Theorems · Theorem · commutative algebra
Polynomial.lifts_iff_coeff_lifts
∀ {R : Type u} [inst : Semiring R] {S : Type v} [inst_1 : Semiring S] {f : R →+* S} (p : Polynomial S),
p ∈ Polynomial.lifts f ↔ ∀ (n : ℕ), p.coeff n ∈ Set.range ⇑f- Defined in
- Mathlib.Algebra.Polynomial.Lifts
- Cited by
- 10 results in Mathlib
- Foundations
- Depth 108 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Setstatement · cited by 53,352
- Semiringstatement and proof · cited by 13,802
- RingHomstatement and proof · cited by 10,189
- Polynomialstatement and proof · cited by 5,681
- Set.rangestatement and proof · cited by 4,705
- Polynomial.coeffstatement and proof · cited by 1,045
- Subsemiringstatement · cited by 456
- Polynomial.liftsstatement · cited by 35
- Polynomial.mem_map_rangeSproof · cited by 4
- Polynomial.lifts_iff_ringHom_rangeSproof · cited by 2
Cited by10
Results whose statement or proof uses this declaration.
- Algebra.IsInvariant.charpoly_mem_liftsproof · cited by 3
- Polynomial.lifts_and_degree_eq_and_monicproof · cited by 3
- Polynomial.exists_support_eq_of_mem_liftsproof · cited by 1
- isIntegral_of_isIntegral_adjoin_of_mul_eq_oneproof · cited by 1
- IsLocalization.scaleRoots_commonDenom_mem_liftsproof · cited by 1
- IsIntegral.of_aeval_monic_of_isIntegral_coeffproof · cited by 1
- Polynomial.lifts_iff_coeffs_subset_rangeproof · cited by 1
- integralClosure.mem_lifts_of_monic_of_dvd_mapproof · cited by 1
- Polynomial.monomial_mem_lifts_and_degree_eqproof · cited by 0
- Polynomial.mem_lifts_of_surjectiveproof · cited by 0