Theorems · Theorem · commutative algebra
IsIntegral.inv_mem_adjoin
∀ {R : Type u_1} {S : Type u_2} [inst : Field R] [inst_1 : DivisionRing S] [inst_2 : Algebra R S] {x : S},
IsIntegral R x → x⁻¹ ∈ R[x]- Cited by
- 2 results in Mathlib
- Foundations
- Depth 122 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- FieldDivisionRingAlgebra
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites17
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- Algebrastatement and proof · cited by 11,388
- Fieldstatement and proof · cited by 7,404
- FiniteDimensionalproof · cited by 1,854
- Subalgebrastatement · cited by 1,353
- eq_or_neproof · cited by 1,117
- DivisionRingstatement and proof · cited by 1,062
- Algebra.adjoinstatement and proof · cited by 535
- IsIntegralstatement and proof · cited by 427
- mul_inv_cancel₀proof · cited by 210
- inv_zeroproof · cited by 184
- Algebra.subset_adjoinproof · cited by 109
Cited by2
Results whose statement or proof uses this declaration.
- IsIntegral.invproof · cited by 2
- IsIntegral.inv_memproof · cited by 2