Theorems · Theorem · linear algebra
Submodule.comap_bot
∀ {R : Type u_1} {R₂ : Type u_2} {M : Type u_5} {M₂ : Type u_7} [inst : Semiring R] [inst_1 : Semiring R₂]
[inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] [inst_4 : Module R M] [inst_5 : Module R₂ M₂] {τ₁₂ : R →+* R₂}
(f : M →ₛₗ[τ₁₂] M₂), Submodule.comap f ⊥ = f.ker- Defined in
- Mathlib.Algebra.Module.Submodule.Ker
- Cited by
- 13 results in Mathlib
- Foundations
- Depth 26 from the axioms · uses propext, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
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
- LinearMapstatement and proof · cited by 10,215
- RingHomstatement and proof · cited by 10,189
- Submodulestatement · cited by 7,192
- Bot.botstatement · cited by 4,720
- LinearMap.kerstatement · cited by 848
- Submodule.comapstatement · cited by 347
Cited by13
Results whose statement or proof uses this declaration.
- Submodule.mkQ_map_selfproof · cited by 8
- Module.length_eq_add_of_exactproof · cited by 6
- LinearMap.ker_comp_of_ker_eq_botproof · cited by 6
- IsIntegralClosure.isNoetherianproof · cited by 3
- Module.Relations.Solution.injective_fromQuotient_iff_ker_π_eq_spanproof · cited by 2
- ContinuousLinearMap.IsFredholm.of_isInvertible_restrictproof · cited by 1
- Submodule.disjoint_map_of_ker_le_rightproof · cited by 1
- KaehlerDifferential.ker_map_of_surjectiveproof · cited by 1
- Matrix.ker_diagonal_toLin'proof · cited by 0
- Algebra.Extension.H1Cotangent.map_toExtendScalars_injectiveproof · cited by 0
- Module.finitePresentation_of_kerproof · cited by 0