feat(algebra/ring/basic): all non-zero elements in a non-trivial ring with no non-zero zero divisors are regular (#12947)

Besides what the PR description says, I also golfed two earlier proofs.

Author: adomani

src/algebra/ring/basic.lean

@@ -1147,9 +1147,7 @@ The typeclass that restricts all terms of `α` to have this property is `no_zero
 lemma is_left_regular_of_non_zero_divisor [ring α] (k : α)
   (h : ∀ (x : α), k * x = 0 → x = 0) : is_left_regular k :=
 begin
-  intros x y h',
-  rw ←sub_eq_zero,
-  refine h _ _,
+  refine λ x y (h' : k * x = k * y), sub_eq_zero.mp (h _ _),
   rw [mul_sub, sub_eq_zero, h']
 end
 
@@ -1158,10 +1156,7 @@ The typeclass that restricts all terms of `α` to have this property is `no_zero
 lemma is_right_regular_of_non_zero_divisor [ring α] (k : α)
   (h : ∀ (x : α), x * k = 0 → x = 0) : is_right_regular k :=
 begin
-  intros x y h',
-  simp only at h',
-  rw ←sub_eq_zero,
-  refine h _ _,
+  refine λ x y (h' : x * k = y * k), sub_eq_zero.mp (h _ _),
   rw [sub_mul, sub_eq_zero, h']
 end
 
@@ -1172,6 +1167,10 @@ lemma is_regular_of_ne_zero' [ring α] [no_zero_divisors α] {k : α} (hk : k 
  is_right_regular_of_non_zero_divisor k
   (λ x h, (no_zero_divisors.eq_zero_or_eq_zero_of_mul_eq_zero h).resolve_right hk)⟩
 
+lemma is_regular_iff_ne_zero' [nontrivial α] [ring α] [no_zero_divisors α] {k : α} :
+  is_regular k ↔ k ≠ 0 :=
+⟨λ h, by { rintro rfl, exact not_not.mpr h.left not_is_left_regular_zero }, is_regular_of_ne_zero'⟩
+
 /-- A ring with no zero divisors is a cancel_monoid_with_zero.
 
 Note this is not an instance as it forms a typeclass loop. -/