English
In an integral domain, if b1 ≠ 0 and b2 is not a unit, then span{b1 b2} is strictly contained in span{b1}.
Русский
В整иваномDomain, если b1 ≠ 0 и b2 не единица, то span{b1 b2} строго меньше span{b1}.
LaTeX
$$$$ \operatorname{span}(\{ b_1 b_2 \}) < \operatorname{span}(\{ b_1 \}) \quad\text{if } b_1 \neq 0 \text{ and } \neg \operatorname{IsUnit}(b_2) $$$$
Lean4
theorem factors_decreasing [IsDomain α] (b₁ b₂ : α) (h₁ : b₁ ≠ 0) (h₂ : ¬IsUnit b₂) :
span ({b₁ * b₂} : Set α) < span { b₁ } :=
lt_of_le_not_ge (Ideal.span_le.2 <| singleton_subset_iff.2 <| Ideal.mem_span_singleton.2 ⟨b₂, rfl⟩) fun h =>
h₂ <| isUnit_of_dvd_one <| (mul_dvd_mul_iff_left h₁).1 <| by rwa [mul_one, ← Ideal.span_singleton_le_span_singleton]
