Theorems · Theorem · field theory
Subfield.mem_closure
∀ {K : Type u} [inst : DivisionRing K] {x : K} {s : Set K}, x ∈ Subfield.closure s ↔ ∀ (S : Subfield K), s ⊆ ↑S → x ∈ S- Defined in
- Mathlib.Algebra.Field.Subfield.Basic
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 74 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- DivisionRing
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- SetLike.coestatement · cited by 8,199
- DivisionRingstatement and proof · cited by 1,062
- Subfieldstatement · cited by 303
- Subfield.closurestatement · cited by 39
- Subfield.mem_sInfproof · cited by 1
Cited by2
Results whose statement or proof uses this declaration.
- Subfield.subset_closureproof · cited by 15
- Subfield.closure_leproof · cited by 8