Theorems · Theorem · linear algebra
Submodule.span_mono
∀ {R : Type u_1} {M : Type u_4} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {s t : Set M},
s ⊆ t → Submodule.span R s ≤ Submodule.span R t- Defined in
- Mathlib.LinearAlgebra.Span.Defs
- Cited by
- 85 results in Mathlib
- Foundations
- Depth 25 from the axioms · uses propext, 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
- Submodule.subset_spanproof · cited by 234
- Set.Subset.transproof · cited by 218
- Submodule.span_leproof · cited by 164
Cited by85
Results whose statement or proof uses this declaration.
- Submodule.span_span_of_towerproof · cited by 18
- Ideal.span_monoproof · cited by 16
- Submodule.span_insert_zeroproof · cited by 11
- UniqueDiffWithinAt.monoproof · cited by 8
- vectorSpan_eq_span_vsub_set_rightproof · cited by 7
- vectorSpan_monoproof · cited by 7
- exists_linearIndependentproof · cited by 6
- ZLattice.rankproof · cited by 5
- Submodule.span_insert_eq_spanproof · cited by 5
- UniqueDiffWithinAt.prodproof · cited by 4
- vectorSpan_eq_span_vsub_set_leftproof · cited by 4
- RootPairing.Base.span_int_root_supportproof · cited by 4