-- Module: Mathlib.LinearAlgebra.RootSystem.Chain
Goal:
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst : Finite ι
inst_1 : CommRing R
inst_2 : CharZero R
inst_3 : IsDomain R
inst_4 : AddCommGroup M
inst_5 : Module R M
inst_6 : AddCommGroup N
inst_7 : Module R N
P : RootPairing ι R M N
inst_8 : P.IsCrystallographic
i j : ι
this : InvolutiveNeg ι := P.indexNeg
h :
  LinearIndependent.{0, u_2, u_3} (ι := Fin (Nat.succ 0).succ) R (M := M)
    (![(P.root : (a : ι) → M) i, (P.root : (a : ι) → M) j] : Fin (Nat.succ 0).succ → M)
h' :
  LinearIndependent.{0, u_2, u_3} (ι := Fin (Nat.succ 0).succ) R (M := M)
    (![(P.root : (a : ι) → M) i, (P.root : (a : ι) → M) (-j)] : Fin (Nat.succ 0).succ → M)
z : ℤ
pos :
  Membership.mem (γ := Set ℤ)
      (Icc (-(↑(chainTopCoeff.{u_1, u_2, u_3, u_4} (R := R) (M := M) (N := N) (P := P) i j) : ℤ))
        (↑(chainBotCoeff.{u_1, u_2, u_3, u_4} (R := R) (M := M) (N := N) (P := P) i j) : ℤ))
      z ↔
    Membership.mem (γ := Set M) (range.{u_3, u_1 + 1} (ι := ι) (⇑P.root : (a : ι) → M))
      (HAdd.hAdd (α := M) ((P.root : (a : ι) → M) (-j) : M) (HSMul.hSMul (β := M) z ((P.root : (a : ι) → M) i : M)))
pos₁ :
  Membership.mem (γ := Set ℤ)
      (Icc (-(↑(chainTopCoeff.{u_1, u_2, u_3, u_4} (R := R) (M := M) (N := N) (P := P) i j) : ℤ))
        (↑(chainBotCoeff.{u_1, u_2, u_3, u_4} (R := R) (M := M) (N := N) (P := P) i j) : ℤ))
      z ↔
    Membership.mem (γ := Set M) (range.{u_3, u_1 + 1} (ι := ι) (⇑P.root : (a : ι) → M))
      (HAdd.hAdd (α := M) ((P.root : (a : ι) → M) ((P.reflectionPerm j : (a : ι) → ι) j : ι) : M)
        (HSMul.hSMul (β := M) z ((P.root : (a : ι) → M) i : M)))
pos₂ :
  Membership.mem (γ := Set ℤ)
      (Icc (-(↑(chainTopCoeff.{u_1, u_2, u_3, u_4} (R := R) (M := M) (N := N) (P := P) i j) : ℤ))
        (↑(chainBotCoeff.{u_1, u_2, u_3, u_4} (R := R) (M := M) (N := N) (P := P) i j) : ℤ))
      z ↔
    Membership.mem (γ := Set M) (range.{u_3, u_1 + 1} (ι := ι) (⇑P.root : (a : ι) → M))
      (HAdd.hAdd (α := M) ((P.reflection j : (a : M) → M) ((P.root : (a : ι) → M) j : M) : M)
        (HSMul.hSMul (β := M) z ((P.root : (a : ι) → M) i : M)))
pos₃ :
  -(↑(chainTopCoeff.{u_1, u_2, u_3, u_4} (R := R) (M := M) (N := N) (P := P) i j) : ℤ) ≤ z ∧
      z ≤ (↑(chainBotCoeff.{u_1, u_2, u_3, u_4} (R := R) (M := M) (N := N) (P := P) i j) : ℤ) ↔
    Membership.mem (γ := Set M) (range.{u_3, u_1 + 1} (ι := ι) (⇑P.root : (a : ι) → M))
      (HAdd.hAdd (α := M) ((P.reflection j : (a : M) → M) ((P.root : (a : ι) → M) j : M) : M)
        (HSMul.hSMul (β := M) z ((P.root : (a : ι) → M) i : M)))
pos₄ :
  -(↑(chainTopCoeff.{u_1, u_2, u_3, u_4} (R := R) (M := M) (N := N) (P := P) i j) : ℤ) ≤ z ∧
      z ≤ (↑(chainBotCoeff.{u_1, u_2, u_3, u_4} (R := R) (M := M) (N := N) (P := P) i j) : ℤ) ↔
    Membership.mem (γ := Set M) (range.{u_3, u_1 + 1} (ι := ι) (⇑P.root : (a : ι) → M))
      (HAdd.hAdd (α := M) (-(P.root : (a : ι) → M) j : M) (HSMul.hSMul (β := M) z ((P.root : (a : ι) → M) i : M)))
