ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- IntermediateField.mem_mk
@[simp]
theorem mem_mk (s : Subsemiring L) (hK : ∀ x, algebraMap K L x ∈ s) (hi) (x : L) :
    x ∈ IntermediateField.mk (Subalgebra.mk s hK) hi ↔ x ∈ s 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (s : Subsemiring L) (hK : ∀ (x : K), (algebraMap K L) x ∈ s) (hi :
  ∀ x ∈ { toSubsemiring := s, algebraMap_mem' := hK }.carrier,
    x⁻¹ ∈ { toSubsemiring := s, algebraMap_mem' := hK }.carrier) (x : L)
Goal: x ∈ { toSubsemiring := s, algebraMap_mem' := hK, inv_mem' := hi } ↔ x ∈ s
-- IntermediateField.mem_toSubalgebra
@[simp]
theorem mem_toSubalgebra (s : IntermediateField K L) (x : L) : x ∈ s.toSubalgebra ↔ x ∈ s 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (s : IntermediateField K L) (x : L)
Goal: x ∈ s.toSubalgebra ↔ x ∈ s
-- IntermediateField.mem_toSubfield
@[simp]
theorem mem_toSubfield (s : IntermediateField K L) (x : L) : x ∈ s.toSubfield ↔ x ∈ s 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (s : IntermediateField K L) (x : L)
Goal: x ∈ s.toSubfield ↔ x ∈ s
-- IntermediateField.coe_copy
@[simp]
theorem coe_copy (S : IntermediateField K L) (s : Set L) (hs : s = ↑S) :
    (S.copy s hs : Set L) = s 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L) (s : Set L) (hs : s = ↑S)
Goal: ↑(S.copy s hs) = s
-- IntermediateField.copy_eq
theorem copy_eq (S : IntermediateField K L) (s : Set L) (hs : s = ↑S) : S.copy s hs = S 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L) (s : Set L) (hs : s = ↑S)
Goal: S.copy s hs = S
-- IntermediateField.algebraMap_mem
theorem algebraMap_mem (x : K) : algebraMap K L x ∈ S 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L) (x : K)
Goal: (algebraMap K L) x ∈ S
-- IntermediateField.smul_mem
theorem smul_mem {y : L} : y ∈ S → ∀ {x : K}, x • y ∈ S 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L) {y : L}
Goal: y ∈ S → ∀ {x : K}, x • y ∈ S
-- IntermediateField.one_mem
protected theorem one_mem : (1 : L) ∈ S 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L)
Goal: 1 ∈ S
-- IntermediateField.zero_mem
protected theorem zero_mem : (0 : L) ∈ S 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L)
Goal: 0 ∈ S
-- IntermediateField.mul_mem
protected theorem mul_mem {x y : L} : x ∈ S → y ∈ S → x * y ∈ S 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L) {x y : L}
Goal: x ∈ S → y ∈ S → x * y ∈ S
-- IntermediateField.add_mem
protected theorem add_mem {x y : L} : x ∈ S → y ∈ S → x + y ∈ S 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L) {x y : L}
Goal: x ∈ S → y ∈ S → x + y ∈ S
-- IntermediateField.sub_mem
protected theorem sub_mem {x y : L} : x ∈ S → y ∈ S → x - y ∈ S 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L) {x y : L}
Goal: x ∈ S → y ∈ S → x - y ∈ S
-- IntermediateField.inv_mem
protected theorem inv_mem {x : L} : x ∈ S → x⁻¹ ∈ S 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L) {x : L}
Goal: x ∈ S → x⁻¹ ∈ S
-- IntermediateField.div_mem
protected theorem div_mem {x y : L} : x ∈ S → y ∈ S → x / y ∈ S 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L) {x y : L}
Goal: x ∈ S → y ∈ S → x / y ∈ S
