Theorems · Theorem · commutative algebra
IsIntegral.inv
∀ {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 → IsIntegral R x⁻¹The inverse of an integral element in a division ring over a field is also integral.
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 132 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.
Cites8
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
- DivisionRingstatement and proof · cited by 1,062
- Algebra.adjoinproof · cited by 535
- IsIntegralstatement and proof · cited by 427
- IsIntegral.of_mem_of_fgproof · cited by 13
- IsIntegral.fg_adjoin_singletonproof · cited by 13
- IsIntegral.inv_mem_adjoinproof · cited by 2
Cited by2
Results whose statement or proof uses this declaration.
- IsFractionRing.isAlgebraic_iff'proof · cited by 0
- IsIntegral.mem_of_inv_memproof · cited by 0