disjWithin_mono — Mathlib

English

If [c,d] ⊆ [a,b], then disjWithin(c,d) ⊆ disjWithin(a,b).

Русский

Если [c,d] ⊆ [a,b], то disjWithin(c,d) ⊆ disjWithin(a,b).

LaTeX

$$$ \text{disjWithin}(c,d) \subseteq \text{disjWithin}(a,b) \quad\text{whenever } uIcc\; c\; d \subseteq uIcc\; a\; b $$$

Lean4

theorem disjWithin_mono {a b c d : ℝ} (habcd : uIcc c d ⊆ uIcc a b) : disjWithin c d ⊆ disjWithin a b :=
  by
  simp +contextual only [disjWithin, Finset.mem_range, setOf_subset_setOf, and_true, and_imp, Prod.forall]
  exact fun (n I h _ i hi) ↦ ⟨habcd (h i hi).left, habcd (h i hi).right⟩

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