ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- AffineMap.linear_bijective_iff
@[simp]
theorem linear_bijective_iff (f : P1 →ᵃ[k] P2) :
    Function.Bijective f.linear ↔ Function.Bijective 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: Function.Bijective ⇑f.linear ↔ Function.Bijective ⇑f
-- AffineMap.image_vsub_image
theorem image_vsub_image {s t : Set P1} (f : P1 →ᵃ[k] P2) :
    f '' s -ᵥ f '' t = f.linear '' (s -ᵥ t) 
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 : Set P1} (f : P1 →ᵃ[k] P2)
Goal: ⇑f '' s -ᵥ ⇑f '' t = ⇑f.linear '' (s -ᵥ t)
-- AffineMap.coe_lineMap
theorem coe_lineMap (p₀ p₁ : P1) : (lineMap p₀ p₁ : k → P1) = fun c => c • (p₁ -ᵥ p₀) +ᵥ p₀ 
Arguments: {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [Ring k] [AddCommGroup V1] [Module k V1] [AffineSpace V1 P1] (p₀ p₁ : P1)
Goal: ⇑(AffineMap.lineMap p₀ p₁) = fun c => c • (p₁ -ᵥ p₀) +ᵥ p₀
-- AffineMap.lineMap_apply
theorem lineMap_apply (p₀ p₁ : P1) (c : k) : lineMap p₀ p₁ c = c • (p₁ -ᵥ p₀) +ᵥ p₀ 
Arguments: {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [Ring k] [AddCommGroup V1] [Module k V1] [AffineSpace V1 P1] (p₀ p₁ : P1) (c : k)
Goal: (AffineMap.lineMap p₀ p₁) c = c • (p₁ -ᵥ p₀) +ᵥ p₀
-- AffineMap.lineMap_apply_module'
theorem lineMap_apply_module' (p₀ p₁ : V1) (c : k) : lineMap p₀ p₁ c = c • (p₁ - p₀) + p₀ 
Arguments: {k : Type u_1} {V1 : Type u_2} [Ring k] [AddCommGroup V1] [Module k V1] (p₀ p₁ : V1) (c : k)
Goal: (AffineMap.lineMap p₀ p₁) c = c • (p₁ - p₀) + p₀
-- AffineMap.lineMap_apply_module
theorem lineMap_apply_module (p₀ p₁ : V1) (c : k) : lineMap p₀ p₁ c = (1 - c) • p₀ + c • p₁ 
Arguments: {k : Type u_1} {V1 : Type u_2} [Ring k] [AddCommGroup V1] [Module k V1] (p₀ p₁ : V1) (c : k)
Goal: (AffineMap.lineMap p₀ p₁) c = (1 - c) • p₀ + c • p₁
-- AffineMap.lineMap_apply_ring'
theorem lineMap_apply_ring' (a b c : k) : lineMap a b c = c * (b - a) + a 
Arguments: {k : Type u_1} [Ring k] (a b c : k)
Goal: (AffineMap.lineMap a b) c = c * (b - a) + a
-- AffineMap.lineMap_apply_ring
theorem lineMap_apply_ring (a b c : k) : lineMap a b c = (1 - c) * a + c * b 
Arguments: {k : Type u_1} [Ring k] (a b c : k)
Goal: (AffineMap.lineMap a b) c = (1 - c) * a + c * b
-- AffineMap.lineMap_vadd_apply
theorem lineMap_vadd_apply (p : P1) (v : V1) (c : k) : lineMap p (v +ᵥ p) c = c • v +ᵥ p 
Arguments: {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [Ring k] [AddCommGroup V1] [Module k V1] [AffineSpace V1 P1] (p : P1) (v : V1) (c : k)
Goal: (AffineMap.lineMap p (v +ᵥ p)) c = c • v +ᵥ p
-- AffineMap.lineMap_linear
@[simp]
theorem lineMap_linear (p₀ p₁ : P1) :
    (lineMap p₀ p₁ : k →ᵃ[k] P1).linear = LinearMap.id.smulRight (p₁ -ᵥ p₀) 
Arguments: {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [Ring k] [AddCommGroup V1] [Module k V1] [AffineSpace V1 P1] (p₀ p₁ : P1)
Goal: (AffineMap.lineMap p₀ p₁).linear = LinearMap.id.smulRight (p₁ -ᵥ p₀)
-- AffineMap.lineMap_same_apply
