Theorems · Theorem · linear algebra
LinearMap.rank_le_of_injective
∀ {R : Type u} {M M₁ : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
[inst_3 : AddCommMonoid M₁] [inst_4 : Module R M₁] (f : M →ₗ[R] M₁),
Function.Injective ⇑f → Module.rank R M ≤ Module.rank R M₁- Defined in
- Mathlib.LinearAlgebra.Dimension.Basic
- Cited by
- 9 results in Mathlib
- Foundations
- Depth 88 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Modulestatement and proof · cited by 20,661
- RingHom.idstatement and proof · cited by 18,349
- Semiringstatement and proof · cited by 13,802
- AddCommMonoidstatement and proof · cited by 12,281
- LinearMapstatement and proof · cited by 10,215
- Cardinalstatement · cited by 2,598
- Module.rankstatement · cited by 496
- Cardinal.lift_leproof · cited by 78
- LinearMap.lift_rank_le_of_injectiveproof · cited by 7
Cited by9
Results whose statement or proof uses this declaration.
- Submodule.rank_monoproof · cited by 11
- Subalgebra.finrank_sup_le_of_freeproof · cited by 2
- IntermediateField.LinearDisjoint.rank_supproof · cited by 1
- MvRatFunc.rank_eq_max_liftproof · cited by 1
- rank_le_of_isSMulRegularproof · cited by 1
- max_aleph0_card_le_rank_fun_natproof · cited by 1
- rank_map_eqproof · cited by 1
- strongRankCondition_iff_forall_zero_lt_finrankproof · cited by 0
- IntermediateField.finrank_sup_leproof · cited by 0