)
state_before: R : Type u_1
M : Type u_2
N : Type u_3
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
e : (Module.Dual R M × Module.Dual R (Module.Dual R M)) ≃ₗ[R] Module.Dual R (Module.Dual R M × M) :=
  LinearEquiv.prodComm R (Module.Dual R M) (Module.Dual R (Module.Dual R M)) ≪≫ₗ
    Module.dualProdDualEquivDual R (Module.Dual R M) M
⊢ Function.Injective ⇑(dualProd R M) ↔ Function.Injective ⇑(Module.Dual.eval R M)
state_after: R : Type u_1
M : Type u_2
N : Type u_3
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
e : (Module.Dual R M × Module.Dual R (Module.Dual R M)) ≃ₗ[R] Module.Dual R (Module.Dual R M × M) :=
  LinearEquiv.prodComm R (Module.Dual R M) (Module.Dual R (Module.Dual R M)) ≪≫ₗ
    Module.dualProdDualEquivDual R (Module.Dual R M) M
h_d : Module.Dual R M × M →ₗ[R] Module.Dual R M × Module.Dual R (Module.Dual R M) := ↑e.symm ∘ₗ dualProd R M
⊢ Function.Injective ⇑(dualProd R M) ↔ Function.Injective ⇑(Module.Dual.eval R M)

tactic: refine (Function.Injective.of_comp_iff e.symm.injective
  (dualProd R M)).symm.trans ?_
state_before: R : Type u_1
M : Type u_2
N : Type u_3
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
e : (Module.Dual R M × Module.Dual R (Module.Dual R M)) ≃ₗ[R] Module.Dual R (Module.Dual R M × M) :=
  LinearEquiv.prodComm R (Module.Dual R M) (Module.Dual R (Module.Dual R M)) ≪≫ₗ
    Module.dualProdDualEquivDual R (Module.Dual R M) M
h_d : Module.Dual R M × M →ₗ[R] Module.Dual R M × Module.Dual R (Module.Dual R M) := ↑e.symm ∘ₗ dualProd R M
⊢ Function.Injective ⇑(dualProd R M) ↔ Function.Injective ⇑(Module.Dual.eval R M)
state_after: R : Type u_1
M : Type u_2
N : Type u_3
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
e : (Module.Dual R M × Module.Dual R (Module.Dual R M)) ≃ₗ[R] Module.Dual R (Module.Dual R M × M) :=
  LinearEquiv.prodComm R (Module.Dual R M) (Module.Dual R (Module.Dual R M)) ≪≫ₗ
    Module.dualProdDualEquivDual R (Module.Dual R M) M
h_d : Module.Dual R M × M →ₗ[R] Module.Dual R M × Module.Dual R (Module.Dual R M) := ↑e.symm ∘ₗ dualProd R M
⊢ Function.Injective (⇑e.symm ∘ ⇑(dualProd R M)) ↔ Function.Injective ⇑(Module.Dual.eval R M)

tactic: rw [← LinearEquiv.coe_toLinearMap, ← coe_comp]
state_before: R : Type u_1
M : Type u_2
N : Type u_3
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
e : (Module.Dual R M × Module.Dual R (Module.Dual R M)) ≃ₗ[R] Module.Dual R (Module.Dual R M × M) :=
  LinearEquiv.prodComm R (Module.Dual R M) (Module.Dual R (Module.Dual R M)) ≪≫ₗ
    Module.dualProdDualEquivDual R (Module.Dual R M) M
h_d : Module.Dual R M × M →ₗ[R] Module.Dual R M × Module.Dual R (Module.Dual R M) := ↑e.symm ∘ₗ dualProd R M
⊢ Function.Injective (⇑e.symm ∘ ⇑(dualProd R M)) ↔ Function.Injective ⇑(Module.Dual.eval R M)
state_after: R : Type u_1
M : Type u_2
N : Type u_3
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
e : (Module.Dual R M × Module.Dual R (Module.Dual R M)) ≃ₗ[R] Module.Dual R (Module.Dual R M × M) :=
  LinearEquiv.prodComm R (Module.Dual R M) (Module.Dual R (Module.Dual R M)) ≪≫ₗ
    Module.dualProdDualEquivDual R (Module.Dual R M) M
h_d : Module.Dual R M × M →ₗ[R] Module.Dual R M × Module.Dual R (Module.Dual R M) := ↑e.symm ∘ₗ dualProd R M
⊢ Function.Injective ⇑(↑e.symm ∘ₗ dualProd R M) ↔ Function.Injective ⇑(Module.Dual.eval R M)

