-- Module: Mathlib.RingTheory.ChainOfDivisors
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q : Associates M
n : ℕ
hn : n ≠ 0
c : Fin (n + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1)), r = c i
hq : q ≠ 0
obtain : ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)), q = c 1 ^ n
h : ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)), n = (↑i : ℕ)
h₃ : ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)), n.succ ≤ (↑i : ℕ).succ
h₄ :
  ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)),
    Finset.image c Finset.univ ⊆ Finset.image (fun (m : Fin ((↑i : ℕ) + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ
h₅ :
  ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)) ⦃r : Associates M⦄ (hr : r ∈ Finset.image c Finset.univ),
    Membership.mem (γ := Finset (Associates M))
      (Finset.image (fun (m : Fin ((↑i : ℕ) + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ) r
h₆ :
  ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)) (j : Fin (n + 1)) (hr : c j ∈ Finset.image c Finset.univ),
    Membership.mem (γ := Finset (Associates M))
      (Finset.image (fun (m : Fin ((↑i : ℕ) + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ) (c j)
h₇ :
  ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)) (j : Fin (n + 1)) (hr : c j ∈ Finset.image c Finset.univ)
    (this : c j ≤ q),
    Membership.mem (γ := Finset (Associates M))
      (Finset.image (fun (m : Fin ((↑i : ℕ) + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ) (c j)
h₈ :
  ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)) (j : Fin (n + 1)) (hr : c j ∈ Finset.image c Finset.univ)
    (this : c j ≤ c 1 ^ (↑i : ℕ)),
    Membership.mem (γ := Finset (Associates M))
      (Finset.image (fun (m : Fin ((↑i : ℕ) + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ) (c j)
h₉ :
  ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)) (j : Fin (n + 1)) (hr : c j ∈ Finset.image c Finset.univ)
    (this : c j ≤ c 1 ^ (↑i : ℕ)) (h : ∃ i_1 ≤ (↑i : ℕ), Associated (c j) (c 1 ^ i_1)),
    Membership.mem (γ := Finset (Associates M))
      (Finset.image (fun (m : Fin ((↑i : ℕ) + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ) (c j)
h₁₀ h₁₁ :
  ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)) (j : Fin (n + 1)) (hr : c j ∈ Finset.image c Finset.univ)
    (this : c j ≤ c 1 ^ (↑i : ℕ)), Prime (c 1)
⊢ q = c 1 ^ n

Tactic:
classical
    obtain ⟨i, hi'⟩ := element_of_chain_eq_pow_second_of_chain hn h₁ (@fun r => h₂) (dvd_refl q) hq
    convert! hi'
    refine (Nat.lt_succ_iff.1 i.prop).antisymm' (Nat.le_of_succ_le_succ ?_)
    calc
      n + 1 = (Finset.univ : Finset (Fin (n + 1))).card := (Finset.card_fin _).symm
      _ = (Finset.univ.image c).card := (Finset.card_image_iff.mpr h₁.injective.injOn).symm
      _ ≤ (Finset.univ.image fun m : Fin (i + 1) => c 1 ^ (m : ℕ)).card :=
        (Finset.card_le_card ?_)
      _ ≤ (Finset.univ : Finset (Fin (i + 1))).card := Finset.card_image_le
      _ = i + 1 := Finset.card_fin _
    intro r hr
    obtain ⟨j, -, rfl⟩ := Finset.mem_image.1 hr
    have := h₂.2 ⟨j, rfl⟩
    rw [hi'] at this
    have h := (dvd_prime_pow (show Prime (c 1) from ?_) i).1 this
    · rcases h with ⟨u, hu, hu'⟩
      refine Finset.mem_image.mpr ⟨⟨u, Nat.lt_succ_of_le hu⟩, Finset.mem_univ _, ?_⟩
      rwa [associated_iff_eq, eq_comm] at hu'
    · rw [← irreducible_iff_prime]
      exact second_of_chain_is_irreducible hn h₁ (@h₂) hq

Premises used: Iff.mpr, Finset.mem_univ, Finset.univ, Finset.mem_image, Function.Injective.injOn, Finset.card_le_card, eq_comm, Finset.card_image_le, associated_iff_eq, StrictMono.injective, Finset.card_fin, And.intro, DivisorChain.second_of_chain_is_irreducible, Finset.card_image_iff, Exists.intro, Finset.card, irreducible_iff_prime, Eq.symm, dvd_refl, DivisorChain.element_of_chain_eq_pow_second_of_chain, rfl