ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- affineSpan_mono
@[mono]
theorem affineSpan_mono {s₁ s₂ : Set P} (h : s₁ ⊆ s₂) : affineSpan k s₁ ≤ affineSpan k s₂ 
Arguments: (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {s₁ s₂ : Set P} (h : s₁ ⊆ s₂)
Goal: affineSpan k s₁ ≤ affineSpan k s₂
-- affineSpan_insert_affineSpan
theorem affineSpan_insert_affineSpan (p : P) (ps : Set P) :
    affineSpan k (insert p (affineSpan k ps : Set P)) = affineSpan k (insert p ps) 
Arguments: (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] (p : P) (ps : Set P)
Goal: affineSpan k (insert p ↑(affineSpan k ps)) = affineSpan k (insert p ps)
-- affineSpan_insert_eq_affineSpan
theorem affineSpan_insert_eq_affineSpan {p : P} {ps : Set P} (h : p ∈ affineSpan k ps) :
    affineSpan k (insert p ps) = affineSpan k ps 
Arguments: (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {p : P} {ps : Set P} (h : p ∈ affineSpan k ps)
Goal: affineSpan k (insert p ps) = affineSpan k ps
-- vectorSpan_insert_eq_vectorSpan
theorem vectorSpan_insert_eq_vectorSpan {p : P} {ps : Set P} (h : p ∈ affineSpan k ps) :
    vectorSpan k (insert p ps) = vectorSpan k ps 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {p : P} {ps : Set P} (h : p ∈ affineSpan k ps)
Goal: vectorSpan k (insert p ps) = vectorSpan k ps
-- affineSpan_le_toAffineSubspace_span
lemma affineSpan_le_toAffineSubspace_span {s : Set V} :
    affineSpan k s ≤ (Submodule.span k s).toAffineSubspace 
Arguments: {k : Type u_1} {V : Type u_2} [Ring k] [AddCommGroup V] [Module k V] {s : Set V}
Goal: affineSpan k s ≤ (Submodule.span k s).toAffineSubspace
-- affineSpan_subset_span
lemma affineSpan_subset_span {s : Set V} :
    (affineSpan k s : Set V) ⊆  Submodule.span k s 
Arguments: {k : Type u_1} {V : Type u_2} [Ring k] [AddCommGroup V] [Module k V] {s : Set V}
Goal: ↑(affineSpan k s) ⊆ ↑(Submodule.span k s)
-- AffineSubspace.direction_sup
theorem direction_sup {s1 s2 : AffineSubspace k P} {p1 p2 : P} (hp1 : p1 ∈ s1) (hp2 : p2 ∈ s2) :
    (s1 ⊔ s2).direction = s1.direction ⊔ s2.direction ⊔ k ∙ p2 -ᵥ p1 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {s1 s2 : AffineSubspace k P} {p1 p2 : P} (hp1 : p1 ∈ s1) (hp2 : p2 ∈ s2)
Goal: (s1 ⊔ s2).direction = s1.direction ⊔ s2.direction ⊔ Submodule.span k {p2 -ᵥ p1}
-- AffineSubspace.direction_affineSpan_insert
theorem direction_affineSpan_insert {s : AffineSubspace k P} {p1 p2 : P} (hp1 : p1 ∈ s) :
    (affineSpan k (insert p2 (s : Set P))).direction =
    Submodule.span k {p2 -ᵥ p1} ⊔ s.direction 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {s : AffineSubspace k P} {p1 p2 : P} (hp1 : p1 ∈ s)
Goal: (affineSpan k (insert p2 ↑s)).direction = Submodule.span k {p2 -ᵥ p1} ⊔ s.direction
-- AffineSubspace.mem_affineSpan_insert_iff
theorem mem_affineSpan_insert_iff {s : AffineSubspace k P} {p1 : P} (hp1 : p1 ∈ s) (p2 p : P) :
    p ∈ affineSpan k (insert p2 (s : Set P)) ↔
      ∃ r : k, ∃ p0 ∈ s, p = r • (p2 -ᵥ p1 : V) +ᵥ p0 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {s : AffineSubspace k P} {p1 : P} (hp1 : p1 ∈ s) (p2 p : P)
