Theorems · Theorem · field theory
AlgebraicIndependent.injective
∀ {ι : Type u} {R : Type u_2} {A : Type v} {x : ι → A} [inst : CommRing R] [inst_1 : CommRing A] [inst_2 : Algebra R A],
AlgebraicIndependent R x → ∀ [Nontrivial R], Function.Injective x- Cited by
- 11 results in Mathlib
- Foundations
- Depth 98 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- Nontrivialstatement and proof · cited by 2,416
- AlgebraicIndependentstatement and proof · cited by 120
- LinearIndependent.injectiveproof · cited by 21
- AlgebraicIndependent.linearIndependentproof · cited by 2
Cited by11
Results whose statement or proof uses this declaration.
- IsTranscendenceBasis.isAlgebraicproof · cited by 8
- IsTranscendenceBasis.lift_cardinalMk_eq_trdegproof · cited by 7
- AlgebraicIndependent.to_subtype_rangeproof · cited by 4
- AlgebraicIndependent.lift_cardinalMk_le_trdegproof · cited by 4
- IsTranscendenceBasis.to_subtype_rangeproof · cited by 3
- IsAlgClosed.cardinal_eq_cardinal_transcendence_basis_of_aleph0_ltproof · cited by 2
- IsTranscendenceBasis.of_isAlgebraic_adjoin_insert_sdiffproof · cited by 2
- AlgebraicIndependent.isTranscendenceBasis_of_lift_trdeg_leproof · cited by 2
- AlgebraicIndependent.image_of_compproof · cited by 1
- lift_trdeg_add_leproof · cited by 1
- AlgebraicIndependent.isTranscendenceBasis_iffproof · cited by 0