Theorems · Theorem · linear algebra
Module.Basis.eq_of_apply_eq
∀ {ι : Type u_10} {R : Type u_11} {M : Type u_12} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
{b₁ b₂ : Module.Basis ι R M}, (∀ (i : ι), b₁ i = b₂ i) → b₁ = b₂Two bases are equal if their basis vectors are the same.
- Defined in
- Mathlib.LinearAlgebra.Basis.Defs
- Cited by
- 15 results in Mathlib
- Foundations
- Depth 79 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.
Cites6
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- Modulestatement and proof · cited by 20,661
- Semiringstatement and proof · cited by 13,802
- AddCommMonoidstatement and proof · cited by 12,281
- Module.Basisstatement and proof · cited by 1,477
- DFunLike.extproof · cited by 240
Cited by15
Results whose statement or proof uses this declaration.
- ZLattice.covolume_comapproof · cited by 3
- LinearMap.BilinForm.dualBasis_dualBasis_flipproof · cited by 3
- Module.Basis.map_equivproof · cited by 2
- Bundle.Trivialization.localFrameCoeff_apply_of_mem_baseSetproof · cited by 2
- Module.Basis.ofZLatticeBasis_comapproof · cited by 1
- AffineBasis.basisOf_reindexproof · cited by 1
- AffineBasis.basisOf_smulproof · cited by 1
- AffineBasis.basisOf_vaddproof · cited by 1
- Module.Basis.span_negproof · cited by 1
- NumberField.mixedEmbedding.euclidean.stdOrthonormalBasis_map_eqproof · cited by 1
- NumberField.Units.basisOfIsMaxRank_fundSystemproof · cited by 1
- Module.Basis.parallelepiped_eq_mapproof · cited by 0