Theorems · Theorem · commutative algebra
Algebra.IsIntegral.inv_mem
∀ {R : Type u_1} {S : Type u_2} [inst : Field R] [inst_1 : DivisionRing S] [inst_2 : Algebra R S] {x : S}
{A : Subalgebra R S} [Algebra.IsIntegral R ↥A], x ∈ A → x⁻¹ ∈ AAn integral subalgebra of a division ring over a field is closed under inverses.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 124 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.
- Algebrastatement and proof · cited by 11,388
- Fieldstatement and proof · cited by 7,404
- Subalgebrastatement and proof · cited by 1,353
- DivisionRingstatement and proof · cited by 1,062
- Subtype.val_injectiveproof · cited by 232
- Algebra.IsIntegralstatement and proof · cited by 224
- Subalgebra.valproof · cited by 104
- Algebra.IsIntegral.isIntegralproof · cited by 86
- isIntegral_algHom_iffproof · cited by 15
- IsIntegral.inv_memproof · cited by 2
Cited by1
Results whose statement or proof uses this declaration.
- IntermediateField.adjoin_toSubalgebra_of_isAlgebraicproof · cited by 10