disjoint_prod — Mathlib

English

Disjointness of products corresponds to disjointness of factors: Disjoint (s1 × t1) (s2 × t2) iff Disjoint s1 s2 or Disjoint t1 t2.

Русский

Бесполезность произведений соответствует бесполезности факторов: Disjoint (s1 × t1) (s2 × t2) ⇔ Disjoint s1 s2 или Disjoint t1 t2.

LaTeX

$$$\\operatorname{Disjoint}(s_1 \\times t_1, s_2 \\times t_2) \\iff \\operatorname{Disjoint}(s_1, s_2) \\lor \\operatorname{Disjoint}(t_1, t_2)$$$

Lean4

@[simp]
theorem disjoint_prod : Disjoint (s₁ ×ˢ t₁) (s₂ ×ˢ t₂) ↔ Disjoint s₁ s₂ ∨ Disjoint t₁ t₂ := by
  simp_rw [disjoint_iff, prod_inf_prod, prod_eq_bot]

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