English
In a Dedekind domain, for I ≠ 0 and I ≠ ⊤, the inverse I^{-1} is not ≤ 1.
Русский
В Дедекинд домене для I ≠ 0 и I ≠ ⊤ обратный I^{-1} не не меньше чем 1.
LaTeX
$$$$I \\neq 0 \\land I \\neq 1 \\Rightarrow \\neg(I^{-1} \\le 1).$$$$
Lean4
/-- This is also available as `_root_.div_eq_mul_inv`, using the
`Semifield` instance defined below.
-/
protected theorem div_eq_mul_inv [IsDedekindDomain A] (I J : FractionalIdeal A⁰ K) : I / J = I * J⁻¹ :=
by
by_cases hJ : J = 0
· rw [hJ, div_zero, inv_zero', mul_zero]
refine le_antisymm ((mul_right_le_iff hJ).mp ?_) ((le_div_iff_mul_le hJ).mpr ?_)
· rw [mul_assoc, mul_comm J⁻¹, FractionalIdeal.mul_inv_cancel hJ, mul_one, mul_le]
intro x hx y hy
rw [mem_div_iff_of_nonzero hJ] at hx
exact hx y hy
rw [mul_assoc, mul_comm J⁻¹, FractionalIdeal.mul_inv_cancel hJ, mul_one]
