Theorems · Theorem · linear algebra
Submodule.rank_le
∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (s : Submodule R M),
Module.rank R ↥s ≤ Module.rank R M- Defined in
- Mathlib.LinearAlgebra.Dimension.Basic
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 91 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.
Cites9
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- 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
- Cardinalstatement and proof · cited by 2,598
- Module.rankstatement and proof · cited by 496
- le_topproof · cited by 411
- Submodule.rank_monoproof · cited by 11
- rank_topproof · cited by 9
Cited by5
Results whose statement or proof uses this declaration.
- Submodule.finrank_leproof · cited by 20
- Submodule.finrank_eq_rankproof · cited by 3
- Matrix.cRank_le_card_heightproof · cited by 1
- LinearMap.lift_rank_comap_leproof · cited by 0
- LinearMap.rank_le_rangeproof · cited by 0