Goal: p ∈ affineSpan k (insert p2 ↑s) ↔ ∃ r, ∃ p0 ∈ s, p = r • (p2 -ᵥ p1) +ᵥ p0
-- AffineMap.vectorSpan_image_eq_submodule_map
@[simp]
theorem AffineMap.vectorSpan_image_eq_submodule_map {s : Set P₁} :
    Submodule.map f.linear (vectorSpan k s) = vectorSpan k (f '' s) 
Arguments: {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} {V₂ : Type u_4} {P₂ : Type u_5} [Ring k] [AddCommGroup V₁] [Module k V₁] [AffineSpace V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AffineSpace V₂ P₂] (f : P₁ →ᵃ[k] P₂) {s : Set P₁}
Goal: Submodule.map f.linear (vectorSpan k s) = vectorSpan k (⇑f '' s)
-- AffineSubspace.coe_map
@[simp]
theorem coe_map (s : AffineSubspace k P₁) : (s.map f : Set P₂) = f '' s 
Arguments: {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} {V₂ : Type u_4} {P₂ : Type u_5} [Ring k] [AddCommGroup V₁] [Module k V₁] [AffineSpace V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AffineSpace V₂ P₂] (f : P₁ →ᵃ[k] P₂) (s : AffineSubspace k P₁)
Goal: ↑(AffineSubspace.map f s) = ⇑f '' ↑s
-- AffineSubspace.mem_map
@[simp]
theorem mem_map {f : P₁ →ᵃ[k] P₂} {x : P₂} {s : AffineSubspace k P₁} :
    x ∈ s.map f ↔ ∃ y ∈ s, f y = x 
Arguments: {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} {V₂ : Type u_4} {P₂ : Type u_5} [Ring k] [AddCommGroup V₁] [Module k V₁] [AffineSpace V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AffineSpace V₂ P₂] {f : P₁ →ᵃ[k] P₂} {x : P₂} {s : AffineSubspace k P₁}
Goal: x ∈ AffineSubspace.map f s ↔ ∃ y ∈ s, f y = x
-- AffineSubspace.mem_map_of_mem
theorem mem_map_of_mem {x : P₁} {s : AffineSubspace k P₁} (h : x ∈ s) : f x ∈ s.map f 
Arguments: {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} {V₂ : Type u_4} {P₂ : Type u_5} [Ring k] [AddCommGroup V₁] [Module k V₁] [AffineSpace V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AffineSpace V₂ P₂] (f : P₁ →ᵃ[k] P₂) {x : P₁} {s : AffineSubspace k P₁} (h : x ∈ s)
Goal: f x ∈ AffineSubspace.map f s
-- AffineSubspace.mem_map_iff_mem_of_injective
@[simp 1100, nolint simpNF]
theorem mem_map_iff_mem_of_injective {f : P₁ →ᵃ[k] P₂} {x : P₁} {s : AffineSubspace k P₁}
    (hf : Function.Injective f) : f x ∈ s.map f ↔ x ∈ s 
Arguments: {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} {V₂ : Type u_4} {P₂ : Type u_5} [Ring k] [AddCommGroup V₁] [Module k V₁] [AffineSpace V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AffineSpace V₂ P₂] {f : P₁ →ᵃ[k] P₂} {x : P₁} {s : AffineSubspace k P₁} (hf : Function.Injective ⇑f)
Goal: f x ∈ AffineSubspace.map f s ↔ x ∈ s
-- AffineSubspace.map_bot
@[simp]
theorem map_bot : (⊥ : AffineSubspace k P₁).map f = ⊥ 
Arguments: {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} {V₂ : Type u_4} {P₂ : Type u_5} [Ring k] [AddCommGroup V₁] [Module k V₁] [AffineSpace V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AffineSpace V₂ P₂] (f : P₁ →ᵃ[k] P₂)
Goal: AffineSubspace.map f ⊥ = ⊥
-- AffineSubspace.map_eq_bot_iff
@[simp]
