Theorems · Theorem · field theory
Subfield.closure_le
∀ {K : Type u} [inst : DivisionRing K] {s : Set K} {t : Subfield K}, Subfield.closure s ≤ t ↔ s ⊆ ↑tA subfield t includes closure s if and only if it includes s.
- Defined in
- Mathlib.Algebra.Field.Subfield.Basic
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 76 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.
Cites8
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 and proof · cited by 8,199
- DivisionRingstatement and proof · cited by 1,062
- Subfieldstatement and proof · cited by 303
- Set.Subset.transproof · cited by 218
- Subfield.closurestatement and proof · cited by 39
- Subfield.subset_closureproof · cited by 15
- Subfield.mem_closureproof · cited by 2
Cited by9
Results whose statement or proof uses this declaration.
- IntermediateField.adjoin_le_iffproof · cited by 28
- Subfield.giproof · cited by 6
- Subfield.closure_inductionproof · cited by 1
- IntermediateField.adjoin_subset_adjoin_iffproof · cited by 1
- RingHom.eqOn_field_closureproof · cited by 1
- Subfield.closure_monoproof · cited by 0
- Subfield.closure_preimage_leproof · cited by 0
- RingHom.field_closure_preimage_leproof · cited by 0
- Subfield.closure_eq_of_leproof · cited by 0