Theorem: prod_mem_ideal_map_of_mem_conductor
File: Mathlib/NumberTheory/KummerDedekind.lean

Proof steps:
state_before: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
p : R
z : S
hp : p ∈ comap (algebraMap R S) (conductor R x)
hz' : z ∈ Ideal.map (algebraMap R S) I
⊢ (algebraMap R S) p * z ∈ ⇑(algebraMap (↥(adjoin R {x})) S) '' ↑(Ideal.map (algebraMap R ↥(adjoin R {x})) I)
state_after: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
p : R
z : S
hp : p ∈ comap (algebraMap R S) (conductor R x)
hz' : ∃ l ∈ Finsupp.supported S S ↑I, (Finsupp.total R S S ⇑(algebraMap R S)) l = z
⊢ (algebraMap R S) p * z ∈ ⇑(algebraMap (↥(adjoin R {x})) S) '' ↑(Ideal.map (algebraMap R ↥(adjoin R {x})) I)
tactic: rw [Ideal.map, Ideal.span, Finsupp.mem_span_image_iff_total] at hz'
state_before: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
p : R
z : S
hp : p ∈ comap (algebraMap R S) (conductor R x)
hz' : ∃ l ∈ Finsupp.supported S S ↑I, (Finsupp.total R S S ⇑(algebraMap R S)) l = z
⊢ (algebraMap R S) p * z ∈ ⇑(algebraMap (↥(adjoin R {x})) S) '' ↑(Ideal.map (algebraMap R ↥(adjoin R {x})) I)
state_after: case intro.intro
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
p : R
z : S
hp : p ∈ comap (algebraMap R S) (conductor R x)
l : R →₀ S
H : l ∈ Finsupp.supported S S ↑I
H' : (Finsupp.total R S S ⇑(algebraMap R S)) l = z
⊢ (algebraMap R S) p * z ∈ ⇑(algebraMap (↥(adjoin R {x})) S) '' ↑(Ideal.map (algebraMap R ↥(adjoin R {x})) I)
tactic: obtain ⟨l, H, H'⟩ := hz'
state_before: case intro.intro
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
p : R
z : S
hp : p ∈ comap (algebraMap R S) (conductor R x)
l : R →₀ S
H : l ∈ Finsupp.supported S S ↑I
H' : (Finsupp.total R S S ⇑(algebraMap R S)) l = z
⊢ (algebraMap R S) p * z ∈ ⇑(algebraMap (↥(adjoin R {x})) S) '' ↑(Ideal.map (algebraMap R ↥(adjoin R {x})) I)
state_after: case intro.intro
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
p : R
z : S
hp : p ∈ comap (algebraMap R S) (conductor R x)
l : R →₀ S
H : l ∈ Finsupp.supported S S ↑I
H' : (l.sum fun i a => a • (algebraMap R S) i) = z
⊢ (algebraMap R S) p * z ∈ ⇑(algebraMap (↥(adjoin R {x})) S) '' ↑(Ideal.map (algebraMap R ↥(adjoin R {x})) I)
tactic: rw [Finsupp.total_apply] at H'
state_before: case intro.intro
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
p : R
z : S
hp : p ∈ comap (algebraMap R S) (conductor R x)
l : R →₀ S
H : l ∈ Finsupp.supported S S ↑I
H' : (l.sum fun i a => a • (algebraMap R S) i) = z
⊢ (algebraMap R S) p * z ∈ ⇑(algebraMap (↥(adjoin R {x})) S) '' ↑(Ideal.map (algebraMap R ↥(adjoin R {x})) I)
state_after: case intro.intro
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
p : R
z : S
hp : p ∈ comap (algebraMap R S) (conductor R x)
l : R →₀ S
H : l ∈ Finsupp.supported S S ↑I
H' : (l.sum fun i a => a • (algebraMap R S) i) = z
⊢ (l.sum fun a c => c • (algebraMap R S) a * (algebraMap R S) p) ∈
    ⇑(algebraMap (↥(adjoin R {x})) S) '' ↑(Ideal.map (algebraMap R ↥(adjoin R {x})) I)
tactic: rw [← H', mul_comm, Finsupp.sum_mul]
state_before: case intro.intro
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
p : R
z : S
hp : p ∈ comap (algebraMap R S) (conductor R x)
l : R →₀ S
H : l ∈ Finsupp.supported S S ↑I
H' : (l.sum fun i a => a • (algebraMap R S) i) = z
lem :
  ∀ {a : R},
    a ∈ I →
      l a • (algebraMap R S) a * (algebraMap R S) p ∈
        ⇑(algebraMap (↥(adjoin R {x})) S) '' ↑(Ideal.map (algebraMap R ↥(adjoin R {x})) I)
⊢ (l.sum fun a c => c • (algebraMap R S) a * (algebraMap R S) p) ∈
    ⇑(algebraMap (↥(adjoin R {x})) S) '' ↑(Ideal.map (algebraMap R ↥(adjoin R {x})) I)
state_after: case intro.intro.refine_1
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
p : R
z : S
hp : p ∈ comap (algebraMap R S) (conductor R x)
l : R →₀ S
H : l ∈ Finsupp.supported S S ↑I
H' : (l.sum fun i a => a • (algebraMap R S) i) = z
lem :
  ∀ {a : R},
    a ∈ I →
      l a • (algebraMap R S) a * (algebraMap R S) p ∈
        ⇑(algebraMap (↥(adjoin R {x})) S) '' ↑(Ideal.map (algebraMap R ↥(adjoin R {x})) I)
a b : S
⊢ (fun u => u ∈ ⇑(algebraMap (↥(adjoin R {x})) S) '' ↑(Ideal.map (algebraMap R ↥(adjoin R {x})) I)) a →
    (fun u => u ∈ ⇑(algebraMap (↥(adjoin R {x})) S) '' ↑(Ideal.map (algebraMap R ↥(adjoin R {x})) I)) b →
      (fun u => u ∈ ⇑(algebraMap (↥(adjoin R {x})) S) '' ↑(Ideal.map (algebraMap R ↥(adjoin R {x})) I)) (a + b)

