Theorems · Theorem · order theory
Submodule.eq_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
- 31 results in Mathlib
- Foundations
- Depth 25 from the axioms · uses propext, 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
- eq_bot_iffproof · cited by 159
- Submodule.mem_botproof · cited by 55
Cited by31
Results whose statement or proof uses this declaration.
- Submodule.ne_bot_iffproof · cited by 28
- Ideal.map_eq_bot_iff_of_injectiveproof · cited by 14
- Submodule.subsingleton_iff_eq_botproof · cited by 10
- Module.End.independent_genEigenspaceproof · cited by 4
- Algebra.Extension.subsingleton_h1Cotangentproof · cited by 3
- maximal_orthonormal_iff_orthogonalComplement_eq_botproof · cited by 3
- LieAlgebra.IsKilling.disjoint_ker_weight_corootSpaceproof · cited by 2
- Submodule.rank_eq_zeroproof · cited by 2
- FractionalIdeal.zero_of_num_eq_botproof · cited by 2
- Module.Dual.eq_of_ker_eq_of_apply_eqproof · cited by 1
- Submodule.sup_span_singleton_eq_top_iffproof · cited by 1
- QuadraticForm.sigPos_add_finrank_le_of_nonposproof · cited by 1