theorem map_eq_bot_iff {s : AffineSubspace k P₁} : s.map f = ⊥ ↔ s = ⊥ 
Arguments: {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} {V₂ : Type u_4} {P₂ : Type u_5} [Ring k] [AddCommGroup V₁] [Module k V₁] [AffineSpace V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AffineSpace V₂ P₂] (f : P₁ →ᵃ[k] P₂) {s : AffineSubspace k P₁}
Goal: AffineSubspace.map f s = ⊥ ↔ s = ⊥
-- AffineSubspace.map_id
@[simp]
theorem map_id (s : AffineSubspace k P₁) : s.map (AffineMap.id k P₁) = s 
Arguments: {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} [Ring k] [AddCommGroup V₁] [Module k V₁] [AffineSpace V₁ P₁] (s : AffineSubspace k P₁)
Goal: AffineSubspace.map (AffineMap.id k P₁) s = s
-- AffineSubspace.map_map
theorem map_map (s : AffineSubspace k P₁) (f : P₁ →ᵃ[k] P₂) (g : P₂ →ᵃ[k] P₃) :
    (s.map f).map g = s.map (g.comp f) 
Arguments: {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} {V₂ : Type u_4} {P₂ : Type u_5} {V₃ : Type u_6} {P₃ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AffineSpace V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AffineSpace V₂ P₂] [AddCommGroup V₃] [Module k V₃] [AffineSpace V₃ P₃] (s : AffineSubspace k P₁) (f : P₁ →ᵃ[k] P₂) (g : P₂ →ᵃ[k] P₃)
Goal: AffineSubspace.map g (AffineSubspace.map f s) = AffineSubspace.map (g.comp f) s
-- AffineSubspace.map_direction
@[simp]
theorem map_direction (s : AffineSubspace k P₁) :
    (s.map f).direction = s.direction.map f.linear 
Arguments: {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} {V₂ : Type u_4} {P₂ : Type u_5} [Ring k] [AddCommGroup V₁] [Module k V₁] [AffineSpace V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AffineSpace V₂ P₂] (f : P₁ →ᵃ[k] P₂) (s : AffineSubspace k P₁)
Goal: (AffineSubspace.map f s).direction = Submodule.map f.linear s.direction
-- AffineSubspace.map_span
theorem map_span (s : Set P₁) : (affineSpan k s).map f = affineSpan k (f '' s) 
Arguments: {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} {V₂ : Type u_4} {P₂ : Type u_5} [Ring k] [AddCommGroup V₁] [Module k V₁] [AffineSpace V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AffineSpace V₂ P₂] (f : P₁ →ᵃ[k] P₂) (s : Set P₁)
Goal: AffineSubspace.map f (affineSpan k s) = affineSpan k (⇑f '' s)
-- AffineSubspace.inclusion_linear
@[simps linear]
def inclusion (h : S₁ ≤ S₂) : S₁ →ᵃ[k] S₂ 
Arguments: {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} [Ring k] [AddCommGroup V₁] [Module k V₁] [AffineSpace V₁ P₁] {S₁ S₂ : AffineSubspace k P₁} [Nonempty ↥S₁] [Nonempty ↥S₂] (h : S₁ ≤ S₂)
Goal: (AffineSubspace.inclusion h).linear = Submodule.inclusion ⋯
-- AffineSubspace.coe_inclusion_apply
@[simp]
theorem coe_inclusion_apply (h : S₁ ≤ S₂) (x : S₁) : (inclusion h x : P₁) = x 
Arguments: {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} [Ring k] [AddCommGroup V₁] [Module k V₁] [AffineSpace V₁ P₁] {S₁ S₂ : AffineSubspace k P₁} [Nonempty ↥S₁] [Nonempty ↥S₂] (h : S₁ ≤ S₂) (x : ↥S₁)
Goal: ↑((AffineSubspace.inclusion h) x) = ↑x
-- AffineSubspace.inclusion_rfl
@[simp]