theorem lineMap_same_apply (p : P1) (c : k) : lineMap p p c = p 
Arguments: {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [Ring k] [AddCommGroup V1] [Module k V1] [AffineSpace V1 P1] (p : P1) (c : k)
Goal: (AffineMap.lineMap p p) c = p
-- AffineMap.lineMap_same
@[simp]
theorem lineMap_same (p : P1) : lineMap p p = const k k p 
Arguments: {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [Ring k] [AddCommGroup V1] [Module k V1] [AffineSpace V1 P1] (p : P1)
Goal: AffineMap.lineMap p p = AffineMap.const k k p
-- AffineMap.lineMap_apply_zero
@[simp]
theorem lineMap_apply_zero (p₀ p₁ : P1) : lineMap p₀ p₁ (0 : k) = p₀ 
Arguments: {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [Ring k] [AddCommGroup V1] [Module k V1] [AffineSpace V1 P1] (p₀ p₁ : P1)
Goal: (AffineMap.lineMap p₀ p₁) 0 = p₀
-- AffineMap.lineMap_apply_one
@[simp]
theorem lineMap_apply_one (p₀ p₁ : P1) : lineMap p₀ p₁ (1 : k) = p₁ 
Arguments: {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [Ring k] [AddCommGroup V1] [Module k V1] [AffineSpace V1 P1] (p₀ p₁ : P1)
Goal: (AffineMap.lineMap p₀ p₁) 1 = p₁
-- AffineMap.lineMap_eq_lineMap_iff
@[simp]
theorem lineMap_eq_lineMap_iff [NoZeroSMulDivisors k V1] {p₀ p₁ : P1} {c₁ c₂ : k} :
    lineMap p₀ p₁ c₁ = lineMap p₀ p₁ c₂ ↔ p₀ = p₁ ∨ c₁ = c₂ 
Arguments: {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [Ring k] [AddCommGroup V1] [Module k V1] [AffineSpace V1 P1] [NoZeroSMulDivisors k V1] {p₀ p₁ : P1} {c₁ c₂ : k}
Goal: (AffineMap.lineMap p₀ p₁) c₁ = (AffineMap.lineMap p₀ p₁) c₂ ↔ p₀ = p₁ ∨ c₁ = c₂
-- AffineMap.lineMap_eq_left_iff
@[simp]
theorem lineMap_eq_left_iff [NoZeroSMulDivisors k V1] {p₀ p₁ : P1} {c : k} :
    lineMap p₀ p₁ c = p₀ ↔ p₀ = p₁ ∨ c = 0 
Arguments: {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [Ring k] [AddCommGroup V1] [Module k V1] [AffineSpace V1 P1] [NoZeroSMulDivisors k V1] {p₀ p₁ : P1} {c : k}
Goal: (AffineMap.lineMap p₀ p₁) c = p₀ ↔ p₀ = p₁ ∨ c = 0
-- AffineMap.lineMap_eq_right_iff
@[simp]
theorem lineMap_eq_right_iff [NoZeroSMulDivisors k V1] {p₀ p₁ : P1} {c : k} :
    lineMap p₀ p₁ c = p₁ ↔ p₀ = p₁ ∨ c = 1 
Arguments: {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [Ring k] [AddCommGroup V1] [Module k V1] [AffineSpace V1 P1] [NoZeroSMulDivisors k V1] {p₀ p₁ : P1} {c : k}
Goal: (AffineMap.lineMap p₀ p₁) c = p₁ ↔ p₀ = p₁ ∨ c = 1
-- AffineMap.lineMap_injective
theorem lineMap_injective [NoZeroSMulDivisors k V1] {p₀ p₁ : P1} (h : p₀ ≠ p₁) :
    Function.Injective (lineMap p₀ p₁ : k → P1) 
Arguments: (k : Type u_1) {V1 : Type u_2} {P1 : Type u_3} [Ring k] [AddCommGroup V1] [Module k V1] [AffineSpace V1 P1] [NoZeroSMulDivisors k V1] {p₀ p₁ : P1} (h : p₀ ≠ p₁)
Goal: Function.Injective ⇑(AffineMap.lineMap p₀ p₁)
-- AffineMap.apply_lineMap
@[simp]
theorem apply_lineMap (f : P1 →ᵃ[k] P2) (p₀ p₁ : P1) (c : k) :
    f (lineMap p₀ p₁ c) = lineMap (f p₀) (f p₁) c 
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) (p₀ p₁ : P1) (c : k)
