Theorems · Theorem · order theory
sq_pos_of_ne_zero
∀ {R : Type u_3} [inst : Semiring R] [inst_1 : LinearOrder R] [IsStrictOrderedRing R] [ExistsAddOfLE R] {a : R},
a ≠ 0 → 0 < a ^ 2Alias of the reverse direction of sq_pos_iff.
- Defined in
- Mathlib.Algebra.Order.Ring.Basic
- Cited by
- 9 results in Mathlib
- Foundations
- Depth 22 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites5
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
- IsStrictOrderedRingstatement and proof · cited by 2,490
- ExistsAddOfLEstatement and proof · cited by 330
- sq_pos_iffproof · cited by 3
Cited by9
Results whose statement or proof uses this declaration.
- Fermat42.ne_zeroproof · cited by 2
- HurwitzKernelBounds.isBigO_atTop_F_nat_zero_subproof · cited by 2
- ProbabilityTheory.meas_ge_le_variance_div_sqproof · cited by 1
- Pell.Solution₁.eq_zero_of_d_negproof · cited by 1
- Pell.Solution₁.exists_pos_of_not_isSquareproof · cited by 1
- SzemerediRegularity.add_div_le_sum_sq_div_cardproof · cited by 1
- PythagoreanTriple.ne_zero_of_coprimeproof · cited by 1
- Fermat42.not_minimalproof · cited by 1
- pow_two_pos_of_ne_zeroproof · cited by 0