Theorems · Theorem · linear algebra
Module.Projective.exists_dual_ne_zero
∀ {V : Type uV} [inst : AddCommMonoid V] (R : Type u_1) [inst_1 : Semiring R] [inst_2 : Module R V]
[Module.Projective R V] {x : V}, x ≠ 0 → ∃ f, f x ≠ 0This is a linear map version of SeparatingDual.exists_ne_zero in a projective module.
- Defined in
- Mathlib.LinearAlgebra.Dual.Lemmas
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 92 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites22
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Modulestatement and proof · cited by 20,661
- RingHom.idstatement and proof · cited by 18,349
- Semiringstatement and proof · cited by 13,802
- AddCommMonoidstatement and proof · cited by 12,281
- LinearMapproof · cited by 10,215
- LinearMap.compproof · cited by 1,642
- Module.Basisproof · cited by 1,477
- LinearMap.idproof · cited by 625
- Module.Freeproof · cited by 597
- Module.Dualstatement and proof · cited by 583
- Module.Basis.reprproof · cited by 498
Cited by4
Results whose statement or proof uses this declaration.
- Submodule.exists_dual_map_eq_bot_of_notMemproof · cited by 4
- AlgEquiv.eq_linearEquivConjAlgEquivproof · cited by 4
- Module.forall_dual_apply_eq_zero_iffproof · cited by 4
- Module.Projective.exists_dual_eq_oneproof · cited by 1