pos₅ :
  -(↑(chainTopCoeff.{u_1, u_2, u_3, u_4} (R := R) (M := M) (N := N) (P := P) i j) : ℤ) ≤ z ∧
      z ≤ (↑(chainBotCoeff.{u_1, u_2, u_3, u_4} (R := R) (M := M) (N := N) (P := P) i j) : ℤ) ↔
    Membership.mem (γ := Set M) (range.{u_3, u_1 + 1} (ι := ι) (⇑P.root : (a : ι) → M))
      (HSub.hSub (α := M) (-(P.root : (a : ι) → M) j : M) (-HSMul.hSMul (β := M) z ((P.root : (a : ι) → M) i : M)))
pos₆ :
  -(↑(chainTopCoeff.{u_1, u_2, u_3, u_4} (R := R) (M := M) (N := N) (P := P) i j) : ℤ) ≤ z ∧
      z ≤ (↑(chainBotCoeff.{u_1, u_2, u_3, u_4} (R := R) (M := M) (N := N) (P := P) i j) : ℤ) ↔
    Membership.mem (γ := Set M) (range.{u_3, u_1 + 1} (ι := ι) (⇑P.root : (a : ι) → M))
      (-HSub.hSub (α := M) ((P.root : (a : ι) → M) j : M) (HSMul.hSMul (β := M) z ((P.root : (a : ι) → M) i : M)))
pos₇ :
  -(↑(chainTopCoeff.{u_1, u_2, u_3, u_4} (R := R) (M := M) (N := N) (P := P) i j) : ℤ) ≤ z ∧
      z ≤ (↑(chainBotCoeff.{u_1, u_2, u_3, u_4} (R := R) (M := M) (N := N) (P := P) i j) : ℤ) ↔
    Membership.mem (γ := Set M) (range.{u_3, u_1 + 1} (ι := ι) (⇑P.root : (a : ι) → M))
      (HSub.hSub (α := M) ((P.root : (a : ι) → M) j : M) (HSMul.hSMul (β := M) z ((P.root : (a : ι) → M) i : M)))
pos₈ :
  -(↑(chainTopCoeff.{u_1, u_2, u_3, u_4} (R := R) (M := M) (N := N) (P := P) i j) : ℤ) ≤ z ∧
      z ≤ (↑(chainBotCoeff.{u_1, u_2, u_3, u_4} (R := R) (M := M) (N := N) (P := P) i j) : ℤ) ↔
    Membership.mem (γ := Set ℤ)
      (Icc (-(↑(chainTopCoeff.{u_1, u_2, u_3, u_4} (R := R) (M := M) (N := N) (P := P) i j) : ℤ))
        (↑(chainBotCoeff.{u_1, u_2, u_3, u_4} (R := R) (M := M) (N := N) (P := P) i j) : ℤ))
      z
pos₉ pos₁₀ :
  -(↑(chainTopCoeff.{u_1, u_2, u_3, u_4} (R := R) (M := M) (N := N) (P := P) i j) : ℤ) ≤ z ∧
      z ≤ (↑(chainBotCoeff.{u_1, u_2, u_3, u_4} (R := R) (M := M) (N := N) (P := P) i j) : ℤ) ↔
    -(↑(chainTopCoeff.{u_1, u_2, u_3, u_4} (R := R) (M := M) (N := N) (P := P) i j) : ℤ) ≤ z ∧
      z ≤ (↑(chainBotCoeff.{u_1, u_2, u_3, u_4} (R := R) (M := M) (N := N) (P := P) i j) : ℤ)
⊢ Membership.mem (γ := Set ℤ)
      (Icc (-(↑(chainTopCoeff.{u_1, u_2, u_3, u_4} (R := R) (M := M) (N := N) (P := P) i j) : ℤ))
        (↑(chainBotCoeff.{u_1, u_2, u_3, u_4} (R := R) (M := M) (N := N) (P := P) i j) : ℤ))
      z ↔
    Membership.mem (γ := Set ℤ)
      (Icc (-(↑(chainBotCoeff.{u_1, u_2, u_3, u_4} (R := R) (M := M) (N := N) (P := P) i (-j)) : ℤ))
        (↑(chainTopCoeff.{u_1, u_2, u_3, u_4} (R := R) (M := M) (N := N) (P := P) i (-j)) : ℤ))
      z

Tactic:
rw [← P.root_add_zsmul_mem_range_iff h', indexNeg_neg, root_reflectionPerm, mem_Icc,
      reflection_apply_self, ← sub_neg_eq_add, ← neg_sub', neg_mem_range_root_iff,
      P.root_sub_zsmul_mem_range_iff h, mem_Icc]

Premises used: neg_sub', RootPairing.reflection_apply_self, RootPairing.root_add_zsmul_mem_range_iff, Set.mem_Icc, RootPairing.root_sub_zsmul_mem_range_iff, RootPairing.neg_mem_range_root_iff, RootPairing.indexNeg_neg, sub_neg_eq_add, RootPairing.root_reflectionPerm