tactic: change Function.Injective h_d ↔ _
state_before: R : Type u_1
M : Type u_2
N : Type u_3
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
e : (Module.Dual R M × Module.Dual R (Module.Dual R M)) ≃ₗ[R] Module.Dual R (Module.Dual R M × M) :=
  LinearEquiv.prodComm R (Module.Dual R M) (Module.Dual R (Module.Dual R M)) ≪≫ₗ
    Module.dualProdDualEquivDual R (Module.Dual R M) M
h_d : Module.Dual R M × M →ₗ[R] Module.Dual R M × Module.Dual R (Module.Dual R M) := ↑e.symm ∘ₗ dualProd R M
⊢ Function.Injective ⇑(↑e.symm ∘ₗ dualProd R M) ↔ Function.Injective ⇑(Module.Dual.eval R M)
state_after: R : Type u_1
M : Type u_2
N : Type u_3
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
e : (Module.Dual R M × Module.Dual R (Module.Dual R M)) ≃ₗ[R] Module.Dual R (Module.Dual R M × M) :=
  LinearEquiv.prodComm R (Module.Dual R M) (Module.Dual R (Module.Dual R M)) ≪≫ₗ
    Module.dualProdDualEquivDual R (Module.Dual R M) M
h_d : Module.Dual R M × M →ₗ[R] Module.Dual R M × Module.Dual R (Module.Dual R M) := ↑e.symm ∘ₗ dualProd R M
⊢ Function.Injective ⇑h_d ↔ Function.Injective ⇑(Module.Dual.eval R M)

tactic: rw [this, coe_prodMap]
state_before: R : Type u_1
M : Type u_2
N : Type u_3
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
e : (Module.Dual R M × Module.Dual R (Module.Dual R M)) ≃ₗ[R] Module.Dual R (Module.Dual R M × M) :=
  LinearEquiv.prodComm R (Module.Dual R M) (Module.Dual R (Module.Dual R M)) ≪≫ₗ
    Module.dualProdDualEquivDual R (Module.Dual R M) M
h_d : Module.Dual R M × M →ₗ[R] Module.Dual R M × Module.Dual R (Module.Dual R M) := ↑e.symm ∘ₗ dualProd R M
this : h_d = id.prodMap (Module.Dual.eval R M)
⊢ Function.Injective ⇑h_d ↔ Function.Injective ⇑(Module.Dual.eval R M)
state_after: R : Type u_1
M : Type u_2
N : Type u_3
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
e : (Module.Dual R M × Module.Dual R (Module.Dual R M)) ≃ₗ[R] Module.Dual R (Module.Dual R M × M) :=
  LinearEquiv.prodComm R (Module.Dual R M) (Module.Dual R (Module.Dual R M)) ≪≫ₗ
    Module.dualProdDualEquivDual R (Module.Dual R M) M
h_d : Module.Dual R M × M →ₗ[R] Module.Dual R M × Module.Dual R (Module.Dual R M) := ↑e.symm ∘ₗ dualProd R M
this : h_d = id.prodMap (Module.Dual.eval R M)
⊢ Function.Injective (Prod.map ⇑id ⇑(Module.Dual.eval R M)) ↔ Function.Injective ⇑(Module.Dual.eval R M)

