Theorem: Ordinal.isClosed_iff_bsup
File: Mathlib/SetTheory/Ordinal/Topology.lean

tactic: rw [isClosed_iff_sup]
state_before: s : Set Ordinal.{u}
a : Ordinal.{u}
⊢ IsClosed s ↔
    ∀ {o : Ordinal.{u}},
      o ≠ 0 →
        ∀ (f : (a : Ordinal.{u}) → a < o → Ordinal.{u}), (∀ (i : Ordinal.{u}) (hi : i < o), f i hi ∈ s) → o.bsup f ∈ s
state_after: s : Set Ordinal.{u}
a : Ordinal.{u}
⊢ (∀ {ι : Type u}, Nonempty ι → ∀ (f : ι → Ordinal.{u}), (∀ (i : ι), f i ∈ s) → sup f ∈ s) ↔
    ∀ {o : Ordinal.{u}},
      o ≠ 0 →
        ∀ (f : (a : Ordinal.{u}) → a < o → Ordinal.{u}), (∀ (i : Ordinal.{u}) (hi : i < o), f i hi ∈ s) → o.bsup f ∈ s

tactic: refine ⟨fun H o ho f hf => H (out_nonempty_iff_ne_zero.2 ho) _ ?_, fun H ι hι f hf => ?_⟩
state_before: s : Set Ordinal.{u}
a : Ordinal.{u}
⊢ (∀ {ι : Type u}, Nonempty ι → ∀ (f : ι → Ordinal.{u}), (∀ (i : ι), f i ∈ s) → sup f ∈ s) ↔
    ∀ {o : Ordinal.{u}},
      o ≠ 0 →
        ∀ (f : (a : Ordinal.{u}) → a < o → Ordinal.{u}), (∀ (i : Ordinal.{u}) (hi : i < o), f i hi ∈ s) → o.bsup f ∈ s
state_after: case refine_1
s : Set Ordinal.{u}
a : Ordinal.{u}
H : ∀ {ι : Type u}, Nonempty ι → ∀ (f : ι → Ordinal.{u}), (∀ (i : ι), f i ∈ s) → sup f ∈ s
o : Ordinal.{u}
ho : o ≠ 0
f : (a : Ordinal.{u}) → a < o → Ordinal.{u}
hf : ∀ (i : Ordinal.{u}) (hi : i < o), f i hi ∈ s
⊢ ∀ (i : (Quotient.out o).α), o.familyOfBFamily f i ∈ s

case refine_2
s : Set Ordinal.{u}
a : Ordinal.{u}
H :
  ∀ {o : Ordinal.{u}},
    o ≠ 0 →
      ∀ (f : (a : Ordinal.{u}) → a < o → Ordinal.{u}), (∀ (i : Ordinal.{u}) (hi : i < o), f i hi ∈ s) → o.bsup f ∈ s
ι : Type u
hι : Nonempty ι
f : ι → Ordinal.{u}
hf : ∀ (i : ι), f i ∈ s
⊢ sup f ∈ s

tactic: exact fun i => hf _ _
state_before: case refine_1
s : Set Ordinal.{u}
a : Ordinal.{u}
H : ∀ {ι : Type u}, Nonempty ι → ∀ (f : ι → Ordinal.{u}), (∀ (i : ι), f i ∈ s) → sup f ∈ s
o : Ordinal.{u}
ho : o ≠ 0
f : (a : Ordinal.{u}) → a < o → Ordinal.{u}
hf : ∀ (i : Ordinal.{u}) (hi : i < o), f i hi ∈ s
⊢ ∀ (i : (Quotient.out o).α), o.familyOfBFamily f i ∈ s
state_after: no goals

tactic: rw [← bsup_eq_sup]
state_before: case refine_2
s : Set Ordinal.{u}
a : Ordinal.{u}
H :
  ∀ {o : Ordinal.{u}},
    o ≠ 0 →
      ∀ (f : (a : Ordinal.{u}) → a < o → Ordinal.{u}), (∀ (i : Ordinal.{u}) (hi : i < o), f i hi ∈ s) → o.bsup f ∈ s
ι : Type u
hι : Nonempty ι
f : ι → Ordinal.{u}
hf : ∀ (i : ι), f i ∈ s
⊢ sup f ∈ s
state_after: case refine_2
s : Set Ordinal.{u}
a : Ordinal.{u}
H :
  ∀ {o : Ordinal.{u}},
    o ≠ 0 →
      ∀ (f : (a : Ordinal.{u}) → a < o → Ordinal.{u}), (∀ (i : Ordinal.{u}) (hi : i < o), f i hi ∈ s) → o.bsup f ∈ s
ι : Type u
hι : Nonempty ι
f : ι → Ordinal.{u}
hf : ∀ (i : ι), f i ∈ s
⊢ (type WellOrderingRel).bsup (bfamilyOfFamily f) ∈ s

tactic: apply H (type_ne_zero_iff_nonempty.2 hι)
state_before: case refine_2
s : Set Ordinal.{u}
a : Ordinal.{u}
H :
  ∀ {o : Ordinal.{u}},
    o ≠ 0 →
      ∀ (f : (a : Ordinal.{u}) → a < o → Ordinal.{u}), (∀ (i : Ordinal.{u}) (hi : i < o), f i hi ∈ s) → o.bsup f ∈ s
ι : Type u
hι : Nonempty ι
f : ι → Ordinal.{u}
hf : ∀ (i : ι), f i ∈ s
⊢ (type WellOrderingRel).bsup (bfamilyOfFamily f) ∈ s
state_after: case refine_2.a
s : Set Ordinal.{u}
a : Ordinal.{u}
H :
  ∀ {o : Ordinal.{u}},
    o ≠ 0 →
      ∀ (f : (a : Ordinal.{u}) → a < o → Ordinal.{u}), (∀ (i : Ordinal.{u}) (hi : i < o), f i hi ∈ s) → o.bsup f ∈ s
ι : Type u
hι : Nonempty ι
f : ι → Ordinal.{u}
hf : ∀ (i : ι), f i ∈ s
⊢ ∀ (i : Ordinal.{u}) (hi : i < type WellOrderingRel), bfamilyOfFamily f i hi ∈ s

tactic: exact fun i hi => hf _
state_before: case refine_2.a
s : Set Ordinal.{u}
a : Ordinal.{u}
H :
  ∀ {o : Ordinal.{u}},
    o ≠ 0 →
      ∀ (f : (a : Ordinal.{u}) → a < o → Ordinal.{u}), (∀ (i : Ordinal.{u}) (hi : i < o), f i hi ∈ s) → o.bsup f ∈ s
ι : Type u
hι : Nonempty ι
f : ι → Ordinal.{u}
hf : ∀ (i : ι), f i ∈ s
⊢ ∀ (i : Ordinal.{u}) (hi : i < type WellOrderingRel), bfamilyOfFamily f i hi ∈ s
state_after: no goals