feat(algebra/order_functions): (min|max)_eq_iff (#8911)

Co-authored-by: Yakov Pechersky <pechersky@users.noreply.github.com>

Author: pechersky

src/algebra/order_functions.lean

@@ -48,6 +48,19 @@ lemma min_le_max : min a b ≤ max a b := le_trans (min_le_left a b) (le_max_lef
 @[simp] lemma max_eq_left_iff : max a b = a ↔ b ≤ a := sup_eq_left
 @[simp] lemma max_eq_right_iff : max a b = b ↔ a ≤ b := sup_eq_right
 
+lemma min_eq_iff : min a b = c ↔ a = c ∧ a ≤ b ∨ b = c ∧ b ≤ a :=
+begin
+  split,
+  { intro h,
+    refine or.imp (λ h', _) (λ h', _) (le_total a b);
+    exact ⟨by simpa [h'] using h, h'⟩ },
+  { rintro (⟨rfl, h⟩|⟨rfl, h⟩);
+    simp [h] }
+end
+
+lemma max_eq_iff : max a b = c ↔ a = c ∧ b ≤ a ∨ b = c ∧ a ≤ b :=
+@min_eq_iff (order_dual α) _ a b c
+
 /-- An instance asserting that `max a a = a` -/
 instance max_idem : is_idempotent α max := by apply_instance -- short-circuit type class inference