Theorems · Theorem · group theory
DirectSum.of_eq_of_ne
∀ {ι : Type v} {β : ι → Type w} [inst : (i : ι) → AddCommMonoid (β i)] [inst_1 : DecidableEq ι] (i j : ι) (x : β i),
j ≠ i → ((DirectSum.of β i) x) j = 0- Defined in
- Mathlib.Algebra.DirectSum.Basic
- Cited by
- 9 results in Mathlib
- Foundations
- Depth 32 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- AddCommMonoidDecidableEq
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- AddCommMonoidstatement and proof · cited by 12,281
- AddMonoidHomstatement · cited by 3,230
- DFinsuppstatement · cited by 694
- DirectSumstatement · cited by 446
- DirectSum.ofstatement · cited by 122
- DFinsupp.single_eq_of_neproof · cited by 12
Cited by9
Results whose statement or proof uses this declaration.
- DirectSum.decompose_of_mem_neproof · cited by 7
- DirectSum.coe_mul_apply_eq_dfinsuppSumproof · cited by 4
- HomogeneousIdeal.mem_irrelevant_of_memproof · cited by 2
- DirectSum.coe_of_applyproof · cited by 2
- IsHomogeneous.subsemiringClosureproof · cited by 1
- DirectSum.lie_of_of_neproof · cited by 1
- DirectSum.IsInternal.ofBijective_coeLinearMap_of_neproof · cited by 1
- DirectSum.completeOrthogonalIdempotents_idempotentproof · cited by 0
- IsHomogeneous.subsemiringClosure_of_isHomogeneousElemproof · cited by 0