theorem inclusion_rfl : inclusion (le_refl S₁) = AffineMap.id k S₁ 
Arguments: {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} [Ring k] [AddCommGroup V₁] [Module k V₁] [AffineSpace V₁ P₁] {S₁ : AffineSubspace k P₁} [Nonempty ↥S₁]
Goal: AffineSubspace.inclusion ⋯ = AffineMap.id k ↥S₁
-- AffineMap.map_top_of_surjective
@[simp]
theorem map_top_of_surjective (hf : Function.Surjective f) : AffineSubspace.map f ⊤ = ⊤ 
Arguments: {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} {V₂ : Type u_4} {P₂ : Type u_5} [Ring k] [AddCommGroup V₁] [Module k V₁] [AffineSpace V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AffineSpace V₂ P₂] (f : P₁ →ᵃ[k] P₂) (hf : Function.Surjective ⇑f)
Goal: AffineSubspace.map f ⊤ = ⊤
-- AffineMap.span_eq_top_of_surjective
theorem span_eq_top_of_surjective {s : Set P₁} (hf : Function.Surjective f)
    (h : affineSpan k s = ⊤) : affineSpan k (f '' s) = ⊤ 
Arguments: {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} {V₂ : Type u_4} {P₂ : Type u_5} [Ring k] [AddCommGroup V₁] [Module k V₁] [AffineSpace V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AffineSpace V₂ P₂] (f : P₁ →ᵃ[k] P₂) {s : Set P₁} (hf : Function.Surjective ⇑f) (h : affineSpan k s = ⊤)
Goal: affineSpan k (⇑f '' s) = ⊤
-- AffineEquiv.linear_ofEq
@[simps linear]
def ofEq (h : S₁ = S₂) : S₁ ≃ᵃ[k] S₂ 
Arguments: {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} [Ring k] [AddCommGroup V₁] [Module k V₁] [AffineSpace V₁ P₁] (S₁ S₂ : AffineSubspace k P₁) [Nonempty ↥S₁] [Nonempty ↥S₂] (h : S₁ = S₂)
Goal: (AffineEquiv.ofEq S₁ S₂ h).linear = LinearEquiv.ofEq S₁.direction S₂.direction ⋯
-- AffineEquiv.coe_ofEq_apply
@[simp]
theorem coe_ofEq_apply (h : S₁ = S₂) (x : S₁) : (ofEq S₁ S₂ h x : P₁) = x 
Arguments: {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} [Ring k] [AddCommGroup V₁] [Module k V₁] [AffineSpace V₁ P₁] (S₁ S₂ : AffineSubspace k P₁) [Nonempty ↥S₁] [Nonempty ↥S₂] (h : S₁ = S₂) (x : ↥S₁)
Goal: ↑((AffineEquiv.ofEq S₁ S₂ h) x) = ↑x
-- AffineEquiv.ofEq_symm
@[simp]
theorem ofEq_symm (h : S₁ = S₂) : (ofEq S₁ S₂ h).symm = ofEq S₂ S₁ h.symm 
Arguments: {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} [Ring k] [AddCommGroup V₁] [Module k V₁] [AffineSpace V₁ P₁] (S₁ S₂ : AffineSubspace k P₁) [Nonempty ↥S₁] [Nonempty ↥S₂] (h : S₁ = S₂)
Goal: (AffineEquiv.ofEq S₁ S₂ h).symm = AffineEquiv.ofEq S₂ S₁ ⋯
-- AffineEquiv.ofEq_rfl
@[simp]
theorem ofEq_rfl : ofEq S₁ S₁ rfl = AffineEquiv.refl k S₁ 
Arguments: {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} [Ring k] [AddCommGroup V₁] [Module k V₁] [AffineSpace V₁ P₁] (S₁ : AffineSubspace k P₁) [Nonempty ↥S₁]
Goal: AffineEquiv.ofEq S₁ S₁ ⋯ = AffineEquiv.refl k ↥S₁
-- AffineEquiv.span_eq_top_iff
theorem span_eq_top_iff {s : Set P₁} (e : P₁ ≃ᵃ[k] P₂) :
    affineSpan k s = ⊤ ↔ affineSpan k (e '' s) = ⊤ 
Arguments: {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} {V₂ : Type u_4} {P₂ : Type u_5} [Ring k] [AddCommGroup V₁] [Module k V₁] [AffineSpace V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AffineSpace V₂ P₂] {s : Set P₁} (e : P₁ ≃ᵃ[k] P₂)
