Theorems · Theorem · commutative algebra
Ideal.pow_mem_pow
∀ {R : Type u} [inst : Semiring R] {I : Ideal R} {x : R}, x ∈ I → ∀ (n : ℕ), x ^ n ∈ I ^ n- Defined in
- Mathlib.RingTheory.Ideal.Operations
- Cited by
- 19 results in Mathlib
- Foundations
- Depth 78 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.
Cites3
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
- Idealstatement and proof · cited by 4,748
- Submodule.pow_mem_powproof · cited by 3
Cited by19
Results whose statement or proof uses this declaration.
- Ideal.pow_eq_zero_of_memproof · cited by 5
- Algebra.FormallyUnramified.of_isSeparableproof · cited by 5
- IsNilpotent.isUnit_quotient_mk_iffproof · cited by 3
- Ideal.FG.isNilpotent_iff_le_nilradicalproof · cited by 2
- PrimeSpectrum.exists_mul_eq_zero_add_eq_one_basicOpen_eq_of_isClopenproof · cited by 2
- WittVector.factorPowSucc_comp_fontaineThetaModPPowproof · cited by 2
- Algebra.FormallyUnramified.bijective_of_isAlgClosed_of_isLocalRingproof · cited by 1
- Algebra.FormallyEtale.of_isSeparable_auxproof · cited by 1
- pow_sub_one_dvd_differentIdeal_auxproof · cited by 1
- exists_maximalIdeal_pow_eq_of_principalproof · cited by 1
- Algebra.FormallyUnramified.pi_iffproof · cited by 1
- Ideal.IsMaximal.exists_inv_powproof · cited by 1