tactic: refine Prod.map_injective.trans ?_
state_before: R : Type u_1
M : Type u_2
N : Type u_3
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
e : (Module.Dual R M × Module.Dual R (Module.Dual R M)) ≃ₗ[R] Module.Dual R (Module.Dual R M × M) :=
  LinearEquiv.prodComm R (Module.Dual R M) (Module.Dual R (Module.Dual R M)) ≪≫ₗ
    Module.dualProdDualEquivDual R (Module.Dual R M) M
h_d : Module.Dual R M × M →ₗ[R] Module.Dual R M × Module.Dual R (Module.Dual R M) := ↑e.symm ∘ₗ dualProd R M
this : h_d = id.prodMap (Module.Dual.eval R M)
⊢ Function.Injective (Prod.map ⇑id ⇑(Module.Dual.eval R M)) ↔ Function.Injective ⇑(Module.Dual.eval R M)
state_after: R : Type u_1
M : Type u_2
N : Type u_3
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
e : (Module.Dual R M × Module.Dual R (Module.Dual R M)) ≃ₗ[R] Module.Dual R (Module.Dual R M × M) :=
  LinearEquiv.prodComm R (Module.Dual R M) (Module.Dual R (Module.Dual R M)) ≪≫ₗ
    Module.dualProdDualEquivDual R (Module.Dual R M) M
h_d : Module.Dual R M × M →ₗ[R] Module.Dual R M × Module.Dual R (Module.Dual R M) := ↑e.symm ∘ₗ dualProd R M
this : h_d = id.prodMap (Module.Dual.eval R M)
⊢ Function.Injective ⇑id ∧ Function.Injective ⇑(Module.Dual.eval R M) ↔ Function.Injective ⇑(Module.Dual.eval R M)

tactic: exact and_iff_right Function.injective_id
state_before: R : Type u_1
M : Type u_2
N : Type u_3
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
e : (Module.Dual R M × Module.Dual R (Module.Dual R M)) ≃ₗ[R] Module.Dual R (Module.Dual R M × M) :=
  LinearEquiv.prodComm R (Module.Dual R M) (Module.Dual R (Module.Dual R M)) ≪≫ₗ
    Module.dualProdDualEquivDual R (Module.Dual R M) M
h_d : Module.Dual R M × M →ₗ[R] Module.Dual R M × Module.Dual R (Module.Dual R M) := ↑e.symm ∘ₗ dualProd R M
this : h_d = id.prodMap (Module.Dual.eval R M)
⊢ Function.Injective ⇑id ∧ Function.Injective ⇑(Module.Dual.eval R M) ↔ Function.Injective ⇑(Module.Dual.eval R M)
state_after: no goals

tactic: refine ext fun x => Prod.ext ?_ ?_
state_before: R : Type u_1
M : Type u_2
N : Type u_3
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
e : (Module.Dual R M × Module.Dual R (Module.Dual R M)) ≃ₗ[R] Module.Dual R (Module.Dual R M × M) :=
  LinearEquiv.prodComm R (Module.Dual R M) (Module.Dual R (Module.Dual R M)) ≪≫ₗ
    Module.dualProdDualEquivDual R (Module.Dual R M) M
h_d : Module.Dual R M × M →ₗ[R] Module.Dual R M × Module.Dual R (Module.Dual R M) := ↑e.symm ∘ₗ dualProd R M
⊢ h_d = id.prodMap (Module.Dual.eval R M)
state_after: case refine_1
R : Type u_1
M : Type u_2
N : Type u_3
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
e : (Module.Dual R M × Module.Dual R (Module.Dual R M)) ≃ₗ[R] Module.Dual R (Module.Dual R M × M) :=
  LinearEquiv.prodComm R (Module.Dual R M) (Module.Dual R (Module.Dual R M)) ≪≫ₗ
    Module.dualProdDualEquivDual R (Module.Dual R M) M
h_d : Module.Dual R M × M →ₗ[R] Module.Dual R M × Module.Dual R (Module.Dual R M) := ↑e.symm ∘ₗ dualProd R M
x : Module.Dual R M × M
⊢ (h_d x).1 = ((id.prodMap (Module.Dual.eval R M)) x).1

case refine_2
R : Type u_1
M : Type u_2
N : Type u_3
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
e : (Module.Dual R M × Module.Dual R (Module.Dual R M)) ≃ₗ[R] Module.Dual R (Module.Dual R M × M) :=
  LinearEquiv.prodComm R (Module.Dual R M) (Module.Dual R (Module.Dual R M)) ≪≫ₗ
    Module.dualProdDualEquivDual R (Module.Dual R M) M
h_d : Module.Dual R M × M →ₗ[R] Module.Dual R M × Module.Dual R (Module.Dual R M) := ↑e.symm ∘ₗ dualProd R M
x : Module.Dual R M × M
⊢ (h_d x).2 = ((id.prodMap (Module.Dual.eval R M)) x).2

