Theorems · Theorem · linear algebra
Submodule.span_union
∀ {R : Type u_1} {M : Type u_4} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (s t : Set M),
Submodule.span R (s ∪ t) = Submodule.span R s ⊔ Submodule.span R t- Defined in
- Mathlib.LinearAlgebra.Span.Defs
- Cited by
- 23 results in Mathlib
- Foundations
- Depth 64 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.
Cites9
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- Modulestatement and proof · cited by 20,661
- Semiringstatement and proof · cited by 13,802
- AddCommMonoidstatement and proof · cited by 12,281
- Submodulestatement · cited by 7,192
- Submodule.spanstatement · cited by 1,504
- GaloisInsertion.gcproof · cited by 137
- GaloisConnection.l_supproof · cited by 81
- Submodule.giproof · cited by 12
Cited by23
Results whose statement or proof uses this declaration.
- Submodule.mem_supproof · cited by 73
- Ideal.span_unionproof · cited by 11
- Submodule.FG.supproof · cited by 9
- Submodule.fg_of_fg_map_of_fg_inf_kerproof · cited by 7
- Submodule.span_insertproof · cited by 6
- Submodule.mem_span_finite_of_mem_spanproof · cited by 6
- Submodule.exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smulproof · cited by 3
- StarAlgebra.adjoin_nonUnitalStarSubalgebra_eq_spanproof · cited by 2
- LinearMap.span_inl_union_inrproof · cited by 2
- Submodule.sup_spanproof · cited by 1
- Module.exists_isPrincipal_quotient_of_finiteproof · cited by 1
- RootPairing.invtRootSubmodule.eq_span_rootproof · cited by 1