Theorem: Set.MulAntidiagonal.finite_of_isPWO
File: Mathlib/Data/Set/MulAntidiagonal.lean

Proof steps:
state_before: α : Type u_1
inst✝ : OrderedCancelCommMonoid α
s t : Set α
a✝ : α
x y : ↑(s.mulAntidiagonal t a✝)
hs : s.IsPWO
ht : t.IsPWO
a : α
⊢ (s.mulAntidiagonal t a).Finite
state_after: α : Type u_1
inst✝ : OrderedCancelCommMonoid α
s t : Set α
a✝ : α
x y : ↑(s.mulAntidiagonal t a✝)
hs : s.IsPWO
ht : t.IsPWO
a : α
h : (s.mulAntidiagonal t a).Infinite
⊢ False
tactic: refine not_infinite.1 fun h => ?_
state_before: α : Type u_1
inst✝ : OrderedCancelCommMonoid α
s t : Set α
a✝ : α
x y : ↑(s.mulAntidiagonal t a✝)
hs : s.IsPWO
ht : t.IsPWO
a : α
h : (s.mulAntidiagonal t a).Infinite
h1 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.fst ⁻¹'o fun x x_1 => x ≤ x_1)
h2 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.snd ⁻¹'o fun x x_1 => x ≤ x_1)
⊢ False
state_after: case intro
α : Type u_1
inst✝ : OrderedCancelCommMonoid α
s t : Set α
a✝ : α
x y : ↑(s.mulAntidiagonal t a✝)
hs : s.IsPWO
ht : t.IsPWO
a : α
h : (s.mulAntidiagonal t a).Infinite
h1 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.fst ⁻¹'o fun x x_1 => x ≤ x_1)
h2 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.snd ⁻¹'o fun x x_1 => x ≤ x_1)
g : ℕ ↪o ℕ
hg :
  ∀ (m n : ℕ),
    m ≤ n →
      (Prod.fst ⁻¹'o fun x x_1 => x ≤ x_1) ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g m))
        ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g n))
⊢ False
tactic: obtain ⟨g, hg⟩ :=
    h1.exists_monotone_subseq (fun n => h.natEmbedding _ n) fun n => (h.natEmbedding _ n).2
state_before: case intro
α : Type u_1
inst✝ : OrderedCancelCommMonoid α
s t : Set α
a✝ : α
x y : ↑(s.mulAntidiagonal t a✝)
hs : s.IsPWO
ht : t.IsPWO
a : α
h : (s.mulAntidiagonal t a).Infinite
h1 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.fst ⁻¹'o fun x x_1 => x ≤ x_1)
h2 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.snd ⁻¹'o fun x x_1 => x ≤ x_1)
g : ℕ ↪o ℕ
hg :
  ∀ (m n : ℕ),
    m ≤ n →
      (Prod.fst ⁻¹'o fun x x_1 => x ≤ x_1) ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g m))
        ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g n))
⊢ False
state_after: case intro.intro.intro.intro
α : Type u_1
inst✝ : OrderedCancelCommMonoid α
s t : Set α
a✝ : α
x y : ↑(s.mulAntidiagonal t a✝)
hs : s.IsPWO
ht : t.IsPWO
a : α
h : (s.mulAntidiagonal t a).Infinite
h1 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.fst ⁻¹'o fun x x_1 => x ≤ x_1)
h2 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.snd ⁻¹'o fun x x_1 => x ≤ x_1)
g : ℕ ↪o ℕ
hg :
  ∀ (m n : ℕ),
    m ≤ n →
      (Prod.fst ⁻¹'o fun x x_1 => x ≤ x_1) ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g m))
        ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g n))
m n : ℕ
mn : m < n
h2' :
  (Prod.snd ⁻¹'o fun x x_1 => x ≤ x_1) ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g m))
    ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g n))
⊢ False
tactic: obtain ⟨m, n, mn, h2'⟩ := h2 (fun x => (h.natEmbedding _) (g x)) fun n => (h.natEmbedding _ _).2
state_before: case intro.intro.intro.intro
α : Type u_1
inst✝ : OrderedCancelCommMonoid α
s t : Set α
a✝ : α
x y : ↑(s.mulAntidiagonal t a✝)
hs : s.IsPWO
ht : t.IsPWO
a : α
h : (s.mulAntidiagonal t a).Infinite
h1 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.fst ⁻¹'o fun x x_1 => x ≤ x_1)
h2 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.snd ⁻¹'o fun x x_1 => x ≤ x_1)
g : ℕ ↪o ℕ
hg :
  ∀ (m n : ℕ),
    m ≤ n →
      (Prod.fst ⁻¹'o fun x x_1 => x ≤ x_1) ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g m))
        ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g n))
m n : ℕ
mn : m < n
h2' :
  (Prod.snd ⁻¹'o fun x x_1 => x ≤ x_1) ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g m))
    ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g n))
⊢ False
state_after: case intro.intro.intro.intro
α : Type u_1
inst✝ : OrderedCancelCommMonoid α
s t : Set α
a✝ : α
x y : ↑(s.mulAntidiagonal t a✝)
hs : s.IsPWO
ht : t.IsPWO
a : α
h : (s.mulAntidiagonal t a).Infinite
h1 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.fst ⁻¹'o fun x x_1 => x ≤ x_1)
h2 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.snd ⁻¹'o fun x x_1 => x ≤ x_1)
g : ℕ ↪o ℕ
hg :
  ∀ (m n : ℕ),
    m ≤ n →
      (Prod.fst ⁻¹'o fun x x_1 => x ≤ x_1) ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g m))
        ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g n))
m n : ℕ
mn : m < n
h2' :
  (Prod.snd ⁻¹'o fun x x_1 => x ≤ x_1) ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g m))
    ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g n))
⊢ (Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g m) = (Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g n)
tactic: refine mn.ne (g.injective <| (h.natEmbedding _).injective ?_)
state_before: case intro.intro.intro.intro
α : Type u_1
inst✝ : OrderedCancelCommMonoid α
s t : Set α
a✝ : α
x y : ↑(s.mulAntidiagonal t a✝)
hs : s.IsPWO
ht : t.IsPWO
a : α
h : (s.mulAntidiagonal t a).Infinite
h1 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.fst ⁻¹'o fun x x_1 => x ≤ x_1)
h2 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.snd ⁻¹'o fun x x_1 => x ≤ x_1)
g : ℕ ↪o ℕ
hg :
  ∀ (m n : ℕ),
    m ≤ n →
      (Prod.fst ⁻¹'o fun x x_1 => x ≤ x_1) ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g m))
        ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g n))
m n : ℕ
mn : m < n
h2' :
  (Prod.snd ⁻¹'o fun x x_1 => x ≤ x_1) ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g m))
    ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g n))
⊢ (Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g m) = (Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g n)
state_after: no goals
tactic: exact eq_of_fst_le_fst_of_snd_le_snd _ _ _ (hg _ _ mn.le) h2'