Theorems · Theorem · commutative algebra
MvPolynomial.mem_range_map_iff_coeffs_subset
∀ {R : Type u} {S₁ : Type v} {σ : Type u_1} [inst : CommSemiring R] [inst_1 : CommSemiring S₁] {f : R →+* S₁}
{x : MvPolynomial σ S₁}, x ∈ Set.range ⇑(MvPolynomial.map f) ↔ ↑x.coeffs ⊆ Set.range ⇑f- Defined in
- Mathlib.Algebra.MvPolynomial.Eval
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 94 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommSemiringCommSemiring
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites32
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
- Finsetstatement and proof · cited by 13,712
- CommSemiringstatement and proof · cited by 10,911
- RingHomstatement and proof · cited by 10,189
- SetLike.coestatement and proof · cited by 8,199
- Finsuppstatement and proof · cited by 5,255
- Set.rangestatement and proof · cited by 4,705
- MvPolynomialstatement and proof · cited by 2,140
- map_zeroproof · cited by 1,614
- map_mulproof · cited by 1,137
- map_addproof · cited by 964
Cited by5
Results whose statement or proof uses this declaration.
- Algebra.SubmersivePresentation.map_invJacobianOfHasCoeffsproof · cited by 2
- TensorProduct.toIntegralClosure_mvPolynomial_bijectiveproof · cited by 1
- Algebra.SubmersivePresentation.map_jacobianRelationsOfHasCoeffsproof · cited by 1
- Algebra.Presentation.HasCoeffs.relation_mem_range_mapproof · cited by 1
- MvPolynomial.range_mapAlgHomproof · cited by 0