Goal: f ((AffineMap.lineMap p₀ p₁) c) = (AffineMap.lineMap (f p₀) (f p₁)) c
-- AffineMap.comp_lineMap
@[simp]
theorem comp_lineMap (f : P1 →ᵃ[k] P2) (p₀ p₁ : P1) :
    f.comp (lineMap p₀ p₁) = lineMap (f p₀) (f p₁) 
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) (p₀ p₁ : P1)
Goal: f.comp (AffineMap.lineMap p₀ p₁) = AffineMap.lineMap (f p₀) (f p₁)
-- AffineMap.fst_lineMap
@[simp]
theorem fst_lineMap (p₀ p₁ : P1 × P2) (c : k) : (lineMap p₀ p₁ c).1 = lineMap p₀.1 p₁.1 c 
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] (p₀ p₁ : P1 × P2) (c : k)
Goal: ((AffineMap.lineMap p₀ p₁) c).1 = (AffineMap.lineMap p₀.1 p₁.1) c
-- AffineMap.snd_lineMap
@[simp]
theorem snd_lineMap (p₀ p₁ : P1 × P2) (c : k) : (lineMap p₀ p₁ c).2 = lineMap p₀.2 p₁.2 c 
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] (p₀ p₁ : P1 × P2) (c : k)
Goal: ((AffineMap.lineMap p₀ p₁) c).2 = (AffineMap.lineMap p₀.2 p₁.2) c
-- AffineMap.lineMap_symm
theorem lineMap_symm (p₀ p₁ : P1) :
    lineMap p₀ p₁ = (lineMap p₁ p₀).comp (lineMap (1 : k) (0 : k)) 
Arguments: {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [Ring k] [AddCommGroup V1] [Module k V1] [AffineSpace V1 P1] (p₀ p₁ : P1)
Goal: AffineMap.lineMap p₀ p₁ = (AffineMap.lineMap p₁ p₀).comp (AffineMap.lineMap 1 0)
-- AffineMap.lineMap_apply_one_sub
theorem lineMap_apply_one_sub (p₀ p₁ : P1) (c : k) : lineMap p₀ p₁ (1 - c) = lineMap p₁ p₀ c 
Arguments: {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [Ring k] [AddCommGroup V1] [Module k V1] [AffineSpace V1 P1] (p₀ p₁ : P1) (c : k)
Goal: (AffineMap.lineMap p₀ p₁) (1 - c) = (AffineMap.lineMap p₁ p₀) c
-- AffineMap.lineMap_vsub_left
@[simp]
theorem lineMap_vsub_left (p₀ p₁ : P1) (c : k) : lineMap p₀ p₁ c -ᵥ p₀ = c • (p₁ -ᵥ p₀) 
Arguments: {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [Ring k] [AddCommGroup V1] [Module k V1] [AffineSpace V1 P1] (p₀ p₁ : P1) (c : k)
Goal: (AffineMap.lineMap p₀ p₁) c -ᵥ p₀ = c • (p₁ -ᵥ p₀)
-- AffineMap.left_vsub_lineMap
@[simp]
theorem left_vsub_lineMap (p₀ p₁ : P1) (c : k) : p₀ -ᵥ lineMap p₀ p₁ c = c • (p₀ -ᵥ p₁) 
Arguments: {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [Ring k] [AddCommGroup V1] [Module k V1] [AffineSpace V1 P1] (p₀ p₁ : P1) (c : k)
Goal: p₀ -ᵥ (AffineMap.lineMap p₀ p₁) c = c • (p₀ -ᵥ p₁)
-- AffineMap.lineMap_vsub_right
@[simp]
theorem lineMap_vsub_right (p₀ p₁ : P1) (c : k) : lineMap p₀ p₁ c -ᵥ p₁ = (1 - c) • (p₀ -ᵥ p₁) 
Arguments: {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [Ring k] [AddCommGroup V1] [Module k V1] [AffineSpace V1 P1] (p₀ p₁ : P1) (c : k)
Goal: (AffineMap.lineMap p₀ p₁) c -ᵥ p₁ = (1 - c) • (p₀ -ᵥ p₁)
-- AffineMap.right_vsub_lineMap
@[simp]
theorem right_vsub_lineMap (p₀ p₁ : P1) (c : k) : p₁ -ᵥ lineMap p₀ p₁ c = (1 - c) • (p₁ -ᵥ p₀) 
Arguments: {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [Ring k] [AddCommGroup V1] [Module k V1] [AffineSpace V1 P1] (p₀ p₁ : P1) (c : k)
