Theorems · Theorem · ring theory
Subalgebra.inclusion_injective
∀ {R : Type u} {A : Type v} [inst : CommSemiring R] [inst_1 : Semiring A] [inst_2 : Algebra R A] {S T : Subalgebra R A}
(h : S ≤ T), Function.Injective ⇑(Subalgebra.inclusion h)- Defined in
- Mathlib.Algebra.Algebra.Subalgebra.Basic
- Cited by
- 9 results in Mathlib
- Foundations
- Depth 32 from the axioms · uses propext, Quot.sound
- Assumes
- CommSemiringSemiringAlgebra
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- Semiringstatement and proof · cited by 13,802
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- AlgHomstatement · cited by 3,236
- Subalgebrastatement and proof · cited by 1,353
- Subalgebra.inclusionstatement · cited by 36
Cited by9
Results whose statement or proof uses this declaration.
- AlgebraicIndependent.extendScalarsproof · cited by 4
- IsTranscendenceBasis.isAlgebraic_fieldproof · cited by 3
- IsAlgebraic.tower_top_of_subalgebra_leproof · cited by 2
- IsAlgebraic.adjoin_of_forall_isAlgebraicproof · cited by 1
- AlgebraicIndependent.sumElim_of_towerproof · cited by 1
- IntermediateField.LinearDisjoint.rank_supproof · cited by 1
- Algebra.TensorProduct.not_isField_of_transcendentalproof · cited by 1
- IntermediateField.inclusion_injectiveproof · cited by 1
- IsTranscendenceBasis.isAlgebraic_iffproof · cited by 0