Theorems · Theorem · linear algebra
Submodule.subset_span_trans
∀ {R : Type u_1} {M : Type u_4} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {U V W : Set M},
U ⊆ ↑(Submodule.span R V) → V ⊆ ↑(Submodule.span R W) → U ⊆ ↑(Submodule.span R W)- Defined in
- Mathlib.LinearAlgebra.Span.Defs
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 62 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.
Cites10
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
- SetLike.coestatement and proof · cited by 8,199
- Submodulestatement · cited by 7,192
- Submodule.spanstatement and proof · cited by 1,504
- GaloisInsertion.gcproof · cited by 137
- Submodule.giproof · cited by 12
- GaloisConnection.le_u_l_transproof · cited by 2
Cited by1
Results whose statement or proof uses this declaration.
- Submodule.mem_span_singleton_transproof · cited by 0