Theorems · Theorem · linear algebra
LinearIndependent.injective
∀ {ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [inst : Semiring R] [inst_1 : AddCommMonoid M]
[inst_2 : Module R M] [Nontrivial R], LinearIndependent R v → Function.Injective v- Cited by
- 21 results in Mathlib
- Foundations
- Depth 79 from the axioms · uses propext, Classical.choice, Quot.sound
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
- Nontrivialstatement and proof · cited by 2,416
- one_smulproof · cited by 1,374
- one_ne_zeroproof · cited by 885
- LinearIndependentstatement and proof · cited by 560
- Finsupp.linearCombination_singleproof · cited by 90
- Finsupp.single_left_injectiveproof · cited by 5
Cited by21
Results whose statement or proof uses this declaration.
- LinearIndependent.cardinal_lift_le_rankproof · cited by 16
- AlgebraicIndependent.injectiveproof · cited by 11
- finrank_span_eq_cardproof · cited by 7
- rank_spanproof · cited by 6
- exists_linearIndependent_cons_of_lt_rankproof · cited by 2
- Subalgebra.LinearDisjoint.adjoin_rank_eq_rank_leftproof · cited by 2
- LinearIndepOn.image_of_compproof · cited by 2
- LinearIndepOn.injOnproof · cited by 2
- Matrix.cRank_diagonalproof · cited by 1
- Matrix.cRank_oneproof · cited by 1
- Orthonormal.exists_orthonormalBasis_extension_of_card_eqproof · cited by 1
- Module.le_rank_iffproof · cited by 1