-- IntermediateField.list_prod_mem
protected theorem list_prod_mem {l : List L} : (∀ x ∈ l, x ∈ S) → l.prod ∈ S 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L) {l : List L}
Goal: (∀ x ∈ l, x ∈ S) → l.prod ∈ S
-- IntermediateField.list_sum_mem
protected theorem list_sum_mem {l : List L} : (∀ x ∈ l, x ∈ S) → l.sum ∈ S 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L) {l : List L}
Goal: (∀ x ∈ l, x ∈ S) → l.sum ∈ S
-- IntermediateField.multiset_prod_mem
protected theorem multiset_prod_mem (m : Multiset L) : (∀ a ∈ m, a ∈ S) → m.prod ∈ S 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L) (m : Multiset L)
Goal: (∀ a ∈ m, a ∈ S) → m.prod ∈ S
-- IntermediateField.multiset_sum_mem
protected theorem multiset_sum_mem (m : Multiset L) : (∀ a ∈ m, a ∈ S) → m.sum ∈ S 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L) (m : Multiset L)
Goal: (∀ a ∈ m, a ∈ S) → m.sum ∈ S
-- IntermediateField.prod_mem
protected theorem prod_mem {ι : Type*} {t : Finset ι} {f : ι → L} (h : ∀ c ∈ t, f c ∈ S) :
    (∏ i ∈ t, f i) ∈ S 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L) {ι : Type u_4} {t : Finset ι} {f : ι → L} (h : ∀ c ∈ t, f c ∈ S)
Goal: ∏ i ∈ t, f i ∈ S
-- IntermediateField.sum_mem
protected theorem sum_mem {ι : Type*} {t : Finset ι} {f : ι → L} (h : ∀ c ∈ t, f c ∈ S) :
    (∑ i ∈ t, f i) ∈ S 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L) {ι : Type u_4} {t : Finset ι} {f : ι → L} (h : ∀ c ∈ t, f c ∈ S)
Goal: ∑ i ∈ t, f i ∈ S
-- IntermediateField.pow_mem
protected theorem pow_mem {x : L} (hx : x ∈ S) (n : ℤ) : x ^ n ∈ S 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L) {x : L} (hx : x ∈ S) (n : ℤ)
Goal: x ^ n ∈ S
-- IntermediateField.zsmul_mem
protected theorem zsmul_mem {x : L} (hx : x ∈ S) (n : ℤ) : n • x ∈ S 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L) {x : L} (hx : x ∈ S) (n : ℤ)
Goal: n • x ∈ S
-- IntermediateField.intCast_mem
protected theorem intCast_mem (n : ℤ) : (n : L) ∈ S 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L) (n : ℤ)
Goal: ↑n ∈ S
-- IntermediateField.coe_add
protected theorem coe_add (x y : S) : (↑(x + y) : L) = ↑x + ↑y 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L) (x y : ↥S)
Goal: ↑(x + y) = ↑x + ↑y
-- IntermediateField.coe_neg
protected theorem coe_neg (x : S) : (↑(-x) : L) = -↑x 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L) (x : ↥S)
Goal: ↑(-x) = -↑x
-- IntermediateField.coe_mul
protected theorem coe_mul (x y : S) : (↑(x * y) : L) = ↑x * ↑y 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L) (x y : ↥S)
Goal: ↑(x * y) = ↑x * ↑y
-- IntermediateField.coe_inv
protected theorem coe_inv (x : S) : (↑x⁻¹ : L) = (↑x)⁻¹ 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L) (x : ↥S)
Goal: ↑x⁻¹ = (↑x)⁻¹
-- IntermediateField.coe_zero
protected theorem coe_zero : ((0 : S) : L) = 0 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L)
Goal: ↑0 = 0
-- IntermediateField.coe_one
protected theorem coe_one : ((1 : S) : L) = 1 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L)
Goal: ↑1 = 1
-- IntermediateField.coe_pow
protected theorem coe_pow (x : S) (n : ℕ) : (↑(x ^ n : S) : L) = (x : L) ^ n 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L) (x : ↥S) (n : ℕ)