Goal: p₁ -ᵥ (AffineMap.lineMap p₀ p₁) c = (1 - c) • (p₁ -ᵥ p₀)
-- AffineMap.lineMap_vadd_lineMap
theorem lineMap_vadd_lineMap (v₁ v₂ : V1) (p₁ p₂ : P1) (c : k) :
    lineMap v₁ v₂ c +ᵥ lineMap p₁ p₂ c = lineMap (v₁ +ᵥ p₁) (v₂ +ᵥ p₂) c 
Arguments: {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [Ring k] [AddCommGroup V1] [Module k V1] [AffineSpace V1 P1] (v₁ v₂ : V1) (p₁ p₂ : P1) (c : k)
Goal: (AffineMap.lineMap v₁ v₂) c +ᵥ (AffineMap.lineMap p₁ p₂) c = (AffineMap.lineMap (v₁ +ᵥ p₁) (v₂ +ᵥ p₂)) c
-- AffineMap.lineMap_vsub_lineMap
theorem lineMap_vsub_lineMap (p₁ p₂ p₃ p₄ : P1) (c : k) :
    lineMap p₁ p₂ c -ᵥ lineMap p₃ p₄ c = lineMap (p₁ -ᵥ p₃) (p₂ -ᵥ p₄) c 
Arguments: {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [Ring k] [AddCommGroup V1] [Module k V1] [AffineSpace V1 P1] (p₁ p₂ p₃ p₄ : P1) (c : k)
Goal: (AffineMap.lineMap p₁ p₂) c -ᵥ (AffineMap.lineMap p₃ p₄) c = (AffineMap.lineMap (p₁ -ᵥ p₃) (p₂ -ᵥ p₄)) c
-- AffineMap.decomp
theorem decomp (f : V1 →ᵃ[k] V2) : (f : V1 → V2) = ⇑f.linear + fun _ => f 0 
Arguments: {k : Type u_1} {V1 : Type u_2} {V2 : Type u_4} [Ring k] [AddCommGroup V1] [Module k V1] [AddCommGroup V2] [Module k V2] (f : V1 →ᵃ[k] V2)
Goal: ⇑f = ⇑f.linear + fun x => f 0
-- AffineMap.decomp'
theorem decomp' (f : V1 →ᵃ[k] V2) : (f.linear : V1 → V2) = ⇑f - fun _ => f 0 
Arguments: {k : Type u_1} {V1 : Type u_2} {V2 : Type u_4} [Ring k] [AddCommGroup V1] [Module k V1] [AddCommGroup V2] [Module k V2] (f : V1 →ᵃ[k] V2)
Goal: ⇑f.linear = ⇑f - fun x => f 0
-- AffineMap.image_uIcc
theorem image_uIcc {k : Type*} [LinearOrderedField k] (f : k →ᵃ[k] k) (a b : k) :
    f '' Set.uIcc a b = Set.uIcc (f a) (f b) 
Arguments: {k : Type u_10} [LinearOrderedField k] (f : k →ᵃ[k] k) (a b : k)
Goal: ⇑f '' Set.uIcc a b = Set.uIcc (f a) (f b)
-- AffineMap.proj_apply
@[simp]
theorem proj_apply (i : ι) (f : ∀ i, P i) : @proj k _ ι V P _ _ _ i f = f i 
Arguments: {k : Type u_1} [Ring k] {ι : Type u_10} {V : ι → Type u_11} {P : ι → Type u_12} [(i : ι) → AddCommGroup (V i)] [(i : ι) → Module k (V i)] [(i : ι) → AffineSpace (V i) (P i)] (i : ι) (f : (i : ι) → P i)
Goal: (AffineMap.proj i) f = f i
-- AffineMap.proj_linear
@[simp]
theorem proj_linear (i : ι) : (@proj k _ ι V P _ _ _ i).linear = @LinearMap.proj k ι _ V _ _ i 
Arguments: {k : Type u_1} [Ring k] {ι : Type u_10} {V : ι → Type u_11} {P : ι → Type u_12} [(i : ι) → AddCommGroup (V i)] [(i : ι) → Module k (V i)] [(i : ι) → AffineSpace (V i) (P i)] (i : ι)
Goal: (AffineMap.proj i).linear = LinearMap.proj i
-- AffineMap.pi_lineMap_apply
theorem pi_lineMap_apply (f g : ∀ i, P i) (c : k) (i : ι) :
    lineMap f g c i = lineMap (f i) (g i) c 
Arguments: {k : Type u_1} [Ring k] {ι : Type u_10} {V : ι → Type u_11} {P : ι → Type u_12} [(i : ι) → AddCommGroup (V i)] [(i : ι) → Module k (V i)] [(i : ι) → AffineSpace (V i) (P i)] (f g : (i : ι) → P i) (c : k) (i : ι)
Goal: (AffineMap.lineMap f g) c i = (AffineMap.lineMap (f i) (g i)) c
