Theorem: DiscreteValuationRing.associated_pow_irreducible
File: Mathlib/RingTheory/DiscreteValuationRing/Basic.lean

Proof steps:
state_before: R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
x : R
hx : x ≠ 0
ϖ : R
hirr : Irreducible ϖ
⊢ ∃ n, Associated x (ϖ ^ n)
state_after: R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
x : R
hx : x ≠ 0
ϖ : R
hirr : Irreducible ϖ
this : WfDvdMonoid R
⊢ ∃ n, Associated x (ϖ ^ n)
tactic: have : WfDvdMonoid R := IsNoetherianRing.wfDvdMonoid
state_before: R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
x : R
hx : x ≠ 0
ϖ : R
hirr : Irreducible ϖ
this : WfDvdMonoid R
⊢ ∃ n, Associated x (ϖ ^ n)
state_after: case intro
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
x : R
hx : x ≠ 0
ϖ : R
hirr : Irreducible ϖ
this : WfDvdMonoid R
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
⊢ ∃ n, Associated x (ϖ ^ n)
tactic: cases' WfDvdMonoid.exists_factors x hx with fx hfx
state_before: case intro
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
x : R
hx : x ≠ 0
ϖ : R
hirr : Irreducible ϖ
this : WfDvdMonoid R
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
⊢ ∃ n, Associated x (ϖ ^ n)
state_after: case h
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
x : R
hx : x ≠ 0
ϖ : R
hirr : Irreducible ϖ
this : WfDvdMonoid R
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
⊢ Associated x (ϖ ^ Multiset.card fx)
tactic: use Multiset.card fx
state_before: case h
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
x : R
hx : x ≠ 0
ϖ : R
hirr : Irreducible ϖ
this : WfDvdMonoid R
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
⊢ Associated x (ϖ ^ Multiset.card fx)
state_after: case h
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
x : R
hx : x ≠ 0
ϖ : R
hirr : Irreducible ϖ
this : WfDvdMonoid R
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
H : Associated fx.prod x
⊢ Associated x (ϖ ^ Multiset.card fx)
tactic: have H := hfx.2
state_before: case h
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
x : R
hx : x ≠ 0
ϖ : R
hirr : Irreducible ϖ
this : WfDvdMonoid R
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
H : Associated fx.prod x
⊢ Associated x (ϖ ^ Multiset.card fx)
state_after: case h
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
x : R
hx : x ≠ 0
ϖ : R
hirr : Irreducible ϖ
this : WfDvdMonoid R
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
H : Associates.mk fx.prod = Associates.mk x
⊢ Associates.mk x = Associates.mk (ϖ ^ Multiset.card fx)
tactic: rw [← Associates.mk_eq_mk_iff_associated] at H ⊢
state_before: case h
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
x : R
hx : x ≠ 0
ϖ : R
hirr : Irreducible ϖ
this : WfDvdMonoid R
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
H : Associates.mk fx.prod = Associates.mk x
⊢ Associates.mk x = Associates.mk (ϖ ^ Multiset.card fx)
state_after: case h
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
x : R
hx : x ≠ 0
ϖ : R
hirr : Irreducible ϖ
this : WfDvdMonoid R
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
H : Associates.mk fx.prod = Associates.mk x
⊢ (Multiset.map Associates.mk fx).prod = (Multiset.replicate (Multiset.card fx) (Associates.mk ϖ)).prod
tactic: rw [← H, ← Associates.prod_mk, Associates.mk_pow, ← Multiset.prod_replicate]
state_before: case h
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
x : R
hx : x ≠ 0
ϖ : R
hirr : Irreducible ϖ
this : WfDvdMonoid R
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
H : Associates.mk fx.prod = Associates.mk x
⊢ (Multiset.map Associates.mk fx).prod = (Multiset.replicate (Multiset.card fx) (Associates.mk ϖ)).prod
state_after: case h.e_a
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
x : R
hx : x ≠ 0
ϖ : R
hirr : Irreducible ϖ
this : WfDvdMonoid R
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
H : Associates.mk fx.prod = Associates.mk x
⊢ Multiset.map Associates.mk fx = Multiset.replicate (Multiset.card fx) (Associates.mk ϖ)
tactic: congr 1
state_before: case h.e_a
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
x : R
hx : x ≠ 0
ϖ : R
hirr : Irreducible ϖ
this : WfDvdMonoid R
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
