Theorems · Theorem · commutative algebra
LinearMap.range.congr_simp
∀ {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 f_1 : M →ₛₗ[τ₁₂] M₂), f = f_1 → f.range = f_1.range- Defined in
- Mathlib.Algebra.Module.Submodule.Range
- Cited by
- 18 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.
Cites8
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
- LinearMap.rangestatement and proof · cited by 893
- RingHomSurjectivestatement and proof · cited by 220
Cited by18
Results whose statement or proof uses this declaration.
- groupHomology.map_chainsFunctor_shortExactproof · cited by 8
- IsArtinian.surjective_of_injective_endomorphismproof · cited by 4
- Module.FinitePresentation.exists_finproof · cited by 4
- ContinuousLinearMap.orthogonal_rangeproof · cited by 3
- ContinuousLinearMap.isStrictMap_isClosed_range_iff_restrictproof · cited by 2
- Submodule.map_le_isotypicComponentproof · cited by 2
- LinearMap.range_adjoint_comp_selfproof · cited by 2
- Rep.FiniteCyclicGroup.groupHomologyπEven_eq_zero_iffproof · cited by 1
- Ideal.exact_mulQuot_quotOfMulproof · cited by 1
- Submodule.isQuotientEquivQuotientPrime_iffproof · cited by 1
- range_mvfderiv_subtypeValproof · cited by 1