Theorems · Theorem · combinatorics
Commute.add_pow
∀ {R : Type u_1} [inst : Semiring R] {x y : R},
Commute x y → ∀ (n : ℕ), (x + y) ^ n = ∑ m ∈ Finset.range (n + 1), x ^ m * y ^ (n - m) * ↑(n.choose m)A version of the binomial theorem for commuting elements in noncommutative semirings.
- Defined in
- Mathlib.Data.Nat.Choose.Sum
- Cited by
- 12 results in Mathlib
- Foundations
- Depth 71 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- Semiring
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites36
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
- Finsetproof · cited by 13,712
- Finset.sumstatement and proof · cited by 5,195
- mul_oneproof · cited by 3,885
- one_mulproof · cited by 2,841
- add_zeroproof · cited by 2,707
- Nat.cast_oneproof · cited by 2,501
- zero_addproof · cited by 2,366
- Finset.sum_congrproof · cited by 2,323
- MulZeroClass.mul_zeroproof · cited by 2,091
- Nat.cast_zeroproof · cited by 1,870
- mul_assocproof · cited by 1,667
Cited by12
Results whose statement or proof uses this declaration.
- add_powproof · cited by 24
- Commute.add_pow'proof · cited by 6
- fwdDiff_iter_eq_sum_shiftproof · cited by 5
- Polynomial.taylor_coeffproof · cited by 4
- Commute.add_pow_prime_pow_eq'proof · cited by 3
- Ideal.add_pow_mem_of_pow_mem_of_le_of_commuteproof · cited by 3
- IsNilpotent.exp_add_of_commuteproof · cited by 3
- IsUltrametricDist.isUltrametricDist_of_forall_norm_natCast_le_oneproof · cited by 1
- shift_eq_sum_fwdDiff_iterproof · cited by 1
- Polynomial.coeff_X_add_C_powproof · cited by 1
- Finset.sum_pow_eq_sum_piAntidiag_of_commuteproof · cited by 1
- Finset.sum_pow_of_commuteproof · cited by 1