Theorems · Theorem · field theory
Algebra.IsAlgebraic.cardinalMk_le_max
∀ (R : Type u) [inst : CommRing R] [IsDomain R] (L : Type u) [inst_2 : CommRing L] [IsDomain L] [inst_4 : Algebra R L] [Module.IsTorsionFree R L] [Algebra.IsAlgebraic R L], Cardinal.mk L ≤ max (Cardinal.mk R) Cardinal.aleph0
The cardinality of an algebraic extension is at most the maximum of the cardinality
of the base ring or ℵ₀.
- Defined in
- Mathlib.RingTheory.Algebraic.Cardinality
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 136 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.
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- Cardinalstatement · cited by 2,598
- IsDomainstatement and proof · cited by 2,196
- Cardinal.mkstatement and proof · cited by 942
- Module.IsTorsionFreestatement and proof · cited by 600
- Cardinal.aleph0statement and proof · cited by 521
- Algebra.IsAlgebraicstatement and proof · cited by 322
- Cardinal.lift_idproof · cited by 163
- Algebra.IsAlgebraic.lift_cardinalMk_le_maxproof · cited by 1
Cited by3
Results whose statement or proof uses this declaration.
- IsAlgClosed.cardinal_le_max_transcendence_basisproof · cited by 2
- IsTranscendenceBasis.lift_cardinalMk_eq_max_liftproof · cited by 1
- IsTranscendenceBasis.lift_rank_eq_max_liftproof · cited by 1