Theorem: IsArtinian.disjoint_partial_infs_eventually_top
File: Mathlib/RingTheory/Artinian.lean

Proof steps:
state_before: R : Type u_1
M : Type u_2
inst✝³ : Ring R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsArtinian R M
f : ℕ → Submodule R M
h : ∀ (n : ℕ), Disjoint ((partialSups (⇑OrderDual.toDual ∘ f)) n) (OrderDual.toDual (f (n + 1)))
⊢ ∃ n, ∀ (m : ℕ), n ≤ m → f m = ⊤
state_after: case intro
R : Type u_1
M : Type u_2
inst✝³ : Ring R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsArtinian R M
f : ℕ → Submodule R M
h : ∀ (n : ℕ), Disjoint ((partialSups (⇑OrderDual.toDual ∘ f)) n) (OrderDual.toDual (f (n + 1)))
n : ℕ
w : ∀ (m : ℕ), n ≤ m → OrderDual.toDual f (m + 1) = ⊤
⊢ ∃ n, ∀ (m : ℕ), n ≤ m → f m = ⊤

R : Type u_1
M : Type u_2
inst✝³ : Ring R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsArtinian R M
f : ℕ → Submodule R M
h : ∀ (n : ℕ), Disjoint ((partialSups (⇑OrderDual.toDual ∘ f)) n) (OrderDual.toDual (f (n + 1)))
⊢ ∃ n, ∀ (m : ℕ), n ≤ m → OrderDual.toDual f (m + 1) = ⊤
tactic: rsuffices ⟨n, w⟩ : ∃ n : ℕ, ∀ m, n ≤ m → OrderDual.toDual f (m + 1) = ⊤
state_before: R : Type u_1
M : Type u_2
inst✝³ : Ring R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsArtinian R M
f : ℕ → Submodule R M
h : ∀ (n : ℕ), Disjoint ((partialSups (⇑OrderDual.toDual ∘ f)) n) (OrderDual.toDual (f (n + 1)))
⊢ ∃ n, ∀ (m : ℕ), n ≤ m → OrderDual.toDual f (m + 1) = ⊤
state_after: case intro
R : Type u_1
M : Type u_2
inst✝³ : Ring R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsArtinian R M
f : ℕ → Submodule R M
h : ∀ (n : ℕ), Disjoint ((partialSups (⇑OrderDual.toDual ∘ f)) n) (OrderDual.toDual (f (n + 1)))
n : ℕ
w : ∀ (m : ℕ), n ≤ m → (partialSups (⇑OrderDual.toDual ∘ f)) n = (partialSups (⇑OrderDual.toDual ∘ f)) m
⊢ ∃ n, ∀ (m : ℕ), n ≤ m → OrderDual.toDual f (m + 1) = ⊤
tactic: obtain ⟨n, w⟩ := monotone_stabilizes (partialSups (OrderDual.toDual ∘ f))
state_before: case intro
R : Type u_1
M : Type u_2
inst✝³ : Ring R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsArtinian R M
f : ℕ → Submodule R M
h : ∀ (n : ℕ), Disjoint ((partialSups (⇑OrderDual.toDual ∘ f)) n) (OrderDual.toDual (f (n + 1)))
n : ℕ
w : ∀ (m : ℕ), n ≤ m → (partialSups (⇑OrderDual.toDual ∘ f)) n = (partialSups (⇑OrderDual.toDual ∘ f)) m
⊢ ∃ n, ∀ (m : ℕ), n ≤ m → OrderDual.toDual f (m + 1) = ⊤
state_after: case intro
R : Type u_1
M : Type u_2
inst✝³ : Ring R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsArtinian R M
f : ℕ → Submodule R M
h : ∀ (n : ℕ), Disjoint ((partialSups (⇑OrderDual.toDual ∘ f)) n) (OrderDual.toDual (f (n + 1)))
n : ℕ
w : ∀ (m : ℕ), n ≤ m → (partialSups (⇑OrderDual.toDual ∘ f)) n = (partialSups (⇑OrderDual.toDual ∘ f)) m
m : ℕ
p : n ≤ m
⊢ OrderDual.toDual f (m + 1) = ⊤
tactic: refine ⟨n, fun m p => ?_⟩
state_before: case intro
R : Type u_1
M : Type u_2
inst✝³ : Ring R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsArtinian R M
f : ℕ → Submodule R M
h : ∀ (n : ℕ), Disjoint ((partialSups (⇑OrderDual.toDual ∘ f)) n) (OrderDual.toDual (f (n + 1)))
n : ℕ
w : ∀ (m : ℕ), n ≤ m → (partialSups (⇑OrderDual.toDual ∘ f)) n = (partialSups (⇑OrderDual.toDual ∘ f)) m
m : ℕ
p : n ≤ m
⊢ OrderDual.toDual f (m + 1) = ⊤
state_after: no goals
tactic: exact (h m).eq_bot_of_ge (sup_eq_left.1 <| (w (m + 1) <| le_add_right p).symm.trans <| w m p)