tactic: ext
state_before: case refine_1
R : Type u_1
M : Type u_2
N : Type u_3
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
e : (Module.Dual R M × Module.Dual R (Module.Dual R M)) ≃ₗ[R] Module.Dual R (Module.Dual R M × M) :=
  LinearEquiv.prodComm R (Module.Dual R M) (Module.Dual R (Module.Dual R M)) ≪≫ₗ
    Module.dualProdDualEquivDual R (Module.Dual R M) M
h_d : Module.Dual R M × M →ₗ[R] Module.Dual R M × Module.Dual R (Module.Dual R M) := ↑e.symm ∘ₗ dualProd R M
x : Module.Dual R M × M
⊢ (h_d x).1 = ((id.prodMap (Module.Dual.eval R M)) x).1
state_after: case refine_1.h
R : Type u_1
M : Type u_2
N : Type u_3
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
e : (Module.Dual R M × Module.Dual R (Module.Dual R M)) ≃ₗ[R] Module.Dual R (Module.Dual R M × M) :=
  LinearEquiv.prodComm R (Module.Dual R M) (Module.Dual R (Module.Dual R M)) ≪≫ₗ
    Module.dualProdDualEquivDual R (Module.Dual R M) M
h_d : Module.Dual R M × M →ₗ[R] Module.Dual R M × Module.Dual R (Module.Dual R M) := ↑e.symm ∘ₗ dualProd R M
x : Module.Dual R M × M
x✝ : M
⊢ (h_d x).1 x✝ = ((id.prodMap (Module.Dual.eval R M)) x).1 x✝

tactic: dsimp [e, h_d, Module.Dual.eval, LinearEquiv.prodComm]
state_before: case refine_1.h
R : Type u_1
M : Type u_2
N : Type u_3
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
e : (Module.Dual R M × Module.Dual R (Module.Dual R M)) ≃ₗ[R] Module.Dual R (Module.Dual R M × M) :=
  LinearEquiv.prodComm R (Module.Dual R M) (Module.Dual R (Module.Dual R M)) ≪≫ₗ
    Module.dualProdDualEquivDual R (Module.Dual R M) M
h_d : Module.Dual R M × M →ₗ[R] Module.Dual R M × Module.Dual R (Module.Dual R M) := ↑e.symm ∘ₗ dualProd R M
x : Module.Dual R M × M
x✝ : M
⊢ (h_d x).1 x✝ = ((id.prodMap (Module.Dual.eval R M)) x).1 x✝
state_after: case refine_1.h
R : Type u_1
M : Type u_2
N : Type u_3
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
e : (Module.Dual R M × Module.Dual R (Module.Dual R M)) ≃ₗ[R] Module.Dual R (Module.Dual R M × M) :=
  LinearEquiv.prodComm R (Module.Dual R M) (Module.Dual R (Module.Dual R M)) ≪≫ₗ
    Module.dualProdDualEquivDual R (Module.Dual R M) M
h_d : Module.Dual R M × M →ₗ[R] Module.Dual R M × Module.Dual R (Module.Dual R M) := ↑e.symm ∘ₗ dualProd R M
x : Module.Dual R M × M
x✝ : M
⊢ 0 + x.1 x✝ = x.1 x✝

tactic: simp
state_before: case refine_1.h
R : Type u_1
M : Type u_2
N : Type u_3
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
e : (Module.Dual R M × Module.Dual R (Module.Dual R M)) ≃ₗ[R] Module.Dual R (Module.Dual R M × M) :=
  LinearEquiv.prodComm R (Module.Dual R M) (Module.Dual R (Module.Dual R M)) ≪≫ₗ
    Module.dualProdDualEquivDual R (Module.Dual R M) M
h_d : Module.Dual R M × M →ₗ[R] Module.Dual R M × Module.Dual R (Module.Dual R M) := ↑e.symm ∘ₗ dualProd R M
x : Module.Dual R M × M
x✝ : M
⊢ 0 + x.1 x✝ = x.1 x✝
state_after: no goals

