-- Mathlib/RingTheory/Valuation/Integers.lean
lemma not_denselyOrdered_of_isPrincipalIdealRing [IsPrincipalIdealRing O] (hv : Integers v O) :
    ¬ DenselyOrdered (range v)

Proof state:
case intro.intro.intro.intro
F : Type u
Γ₀ : Type v
inst✝⁴ : Field F
inst✝³ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation F Γ₀
O : Type w
inst✝² : CommRing O
inst✝¹ : Algebra O F
inst✝ : IsPrincipalIdealRing O
hv : v.Integers O
H : DenselyOrdered ↑(range ⇑v)
I : Ideal O := { carrier := ⇑v ∘ ⇑(algebraMap O F) ⁻¹' Iio 1, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }
x : O
hx₁ : v ((algebraMap O F) x) < 1
hx : v ((algebraMap O F) x) ∈ upperBounds (Iio 1 ∩ range (⇑v ∘ ⇑(algebraMap O F)))
y : F
hy : v ((algebraMap O F) x) < v y
hy₁ : v y < 1
⊢ False

Tactic:
obtain ⟨z, rfl⟩ := hv.exists_of_le_one hy₁.le

Result:
case intro.intro.intro.intro.intro
F : Type u
Γ₀ : Type v
inst✝⁴ : Field F
inst✝³ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation F Γ₀
O : Type w
inst✝² : CommRing O
inst✝¹ : Algebra O F
inst✝ : IsPrincipalIdealRing O
hv : v.Integers O
H : DenselyOrdered ↑(range ⇑v)
I : Ideal O := { carrier := ⇑v ∘ ⇑(algebraMap O F) ⁻¹' Iio 1, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }
x : O
hx₁ : v ((algebraMap O F) x) < 1
hx : v ((algebraMap O F) x) ∈ upperBounds (Iio 1 ∩ range (⇑v ∘ ⇑(algebraMap O F)))
z : O
hy : v ((algebraMap O F) x) < v ((algebraMap O F) z)
hy₁ : v ((algebraMap O F) z) < 1
⊢ False

Proof state:
case intro.intro.intro.intro.intro
F : Type u
Γ₀ : Type v
inst✝⁴ : Field F
inst✝³ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation F Γ₀
O : Type w
inst✝² : CommRing O
inst✝¹ : Algebra O F
inst✝ : IsPrincipalIdealRing O
hv : v.Integers O
H : DenselyOrdered ↑(range ⇑v)
I : Ideal O := { carrier := ⇑v ∘ ⇑(algebraMap O F) ⁻¹' Iio 1, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }
x : O
hx₁ : v ((algebraMap O F) x) < 1
hx : v ((algebraMap O F) x) ∈ upperBounds (Iio 1 ∩ range (⇑v ∘ ⇑(algebraMap O F)))
z : O
hy : v ((algebraMap O F) x) < v ((algebraMap O F) z)
hy₁ : v ((algebraMap O F) z) < 1
⊢ False

Tactic:
exact hy.not_le <| hx ⟨hy₁, mem_range_self _⟩

Result:
no goals

Proof state:
F : Type u
Γ₀ : Type v
inst✝⁴ : Field F
inst✝³ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation F Γ₀
O : Type w
inst✝² : CommRing O
inst✝¹ : Algebra O F
inst✝ : IsPrincipalIdealRing O
hv : v.Integers O
H : DenselyOrdered ↑(range ⇑v)
⊢ ∀ (c : O) {x : O},
    x ∈ { carrier := ⇑v ∘ ⇑(algebraMap O F) ⁻¹' Iio 1, add_mem' := ⋯, zero_mem' := ⋯ }.carrier →
      c • x ∈ { carrier := ⇑v ∘ ⇑(algebraMap O F) ⁻¹' Iio 1, add_mem' := ⋯, zero_mem' := ⋯ }.carrier

Tactic:
intro c x

Result:
F : Type u
Γ₀ : Type v
inst✝⁴ : Field F
inst✝³ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation F Γ₀
O : Type w
inst✝² : CommRing O
inst✝¹ : Algebra O F
inst✝ : IsPrincipalIdealRing O
hv : v.Integers O
H : DenselyOrdered ↑(range ⇑v)
c x : O
⊢ x ∈ { carrier := ⇑v ∘ ⇑(algebraMap O F) ⁻¹' Iio 1, add_mem' := ⋯, zero_mem' := ⋯ }.carrier →
    c • x ∈ { carrier := ⇑v ∘ ⇑(algebraMap O F) ⁻¹' Iio 1, add_mem' := ⋯, zero_mem' := ⋯ }.carrier

Proof state:
F : Type u
Γ₀ : Type v
inst✝⁴ : Field F
inst✝³ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation F Γ₀
O : Type w
inst✝² : CommRing O
inst✝¹ : Algebra O F
inst✝ : IsPrincipalIdealRing O
hv : v.Integers O
H : DenselyOrdered ↑(range ⇑v)
c x : O
⊢ x ∈ { carrier := ⇑v ∘ ⇑(algebraMap O F) ⁻¹' Iio 1, add_mem' := ⋯, zero_mem' := ⋯ }.carrier →
    c • x ∈ { carrier := ⇑v ∘ ⇑(algebraMap O F) ⁻¹' Iio 1, add_mem' := ⋯, zero_mem' := ⋯ }.carrier

Tactic:
simp only [mem_preimage, Function.comp_apply, mem_Iio, smul_eq_mul, _root_.map_mul]

Result:
F : Type u
Γ₀ : Type v
inst✝⁴ : Field F
inst✝³ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation F Γ₀
O : Type w
inst✝² : CommRing O
inst✝¹ : Algebra O F
inst✝ : IsPrincipalIdealRing O
hv : v.Integers O
H : DenselyOrdered ↑(range ⇑v)
c x : O
⊢ v ((algebraMap O F) x) < 1 → v ((algebraMap O F) c) * v ((algebraMap O F) x) < 1