-- Mathlib/RingTheory/Perfection.lean
theorem v_p_lt_preVal {x : ModP O p} : v p < preVal K v O p x ↔ x ≠ 0

Proof state:
K : Type u₁
inst✝² : Field K
v : Valuation K ℝ≥0
O : Type u₂
inst✝¹ : CommRing O
inst✝ : Algebra O K
hv : v.Integers O
p : ℕ
x : ModP O p
h : x ≠ 0
hp : preVal K v O p x ≤ v ↑p
⊢ x = 0

Tactic:
obtain ⟨r, rfl⟩ := Ideal.Quotient.mk_surjective x

Result:
case intro
K : Type u₁
inst✝² : Field K
v : Valuation K ℝ≥0
O : Type u₂
inst✝¹ : CommRing O
inst✝ : Algebra O K
hv : v.Integers O
p : ℕ
r : O
h : (Ideal.Quotient.mk (Ideal.span {↑p})) r ≠ 0
hp : preVal K v O p ((Ideal.Quotient.mk (Ideal.span {↑p})) r) ≤ v ↑p
⊢ (Ideal.Quotient.mk (Ideal.span {↑p})) r = 0

Proof state:
case intro
K : Type u₁
inst✝² : Field K
v : Valuation K ℝ≥0
O : Type u₂
inst✝¹ : CommRing O
inst✝ : Algebra O K
hv : v.Integers O
p : ℕ
r : O
h : (Ideal.Quotient.mk (Ideal.span {↑p})) r ≠ 0
hp : ↑p ∣ r
⊢ (Ideal.Quotient.mk (Ideal.span {↑p})) r = 0

Tactic:
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]

Result:
case intro
K : Type u₁
inst✝² : Field K
v : Valuation K ℝ≥0
O : Type u₂
inst✝¹ : CommRing O
inst✝ : Algebra O K
hv : v.Integers O
p : ℕ
r : O
h : (Ideal.Quotient.mk (Ideal.span {↑p})) r ≠ 0
hp : ↑p ∣ r
⊢ ↑p ∣ r

Proof state:
K : Type u₁
inst✝² : Field K
v : Valuation K ℝ≥0
O : Type u₂
inst✝¹ : CommRing O
inst✝ : Algebra O K
hv : v.Integers O
p : ℕ
x : ModP O p
h : v ↑p < 0
hx : x = 0
⊢ False

Tactic:
exact not_lt_zero' h

Result:
no goals

Proof state:
K : Type u₁
inst✝² : Field K
v : Valuation K ℝ≥0
O : Type u₂
inst✝¹ : CommRing O
inst✝ : Algebra O K
hv : v.Integers O
p : ℕ
x : ModP O p
h : v ↑p < preVal K v O p x
hx : x = 0
⊢ False

Tactic:
rw [hx, preVal_zero] at h

Result:
K : Type u₁
inst✝² : Field K
v : Valuation K ℝ≥0
O : Type u₂
inst✝¹ : CommRing O
inst✝ : Algebra O K
hv : v.Integers O
p : ℕ
x : ModP O p
h : v ↑p < 0
hx : x = 0
⊢ False

Proof state:
K : Type u₁
inst✝² : Field K
v : Valuation K ℝ≥0
O : Type u₂
inst✝¹ : CommRing O
inst✝ : Algebra O K
hv : v.Integers O
p : ℕ
x : ModP O p
⊢ v ↑p < preVal K v O p x ↔ x ≠ 0

Tactic:
refine ⟨fun h hx => by rw [hx, preVal_zero] at h; exact not_lt_zero' h,
  fun h => lt_of_not_le fun hp => h ?_⟩

Result:
K : Type u₁
inst✝² : Field K
v : Valuation K ℝ≥0
O : Type u₂
inst✝¹ : CommRing O
inst✝ : Algebra O K
hv : v.Integers O
p : ℕ
x : ModP O p
h : x ≠ 0
hp : preVal K v O p x ≤ v ↑p
⊢ x = 0

Proof state:
case intro
K : Type u₁
inst✝² : Field K
v : Valuation K ℝ≥0
O : Type u₂
inst✝¹ : CommRing O
inst✝ : Algebra O K
hv : v.Integers O
p : ℕ
r : O
h : (Ideal.Quotient.mk (Ideal.span {↑p})) r ≠ 0
hp : ↑p ∣ r
⊢ ↑p ∣ r

Tactic:
exact hp

Result:
no goals

Proof state:
case intro
K : Type u₁
inst✝² : Field K
v : Valuation K ℝ≥0
O : Type u₂
inst✝¹ : CommRing O
inst✝ : Algebra O K
hv : v.Integers O
p : ℕ
r : O
h : (Ideal.Quotient.mk (Ideal.span {↑p})) r ≠ 0
hp : preVal K v O p ((Ideal.Quotient.mk (Ideal.span {↑p})) r) ≤ v ↑p
⊢ (Ideal.Quotient.mk (Ideal.span {↑p})) r = 0

Tactic:
rw [preVal_mk hv h, ← map_natCast (algebraMap O K) p, hv.le_iff_dvd] at hp

Result:
case intro
K : Type u₁
inst✝² : Field K
v : Valuation K ℝ≥0
O : Type u₂
inst✝¹ : CommRing O
inst✝ : Algebra O K
hv : v.Integers O
p : ℕ
r : O
h : (Ideal.Quotient.mk (Ideal.span {↑p})) r ≠ 0
hp : ↑p ∣ r
⊢ (Ideal.Quotient.mk (Ideal.span {↑p})) r = 0