Theorems · Theorem · commutative algebra
Submodule.IsPrincipal.dvd_generator_span_iff
∀ {R : Type u} [inst : CommSemiring R] {r : R} {s : Set R} [inst_1 : Submodule.IsPrincipal (Ideal.span s)],
r ∣ Submodule.IsPrincipal.generator (Ideal.span s) ↔ ∀ x ∈ s, r ∣ 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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Cites14
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- CommSemiringstatement and proof · cited by 10,911
- Submodule.spanproof · cited by 1,504
- Ideal.spanstatement and proof · cited by 948
- Dvd.dvd.transproof · cited by 148
- Submodule.IsPrincipalstatement and proof · cited by 129
- Ideal.subset_spanproof · cited by 86
- Submodule.span_inductionproof · cited by 77
- dvd_zeroproof · cited by 63
- Submodule.IsPrincipal.generatorstatement and proof · cited by 56
- dvd_addproof · cited by 25
- Dvd.dvd.mul_leftproof · cited by 14
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