Theorems · Theorem · linear algebra
rank_span_le
∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] [StrongRankCondition R]
(s : Set M), Module.rank R ↥(Submodule.span R s) ≤ Cardinal.mk ↑s- Cited by
- 9 results in Mathlib
- Foundations
- Depth 115 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites20
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 and proof · cited by 7,192
- Set.Elemstatement and proof · cited by 7,166
- LE.le.transproof · cited by 3,151
- Cardinalstatement and proof · cited by 2,598
- Submodule.spanstatement · cited by 1,504
- Cardinal.mkstatement and proof · cited by 942
- Cardinal.liftproof · cited by 583
- Module.rankstatement and proof · cited by 496
Cited by9
Results whose statement or proof uses this declaration.
- finrank_span_le_cardproof · cited by 8
- rank_span_finset_leproof · cited by 2
- LinearEquiv.mem_dilatransvections_iff_rankproof · cited by 2
- rank_le_one_iffproof · cited by 2
- Subalgebra.rank_sup_le_of_freeproof · cited by 2
- Matrix.cRank_le_card_widthproof · cited by 1
- Subalgebra.adjoin_rank_leproof · cited by 1
- rank_le_card_isVisibleproof · cited by 0
- rank_submodule_le_one_iff'proof · cited by 0