add_eq_zero_iff — Mathlib

English

The coe of tropOrderIso equals trop: (tropOrderIso : R → Tropical R) = trop.

Русский

Коэрциа tropOrderIso равна trop: (tropOrderIso : R → Tropical R) = trop.

LaTeX

$$$(\\operatorname{tropOrderIso} : R \\to Tropical R) = \\operatorname{trop}$$$

Lean4

@[simp]
theorem add_eq_zero_iff {a b : Tropical (WithTop R)} : a + b = 0 ↔ a = 0 ∧ b = 0 :=
  by
  rw [add_eq_iff]
  constructor
  · rintro (⟨rfl, h⟩ | ⟨rfl, h⟩)
    · exact ⟨rfl, le_antisymm (le_zero _) h⟩
    · exact ⟨le_antisymm (le_zero _) h, rfl⟩
  · rintro ⟨rfl, rfl⟩
    simp

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