Theorems · Theorem · order theory
Submodule.nontrivial_iff
∀ (R : Type u_1) {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M],
Nontrivial (Submodule R M) ↔ Nontrivial M- Defined in
- Mathlib.Algebra.Module.Submodule.Lattice
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 65 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- SemiringAddCommMonoidModule
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Modulestatement and proof · cited by 20,661
- Semiringstatement and proof · cited by 13,802
- AddCommMonoidstatement and proof · cited by 12,281
- Submodulestatement · cited by 7,192
- Nontrivialstatement · cited by 2,416
- not_iff_notproof · cited by 159
- not_nontrivial_iff_subsingletonproof · cited by 23
- Submodule.subsingleton_iffproof · cited by 7
Cited by2
Results whose statement or proof uses this declaration.
- ModuleCat.exists_isRegular_of_exists_subsingleton_extproof · cited by 1
- isSimpleModule_iff_eq_zero_or_injectiveproof · cited by 0