Theorems · Theorem · order theory
mul_self_add_mul_self_eq_zero
∀ {R : Type u} [inst : Semiring R] [inst_1 : LinearOrder R] {a b : R} [NoZeroDivisors R] [ExistsAddOfLE R]
[PosMulMono R] [AddLeftMono R], a * a + b * b = 0 ↔ a = 0 ∧ b = 0The sum of two squares is zero iff both elements are zero.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 19 from the axioms · uses propext, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Semiringstatement and proof · cited by 13,802
- LinearOrderstatement and proof · cited by 8,572
- AddLeftMonostatement and proof · cited by 687
- NoZeroDivisorsstatement and proof · cited by 545
- ExistsAddOfLEstatement and proof · cited by 330
- PosMulMonostatement and proof · cited by 165
- mul_self_nonnegproof · cited by 35
- add_eq_zero_iff_of_nonnegproof · cited by 8
- mul_self_eq_zeroproof · cited by 7
Cited by1
Results whose statement or proof uses this declaration.
- eq_zero_of_mul_self_add_mul_self_eq_zeroproof · cited by 1