Theorems · Theorem · ring theory
geom_sum_mul
∀ {R : Type u_1} [inst : Ring R] (x : R) (n : ℕ), (∑ i ∈ Finset.range n, x ^ i) * (x - 1) = x ^ n - 1- Defined in
- Mathlib.Algebra.Ring.GeomSum
- Cited by
- 15 results in Mathlib
- Foundations
- Depth 70 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- Ring
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Ringstatement and proof · cited by 7,463
- Finset.sumstatement and proof · cited by 5,195
- Finset.rangestatement and proof · cited by 1,341
- one_powproof · cited by 521
- Commute.one_rightproof · cited by 22
- Commute.geom_sum₂_mulproof · cited by 4
- geom_sum₂_with_oneproof · cited by 2
Cited by15
Results whose statement or proof uses this declaration.
- geom_sum_eqproof · cited by 10
- mul_geom_sumproof · cited by 4
- geom_sum_mul_negproof · cited by 4
- Polynomial.prod_cyclotomic_eq_geom_sumproof · cited by 4
- IsNilpotent.isUnit_sub_oneproof · cited by 3
- Polynomial.cyclotomic_prime_pow_eq_geom_sumproof · cited by 3
- sum_range_powproof · cited by 2
- sum_hom_units_eq_zeroproof · cited by 1
- sub_one_dvd_natCast_of_pow_eq_oneproof · cited by 1
- Polynomial.cyclotomic_prime_pow_mul_X_pow_sub_oneproof · cited by 1
- Projectivization.card_of_finrankproof · cited by 1
- IsPrimitiveRoot.self_sub_one_pow_dvd_orderproof · cited by 1