Theorems · Theorem · linear algebra
Module.Basis.span_eq
∀ {ι : Type u_1} {R : Type u_3} {M : Type u_5} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
(b : Module.Basis ι R M), Submodule.span R (Set.range ⇑b) = ⊤- Defined in
- Mathlib.LinearAlgebra.Basis.Basic
- Cited by
- 44 results in Mathlib
- Foundations
- Depth 86 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.
Cites11
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
- Top.topstatement and proof · cited by 9,680
- Submodulestatement · cited by 7,192
- Set.rangestatement · cited by 4,705
- Submodule.spanstatement · cited by 1,504
- Module.Basisstatement and proof · cited by 1,477
- eq_top_iffproof · cited by 236
- Module.Basis.mem_spanproof · cited by 3
Cited by44
Results whose statement or proof uses this declaration.
- Module.Finite.of_basisproof · cited by 22
- Submodule.eq_top_of_finrank_eqproof · cited by 15
- Bundle.Trivialization.isLocalFrameOn_localFrame_baseSetproof · cited by 7
- Module.Basis.ofZLatticeBasis_spanproof · cited by 6
- ZLattice.rankproof · cited by 5
- linearIndependent_le_basisproof · cited by 4
- PowerBasis.adjoin_gen_eq_topproof · cited by 4
- Module.Basis.isUnit_detproof · cited by 4
- Module.Basis.mem_submodule_iffproof · cited by 4
- Module.finitePresentation_of_free_of_surjectiveproof · cited by 3
- Subalgebra.adjoin_eq_span_basisproof · cited by 3
- finrank_eq_one_iff_of_nonzeroproof · cited by 3