English
Definition fragment: relation used in equivalence for orderIso invariance of transfinite shapes.
Русский
Фрагмент определения: отношение, используемое для эквивалентности инвариантности по порядковому изоморфизму форм трансфинитной композиции.
LaTeX
$$Definition fragment for isStableUnderTransfiniteComposition equivalences$$
Lean4
theorem mem_map_bot_le {j : J} (g : ⊥ ⟶ j) : W (hf.F.map g) :=
by
obtain rfl : g = homOfLE bot_le := rfl
induction j using SuccOrder.limitRecOn with
| isMin j hj =>
obtain rfl := hj.eq_bot
simpa using W.id_mem _
| succ j hj hj' =>
rw [← homOfLE_comp bot_le (Order.le_succ j), hf.F.map_comp]
exact W.comp_mem _ _ hj' (hf.map_mem j hj)
| isSuccLimit j hj
hj' =>
letI : OrderBot (Set.Iio j) :=
{ bot := ⟨⊥, Order.IsSuccLimit.bot_lt hj⟩
bot_le j := bot_le }
exact
MorphismProperty.colimitsOfShape_le _
(.of_isColimit (hf.F.isColimitOfIsWellOrderContinuous j hj) (fun k ↦ hj' _ k.2))
