Theorems · Theorem · linear algebra
Module.Basis.toDual_apply
∀ {R : Type uR} {M : Type uM} {ι : Type uι} [inst : CommSemiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
[inst_3 : DecidableEq ι] (b : Module.Basis ι R M) (i j : ι), (b.toDual (b i)) (b j) = if i = j then 1 else 0- Defined in
- Mathlib.LinearAlgebra.Dual.Basis
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 89 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
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
- Modulestatement and proof · cited by 20,661
- RingHom.idstatement and proof · cited by 18,349
- AddCommMonoidstatement and proof · cited by 12,281
- CommSemiringstatement and proof · cited by 10,911
- LinearMapstatement and proof · cited by 10,215
- Module.Basisstatement and proof · cited by 1,477
- Module.Dualstatement and proof · cited by 583
- Module.Basis.constrproof · cited by 37
- Module.Basis.toDualstatement · cited by 18
- Module.Basis.constr_basisproof · cited by 17
Cited by3
Results whose statement or proof uses this declaration.
- Module.Basis.dualBasis_apply_selfproof · cited by 2
- Module.Basis.toDual_linearCombination_leftproof · cited by 2
- Module.Basis.toDual_linearCombination_rightproof · cited by 1