Goal: affineSpan k s = ⊤ ↔ affineSpan k (⇑e '' s) = ⊤
-- AffineSubspace.coe_comap
@[simp]
theorem coe_comap (f : P₁ →ᵃ[k] P₂) (s : AffineSubspace k P₂) : (s.comap f : Set P₁) = f ⁻¹' ↑s 
Arguments: {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} {V₂ : Type u_4} {P₂ : Type u_5} [Ring k] [AddCommGroup V₁] [Module k V₁] [AffineSpace V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AffineSpace V₂ P₂] (f : P₁ →ᵃ[k] P₂) (s : AffineSubspace k P₂)
Goal: ↑(AffineSubspace.comap f s) = ⇑f ⁻¹' ↑s
-- AffineSubspace.mem_comap
@[simp]
theorem mem_comap {f : P₁ →ᵃ[k] P₂} {x : P₁} {s : AffineSubspace k P₂} : x ∈ s.comap f ↔ f x ∈ s 
Arguments: {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} {V₂ : Type u_4} {P₂ : Type u_5} [Ring k] [AddCommGroup V₁] [Module k V₁] [AffineSpace V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AffineSpace V₂ P₂] {f : P₁ →ᵃ[k] P₂} {x : P₁} {s : AffineSubspace k P₂}
Goal: x ∈ AffineSubspace.comap f s ↔ f x ∈ s
-- AffineSubspace.comap_mono
theorem comap_mono {f : P₁ →ᵃ[k] P₂} {s t : AffineSubspace k P₂} : s ≤ t → s.comap f ≤ t.comap f 
Arguments: {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} {V₂ : Type u_4} {P₂ : Type u_5} [Ring k] [AddCommGroup V₁] [Module k V₁] [AffineSpace V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AffineSpace V₂ P₂] {f : P₁ →ᵃ[k] P₂} {s t : AffineSubspace k P₂}
Goal: s ≤ t → AffineSubspace.comap f s ≤ AffineSubspace.comap f t
-- AffineSubspace.comap_top
@[simp]
theorem comap_top {f : P₁ →ᵃ[k] P₂} : (⊤ : AffineSubspace k P₂).comap f = ⊤ 
Arguments: {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} {V₂ : Type u_4} {P₂ : Type u_5} [Ring k] [AddCommGroup V₁] [Module k V₁] [AffineSpace V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AffineSpace V₂ P₂] {f : P₁ →ᵃ[k] P₂}
Goal: AffineSubspace.comap f ⊤ = ⊤
-- AffineSubspace.comap_bot
@[simp] theorem comap_bot (f : P₁ →ᵃ[k] P₂) : comap f ⊥ = ⊥ 
Arguments: {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} {V₂ : Type u_4} {P₂ : Type u_5} [Ring k] [AddCommGroup V₁] [Module k V₁] [AffineSpace V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AffineSpace V₂ P₂] (f : P₁ →ᵃ[k] P₂)
Goal: AffineSubspace.comap f ⊥ = ⊥
-- AffineSubspace.comap_id
@[simp]
theorem comap_id (s : AffineSubspace k P₁) : s.comap (AffineMap.id k P₁) = s 
Arguments: {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} [Ring k] [AddCommGroup V₁] [Module k V₁] [AffineSpace V₁ P₁] (s : AffineSubspace k P₁)
Goal: AffineSubspace.comap (AffineMap.id k P₁) s = s
-- AffineSubspace.comap_comap
theorem comap_comap (s : AffineSubspace k P₃) (f : P₁ →ᵃ[k] P₂) (g : P₂ →ᵃ[k] P₃) :
    (s.comap g).comap f = s.comap (g.comp f) 
Arguments: {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} {V₂ : Type u_4} {P₂ : Type u_5} {V₃ : Type u_6} {P₃ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AffineSpace V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AffineSpace V₂ P₂] [AddCommGroup V₃] [Module k V₃] [AffineSpace V₃ P₃] (s : AffineSubspace k P₃) (f : P₁ →ᵃ[k] P₂) (g : P₂ →ᵃ[k] P₃)
