Theorems · Theorem · group theory
DirectSum.decompose_of_mem_ne
∀ {ι : Type u_1} {M : Type u_3} {σ : Type u_4} [inst : DecidableEq ι] [inst_1 : AddCommMonoid M] [inst_2 : SetLike σ M]
[inst_3 : AddSubmonoidClass σ M] (ℳ : ι → σ) [inst_4 : DirectSum.Decomposition ℳ] {x : M} {i j : ι},
x ∈ ℳ i → i ≠ j → ↑(((DirectSum.decompose ℳ) x) j) = 0- Defined in
- Mathlib.Algebra.DirectSum.Decomposition
- Cited by
- 7 results in Mathlib
- Foundations
- Depth 91 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.coestatement and proof · cited by 62,936
- AddCommMonoidstatement and proof · cited by 12,281
- Equivstatement · cited by 8,337
- SetLikestatement and proof · cited by 1,084
- DFinsuppstatement · cited by 694
- DirectSumstatement and proof · cited by 446
- AddSubmonoidClassstatement and proof · cited by 346
- DirectSum.decomposestatement · cited by 93
- DirectSum.Decompositionstatement and proof · cited by 57
- ZeroMemClass.coe_zeroproof · cited by 17
- DirectSum.of_eq_of_neproof · cited by 9
- DirectSum.decompose_of_memproof · cited by 7
Cited by7
Results whose statement or proof uses this declaration.
- DirectSum.degree_eq_of_mem_memproof · cited by 5
- HomogeneousLocalization.Away.span_mk_prod_pow_eq_topproof · cited by 1
- AlgebraicGeometry.Proj.iSup_basicOpen_eq_top'proof · cited by 1
- AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.asIdeal.primeproof · cited by 0
- AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.relevantproof · cited by 0