top_mem_range — Mathlib

English

There exists an ordinal o and j in o such that transfiniteIterate(largerSubobject hG) applied to A0 reaches top.

Русский

Существует порядок o и элемент j such that трансфинитная итерация приведёт A0 к верхнему пределу.

LaTeX

$$$\exists(o:Ordinal)(j:o.toType),\; \text{transfiniteIterate}(largerSubobject hG)\; j\; A0 = ⊤$$$

Lean4

theorem top_mem_range (A₀ : Subobject X) {J : Type w} [LinearOrder J] [OrderBot J] [SuccOrder J] [WellFoundedLT J]
    (hJ : HasCardinalLT (Subobject X) (Cardinal.mk J)) : ∃ (j : J), transfiniteIterate (largerSubobject hG) j A₀ = ⊤ :=
  top_mem_range_transfiniteIterate (largerSubobject hG) A₀ (lt_largerSubobject hG) (by simp)
    (fun h ↦ by simpa [hasCardinalLT_iff_cardinal_mk_lt] using hJ.of_injective _ h)

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