Theorems · Theorem · linear algebra
Submodule.span_nat_eq_addSubmonoidClosure
∀ {M : Type u_4} [inst : AddCommMonoid M] (s : Set M), (Submodule.span ℕ s).toAddSubmonoid = AddSubmonoid.closure s- Defined in
- Mathlib.LinearAlgebra.Span.Defs
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 65 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- AddCommMonoid
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
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
- AddCommMonoidstatement and proof · cited by 12,281
- Submodule.spanstatement · cited by 1,504
- AddSubmonoidstatement · cited by 1,178
- OrderIso.symmproof · cited by 475
- Submodule.subset_spanproof · cited by 234
- AddSubmonoid.closurestatement · cited by 224
- Submodule.span_leproof · cited by 164
- Submodule.toAddSubmonoidstatement · cited by 162
- AddSubmonoid.subset_closureproof · cited by 63
- OrderIso.to_galoisConnectionproof · cited by 32
- GaloisConnection.l_leproof · cited by 28
Cited by5
Results whose statement or proof uses this declaration.
- RootPairing.Base.pos_or_neg_of_sum_smul_root_memproof · cited by 2
- Nat.exists_mem_closure_of_geproof · cited by 2
- Nat.one_mem_closure_iffproof · cited by 1
- Nat.finite_setOfPred_setGcd_dvd_and_mem_spanproof · cited by 1
- Submodule.span_nat_eqproof · cited by 0