Goal: ↑(x ^ n) = ↑x ^ n
-- IntermediateField.natCast_mem
theorem natCast_mem (n : ℕ) : (n : L) ∈ S 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L) (n : ℕ)
Goal: ↑n ∈ S
-- toSubalgebra_toIntermediateField
@[simp]
theorem toSubalgebra_toIntermediateField (S : Subalgebra K L) (inv_mem : ∀ x ∈ S, x⁻¹ ∈ S) :
    (S.toIntermediateField inv_mem).toSubalgebra = S 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : Subalgebra K L) (inv_mem : ∀ x ∈ S, x⁻¹ ∈ S)
Goal: (S.toIntermediateField inv_mem).toSubalgebra = S
-- toIntermediateField_toSubalgebra
@[simp]
theorem toIntermediateField_toSubalgebra (S : IntermediateField K L) :
    (S.toSubalgebra.toIntermediateField fun x => S.inv_mem) = S 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L)
Goal: S.toIntermediateField ⋯ = S
-- toSubalgebra_toIntermediateField'
@[simp]
theorem toSubalgebra_toIntermediateField' (S : Subalgebra K L) (hS : IsField S) :
    (S.toIntermediateField' hS).toSubalgebra = S 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : Subalgebra K L) (hS : IsField ↥S)
Goal: (S.toIntermediateField' hS).toSubalgebra = S
-- toIntermediateField'_toSubalgebra
@[simp]
theorem toIntermediateField'_toSubalgebra (S : IntermediateField K L) :
    S.toSubalgebra.toIntermediateField' (Field.toIsField S) = S 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L)
Goal: S.toIntermediateField' ⋯ = S
-- IntermediateField.coe_sum
@[simp, norm_cast]
theorem coe_sum {ι : Type*} [Fintype ι] (f : ι → S) : (↑(∑ i, f i) : L) = ∑ i, (f i : L) 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L) {ι : Type u_4} [Fintype ι] (f : ι → ↥S)
Goal: ↑(∑ i : ι, f i) = ∑ i : ι, ↑(f i)
-- IntermediateField.coe_prod
@[norm_cast] --Porting note (#10618): `simp` can prove it
theorem coe_prod {ι : Type*} [Fintype ι] (f : ι → S) : (↑(∏ i, f i) : L) = ∏ i, (f i : L) 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L) {ι : Type u_4} [Fintype ι] (f : ι → ↥S)
Goal: ↑(∏ i : ι, f i) = ∏ i : ι, ↑(f i)
-- IntermediateField.coe_smul
@[simp]
theorem coe_smul {R} [Semiring R] [SMul R K] [Module R L] [IsScalarTower R K L] (r : R) (x : S) :
    ↑(r • x : S) = (r • (x : L)) 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L) {R : Type u_4} [Semiring R] [SMul R K] [Module R L] [IsScalarTower R K L] (r : R) (x : ↥S)
Goal: ↑(r • x) = r • ↑x
-- IntermediateField.algebraMap_apply
@[simp] lemma algebraMap_apply (x : S) : algebraMap S L x = x 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L) (x : ↥S)
Goal: (algebraMap (↥S) L) x = ↑x
-- IntermediateField.coe_algebraMap_apply
@[simp] lemma coe_algebraMap_apply (x : K) : ↑(algebraMap K S x) = algebraMap K L x 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L) (x : K)
Goal: ↑((algebraMap K ↥S) x) = (algebraMap K L) x
-- IntermediateField.coe_map
@[simp]
theorem coe_map (f : L →ₐ[K] L') : (S.map f : Set L') = f '' S 
Arguments: {K : Type u_1} {L : Type u_2} {L' : Type u_3} [Field K] [Field L] [Field L'] [Algebra K L] [Algebra K L'] (S : IntermediateField K L) (f : L →ₐ[K] L')
Goal: ↑(IntermediateField.map f S) = ⇑f '' ↑S
