Theorems · Theorem · order theory
one_add_mul_le_pow_of_sq_nonneg
∀ {R : Type u_1} [inst : Semiring R] [inst_1 : PartialOrder R] [IsOrderedRing R] {a : R},
0 ≤ a ^ 2 → 0 ≤ (1 + a) ^ 2 → 0 ≤ 2 + a → ∀ (n : ℕ), 1 + ↑n * a ≤ (1 + a) ^ nBernoulli's inequality. This version works for semirings but requires
additional hypotheses 0 ≤ a ^ 2 and 0 ≤ (1 + a) ^ 2.
- Defined in
- Mathlib.Algebra.Order.Ring.Pow
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 30 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
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
- PartialOrderstatement and proof · cited by 6,410
- mul_oneproof · cited by 3,885
- IsOrderedRingstatement and proof · cited by 777
- one_powproof · cited by 521
- zero_le_oneproof · cited by 316
- Commute.one_leftproof · cited by 15
- Commute.pow_add_mul_le_add_pow_of_sq_nonnegproof · cited by 3
Cited by2
Results whose statement or proof uses this declaration.
- add_one_pow_unbounded_of_posproof · cited by 3
- one_add_le_pow_of_two_add_nonnegproof · cited by 1