Theorems · Theorem · linear algebra
LinearMap.injOn_iff_surjOn
∀ {K : Type u} {V : Type v} [inst : DivisionRing K] [inst_1 : AddCommGroup V] [inst_2 : Module K V] {p : Submodule K V}
[FiniteDimensional K ↥p] {f : V →ₗ[K] V}, (∀ x ∈ p, f x ∈ p) → (Set.InjOn ⇑f ↑p ↔ Set.SurjOn ⇑f ↑p ↑p)- Cited by
- 1 results in Mathlib
- Foundations
- Depth 121 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites16
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
- AddCommGroupstatement and proof · cited by 12,871
- LinearMapstatement and proof · cited by 10,215
- SetLike.coestatement and proof · cited by 8,199
- Submodulestatement and proof · cited by 7,192
- Bot.botproof · cited by 4,720
- FiniteDimensionalstatement and proof · cited by 1,854
- DivisionRingstatement and proof · cited by 1,062
- LinearMap.kerproof · cited by 848
- Set.InjOnstatement · cited by 543
Cited by1
Results whose statement or proof uses this declaration.
- Submodule.inf_iSup_genEigenspaceproof · cited by 1