Goal: AffineSubspace.comap f (AffineSubspace.comap g s) = AffineSubspace.comap (g.comp f) s
-- AffineSubspace.map_le_iff_le_comap
theorem map_le_iff_le_comap {f : P₁ →ᵃ[k] P₂} {s : AffineSubspace k P₁} {t : AffineSubspace k P₂} :
    s.map f ≤ t ↔ s ≤ t.comap f 
Arguments: {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} {V₂ : Type u_4} {P₂ : Type u_5} [Ring k] [AddCommGroup V₁] [Module k V₁] [AffineSpace V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AffineSpace V₂ P₂] {f : P₁ →ᵃ[k] P₂} {s : AffineSubspace k P₁} {t : AffineSubspace k P₂}
Goal: AffineSubspace.map f s ≤ t ↔ s ≤ AffineSubspace.comap f t
-- AffineSubspace.gc_map_comap
theorem gc_map_comap (f : P₁ →ᵃ[k] P₂) : GaloisConnection (map f) (comap f) 
Arguments: {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} {V₂ : Type u_4} {P₂ : Type u_5} [Ring k] [AddCommGroup V₁] [Module k V₁] [AffineSpace V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AffineSpace V₂ P₂] (f : P₁ →ᵃ[k] P₂)
Goal: GaloisConnection (AffineSubspace.map f) (AffineSubspace.comap f)
-- AffineSubspace.map_comap_le
theorem map_comap_le (f : P₁ →ᵃ[k] P₂) (s : AffineSubspace k P₂) : (s.comap f).map f ≤ s 
Arguments: {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} {V₂ : Type u_4} {P₂ : Type u_5} [Ring k] [AddCommGroup V₁] [Module k V₁] [AffineSpace V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AffineSpace V₂ P₂] (f : P₁ →ᵃ[k] P₂) (s : AffineSubspace k P₂)
Goal: AffineSubspace.map f (AffineSubspace.comap f s) ≤ s
-- AffineSubspace.le_comap_map
theorem le_comap_map (f : P₁ →ᵃ[k] P₂) (s : AffineSubspace k P₁) : s ≤ (s.map f).comap f 
Arguments: {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} {V₂ : Type u_4} {P₂ : Type u_5} [Ring k] [AddCommGroup V₁] [Module k V₁] [AffineSpace V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AffineSpace V₂ P₂] (f : P₁ →ᵃ[k] P₂) (s : AffineSubspace k P₁)
Goal: s ≤ AffineSubspace.comap f (AffineSubspace.map f s)
-- AffineSubspace.map_sup
theorem map_sup (s t : AffineSubspace k P₁) (f : P₁ →ᵃ[k] P₂) : (s ⊔ t).map f = s.map f ⊔ t.map f 
Arguments: {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} {V₂ : Type u_4} {P₂ : Type u_5} [Ring k] [AddCommGroup V₁] [Module k V₁] [AffineSpace V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AffineSpace V₂ P₂] (s t : AffineSubspace k P₁) (f : P₁ →ᵃ[k] P₂)
Goal: AffineSubspace.map f (s ⊔ t) = AffineSubspace.map f s ⊔ AffineSubspace.map f t
-- AffineSubspace.map_iSup
theorem map_iSup {ι : Sort*} (f : P₁ →ᵃ[k] P₂) (s : ι → AffineSubspace k P₁) :
    (iSup s).map f = ⨆ i, (s i).map f 
Arguments: {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} {V₂ : Type u_4} {P₂ : Type u_5} [Ring k] [AddCommGroup V₁] [Module k V₁] [AffineSpace V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AffineSpace V₂ P₂] {ι : Sort u_8} (f : P₁ →ᵃ[k] P₂) (s : ι → AffineSubspace k P₁)
Goal: AffineSubspace.map f (iSup s) = ⨆ i, AffineSubspace.map f (s i)
