exists_ne_zero_and_lt — Mathlib

English

If f is a strictly monotone map between α and β, then map' f is strictly monotone on WithZero α.

Русский

Если f строго монотонна, то map' f строго монотонна на WithZero α.

LaTeX

$$$ \text{map' } f \text{ is strictly monotone given } f \text{ is strictly monotone}$$$

Lean4

theorem exists_ne_zero_and_lt [NoMinOrder α] (hx : x ≠ 0) : ∃ y, y ≠ 0 ∧ y < x :=
  by
  obtain ⟨z, hlt⟩ := exists_lt (WithZero.unzero hx)
  rw [← WithZero.coe_lt_coe, WithZero.coe_unzero hx] at hlt
  exact ⟨z, WithZero.coe_ne_zero, hlt⟩

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