H : Associates.mk fx.prod = Associates.mk x
⊢ Multiset.map Associates.mk fx = Multiset.replicate (Multiset.card fx) (Associates.mk ϖ)
state_after: case h.e_a
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
x : R
hx : x ≠ 0
ϖ : R
hirr : Irreducible ϖ
this : WfDvdMonoid R
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
H : Associates.mk fx.prod = Associates.mk x
⊢ Multiset.card (Multiset.map Associates.mk fx) = Multiset.card fx ∧
    ∀ b ∈ Multiset.map Associates.mk fx, b = Associates.mk ϖ
tactic: rw [Multiset.eq_replicate]
state_before: case h.e_a
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
x : R
hx : x ≠ 0
ϖ : R
hirr : Irreducible ϖ
this : WfDvdMonoid R
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
H : Associates.mk fx.prod = Associates.mk x
⊢ Multiset.card (Multiset.map Associates.mk fx) = Multiset.card fx ∧
    ∀ b ∈ Multiset.map Associates.mk fx, b = Associates.mk ϖ
state_after: case h.e_a
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
x : R
hx : x ≠ 0
ϖ : R
hirr : Irreducible ϖ
this : WfDvdMonoid R
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
H : Associates.mk fx.prod = Associates.mk x
⊢ ∀ (b : Associates R), ∀ x ∈ fx, Associates.mk x = b → b = Associates.mk ϖ
tactic: simp only [true_and_iff, and_imp, Multiset.card_map, eq_self_iff_true, Multiset.mem_map,
    exists_imp]
state_before: case h.e_a
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
x : R
hx : x ≠ 0
ϖ : R
hirr : Irreducible ϖ
this : WfDvdMonoid R
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
H : Associates.mk fx.prod = Associates.mk x
⊢ ∀ (b : Associates R), ∀ x ∈ fx, Associates.mk x = b → b = Associates.mk ϖ
state_after: case h.e_a
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
x : R
hx : x ≠ 0
ϖ : R
hirr : Irreducible ϖ
this : WfDvdMonoid R
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
H : Associates.mk fx.prod = Associates.mk x
x✝ : R
a✝ : x✝ ∈ fx
⊢ Associates.mk x✝ = Associates.mk ϖ
tactic: rintro _ _ _ rfl
state_before: case h.e_a
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
x : R
hx : x ≠ 0
ϖ : R
hirr : Irreducible ϖ
this : WfDvdMonoid R
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
H : Associates.mk fx.prod = Associates.mk x
x✝ : R
a✝ : x✝ ∈ fx
⊢ Associates.mk x✝ = Associates.mk ϖ
state_after: case h.e_a
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
x : R
hx : x ≠ 0
ϖ : R
hirr : Irreducible ϖ
this : WfDvdMonoid R
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
H : Associates.mk fx.prod = Associates.mk x
x✝ : R
a✝ : x✝ ∈ fx
⊢ Associated x✝ ϖ
tactic: rw [Associates.mk_eq_mk_iff_associated]
state_before: case h.e_a
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
x : R
hx : x ≠ 0
ϖ : R
hirr : Irreducible ϖ
this : WfDvdMonoid R
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
H : Associates.mk fx.prod = Associates.mk x
x✝ : R
a✝ : x✝ ∈ fx
⊢ Associated x✝ ϖ
state_after: case h.e_a
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
x : R
hx : x ≠ 0
ϖ : R
hirr : Irreducible ϖ
this : WfDvdMonoid R
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
H : Associates.mk fx.prod = Associates.mk x
x✝ : R
a✝ : x✝ ∈ fx
⊢ Irreducible x✝
tactic: refine associated_of_irreducible _ ?_ hirr
state_before: case h.e_a
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
x : R
hx : x ≠ 0
ϖ : R
hirr : Irreducible ϖ
this : WfDvdMonoid R
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
H : Associates.mk fx.prod = Associates.mk x
x✝ : R
a✝ : x✝ ∈ fx
⊢ Irreducible x✝
state_after: case h.e_a.a
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
x : R
hx : x ≠ 0
ϖ : R
hirr : Irreducible ϖ
this : WfDvdMonoid R
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
H : Associates.mk fx.prod = Associates.mk x
x✝ : R
a✝ : x✝ ∈ fx
⊢ x✝ ∈ fx
tactic: apply hfx.1
state_before: case h.e_a.a
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
x : R
hx : x ≠ 0
ϖ : R
hirr : Irreducible ϖ
this : WfDvdMonoid R
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
H : Associates.mk fx.prod = Associates.mk x
x✝ : R
a✝ : x✝ ∈ fx
⊢ x✝ ∈ fx
state_after: no goals
tactic: assumption