-- AffineSubspace.comap_inf
theorem comap_inf (s t : AffineSubspace k P₂) (f : P₁ →ᵃ[k] P₂) :
    (s ⊓ t).comap f = s.comap f ⊓ t.comap f 
Arguments: {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} {V₂ : Type u_4} {P₂ : Type u_5} [Ring k] [AddCommGroup V₁] [Module k V₁] [AffineSpace V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AffineSpace V₂ P₂] (s t : AffineSubspace k P₂) (f : P₁ →ᵃ[k] P₂)
Goal: AffineSubspace.comap f (s ⊓ t) = AffineSubspace.comap f s ⊓ AffineSubspace.comap f t
-- AffineSubspace.comap_supr
theorem comap_supr {ι : Sort*} (f : P₁ →ᵃ[k] P₂) (s : ι → AffineSubspace k P₂) :
    (iInf s).comap f = ⨅ i, (s i).comap f 
Arguments: {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} {V₂ : Type u_4} {P₂ : Type u_5} [Ring k] [AddCommGroup V₁] [Module k V₁] [AffineSpace V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AffineSpace V₂ P₂] {ι : Sort u_8} (f : P₁ →ᵃ[k] P₂) (s : ι → AffineSubspace k P₂)
Goal: AffineSubspace.comap f (iInf s) = ⨅ i, AffineSubspace.comap f (s i)
-- AffineSubspace.comap_symm
@[simp]
theorem comap_symm (e : P₁ ≃ᵃ[k] P₂) (s : AffineSubspace k P₁) :
    s.comap (e.symm : P₂ →ᵃ[k] P₁) = s.map e 
Arguments: {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} {V₂ : Type u_4} {P₂ : Type u_5} [Ring k] [AddCommGroup V₁] [Module k V₁] [AffineSpace V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AffineSpace V₂ P₂] (e : P₁ ≃ᵃ[k] P₂) (s : AffineSubspace k P₁)
Goal: AffineSubspace.comap (↑e.symm) s = AffineSubspace.map (↑e) s
-- AffineSubspace.map_symm
@[simp]
theorem map_symm (e : P₁ ≃ᵃ[k] P₂) (s : AffineSubspace k P₂) :
    s.map (e.symm : P₂ →ᵃ[k] P₁) = s.comap e 
Arguments: {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} {V₂ : Type u_4} {P₂ : Type u_5} [Ring k] [AddCommGroup V₁] [Module k V₁] [AffineSpace V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AffineSpace V₂ P₂] (e : P₁ ≃ᵃ[k] P₂) (s : AffineSubspace k P₂)
Goal: AffineSubspace.map (↑e.symm) s = AffineSubspace.comap (↑e) s
-- AffineSubspace.comap_span
theorem comap_span (f : P₁ ≃ᵃ[k] P₂) (s : Set P₂) :
    (affineSpan k s).comap (f : P₁ →ᵃ[k] P₂) = affineSpan k (f ⁻¹' s) 
Arguments: {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} {V₂ : Type u_4} {P₂ : Type u_5} [Ring k] [AddCommGroup V₁] [Module k V₁] [AffineSpace V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AffineSpace V₂ P₂] (f : P₁ ≃ᵃ[k] P₂) (s : Set P₂)
Goal: AffineSubspace.comap (↑f) (affineSpan k s) = affineSpan k (⇑f ⁻¹' s)
-- AffineSubspace.Parallel.symm
@[symm]
theorem Parallel.symm {s₁ s₂ : AffineSubspace k P} (h : s₁ ∥ s₂) : s₂ ∥ s₁ 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {s₁ s₂ : AffineSubspace k P} (h : s₁ ∥ s₂)
Goal: s₂ ∥ s₁
-- AffineSubspace.parallel_comm
theorem parallel_comm {s₁ s₂ : AffineSubspace k P} : s₁ ∥ s₂ ↔ s₂ ∥ s₁ 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {s₁ s₂ : AffineSubspace k P}
Goal: s₁ ∥ s₂ ↔ s₂ ∥ s₁
-- AffineSubspace.Parallel.refl
@[refl]
theorem Parallel.refl (s : AffineSubspace k P) : s ∥ s 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] (s : AffineSubspace k P)