tactic: ext
state_before: case refine_2
R : Type u_1
M : Type u_2
N : Type u_3
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
e : (Module.Dual R M × Module.Dual R (Module.Dual R M)) ≃ₗ[R] Module.Dual R (Module.Dual R M × M) :=
  LinearEquiv.prodComm R (Module.Dual R M) (Module.Dual R (Module.Dual R M)) ≪≫ₗ
    Module.dualProdDualEquivDual R (Module.Dual R M) M
h_d : Module.Dual R M × M →ₗ[R] Module.Dual R M × Module.Dual R (Module.Dual R M) := ↑e.symm ∘ₗ dualProd R M
x : Module.Dual R M × M
⊢ (h_d x).2 = ((id.prodMap (Module.Dual.eval R M)) x).2
state_after: case refine_2.h
R : Type u_1
M : Type u_2
N : Type u_3
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
e : (Module.Dual R M × Module.Dual R (Module.Dual R M)) ≃ₗ[R] Module.Dual R (Module.Dual R M × M) :=
  LinearEquiv.prodComm R (Module.Dual R M) (Module.Dual R (Module.Dual R M)) ≪≫ₗ
    Module.dualProdDualEquivDual R (Module.Dual R M) M
h_d : Module.Dual R M × M →ₗ[R] Module.Dual R M × Module.Dual R (Module.Dual R M) := ↑e.symm ∘ₗ dualProd R M
x : Module.Dual R M × M
x✝ : Module.Dual R M
⊢ (h_d x).2 x✝ = ((id.prodMap (Module.Dual.eval R M)) x).2 x✝

tactic: dsimp [e, h_d, Module.Dual.eval, LinearEquiv.prodComm]
state_before: case refine_2.h
R : Type u_1
M : Type u_2
N : Type u_3
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
e : (Module.Dual R M × Module.Dual R (Module.Dual R M)) ≃ₗ[R] Module.Dual R (Module.Dual R M × M) :=
  LinearEquiv.prodComm R (Module.Dual R M) (Module.Dual R (Module.Dual R M)) ≪≫ₗ
    Module.dualProdDualEquivDual R (Module.Dual R M) M
h_d : Module.Dual R M × M →ₗ[R] Module.Dual R M × Module.Dual R (Module.Dual R M) := ↑e.symm ∘ₗ dualProd R M
x : Module.Dual R M × M
x✝ : Module.Dual R M
⊢ (h_d x).2 x✝ = ((id.prodMap (Module.Dual.eval R M)) x).2 x✝
state_after: case refine_2.h
R : Type u_1
M : Type u_2
N : Type u_3
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
e : (Module.Dual R M × Module.Dual R (Module.Dual R M)) ≃ₗ[R] Module.Dual R (Module.Dual R M × M) :=
  LinearEquiv.prodComm R (Module.Dual R M) (Module.Dual R (Module.Dual R M)) ≪≫ₗ
    Module.dualProdDualEquivDual R (Module.Dual R M) M
h_d : Module.Dual R M × M →ₗ[R] Module.Dual R M × Module.Dual R (Module.Dual R M) := ↑e.symm ∘ₗ dualProd R M
x : Module.Dual R M × M
x✝ : Module.Dual R M
⊢ x✝ x.2 + x.1 0 = x✝ x.2

tactic: simp
state_before: case refine_2.h
R : Type u_1
M : Type u_2
N : Type u_3
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
e : (Module.Dual R M × Module.Dual R (Module.Dual R M)) ≃ₗ[R] Module.Dual R (Module.Dual R M × M) :=
  LinearEquiv.prodComm R (Module.Dual R M) (Module.Dual R (Module.Dual R M)) ≪≫ₗ
    Module.dualProdDualEquivDual R (Module.Dual R M) M
h_d : Module.Dual R M × M →ₗ[R] Module.Dual R M × Module.Dual R (Module.Dual R M) := ↑e.symm ∘ₗ dualProd R M
x : Module.Dual R M × M
x✝ : Module.Dual R M
⊢ x✝ x.2 + x.1 0 = x✝ x.2
state_after: no goals