-- IntermediateField.toSubalgebra_map
@[simp]
theorem toSubalgebra_map (f : L →ₐ[K] L') : (S.map f).toSubalgebra = S.toSubalgebra.map f 
Arguments: {K : Type u_1} {L : Type u_2} {L' : Type u_3} [Field K] [Field L] [Field L'] [Algebra K L] [Algebra K L'] (S : IntermediateField K L) (f : L →ₐ[K] L')
Goal: (IntermediateField.map f S).toSubalgebra = Subalgebra.map f S.toSubalgebra
-- IntermediateField.toSubfield_map
@[simp]
theorem toSubfield_map (f : L →ₐ[K] L') : (S.map f).toSubfield = S.toSubfield.map f 
Arguments: {K : Type u_1} {L : Type u_2} {L' : Type u_3} [Field K] [Field L] [Field L'] [Algebra K L] [Algebra K L'] (S : IntermediateField K L) (f : L →ₐ[K] L')
Goal: (IntermediateField.map f S).toSubfield = Subfield.map (↑f) S.toSubfield
-- IntermediateField.map_map
theorem map_map {K L₁ L₂ L₃ : Type*} [Field K] [Field L₁] [Algebra K L₁] [Field L₂] [Algebra K L₂]
    [Field L₃] [Algebra K L₃] (E : IntermediateField K L₁) (f : L₁ →ₐ[K] L₂) (g : L₂ →ₐ[K] L₃) :
    (E.map f).map g = E.map (g.comp f) 
Arguments: {K : Type u_4} {L₁ : Type u_5} {L₂ : Type u_6} {L₃ : Type u_7} [Field K] [Field L₁] [Algebra K L₁] [Field L₂] [Algebra K L₂] [Field L₃] [Algebra K L₃] (E : IntermediateField K L₁) (f : L₁ →ₐ[K] L₂) (g : L₂ →ₐ[K] L₃)
Goal: IntermediateField.map g (IntermediateField.map f E) = IntermediateField.map (g.comp f) E
-- IntermediateField.map_mono
theorem map_mono (f : L →ₐ[K] L') {S T : IntermediateField K L} (h : S ≤ T) :
    S.map f ≤ T.map f 
Arguments: {K : Type u_1} {L : Type u_2} {L' : Type u_3} [Field K] [Field L] [Field L'] [Algebra K L] [Algebra K L'] (f : L →ₐ[K] L') {S T : IntermediateField K L} (h : S ≤ T)
Goal: IntermediateField.map f S ≤ IntermediateField.map f T
-- IntermediateField.map_le_iff_le_comap
theorem map_le_iff_le_comap {f : L →ₐ[K] L'}
    {s : IntermediateField K L} {t : IntermediateField K L'} :
    s.map f ≤ t ↔ s ≤ t.comap f 
Arguments: {K : Type u_1} {L : Type u_2} {L' : Type u_3} [Field K] [Field L] [Field L'] [Algebra K L] [Algebra K L'] {f : L →ₐ[K] L'} {s : IntermediateField K L} {t : IntermediateField K L'}
Goal: IntermediateField.map f s ≤ t ↔ s ≤ IntermediateField.comap f t
-- IntermediateField.gc_map_comap
theorem gc_map_comap (f : L →ₐ[K] L') : GaloisConnection (map f) (comap f) 
Arguments: {K : Type u_1} {L : Type u_2} {L' : Type u_3} [Field K] [Field L] [Field L'] [Algebra K L] [Algebra K L'] (f : L →ₐ[K] L')
Goal: GaloisConnection (IntermediateField.map f) (IntermediateField.comap f)
-- IntermediateField.intermediateFieldMap_apply_coe
@[simp, nolint simpNF]
theorem intermediateFieldMap_apply_coe (e : L ≃ₐ[K] L') (E : IntermediateField K L) (a : E) :
    ↑(intermediateFieldMap e E a) = e a 
Arguments: {K : Type u_1} {L : Type u_2} {L' : Type u_3} [Field K] [Field L] [Field L'] [Algebra K L] [Algebra K L'] (e : L ≃ₐ[K] L') (E : IntermediateField K L) (a : ↥E)
Goal: ↑((IntermediateField.intermediateFieldMap e E) a) = e ↑a
-- IntermediateField.intermediateFieldMap_symm_apply_coe
@[simp, nolint simpNF]
theorem intermediateFieldMap_symm_apply_coe (e : L ≃ₐ[K] L') (E : IntermediateField K L)
    (a : E.map e.toAlgHom) : ↑((intermediateFieldMap e E).symm a) = e.symm a 
