Ico_filter_lt_of_le_right — Mathlib

English

If a2 ≤ b2, a2 < a1, and b1 ≤ b2, then Icc(a1, b1) is strictly contained in Icc(a2, b2).

Русский

Если a2 ≤ b2, a2 < a1 и b1 ≤ b2, то Icc(a1,b1) ⊂ Icc(a2,b2).

LaTeX

$$$ (a_2 \le b_2) \land (a_2 < a_1) \land (b_1 \le b_2) \Rightarrow Icc(a_1,b_1) \subset Icc(a_2,b_2) $$$

Lean4

theorem Ico_filter_lt_of_le_right [DecidablePred (· < c)] (hcb : c ≤ b) : {x ∈ Ico a b | x < c} = Ico a c :=
  by
  ext x
  rw [mem_filter, mem_Ico, mem_Ico, and_right_comm]
  exact and_iff_left_of_imp fun h => h.2.trans_le hcb

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