-- Mathlib/LinearAlgebra/PerfectPairing/Basic.lean
@[simp]
lemma map_dualCoannihilator_linearEquiv_flip (p : Submodule R (Dual R M)) :
    p.dualCoannihilator.map e.flip = (p.map e.symm).dualAnnihilator

Proof state:
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : IsReflexive R M
e : N ≃ₗ[R] Dual R M
p : Submodule R (Dual R M)
this✝ : IsReflexive R N
this : map e.flip.symm (map e.symm p).dualAnnihilator = map e.flip.symm (map e.flip p.dualCoannihilator)
⊢ map e.flip p.dualCoannihilator = (map e.symm p).dualAnnihilator

Tactic:
exact (Submodule.map_injective_of_injective e.flip.symm.injective this).symm

Result:
no goals

Proof state:
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : IsReflexive R M
e : N ≃ₗ[R] Dual R M
p : Submodule R (Dual R M)
this : IsReflexive R N
⊢ (map (↑e ∘ₗ ↑e.symm) p).dualCoannihilator = map (↑e.flip.symm ∘ₗ ↑e.flip) p.dualCoannihilator

Tactic:
simp [-coe_toLinearMap_flip]

Result:
no goals

Proof state:
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : IsReflexive R M
e : N ≃ₗ[R] Dual R M
p : Submodule R (Dual R M)
⊢ map e.flip p.dualCoannihilator = (map e.symm p).dualAnnihilator

Tactic:
have : IsReflexive R N := e.isReflexive_of_equiv_dual_of_isReflexive

Result:
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : IsReflexive R M
e : N ≃ₗ[R] Dual R M
p : Submodule R (Dual R M)
this : IsReflexive R N
⊢ map e.flip p.dualCoannihilator = (map e.symm p).dualAnnihilator

Proof state:
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : IsReflexive R M
e : N ≃ₗ[R] Dual R M
p : Submodule R (Dual R M)
this : IsReflexive R N
⊢ map e.flip.symm (map e.symm p).dualAnnihilator = map e.flip.symm (map e.flip p.dualCoannihilator)

Tactic:
erw [← dualCoannihilator_map_linearEquiv_flip, flip_flip, ← map_comp, ← map_comp]

Result:
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : IsReflexive R M
e : N ≃ₗ[R] Dual R M
p : Submodule R (Dual R M)
this : IsReflexive R N
⊢ (map (↑e ∘ₗ ↑e.symm) p).dualCoannihilator = map (↑e.flip.symm ∘ₗ ↑e.flip) p.dualCoannihilator

Proof state:
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : IsReflexive R M
e : N ≃ₗ[R] Dual R M
p : Submodule R (Dual R M)
this : IsReflexive R N
⊢ map e.flip p.dualCoannihilator = (map e.symm p).dualAnnihilator

Tactic:
suffices (p.map e.symm).dualAnnihilator.map e.flip.symm =
    (p.dualCoannihilator.map e.flip).map e.flip.symm by
  exact (Submodule.map_injective_of_injective e.flip.symm.injective this).symm

Result:
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : IsReflexive R M
e : N ≃ₗ[R] Dual R M
p : Submodule R (Dual R M)
this : IsReflexive R N
⊢ map e.flip.symm (map e.symm p).dualAnnihilator = map e.flip.symm (map e.flip p.dualCoannihilator)