Theorems · Theorem · linear algebra
Submodule.span_eq_of_le
∀ {R : Type u_1} {M : Type u_4} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (p : Submodule R M)
{s : Set M}, s ⊆ ↑p → p ≤ Submodule.span R s → Submodule.span R s = p- Defined in
- Mathlib.LinearAlgebra.Span.Defs
- Cited by
- 11 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
- SetLike.coestatement and proof · cited by 8,199
- Submodulestatement and proof · cited by 7,192
- le_antisymmproof · cited by 2,068
- Submodule.spanstatement and proof · cited by 1,504
- Submodule.span_leproof · cited by 164
Cited by11
Results whose statement or proof uses this declaration.
- Submodule.map_spanproof · cited by 84
- Ideal.map_spanproof · cited by 43
- Submodule.span_eqproof · cited by 26
- Finsupp.span_image_eq_map_linearCombinationproof · cited by 7
- exists_linearIndependentproof · cited by 6
- Submodule.span_insert_eq_spanproof · cited by 5
- Polynomial.Sequence.span_degreeLTproof · cited by 3
- Submodule.fg_unitproof · cited by 3
- Finsupp.supported_eq_span_singleproof · cited by 3
- groupHomology.range_d₁₀_eq_coinvariantsKerproof · cited by 2
- TensorProduct.AlgebraTensorModule.ker_mapOfCompatibleSMulproof · cited by 0