Theorems · Theorem · commutative algebra
LinearMap.range_eq_top_of_surjective
∀ {R : Type u_1} {R₂ : Type u_2} {M : Type u_5} {M₂ : Type u_6} [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}
[inst_6 : RingHomSurjective τ₁₂] (f : M →ₛₗ[τ₁₂] M₂), Function.Surjective ⇑f → f.range = ⊤- Defined in
- Mathlib.Algebra.Module.Submodule.Range
- Cited by
- 12 results in Mathlib
- Foundations
- Depth 28 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
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
- 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
- Top.topstatement · cited by 9,680
- Submodulestatement · cited by 7,192
- LinearMap.rangestatement · cited by 893
- RingHomSurjectivestatement and proof · cited by 220
- LinearMap.range_eq_topproof · cited by 107
Cited by12
Results whose statement or proof uses this declaration.
- AdicCompletion.pow_smul_top_eq_ker_evalproof · cited by 4
- ContinuousLinearMap.isTopCompl_range_ker_of_leftInverseproof · cited by 3
- SymmetricPower.range_mkproof · cited by 1
- LinearMap.surjective_of_surjective_comp_mkQproof · cited by 1
- Submodule.codisjoint_mapproof · cited by 1
- Module.finite_of_surjective_of_ker_le_nilradicalproof · cited by 0
- Function.Surjective.surjective_linearMapComp_leftproof · cited by 0
- ContinuousLinearMap.HasRightInverse.of_surjective_of_finiteDimensionalproof · cited by 0
- MonoidAlgebra.span_isGroupLikeElemproof · cited by 0
- Algebra.Extension.Cotangent.span_eq_top_of_span_eq_kerproof · cited by 0
- AddMonoidAlgebra.span_isGroupLikeElemproof · cited by 0