Theorems · Theorem · field theory
Subfield.subset_closure
∀ {K : Type u} [inst : DivisionRing K] {s : Set K}, s ⊆ ↑(Subfield.closure s)The subfield generated by a set includes the set.
- Defined in
- Mathlib.Algebra.Field.Subfield.Basic
- Cited by
- 15 results in Mathlib
- Foundations
- Depth 75 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 and proof · cited by 8,199
- DivisionRingstatement and proof · cited by 1,062
- Subfieldstatement and proof · cited by 303
- Subfield.closurestatement · cited by 39
- Subfield.mem_closureproof · cited by 2
Cited by15
Results whose statement or proof uses this declaration.
- IntermediateField.subset_adjoinproof · cited by 59
- IntermediateField.adjoin_le_iffproof · cited by 28
- Subfield.closure_leproof · cited by 8
- IntermediateField.adjoin_adjoin_leftproof · cited by 6
- IntermediateField.adjoin_inductionproof · cited by 2
- Subfield.closure_inductionstatement and proof · cited by 1
- IsFractionRing.closure_range_algebraMapproof · cited by 1
- max_aleph0_card_le_rank_fun_natproof · cited by 1
- Subfield.closure_monoproof · cited by 0
- Subfield.closure_preimage_leproof · cited by 0
- Subfield.subring_closure_leproof · cited by 0
- RingHom.field_closure_preimage_leproof · cited by 0