-- AffineMap.toConstProdLinearMap_apply
@[simps]
def toConstProdLinearMap : (V1 →ᵃ[k] V2) ≃ₗ[R] V2 × (V1 →ₗ[k] V2) 
Arguments: (R : Type u_1) {k : Type u_2} {V1 : Type u_3} {V2 : Type u_5} [Ring k] [AddCommGroup V1] [AddCommGroup V2] [Module k V1] [Module k V2] [Semiring R] [Module R V2] [SMulCommClass k R V2] (f : V1 →ᵃ[k] V2)
Goal: (AffineMap.toConstProdLinearMap R) f = (f 0, f.linear)
-- AffineMap.toConstProdLinearMap_symm_apply
@[simps]
def toConstProdLinearMap : (V1 →ᵃ[k] V2) ≃ₗ[R] V2 × (V1 →ₗ[k] V2) 
Arguments: (R : Type u_1) {k : Type u_2} {V1 : Type u_3} {V2 : Type u_5} [Ring k] [AddCommGroup V1] [AddCommGroup V2] [Module k V1] [Module k V2] [Semiring R] [Module R V2] [SMulCommClass k R V2] (p : V2 × (V1 →ₗ[k] V2))
Goal: (AffineMap.toConstProdLinearMap R).symm p = p.2.toAffineMap + AffineMap.const k V1 p.1
-- AffineMap.pi_apply
@[simp]
theorem pi_apply (c : P1) (i : ι) : pi fp c i = fp i c 
Arguments: {k : Type u_2} {V1 : Type u_3} {P1 : Type u_4} [Ring k] [AddCommGroup V1] [AffineSpace V1 P1] [Module k V1] {ι : Type u_9} {φv : ι → Type u_10} {φp : ι → Type u_11} [(i : ι) → AddCommGroup (φv i)] [(i : ι) → Module k (φv i)] [(i : ι) → AffineSpace (φv i) (φp i)] (fp : (i : ι) → P1 →ᵃ[k] φp i) (c : P1) (i : ι)
Goal: (AffineMap.pi fp) c i = (fp i) c
-- AffineMap.pi_comp
theorem pi_comp (g : P3 →ᵃ[k] P1) : (pi fp).comp g = pi (fun i => (fp i).comp g) 
Arguments: {k : Type u_2} {V1 : Type u_3} {P1 : Type u_4} {V3 : Type u_7} {P3 : Type u_8} [Ring k] [AddCommGroup V1] [AffineSpace V1 P1] [AddCommGroup V3] [AffineSpace V3 P3] [Module k V1] [Module k V3] {ι : Type u_9} {φv : ι → Type u_10} {φp : ι → Type u_11} [(i : ι) → AddCommGroup (φv i)] [(i : ι) → Module k (φv i)] [(i : ι) → AffineSpace (φv i) (φp i)] (fp : (i : ι) → P1 →ᵃ[k] φp i) (g : P3 →ᵃ[k] P1)
Goal: (AffineMap.pi fp).comp g = AffineMap.pi fun i => (fp i).comp g
-- AffineMap.pi_eq_zero
theorem pi_eq_zero : pi fv = 0 ↔ ∀ i, fv i = 0 
Arguments: {k : Type u_2} {V1 : Type u_3} {P1 : Type u_4} [Ring k] [AddCommGroup V1] [AffineSpace V1 P1] [Module k V1] {ι : Type u_9} {φv : ι → Type u_10} [(i : ι) → AddCommGroup (φv i)] [(i : ι) → Module k (φv i)] (fv : (i : ι) → P1 →ᵃ[k] φv i)
Goal: AffineMap.pi fv = 0 ↔ ∀ (i : ι), fv i = 0
-- AffineMap.pi_zero
theorem pi_zero : pi (fun _ ↦ 0 : (i : ι) → P1 →ᵃ[k] φv i) = 0 
Arguments: {k : Type u_2} {V1 : Type u_3} {P1 : Type u_4} [Ring k] [AddCommGroup V1] [AffineSpace V1 P1] [Module k V1] {ι : Type u_9} {φv : ι → Type u_10} [(i : ι) → AddCommGroup (φv i)] [(i : ι) → Module k (φv i)]
Goal: (AffineMap.pi fun x => 0) = 0
-- AffineMap.proj_pi
theorem proj_pi (i : ι) : (proj i).comp (pi fp) = fp i 
Arguments: {k : Type u_2} {V1 : Type u_3} {P1 : Type u_4} [Ring k] [AddCommGroup V1] [AffineSpace V1 P1] [Module k V1] {ι : Type u_9} {φv : ι → Type u_10} {φp : ι → Type u_11} [(i : ι) → AddCommGroup (φv i)] [(i : ι) → Module k (φv i)] [(i : ι) → AffineSpace (φv i) (φp i)] (fp : (i : ι) → P1 →ᵃ[k] φp i) (i : ι)
