Theorems · Theorem · linear algebra
Module.Basis.apply_eq_iff
∀ {ι : Type u_10} {R : Type u_11} {M : Type u_12} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
{b : Module.Basis ι R M} {x : M} {i : ι}, b i = x ↔ b.repr x = fun₀ | i => 1- Defined in
- Mathlib.LinearAlgebra.Basis.Defs
- Cited by
- 12 results in Mathlib
- Foundations
- Depth 80 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- SemiringAddCommMonoidModule
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
- Modulestatement and proof · cited by 20,661
- RingHom.idstatement · cited by 18,349
- Semiringstatement and proof · cited by 13,802
- AddCommMonoidstatement and proof · cited by 12,281
- Finsuppstatement · cited by 5,255
- LinearEquivstatement · cited by 3,317
- Module.Basisstatement and proof · cited by 1,477
- Finsupp.singlestatement and proof · cited by 943
- Module.Basis.reprstatement and proof · cited by 498
- LinearEquiv.injectiveproof · cited by 162
- Module.Basis.repr_selfproof · cited by 82
Cited by12
Results whose statement or proof uses this declaration.
- Finsupp.coe_basisSingleOneproof · cited by 7
- Complex.coe_basisOneIproof · cited by 5
- Module.Basis.singleton_applyproof · cited by 4
- Polynomial.degreeLT.basis_valproof · cited by 4
- Pi.basis_applyproof · cited by 3
- Module.Basis.mapCoeffs_applyproof · cited by 2
- Module.Basis.sumQuot_repr_leftproof · cited by 2
- Module.Basis.smulTower_applyproof · cited by 2
- IsAdjoinRootMonic.basis_applyproof · cited by 2
- Module.DualBases.coe_basisproof · cited by 1
- ModN.basis_apply_eq_mkQproof · cited by 0
- Finsupp.coe_basisproof · cited by 0