Theorems · Theorem · order theory
Submodule.ne_bot_iff
∀ {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 ≠ ⊥ ↔ ∃ x ∈ p, x ≠ 0- Defined in
- Mathlib.Algebra.Module.Submodule.Lattice
- Cited by
- 28 results in Mathlib
- Foundations
- Depth 26 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.
Cites6
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 · cited by 4,720
- Submodule.eq_bot_iffproof · cited by 31
Cited by28
Results whose statement or proof uses this declaration.
- Submodule.exists_mem_ne_zero_of_ne_botproof · cited by 9
- LieAlgebra.IsKilling.finrank_rootSpace_eq_oneproof · cited by 6
- Submodule.nonzero_mem_of_bot_ltproof · cited by 5
- Ideal.IsIntegral.comap_ne_botproof · cited by 4
- LinearMap.det_eq_zero_iff_ker_ne_botproof · cited by 3
- Submodule.isInternal_prime_power_torsionproof · cited by 2
- IsLocalization.AtPrime.not_isFieldproof · cited by 2
- isSimpleModule_iff_toSpanSingleton_surjectiveproof · cited by 1
- IsCyclotomicExtension.Rat.span_zeta_sub_one_ne_botproof · cited by 1
- UniqueFactorizationMonoid.of_forall_isPrincipal_of_height_eq_oneproof · cited by 1
- maximalIdeal_isPrincipal_of_isDedekindDomainproof · cited by 1
- iSupIndep.subtype_ne_bot_le_rankproof · cited by 1