Goal: s ∥ s
-- AffineSubspace.Parallel.trans
@[trans]
theorem Parallel.trans {s₁ s₂ s₃ : AffineSubspace k P} (h₁₂ : s₁ ∥ s₂) (h₂₃ : s₂ ∥ s₃) :
    s₁ ∥ s₃ 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {s₁ s₂ s₃ : AffineSubspace k P} (h₁₂ : s₁ ∥ s₂) (h₂₃ : s₂ ∥ s₃)
Goal: s₁ ∥ s₃
-- AffineSubspace.Parallel.direction_eq
theorem Parallel.direction_eq {s₁ s₂ : AffineSubspace k P} (h : s₁ ∥ s₂) :
    s₁.direction = s₂.direction 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {s₁ s₂ : AffineSubspace k P} (h : s₁ ∥ s₂)
Goal: s₁.direction = s₂.direction
-- AffineSubspace.parallel_bot_iff_eq_bot
@[simp]
theorem parallel_bot_iff_eq_bot {s : AffineSubspace k P} : s ∥ ⊥ ↔ s = ⊥ 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {s : AffineSubspace k P}
Goal: s ∥ ⊥ ↔ s = ⊥
-- AffineSubspace.bot_parallel_iff_eq_bot
@[simp]
theorem bot_parallel_iff_eq_bot {s : AffineSubspace k P} : ⊥ ∥ s ↔ s = ⊥ 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {s : AffineSubspace k P}
Goal: ⊥ ∥ s ↔ s = ⊥
-- AffineSubspace.parallel_iff_direction_eq_and_eq_bot_iff_eq_bot
theorem parallel_iff_direction_eq_and_eq_bot_iff_eq_bot {s₁ s₂ : AffineSubspace k P} :
    s₁ ∥ s₂ ↔ s₁.direction = s₂.direction ∧ (s₁ = ⊥ ↔ s₂ = ⊥) 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {s₁ s₂ : AffineSubspace k P}
Goal: s₁ ∥ s₂ ↔ s₁.direction = s₂.direction ∧ (s₁ = ⊥ ↔ s₂ = ⊥)
-- AffineSubspace.Parallel.vectorSpan_eq
theorem Parallel.vectorSpan_eq {s₁ s₂ : Set P} (h : affineSpan k s₁ ∥ affineSpan k s₂) :
    vectorSpan k s₁ = vectorSpan k s₂ 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {s₁ s₂ : Set P} (h : affineSpan k s₁ ∥ affineSpan k s₂)
Goal: vectorSpan k s₁ = vectorSpan k s₂
-- AffineSubspace.affineSpan_parallel_iff_vectorSpan_eq_and_eq_empty_iff_eq_empty
theorem affineSpan_parallel_iff_vectorSpan_eq_and_eq_empty_iff_eq_empty {s₁ s₂ : Set P} :
    affineSpan k s₁ ∥ affineSpan k s₂ ↔ vectorSpan k s₁ = vectorSpan k s₂ ∧ (s₁ = ∅ ↔ s₂ = ∅) 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {s₁ s₂ : Set P}
Goal: affineSpan k s₁ ∥ affineSpan k s₂ ↔ vectorSpan k s₁ = vectorSpan k s₂ ∧ (s₁ = ∅ ↔ s₂ = ∅)
-- AffineSubspace.affineSpan_pair_parallel_iff_vectorSpan_eq
theorem affineSpan_pair_parallel_iff_vectorSpan_eq {p₁ p₂ p₃ p₄ : P} :
    line[k, p₁, p₂] ∥ line[k, p₃, p₄] ↔
      vectorSpan k ({p₁, p₂} : Set P) = vectorSpan k ({p₃, p₄} : Set P) 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {p₁ p₂ p₃ p₄ : P}
Goal: affineSpan k {p₁, p₂} ∥ affineSpan k {p₃, p₄} ↔ vectorSpan k {p₁, p₂} = vectorSpan k {p₃, p₄}
-- Matrix.GeneralLinearGroup.val_inv_det_apply
@[simps]
def det : GL n R →* Rˣ 
Arguments: {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] (A : GL n R)
Goal: ↑(Matrix.GeneralLinearGroup.det A)⁻¹ = (↑A⁻¹).det
-- Matrix.GeneralLinearGroup.val_det_apply
@[simps]
def det : GL n R →* Rˣ 
Arguments: {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] (A : GL n R)
Goal: ↑(Matrix.GeneralLinearGroup.det A) = (↑A).det