Arguments: {K : Type u_1} {L : Type u_2} {L' : Type u_3} [Field K] [Field L] [Field L'] [Algebra K L] [Algebra K L'] (e : L ≃ₐ[K] L') (E : IntermediateField K L) (a : ↥(IntermediateField.map (↑e) E))
Goal: ↑((IntermediateField.intermediateFieldMap e E).symm a) = e.symm ↑a
-- AlgHom.fieldRange_toSubalgebra
@[simps toSubalgebra]
def fieldRange : IntermediateField K L' 
Arguments: {K : Type u_1} {L : Type u_2} {L' : Type u_3} [Field K] [Field L] [Field L'] [Algebra K L] [Algebra K L'] (f : L →ₐ[K] L')
Goal: f.fieldRange.toSubalgebra = f.range
-- AlgHom.coe_fieldRange
@[simp]
theorem coe_fieldRange : ↑f.fieldRange = Set.range f 
Arguments: {K : Type u_1} {L : Type u_2} {L' : Type u_3} [Field K] [Field L] [Field L'] [Algebra K L] [Algebra K L'] (f : L →ₐ[K] L')
Goal: ↑f.fieldRange = Set.range ⇑f
-- AlgHom.fieldRange_toSubfield
@[simp]
theorem fieldRange_toSubfield : f.fieldRange.toSubfield = (f : L →+* L').fieldRange 
Arguments: {K : Type u_1} {L : Type u_2} {L' : Type u_3} [Field K] [Field L] [Field L'] [Algebra K L] [Algebra K L'] (f : L →ₐ[K] L')
Goal: f.fieldRange.toSubfield = (↑f).fieldRange
-- AlgHom.mem_fieldRange
@[simp]
theorem mem_fieldRange {y : L'} : y ∈ f.fieldRange ↔ ∃ x, f x = y 
Arguments: {K : Type u_1} {L : Type u_2} {L' : Type u_3} [Field K] [Field L] [Field L'] [Algebra K L] [Algebra K L'] {f : L →ₐ[K] L'} {y : L'}
Goal: y ∈ f.fieldRange ↔ ∃ x, f x = y
-- IntermediateField.coe_val
@[simp]
theorem coe_val : ⇑S.val = ((↑) : S → L) 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L)
Goal: ⇑S.val = Subtype.val
-- IntermediateField.val_mk
@[simp]
theorem val_mk {x : L} (hx : x ∈ S) : S.val ⟨x, hx⟩ = x 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L) {x : L} (hx : x ∈ S)
Goal: S.val ⟨x, hx⟩ = x
-- IntermediateField.range_val
theorem range_val : S.val.range = S.toSubalgebra 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L)
Goal: S.val.range = S.toSubalgebra
-- IntermediateField.fieldRange_val
@[simp]
theorem fieldRange_val : S.val.fieldRange = S 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L)
Goal: S.val.fieldRange = S
-- IntermediateField.aeval_coe
theorem aeval_coe {R : Type*} [CommRing R] [Algebra R K] [Algebra R L] [IsScalarTower R K L]
    (x : S) (P : R[X]) : aeval (x : L) P = aeval x P 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L) {R : Type u_4} [CommRing R] [Algebra R K] [Algebra R L] [IsScalarTower R K L] (x : ↥S) (P : R[X])
Goal: (Polynomial.aeval ↑x) P = ↑((Polynomial.aeval x) P)
-- IntermediateField.coe_isIntegral_iff
theorem coe_isIntegral_iff {R : Type*} [CommRing R] [Algebra R K] [Algebra R L]
    [IsScalarTower R K L] {x : S} : IsIntegral R (x : L) ↔ IsIntegral R x 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L) {R : Type u_4} [CommRing R] [Algebra R K] [Algebra R L] [IsScalarTower R K L] {x : ↥S}
Goal: IsIntegral R ↑x ↔ IsIntegral R x
-- IntermediateField.inclusion_injective
theorem inclusion_injective {E F : IntermediateField K L} (hEF : E ≤ F) :
    Function.Injective (inclusion hEF) 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] {E F : IntermediateField K L} (hEF : E ≤ F)
