Theorems · Theorem · field theory
map_mem_perfectClosure_iff
∀ {F : Type u} {E : Type v} [inst : Field F] [inst_1 : Field E] [inst_2 : Algebra F E] {K : Type w} [inst_3 : Field K]
[inst_4 : Algebra F K] (i : E →ₐ[F] K) {x : E}, i x ∈ perfectClosure F K ↔ x ∈ perfectClosure F EIf i is an F-algebra homomorphism from E to K, then i x is contained in
perfectClosure F K if and only if x is contained in perfectClosure F E.
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 100 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Algebrastatement and proof · cited by 11,388
- Fieldstatement and proof · cited by 7,404
- Algebra.algebraMapproof · cited by 4,706
- AlgHomstatement and proof · cited by 3,236
- IntermediateFieldstatement · cited by 988
- map_powproof · cited by 503
- AlgHom.toRingHomproof · cited by 490
- RingHom.injectiveproof · cited by 187
- RingHom.rangeproof · cited by 138
- AlgHom.commutesproof · cited by 96
- ringExpCharproof · cited by 27
Cited by2
Results whose statement or proof uses this declaration.
- perfectClosure.comap_eq_of_algHomproof · cited by 1
- perfectClosure.map_eq_of_algEquivproof · cited by 0