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Theorems · Theorem · general algebraic systems

Pi.single_apply

∀ {ι : Type u_1} [inst : DecidableEq ι] {M : Type u_9} [inst_1 : Zero M] (i : ι) (x : M) (i' : ι),
  Pi.single i x i' = if i' = i then x else 0

On non-dependent functions, Pi.single can be expressed as an ite

Defined in
Mathlib.Algebra.Notation.Pi.Basic
Cited by
78 results in Mathlib
Foundations
Depth 11 from the axioms · uses propext
Assumes
DecidableEqZero
Mathlib docsSourceLoogleLeanSearch

Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

Finsupp.single_apply · cited by 99Finsupp.single_applyMvPowerSeries.coeff_monomial · cited by 17MvPowerSeries.coeff_monom…Finsupp.single_eq_pi_single · cited by 15Finsupp.single_eq_pi_sing…PiLp.single_apply · cited by 11PiLp.single_applyMatrix.mul_adjugate · cited by 10Matrix.mul_adjugateFunction.locallyFinsuppWithin.single_apply · cited by 8locallyFinsuppWithin.sing…LinearMap.toMatrix₂_apply · cited by 8LinearMap.toMatrix₂_applySet.indicator_singleton · cited by 6Set.indicator_singletonLinearMap.toMatrix'_id · cited by 5LinearMap.toMatrix'_idMatrix.single_mulVec_eq · cited by 5Matrix.single_mulVec_eqAlgebra.Generators.CotangentSpace.compEquiv_symm_inr · cited by 4CotangentSpace.compEquiv_…Algebra.Generators.CotangentSpace.fst_compEquiv · cited by 4CotangentSpace.fst_compEq…Matrix.one_eq_pi_single · cited by 4Matrix.one_eq_pi_singleMatrix.col_diagonal · cited by 3Matrix.col_diagonalPolynomial.fourierCoeff_toAddCircle · cited by 3Polynomial.fourierCoeff_t…Pi.single · cited by 518Pi.singleFunction.update_apply · cited by 23Function.update_applyPi.single_applyCITED BYCITES

Cites2

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Cited by78

Results whose statement or proof uses this declaration.