Goal: Function.Injective ⇑(IntermediateField.inclusion hEF)
-- IntermediateField.inclusion_self
@[simp]
theorem inclusion_self {E : IntermediateField K L} : inclusion (le_refl E) = AlgHom.id K E 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] {E : IntermediateField K L}
Goal: IntermediateField.inclusion ⋯ = AlgHom.id K ↥E
-- IntermediateField.inclusion_inclusion
@[simp]
theorem inclusion_inclusion {E F G : IntermediateField K L} (hEF : E ≤ F) (hFG : F ≤ G) (x : E) :
    inclusion hFG (inclusion hEF x) = inclusion (le_trans hEF hFG) x 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] {E F G : IntermediateField K L} (hEF : E ≤ F) (hFG : F ≤ G) (x : ↥E)
Goal: (IntermediateField.inclusion hFG) ((IntermediateField.inclusion hEF) x) = (IntermediateField.inclusion ⋯) x
-- IntermediateField.coe_inclusion
@[simp]
theorem coe_inclusion {E F : IntermediateField K L} (hEF : E ≤ F) (e : E) :
    (inclusion hEF e : L) = e 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] {E F : IntermediateField K L} (hEF : E ≤ F) (e : ↥E)
Goal: ↑((IntermediateField.inclusion hEF) e) = ↑e
-- IntermediateField.toSubalgebra_injective
theorem toSubalgebra_injective :
    Function.Injective (IntermediateField.toSubalgebra : IntermediateField K L → _) 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L]
Goal: Function.Injective IntermediateField.toSubalgebra
-- IntermediateField.map_injective
theorem map_injective (f : L →ₐ[K] L') :
    Function.Injective (map f) 
Arguments: {K : Type u_1} {L : Type u_2} {L' : Type u_3} [Field K] [Field L] [Field L'] [Algebra K L] [Algebra K L'] (f : L →ₐ[K] L')
Goal: Function.Injective (IntermediateField.map f)
-- IntermediateField.set_range_subset
theorem set_range_subset : Set.range (algebraMap K L) ⊆ S 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L)
Goal: Set.range ⇑(algebraMap K L) ⊆ ↑S
-- IntermediateField.fieldRange_le
theorem fieldRange_le : (algebraMap K L).fieldRange ≤ S.toSubfield 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L)
Goal: (algebraMap K L).fieldRange ≤ S.toSubfield
-- IntermediateField.toSubalgebra_le_toSubalgebra
@[simp]
theorem toSubalgebra_le_toSubalgebra {S S' : IntermediateField K L} :
    S.toSubalgebra ≤ S'.toSubalgebra ↔ S ≤ S' 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] {S S' : IntermediateField K L}
Goal: S.toSubalgebra ≤ S'.toSubalgebra ↔ S ≤ S'
-- IntermediateField.toSubalgebra_lt_toSubalgebra
@[simp]
theorem toSubalgebra_lt_toSubalgebra {S S' : IntermediateField K L} :
    S.toSubalgebra < S'.toSubalgebra ↔ S < S' 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] {S S' : IntermediateField K L}
Goal: S.toSubalgebra < S'.toSubalgebra ↔ S < S'
-- IntermediateField.lift_injective
theorem lift_injective (F : IntermediateField K L) : Function.Injective F.lift 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (F : IntermediateField K L)
Goal: Function.Injective IntermediateField.lift
-- IntermediateField.lift_le
theorem lift_le {F : IntermediateField K L} (E : IntermediateField K F) : lift E ≤ F 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] {F : IntermediateField K L} (E : IntermediateField K ↥F)