Goal: (AffineMap.proj i).comp (AffineMap.pi fp) = fp i
-- AffineMap.pi_ext_zero
theorem pi_ext_zero (h : ∀ i x, f (Pi.single i x) = g (Pi.single i x)) (h₂ : f 0 = g 0) :
    f = g 
Arguments: {k : Type u_2} {V2 : Type u_5} {P2 : Type u_6} [Ring k] [AddCommGroup V2] [AffineSpace V2 P2] [Module k V2] {ι : Type u_9} {φv : ι → Type u_10} [(i : ι) → AddCommGroup (φv i)] [(i : ι) → Module k (φv i)] [Finite ι] [DecidableEq ι] {f g : ((i : ι) → φv i) →ᵃ[k] P2} (h : ∀ (i : ι) (x : φv i), f (Pi.single i x) = g (Pi.single i x)) (h₂ : f 0 = g 0)
Goal: f = g
-- AffineMap.pi_ext_nonempty
theorem pi_ext_nonempty [Nonempty ι] (h : ∀ i x, f (Pi.single i x) = g (Pi.single i x)) :
    f = g 
Arguments: {k : Type u_2} {V2 : Type u_5} {P2 : Type u_6} [Ring k] [AddCommGroup V2] [AffineSpace V2 P2] [Module k V2] {ι : Type u_9} {φv : ι → Type u_10} [(i : ι) → AddCommGroup (φv i)] [(i : ι) → Module k (φv i)] [Finite ι] [DecidableEq ι] {f g : ((i : ι) → φv i) →ᵃ[k] P2} [Nonempty ι] (h : ∀ (i : ι) (x : φv i), f (Pi.single i x) = g (Pi.single i x))
Goal: f = g
-- AffineMap.pi_ext_nonempty'
@[ext]
theorem pi_ext_nonempty' [Nonempty ι] (h : ∀ i, f.comp (LinearMap.single i).toAffineMap =
    g.comp (LinearMap.single i).toAffineMap) : f = g 
Arguments: {k : Type u_2} {V2 : Type u_5} {P2 : Type u_6} [Ring k] [AddCommGroup V2] [AffineSpace V2 P2] [Module k V2] {ι : Type u_9} {φv : ι → Type u_10} [(i : ι) → AddCommGroup (φv i)] [(i : ι) → Module k (φv i)] [Finite ι] [DecidableEq ι] {f g : ((i : ι) → φv i) →ᵃ[k] P2} [Nonempty ι] (h : ∀ (i : ι), f.comp (LinearMap.single i).toAffineMap = g.comp (LinearMap.single i).toAffineMap)
Goal: f = g
-- AffineMap.homothety_def
theorem homothety_def (c : P1) (r : k) :
    homothety c r = r • (id k P1 -ᵥ const k P1 c) +ᵥ const k P1 c 
Arguments: {k : Type u_2} {V1 : Type u_3} {P1 : Type u_4} [CommRing k] [AddCommGroup V1] [AffineSpace V1 P1] [Module k V1] (c : P1) (r : k)
Goal: AffineMap.homothety c r = r • (AffineMap.id k P1 -ᵥ AffineMap.const k P1 c) +ᵥ AffineMap.const k P1 c
-- AffineMap.homothety_apply
theorem homothety_apply (c : P1) (r : k) (p : P1) : homothety c r p = r • (p -ᵥ c : V1) +ᵥ c 
Arguments: {k : Type u_2} {V1 : Type u_3} {P1 : Type u_4} [CommRing k] [AddCommGroup V1] [AffineSpace V1 P1] [Module k V1] (c : P1) (r : k) (p : P1)
Goal: (AffineMap.homothety c r) p = r • (p -ᵥ c) +ᵥ c
-- AffineMap.homothety_eq_lineMap
theorem homothety_eq_lineMap (c : P1) (r : k) (p : P1) : homothety c r p = lineMap c p r 
Arguments: {k : Type u_2} {V1 : Type u_3} {P1 : Type u_4} [CommRing k] [AddCommGroup V1] [AffineSpace V1 P1] [Module k V1] (c : P1) (r : k) (p : P1)
Goal: (AffineMap.homothety c r) p = (AffineMap.lineMap c p) r
-- AffineMap.homothety_one
@[simp]
theorem homothety_one (c : P1) : homothety c (1 : k) = id k P1 
Arguments: {k : Type u_2} {V1 : Type u_3} {P1 : Type u_4} [CommRing k] [AddCommGroup V1] [AffineSpace V1 P1] [Module k V1] (c : P1)
