Theorems · Theorem · commutative algebra
LinearMap.range_zero
∀ {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], LinearMap.range 0 = ⊥- Defined in
- Mathlib.Algebra.Module.Submodule.Range
- Cited by
- 11 results in Mathlib
- Foundations
- Depth 29 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.
- Modulestatement and proof · cited by 20,661
- Semiringstatement and proof · cited by 13,802
- AddCommMonoidstatement and proof · cited by 12,281
- LinearMapstatement · cited by 10,215
- RingHomstatement and proof · cited by 10,189
- Top.topproof · cited by 9,680
- Submodulestatement · cited by 7,192
- Bot.botstatement and proof · cited by 4,720
- LinearMap.rangestatement · cited by 893
- RingHomSurjectivestatement and proof · cited by 220
- LinearMap.range_eq_mapproof · cited by 27
- Submodule.map_zeroproof · cited by 3
Cited by11
Results whose statement or proof uses this declaration.
- Module.FaithfullyFlat.zero_iff_lTensor_zeroproof · cited by 5
- Submodule.map_le_isotypicComponentproof · cited by 2
- LinearMap.exact_zero_iff_injectiveproof · cited by 2
- LinearMap.ker_eq_bot_of_cancelproof · cited by 1
- LinearMap.HasNoetherianRange.zeroproof · cited by 1
- Submodule.isTopCompl_bot_topproof · cited by 1
- LinearMap.singularValues_zeroproof · cited by 1
- LinearMap.rank_zeroproof · cited by 1
- LinearMap.index_zeroproof · cited by 0
- LinearMap.IsProj.submodule_eq_bot_iffproof · cited by 0
- Matrix.rank_zeroproof · cited by 0