Theorems · Theorem · field theory
IsTranscendenceBasis.isAlgebraic
∀ {ι : Type u} {R : Type u_1} {A : Type w} {x : ι → A} [inst : CommRing R] [inst_1 : CommRing A] [inst_2 : Algebra R A]
[Nontrivial R], IsTranscendenceBasis R x → Algebra.IsAlgebraic (↥(Algebra.adjoin R (Set.range x))) A- Cited by
- 8 results in Mathlib
- Foundations
- Depth 116 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setproof · cited by 53,352
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- Set.rangestatement and proof · cited by 4,705
- Nontrivialstatement and proof · cited by 2,416
- Subalgebrastatement · cited by 1,353
- Algebra.adjoinstatement · cited by 535
- Algebra.IsAlgebraicstatement · cited by 322
- AlgebraicIndependentproof · cited by 120
- IsTranscendenceBasisstatement and proof · cited by 74
- not_iff_commproof · cited by 32
- AlgebraicIndependent.injectiveproof · cited by 11
Cited by8
Results whose statement or proof uses this declaration.
- IsTranscendenceBasis.isAlgebraic_fieldproof · cited by 3
- AlgebraicIndependent.isTranscendenceBasis_iff_isAlgebraicproof · cited by 3
- IsTranscendenceBasis.isEmpty_iff_isAlgebraicproof · cited by 1
- IsTranscendenceBasis.lift_cardinalMk_eq_max_liftproof · cited by 1
- IsTranscendenceBasis.sumElim_compproof · cited by 1
- IsAlgClosed.isAlgClosure_of_transcendence_basisproof · cited by 1
- IsTranscendenceBasis.isAlgebraic_iffproof · cited by 0
- IsTranscendenceBasis.algebraMap_compproof · cited by 0