Theorems · Theorem · commutative algebra
Submodule.FG.exists_span_set_encard_eq_spanFinrank
∀ {R : Type u_1} {M : Type u} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {p : Submodule R M},
p.FG → ∃ s, s.encard = ↑p.spanFinrank ∧ Submodule.span R s = pConstructs a generating set with cardinality equal to the spanFinrank of the submodule when
the submodule is finitely generated.
- Defined in
- Mathlib.Algebra.Module.SpanRank
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 94 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.
Cites21
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- 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 and proof · cited by 7,192
- Set.Elemproof · cited by 7,166
- ENatstatement and proof · cited by 4,985
- Cardinalproof · cited by 2,598
- Submodule.spanstatement and proof · cited by 1,504
- Cardinal.mkproof · cited by 942
- Set.encardstatement · cited by 327
Cited by1
Results whose statement or proof uses this declaration.
- Submodule.FG.exists_span_finset_card_eq_spanFinrankproof · cited by 1