Theorems · Theorem · order theory
Submodule.nonzero_mem_of_bot_lt
∀ {R : Type u_1} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
{p : Submodule R M}, ⊥ < p → ∃ a, a ≠ 0- Defined in
- Mathlib.Algebra.Module.Submodule.Lattice
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 27 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.
Cites7
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 and proof · cited by 7,192
- Bot.botstatement and proof · cited by 4,720
- LT.lt.ne'proof · cited by 1,417
- Submodule.ne_bot_iffproof · cited by 28
Cited by5
Results whose statement or proof uses this declaration.
- FractionalIdeal.not_inv_le_one_of_ne_botproof · cited by 2
- Ideal.rank_eqproof · cited by 1
- Polynomial.isMaximal_comap_C_of_isMaximalproof · cited by 0
- IsDiscreteValuationRing.RingEquivClass.isDiscreteValuationRingproof · cited by 0
- Ideal.exists_nonzero_mem_of_ne_botproof · cited by 0