File: Mathlib/Deprecated/Subfield.lean

full_name: IsSubfield.inv_mem
code: /-- `IsSubfield (S : Set F)` is the predicate saying that a given subset of a field is
the set underlying a subfield. This structure is deprecated; use the bundled variant
`Subfield F` to model subfields of a field. -/
structure IsSubfield extends IsSubring S : Prop where
  inv_mem : ∀ {x : F}, x ∈ S → x⁻¹ ∈ S
goal: x ∈ S → x⁻¹ ∈ S

full_name: IsSubfield
code: /-- `IsSubfield (S : Set F)` is the predicate saying that a given subset of a field is
the set underlying a subfield. This structure is deprecated; use the bundled variant
`Subfield F` to model subfields of a field. -/
structure IsSubfield extends IsSubring S : Prop where
  inv_mem : ∀ {x : F}, x ∈ S → x⁻¹ ∈ S
goal: Prop

full_name: IsSubfield.div_mem
code: theorem IsSubfield.div_mem {S : Set F} (hS : IsSubfield S) {x y : F} (hx : x ∈ S) (hy : y ∈ S) :
    x / y ∈ S 
goal: x / y ∈ S

full_name: IsSubfield.pow_mem
code: theorem IsSubfield.pow_mem {a : F} {n : ℤ} {s : Set F} (hs : IsSubfield s) (h : a ∈ s) :
    a ^ n ∈ s 
goal: a ^ n ∈ s

full_name: Univ.isSubfield
code: theorem Univ.isSubfield : IsSubfield (@Set.univ F) 
goal: IsSubfield Set.univ

full_name: Preimage.isSubfield
code: theorem Preimage.isSubfield {K : Type*} [Field K] (f : F →+* K) {s : Set K} (hs : IsSubfield s) :
    IsSubfield (f ⁻¹' s) 
goal: IsSubfield (⇑f ⁻¹' s)

full_name: Image.isSubfield
code: theorem Image.isSubfield {K : Type*} [Field K] (f : F →+* K) {s : Set F} (hs : IsSubfield s) :
    IsSubfield (f '' s) 
goal: IsSubfield (⇑f '' s)

full_name: Range.isSubfield
code: theorem Range.isSubfield {K : Type*} [Field K] (f : F →+* K) : IsSubfield (Set.range f) 
goal: IsSubfield (Set.range ⇑f)

full_name: Field.closure
code: /-- `Field.closure s` is the minimal subfield that includes `s`. -/
def closure : Set F 
goal: Set F

full_name: Field.ring_closure_subset
code: theorem ring_closure_subset : Ring.closure S ⊆ closure S 
goal: Ring.closure S ⊆ Field.closure S

full_name: Field.closure.isSubmonoid
code: theorem closure.isSubmonoid : IsSubmonoid (closure S) 
goal: IsSubmonoid (Field.closure S)

full_name: Field.closure.isSubfield
code: theorem closure.isSubfield : IsSubfield (closure S) 
goal: IsSubfield (Field.closure S)

full_name: Field.mem_closure
code: theorem mem_closure {a : F} (ha : a ∈ S) : a ∈ closure S 
goal: a ∈ Field.closure S

full_name: Field.subset_closure
code: theorem subset_closure : S ⊆ closure S 
goal: S ⊆ Field.closure S

full_name: Field.closure_subset
code: theorem closure_subset {T : Set F} (hT : IsSubfield T) (H : S ⊆ T) : closure S ⊆ T 
goal: Field.closure S ⊆ T

full_name: Field.closure_subset_iff
code: theorem closure_subset_iff {s t : Set F} (ht : IsSubfield t) : closure s ⊆ t ↔ s ⊆ t 
goal: Field.closure s ⊆ t ↔ s ⊆ t

full_name: Field.closure_mono
code: theorem closure_mono {s t : Set F} (H : s ⊆ t) : closure s ⊆ closure t 
goal: Field.closure s ⊆ Field.closure t

full_name: isSubfield_iUnion_of_directed
code: theorem isSubfield_iUnion_of_directed {ι : Type*} [Nonempty ι] {s : ι → Set F}
    (hs : ∀ i, IsSubfield (s i)) (directed : ∀ i j, ∃ k, s i ⊆ s k ∧ s j ⊆ s k) :
    IsSubfield (⋃ i, s i) 
goal: IsSubfield (⋃ i, s i)

full_name: IsSubfield.inter
code: theorem IsSubfield.inter {S₁ S₂ : Set F} (hS₁ : IsSubfield S₁) (hS₂ : IsSubfield S₂) :
    IsSubfield (S₁ ∩ S₂) 
goal: IsSubfield (S₁ ∩ S₂)

full_name: IsSubfield.iInter
code: theorem IsSubfield.iInter {ι : Sort*} {S : ι → Set F} (h : ∀ y : ι, IsSubfield (S y)) :
    IsSubfield (Set.iInter S) 
goal: IsSubfield (Set.iInter S)