Goal: AffineMap.homothety c 1 = AffineMap.id k P1
-- AffineMap.homothety_apply_same
@[simp]
theorem homothety_apply_same (c : P1) (r : k) : homothety c r c = c 
Arguments: {k : Type u_2} {V1 : Type u_3} {P1 : Type u_4} [CommRing k] [AddCommGroup V1] [AffineSpace V1 P1] [Module k V1] (c : P1) (r : k)
Goal: (AffineMap.homothety c r) c = c
-- AffineMap.homothety_mul_apply
theorem homothety_mul_apply (c : P1) (r₁ r₂ : k) (p : P1) :
    homothety c (r₁ * r₂) p = homothety c r₁ (homothety c r₂ p) 
Arguments: {k : Type u_2} {V1 : Type u_3} {P1 : Type u_4} [CommRing k] [AddCommGroup V1] [AffineSpace V1 P1] [Module k V1] (c : P1) (r₁ r₂ : k) (p : P1)
Goal: (AffineMap.homothety c (r₁ * r₂)) p = (AffineMap.homothety c r₁) ((AffineMap.homothety c r₂) p)
-- AffineMap.homothety_mul
theorem homothety_mul (c : P1) (r₁ r₂ : k) :
    homothety c (r₁ * r₂) = (homothety c r₁).comp (homothety c r₂) 
Arguments: {k : Type u_2} {V1 : Type u_3} {P1 : Type u_4} [CommRing k] [AddCommGroup V1] [AffineSpace V1 P1] [Module k V1] (c : P1) (r₁ r₂ : k)
Goal: AffineMap.homothety c (r₁ * r₂) = (AffineMap.homothety c r₁).comp (AffineMap.homothety c r₂)
-- AffineMap.homothety_zero
@[simp]
theorem homothety_zero (c : P1) : homothety c (0 : k) = const k P1 c 
Arguments: {k : Type u_2} {V1 : Type u_3} {P1 : Type u_4} [CommRing k] [AddCommGroup V1] [AffineSpace V1 P1] [Module k V1] (c : P1)
Goal: AffineMap.homothety c 0 = AffineMap.const k P1 c
-- AffineMap.homothety_add
@[simp]
theorem homothety_add (c : P1) (r₁ r₂ : k) :
    homothety c (r₁ + r₂) = r₁ • (id k P1 -ᵥ const k P1 c) +ᵥ homothety c r₂ 
Arguments: {k : Type u_2} {V1 : Type u_3} {P1 : Type u_4} [CommRing k] [AddCommGroup V1] [AffineSpace V1 P1] [Module k V1] (c : P1) (r₁ r₂ : k)
Goal: AffineMap.homothety c (r₁ + r₂) = r₁ • (AffineMap.id k P1 -ᵥ AffineMap.const k P1 c) +ᵥ AffineMap.homothety c r₂
-- AffineMap.coe_homothetyHom
@[simp]
theorem coe_homothetyHom (c : P1) : ⇑(homothetyHom c : k →* _) = homothety c 
Arguments: {k : Type u_2} {V1 : Type u_3} {P1 : Type u_4} [CommRing k] [AddCommGroup V1] [AffineSpace V1 P1] [Module k V1] (c : P1)
Goal: ⇑(AffineMap.homothetyHom c) = AffineMap.homothety c
-- AffineMap.coe_homothetyAffine
@[simp]
theorem coe_homothetyAffine (c : P1) : ⇑(homothetyAffine c : k →ᵃ[k] _) = homothety c 
Arguments: {k : Type u_2} {V1 : Type u_3} {P1 : Type u_4} [CommRing k] [AddCommGroup V1] [AffineSpace V1 P1] [Module k V1] (c : P1)
Goal: ⇑(AffineMap.homothetyAffine c) = AffineMap.homothety c
-- Convex.combo_affine_apply
theorem Convex.combo_affine_apply {x y : E} {a b : 𝕜} {f : E →ᵃ[𝕜] F} (h : a + b = 1) :
    f (a • x + b • y) = a • f x + b • f y 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [Ring 𝕜] [AddCommGroup E] [AddCommGroup F] [Module 𝕜 E] [Module 𝕜 F] {x y : E} {a b : 𝕜} {f : E →ᵃ[𝕜] F} (h : a + b = 1)
Goal: f (a • x + b • y) = a • f x + b • f y
-- Matrix.val_planeConformalMatrix
@[simps! (config := .asFn) val]
