Theorems · Theorem · commutative algebra
LinearMap.mem_range
∀ {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} {x : M₂}, x ∈ f.range ↔ ∃ y, f y = x- Defined in
- Mathlib.Algebra.Module.Submodule.Range
- Cited by
- 16 results in Mathlib
- Foundations
- Depth 26 from the axioms · uses propext, Classical.choice, 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.
- DFunLike.coestatement · 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
- Submodulestatement · cited by 7,192
- LinearMap.rangestatement · cited by 893
- RingHomSurjectivestatement and proof · cited by 220
Cited by16
Results whose statement or proof uses this declaration.
- Finsupp.range_linearCombinationproof · cited by 24
- LinearMap.exists_range_eq_graphproof · cited by 2
- LinearMap.eventually_codisjoint_ker_pow_range_powproof · cited by 2
- Subspace.dualAnnihilator_inf_eqproof · cited by 2
- RootPairing.prod_rootFormIn_smul_coroot_mem_range_PolarizationInproof · cited by 1
- LieModule.exists_nontrivial_weightSpace_of_isSolvableproof · cited by 1
- Submodule.index_smul_leproof · cited by 1
- Ideal.homogeneous_spanproof · cited by 1
- HahnEmbedding.Partial.truncLT_eval_mem_range_extendFunproof · cited by 1
- HahnEmbedding.Partial.truncLT_mem_range_extendFunproof · cited by 1
- Submodule.range_dualMap_mkQ_eqproof · cited by 1
- LieHom.mem_rangeproof · cited by 1