Theorems · Theorem · commutative algebra
Submodule.IsPrincipal.generator_submoduleImage_dvd_of_mem
∀ {R : Type u} {M : Type v} [inst : AddCommMonoid M] [inst_1 : CommSemiring R] [inst_2 : Module R M]
{N O : Submodule R M} (hNO : N ≤ O) (ϕ : ↥O →ₗ[R] R) [inst_3 : (ϕ.submoduleImage N).IsPrincipal] {x : M} (hx : x ∈ N),
Submodule.IsPrincipal.generator (ϕ.submoduleImage N) ∣ ϕ ⟨x, ⋯⟩- Defined in
- Mathlib.RingTheory.PrincipalIdealDomain
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 30 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- Modulestatement and proof · cited by 20,661
- RingHom.idstatement and proof · cited by 18,349
- AddCommMonoidstatement and proof · cited by 12,281
- CommSemiringstatement and proof · cited by 10,911
- LinearMapstatement and proof · cited by 10,215
- Submodulestatement and proof · cited by 7,192
- Submodule.IsPrincipalstatement and proof · cited by 129
- Submodule.IsPrincipal.generatorstatement · cited by 56
- Submodule.IsPrincipal.mem_iff_generator_dvdproof · cited by 11
- LinearMap.submoduleImagestatement and proof · cited by 7
- LinearMap.mem_submoduleImage_of_leproof · cited by 4
Cited by1
Results whose statement or proof uses this declaration.
- Submodule.basis_of_pid_auxproof · cited by 2