-- Module: Mathlib.NumberTheory.RamificationInertia.Basic
Goal:
R : Type u
inst : CommRing R
S : Type v
inst_1 : CommRing S
inst_2 : Algebra R S
p : Ideal R
P : Ideal S
hfp : NeZero e
inst_3 : IsDedekindDomain S
hP0 : P ≠ ⊥
hP : P.IsPrime
i : ℕ
hi : i < e
a : S
a_mem : a ∈ P ^ i
a_notMem : a ∉ P ^ (i + 1)
b :
  HasQuotient.Quotient ↥(map (F := S →+* S ⧸ P ^ e) (Quotient.mk (P ^ e)) (P ^ i)) (B :=
    Submodule.{u, v} (R ⧸ p) ↥(map (F := S →+* S ⧸ P ^ e) (Quotient.mk (P ^ e)) (P ^ i)))
    (LinearMap.range (p.powQuotSuccInclusion P i))
val : S ⧸ P ^ e
x : S
hx_1 :
  Membership.mem (γ := Ideal (S ⧸ P ^ e)) (map (F := S →+* S ⧸ P ^ e) (Quotient.mk (P ^ e)) (P ^ i))
    (Quot.mk (⇑(Submodule.quotientRel (P ^ e)) : S → S → Prop) x)
hx : x ∈ P ^ i
Pe_le_Pi : P ^ e ≤ P ^ i
hx' : x ∈ span {a} ⊔ P ^ (i + 1)
obtain :
  ∀ (y' : S) (hy' : Membership.mem (γ := Submodule S S) (span {a}) y') (z : S) (hz : z ∈ P ^ (i + 1))
    (hx :
      Membership.mem (γ := Ideal (S ⧸ P ^ e)) (map (F := S →+* S ⧸ P ^ e) (Quotient.mk (P ^ e)) (P ^ i))
        (Quot.mk (⇑(Submodule.quotientRel (P ^ e)) : S → S → Prop) (y' + z)))
    (hx_2 : y' + z ∈ P ^ i) (hx' : y' + z ∈ span {a} ⊔ P ^ (i + 1)),
    ∃ (a_1 : S ⧸ P),
      Eq (α :=
        HasQuotient.Quotient ↥(map (F := S →+* S ⧸ P ^ e) (Quotient.mk (P ^ e)) (P ^ i)) (B :=
          Submodule.{u, v} (R ⧸ p) ↥(map (F := S →+* S ⧸ P ^ e) (Quotient.mk (P ^ e)) (P ^ i)))
          (LinearMap.range (p.powQuotSuccInclusion P i)))
        ((p.quotientToQuotientRangePowQuotSucc P a_mem :
              (a : S ⧸ P) → ↥(map (Quotient.mk (P ^ e)) (P ^ i)) ⧸ (p.powQuotSuccInclusion P i).range)
            a_1 :
          HasQuotient.Quotient ↥(map (F := S →+* S ⧸ P ^ e) (Quotient.mk (P ^ e)) (P ^ i)) (B :=
            Submodule.{u, v} (R ⧸ p) ↥(map (F := S →+* S ⧸ P ^ e) (Quotient.mk (P ^ e)) (P ^ i)))
            (LinearMap.range (p.powQuotSuccInclusion P i)))
        (Quot.mk
            (⇑(p.powQuotSuccInclusion P i).range.quotientRel :
              ↥(map (Quotient.mk (P ^ e)) (P ^ i)) → ↥(map (Quotient.mk (P ^ e)) (P ^ i)) → Prop)
            (Subtype.mk (p := fun (x : S ⧸ P ^ e) ↦
              Membership.mem (γ := Ideal (S ⧸ P ^ e)) (map (F := S →+* S ⧸ P ^ e) (Quotient.mk (P ^ e)) (P ^ i)) x)
              (Quot.mk (⇑(Submodule.quotientRel (P ^ e)) : S → S → Prop) (y' + z)) hx) :
          Quot (α := ↥(map (F := S →+* S ⧸ P ^ e) (Quotient.mk (P ^ e)) (P ^ i)))
            (⇑(p.powQuotSuccInclusion P i).range.quotientRel :
              ↥(map (Quotient.mk (P ^ e)) (P ^ i)) → ↥(map (Quotient.mk (P ^ e)) (P ^ i)) → Prop))
⊢ ∃ (a_1 : S ⧸ P),
    Eq (α :=
      HasQuotient.Quotient ↥(map (F := S →+* S ⧸ P ^ e) (Quotient.mk (P ^ e)) (P ^ i)) (B :=
        Submodule.{u, v} (R ⧸ p) ↥(map (F := S →+* S ⧸ P ^ e) (Quotient.mk (P ^ e)) (P ^ i)))
        (LinearMap.range (p.powQuotSuccInclusion P i)))
      ((p.quotientToQuotientRangePowQuotSucc P a_mem :
            (a : S ⧸ P) → ↥(map (Quotient.mk (P ^ e)) (P ^ i)) ⧸ (p.powQuotSuccInclusion P i).range)
          a_1 :
        HasQuotient.Quotient ↥(map (F := S →+* S ⧸ P ^ e) (Quotient.mk (P ^ e)) (P ^ i)) (B :=
          Submodule.{u, v} (R ⧸ p) ↥(map (F := S →+* S ⧸ P ^ e) (Quotient.mk (P ^ e)) (P ^ i)))
          (LinearMap.range (p.powQuotSuccInclusion P i)))
      (Quot.mk
          (⇑(p.powQuotSuccInclusion P i).range.quotientRel :
            ↥(map (Quotient.mk (P ^ e)) (P ^ i)) → ↥(map (Quotient.mk (P ^ e)) (P ^ i)) → Prop)
          (Subtype.mk (p := fun (x : S ⧸ P ^ e) ↦
            Membership.mem (γ := Ideal (S ⧸ P ^ e)) (map (F := S →+* S ⧸ P ^ e) (Quotient.mk (P ^ e)) (P ^ i)) x)
            (Quot.mk (⇑(Submodule.quotientRel (P ^ e)) : S → S → Prop) x) hx_1) :
        Quot (α := ↥(map (F := S →+* S ⧸ P ^ e) (Quotient.mk (P ^ e)) (P ^ i)))
          (⇑(p.powQuotSuccInclusion P i).range.quotientRel :
            ↥(map (Quotient.mk (P ^ e)) (P ^ i)) → ↥(map (Quotient.mk (P ^ e)) (P ^ i)) → Prop))

Tactic:
obtain ⟨y', hy', z, hz, rfl⟩ := Submodule.mem_sup.mp hx'

Premises used: Iff.mp, Submodule.mem_sup, rfl