Goal: IntermediateField.lift E ≤ F
-- IntermediateField.mem_lift
theorem mem_lift {F : IntermediateField K L} {E : IntermediateField K F} (x : F) :
    x.1 ∈ lift E ↔ x ∈ E 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] {F : IntermediateField K L} {E : IntermediateField K ↥F} (x : ↥F)
Goal: ↑x ∈ IntermediateField.lift E ↔ x ∈ E
-- IntermediateField.coe_restrictScalars
@[simp]
theorem coe_restrictScalars {E : IntermediateField L' L} :
    (restrictScalars K E : Set L) = (E : Set L) 
Arguments: (K : Type u_1) {L : Type u_2} {L' : Type u_3} [Field K] [Field L] [Field L'] [Algebra K L] [Algebra K L'] [Algebra L' L] [IsScalarTower K L' L] {E : IntermediateField L' L}
Goal: ↑(IntermediateField.restrictScalars K E) = ↑E
-- IntermediateField.restrictScalars_toSubalgebra
@[simp]
theorem restrictScalars_toSubalgebra {E : IntermediateField L' L} :
    (E.restrictScalars K).toSubalgebra = E.toSubalgebra.restrictScalars K 
Arguments: (K : Type u_1) {L : Type u_2} {L' : Type u_3} [Field K] [Field L] [Field L'] [Algebra K L] [Algebra K L'] [Algebra L' L] [IsScalarTower K L' L] {E : IntermediateField L' L}
Goal: (IntermediateField.restrictScalars K E).toSubalgebra = Subalgebra.restrictScalars K E.toSubalgebra
-- IntermediateField.restrictScalars_toSubfield
@[simp]
theorem restrictScalars_toSubfield {E : IntermediateField L' L} :
    (E.restrictScalars K).toSubfield = E.toSubfield 
Arguments: (K : Type u_1) {L : Type u_2} {L' : Type u_3} [Field K] [Field L] [Field L'] [Algebra K L] [Algebra K L'] [Algebra L' L] [IsScalarTower K L' L] {E : IntermediateField L' L}
Goal: (IntermediateField.restrictScalars K E).toSubfield = E.toSubfield
-- IntermediateField.mem_restrictScalars
@[simp]
theorem mem_restrictScalars {E : IntermediateField L' L} {x : L} :
    x ∈ restrictScalars K E ↔ x ∈ E 
Arguments: (K : Type u_1) {L : Type u_2} {L' : Type u_3} [Field K] [Field L] [Field L'] [Algebra K L] [Algebra K L'] [Algebra L' L] [IsScalarTower K L' L] {E : IntermediateField L' L} {x : L}
Goal: x ∈ IntermediateField.restrictScalars K E ↔ x ∈ E
-- IntermediateField.restrictScalars_injective
theorem restrictScalars_injective :
    Function.Injective (restrictScalars K : IntermediateField L' L → IntermediateField K L) 
Arguments: (K : Type u_1) {L : Type u_2} {L' : Type u_3} [Field K] [Field L] [Field L'] [Algebra K L] [Algebra K L'] [Algebra L' L] [IsScalarTower K L' L]
Goal: Function.Injective (IntermediateField.restrictScalars K)
-- Subfield.coe_extendScalars
@[simp]
theorem coe_extendScalars : (extendScalars h : Set L) = (E : Set L) 
Arguments: {L : Type u_2} [Field L] {F E : Subfield L} (h : F ≤ E)
Goal: ↑(Subfield.extendScalars h) = ↑E
-- Subfield.extendScalars_toSubfield
@[simp]
theorem extendScalars_toSubfield : (extendScalars h).toSubfield = E 
Arguments: {L : Type u_2} [Field L] {F E : Subfield L} (h : F ≤ E)
Goal: (Subfield.extendScalars h).toSubfield = E
-- Subfield.mem_extendScalars
@[simp]
theorem mem_extendScalars : x ∈ extendScalars h ↔ x ∈ E 
Arguments: {L : Type u_2} [Field L] {F E : Subfield L} (h : F ≤ E) {x : L}
Goal: x ∈ Subfield.extendScalars h ↔ x ∈ E