Theorems · Theorem · commutative algebra
LinearMap.range_eq_map
∀ {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), f.range = Submodule.map f ⊤- Defined in
- Mathlib.Algebra.Module.Submodule.Range
- Cited by
- 27 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.
Cites12
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · 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
- Submodule.mapstatement · cited by 614
- RingHomSurjectivestatement and proof · cited by 220
- Submodule.extproof · cited by 204
Cited by27
Results whose statement or proof uses this declaration.
- Submodule.map_topproof · cited by 105
- LinearMap.range_codRestrictproof · cited by 11
- Submodule.range_liftQproof · cited by 11
- LinearMap.range_zeroproof · cited by 11
- LinearMap.range_le_iff_comapproof · cited by 7
- exteriorPower.ιMulti_span_fixedDegree_of_span_eq_topproof · cited by 2
- LinearMap.range_smulproof · cited by 2
- Submodule.mem_of_span_top_of_smul_memproof · cited by 2
- range_vecMulLinearproof · cited by 2
- Algebra.Presentation.differentials.surjective_hom₁proof · cited by 1
- Submodule.fg_rangeproof · cited by 1
- Module.free_of_lTensor_residueField_injectiveproof · cited by 1