English
If a fact about units holds for all ideals, then the natural-number cast of a positive natural is a unit.
Русский
Если верно для всех идеалов, то число n в кольце является единицей.
LaTeX
$$$\\forall n : \\mathbb{N}_{>0}, \\operatorname{IsUnit}(n: R).$$$
Lean4
/-- Internal: Not intended to be used outside this local construction. -/
theorem isUnit_natCast [h : Fact (∀ I : Ideal R, I ≠ ⊤ → CharZero (R ⧸ I))] (n : ℕ+) : IsUnit (n : R) := by
-- `n : R` is a unit iff `(n)` is not a proper ideal in `R`.
rw [← Ideal.span_singleton_eq_top]
-- So by contrapositive, we should show the quotient does not have characteristic zero.
apply not_imp_comm.mp (h.elim (Ideal.span {↑n}))
intro h_char_zero
apply h_char_zero.cast_injective.ne n.ne_zero
rw [← map_natCast (Ideal.Quotient.mk _), Nat.cast_zero, Ideal.Quotient.eq_zero_iff_mem]
exact Ideal.subset_span (Set.mem_singleton _)