case intro.intro.refine_2
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
p : R
z : S
hp : p ∈ comap (algebraMap R S) (conductor R x)
l : R →₀ S
H : l ∈ Finsupp.supported S S ↑I
H' : (l.sum fun i a => a • (algebraMap R S) i) = z
lem :
  ∀ {a : R},
    a ∈ I →
      l a • (algebraMap R S) a * (algebraMap R S) p ∈
        ⇑(algebraMap (↥(adjoin R {x})) S) '' ↑(Ideal.map (algebraMap R ↥(adjoin R {x})) I)
⊢ (fun u => u ∈ ⇑(algebraMap (↥(adjoin R {x})) S) '' ↑(Ideal.map (algebraMap R ↥(adjoin R {x})) I)) 0

case intro.intro.refine_3
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
p : R
z : S
hp : p ∈ comap (algebraMap R S) (conductor R x)
l : R →₀ S
H : l ∈ Finsupp.supported S S ↑I
H' : (l.sum fun i a => a • (algebraMap R S) i) = z
lem :
  ∀ {a : R},
    a ∈ I →
      l a • (algebraMap R S) a * (algebraMap R S) p ∈
        ⇑(algebraMap (↥(adjoin R {x})) S) '' ↑(Ideal.map (algebraMap R ↥(adjoin R {x})) I)
⊢ ∀ x_1 ∈ l.support,
    (fun u => u ∈ ⇑(algebraMap (↥(adjoin R {x})) S) '' ↑(Ideal.map (algebraMap R ↥(adjoin R {x})) I))
      ((fun a c => c • (algebraMap R S) a * (algebraMap R S) p) x_1 (l x_1))
tactic: refine Finset.sum_induction _ (fun u => u ∈ algebraMap R<x> S '' I.map (algebraMap R R<x>))
      (fun a b => ?_) ?_ ?_