def planeConformalMatrix {R} [Field R] (a b : R) (hab : a ^ 2 + b ^ 2 ≠ 0) :
    Matrix.GeneralLinearGroup (Fin 2) R 
Arguments: {R : Type u_1} [Field R] (a b : R) (hab : a ^ 2 + b ^ 2 ≠ 0)
Goal: ↑(Matrix.planeConformalMatrix a b hab) = !![a, -b; b, a]
-- Matrix.transpose_hasOrthogonalRows_iff_hasOrthogonalCols
@[simp]
theorem transpose_hasOrthogonalRows_iff_hasOrthogonalCols [Fintype m] :
    Aᵀ.HasOrthogonalRows ↔ A.HasOrthogonalCols 
Arguments: {α : Type u_1} {n : Type u_2} {m : Type u_3} [Mul α] [AddCommMonoid α] (A : Matrix m n α) [Fintype m]
Goal: A.transpose.HasOrthogonalRows ↔ A.HasOrthogonalCols
-- Matrix.transpose_hasOrthogonalCols_iff_hasOrthogonalRows
@[simp]
theorem transpose_hasOrthogonalCols_iff_hasOrthogonalRows [Fintype n] :
    Aᵀ.HasOrthogonalCols ↔ A.HasOrthogonalRows 
Arguments: {α : Type u_1} {n : Type u_2} {m : Type u_3} [Mul α] [AddCommMonoid α] (A : Matrix m n α) [Fintype n]
Goal: A.transpose.HasOrthogonalCols ↔ A.HasOrthogonalRows
-- Matrix.HasOrthogonalRows.hasOrthogonalCols
theorem HasOrthogonalRows.hasOrthogonalCols [Fintype m] (h : Aᵀ.HasOrthogonalRows) :
    A.HasOrthogonalCols 
Arguments: {α : Type u_1} {n : Type u_2} {m : Type u_3} [Mul α] [AddCommMonoid α] {A : Matrix m n α} [Fintype m] (h : A.transpose.HasOrthogonalRows)
Goal: A.HasOrthogonalCols
-- Matrix.HasOrthogonalCols.transpose_hasOrthogonalRows
theorem HasOrthogonalCols.transpose_hasOrthogonalRows [Fintype m] (h : A.HasOrthogonalCols) :
    Aᵀ.HasOrthogonalRows 
Arguments: {α : Type u_1} {n : Type u_2} {m : Type u_3} [Mul α] [AddCommMonoid α] {A : Matrix m n α} [Fintype m] (h : A.HasOrthogonalCols)
Goal: A.transpose.HasOrthogonalRows
-- Matrix.HasOrthogonalCols.hasOrthogonalRows
theorem HasOrthogonalCols.hasOrthogonalRows [Fintype n] (h : Aᵀ.HasOrthogonalCols) :
    A.HasOrthogonalRows 
Arguments: {α : Type u_1} {n : Type u_2} {m : Type u_3} [Mul α] [AddCommMonoid α] {A : Matrix m n α} [Fintype n] (h : A.transpose.HasOrthogonalCols)
Goal: A.HasOrthogonalRows
-- Matrix.HasOrthogonalRows.transpose_hasOrthogonalCols
theorem HasOrthogonalRows.transpose_hasOrthogonalCols [Fintype n] (h : A.HasOrthogonalRows) :
    Aᵀ.HasOrthogonalCols 
Arguments: {α : Type u_1} {n : Type u_2} {m : Type u_3} [Mul α] [AddCommMonoid α] {A : Matrix m n α} [Fintype n] (h : A.HasOrthogonalRows)
Goal: A.transpose.HasOrthogonalCols
-- LinearMap.toMatrix_transpose
@[simp]
theorem LinearMap.toMatrix_transpose (u : V₁ →ₗ[K] V₂) :
    LinearMap.toMatrix B₂.dualBasis B₁.dualBasis (Module.Dual.transpose (R := K) u) =
      (LinearMap.toMatrix B₁ B₂ u)ᵀ 
Arguments: {K : Type u_1} {V₁ : Type u_2} {V₂ : Type u_3} {ι₁ : Type u_4} {ι₂ : Type u_5} [Field K] [AddCommGroup V₁] [Module K V₁] [AddCommGroup V₂] [Module K V₂] [Fintype ι₁] [Fintype ι₂] [DecidableEq ι₁] [DecidableEq ι₂] {B₁ : Basis ι₁ K V₁} {B₂ : Basis ι₂ K V₂} (u : V₁ →ₗ[K] V₂)
Goal: (LinearMap.toMatrix B₂.dualBasis B₁.dualBasis) (Module.Dual.transpose u) = ((LinearMap.toMatrix B₁ B₂) u)ᵀ