Theorems · Theorem · commutative algebra
Submodule.IsPrincipal.generator_map_dvd_of_mem
∀ {R : Type u} {M : Type v} [inst : AddCommMonoid M] [inst_1 : CommSemiring R] [inst_2 : Module R M] {N : Submodule R M}
(ϕ : M →ₗ[R] R) [inst_3 : (Submodule.map ϕ N).IsPrincipal] {x : M},
x ∈ N → Submodule.IsPrincipal.generator (Submodule.map ϕ N) ∣ ϕ x- Defined in
- Mathlib.RingTheory.PrincipalIdealDomain
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 30 from the axioms · uses propext, Classical.choice, Quot.sound
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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.mapstatement and proof · cited by 614
- Submodule.IsPrincipalstatement and proof · cited by 129
- Submodule.IsPrincipal.generatorstatement · cited by 56
- Submodule.mem_mapproof · cited by 24
- Submodule.IsPrincipal.mem_iff_generator_dvdproof · cited by 11
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