ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- NormedAddGroupHom.coe_inj_iff
theorem coe_inj_iff : f = g ↔ (f : V₁ → V₂) = g 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] {f g : NormedAddGroupHom V₁ V₂}
Goal: f = g ↔ ⇑f = ⇑g
-- NormedAddGroupHom.ext
@[ext]
theorem ext (H : ∀ x, f x = g x) : f = g 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] {f g : NormedAddGroupHom V₁ V₂} (H : ∀ (x : V₁), f x = g x)
Goal: f = g
-- NormedAddGroupHom.ext_iff
protected theorem ext_iff : f = g ↔ ∀ x, f x = g x 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] {f g : NormedAddGroupHom V₁ V₂}
Goal: f = g ↔ ∀ (x : V₁), f x = g x
-- NormedAddGroupHom.toFun_eq_coe
@[simp]
theorem toFun_eq_coe : f.toFun = f 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] (f : NormedAddGroupHom V₁ V₂)
Goal: f.toFun = ⇑f
-- NormedAddGroupHom.coe_mk
theorem coe_mk (f) (h₁) (h₂) (h₃) : ⇑(⟨f, h₁, h₂, h₃⟩ : NormedAddGroupHom V₁ V₂) = f 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] (f : V₁ → V₂) (h₁ : ∀ (v₁ v₂ : V₁), f (v₁ + v₂) = f v₁ + f v₂) (h₂ : ℝ) (h₃ : ∀ (v : V₁), ‖f v‖ ≤ h₂ * ‖v‖)
Goal: ⇑{ toFun := f, map_add' := h₁, bound' := ⋯ } = f
-- NormedAddGroupHom.coe_mkNormedAddGroupHom
@[simp]
theorem coe_mkNormedAddGroupHom (f : V₁ →+ V₂) (C) (hC) : ⇑(f.mkNormedAddGroupHom C hC) = f 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] (f : V₁ →+ V₂) (C : ℝ) (hC : ∀ (v : V₁), ‖f v‖ ≤ C * ‖v‖)
Goal: ⇑(f.mkNormedAddGroupHom C hC) = ⇑f
-- NormedAddGroupHom.coe_mkNormedAddGroupHom'
@[simp]
theorem coe_mkNormedAddGroupHom' (f : V₁ →+ V₂) (C) (hC) : ⇑(f.mkNormedAddGroupHom' C hC) = f 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] (f : V₁ →+ V₂) (C : ℝ≥0) (hC : ∀ (x : V₁), ‖f x‖₊ ≤ C * ‖x‖₊)
Goal: ⇑(f.mkNormedAddGroupHom' C hC) = ⇑f
-- NormedAddGroupHom.coe_toAddMonoidHom
@[simp]
theorem coe_toAddMonoidHom : ⇑f.toAddMonoidHom = f 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] (f : NormedAddGroupHom V₁ V₂)
Goal: ⇑f.toAddMonoidHom = ⇑f
-- NormedAddGroupHom.toAddMonoidHom_injective
theorem toAddMonoidHom_injective :
    Function.Injective (@NormedAddGroupHom.toAddMonoidHom V₁ V₂ _ _) 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂]
Goal: Function.Injective NormedAddGroupHom.toAddMonoidHom
-- NormedAddGroupHom.mk_toAddMonoidHom
@[simp]
theorem mk_toAddMonoidHom (f) (h₁) (h₂) :
    (⟨f, h₁, h₂⟩ : NormedAddGroupHom V₁ V₂).toAddMonoidHom = AddMonoidHom.mk' f h₁ 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] (f : V₁ → V₂) (h₁ : ∀ (v₁ v₂ : V₁), f (v₁ + v₂) = f v₁ + f v₂) (h₂ : ∃ C, ∀ (v : V₁), ‖f v‖ ≤ C * ‖v‖)
Goal: { toFun := f, map_add' := h₁, bound' := h₂ }.toAddMonoidHom = AddMonoidHom.mk' f h₁
-- NormedAddGroupHom.bound
theorem bound : ∃ C, 0 < C ∧ ∀ x, ‖f x‖ ≤ C * ‖x‖ 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] (f : NormedAddGroupHom V₁ V₂)
Goal: ∃ C, 0 < C ∧ ∀ (x : V₁), ‖f x‖ ≤ C * ‖x‖
-- NormedAddGroupHom.antilipschitz_of_norm_ge
theorem antilipschitz_of_norm_ge {K : ℝ≥0} (h : ∀ x, ‖x‖ ≤ K * ‖f x‖) : AntilipschitzWith K f 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] (f : NormedAddGroupHom V₁ V₂) {K : ℝ≥0} (h : ∀ (x : V₁), ‖x‖ ≤ ↑K * ‖f x‖)
Goal: AntilipschitzWith K ⇑f
-- NormedAddGroupHom.SurjectiveOnWith.mono
theorem SurjectiveOnWith.mono {f : NormedAddGroupHom V₁ V₂} {K : AddSubgroup V₂} {C C' : ℝ}
    (h : f.SurjectiveOnWith K C) (H : C ≤ C') : f.SurjectiveOnWith K C' 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] {f : NormedAddGroupHom V₁ V₂} {K : AddSubgroup V₂} {C C' : ℝ} (h : f.SurjectiveOnWith K C) (H : C ≤ C')
Goal: f.SurjectiveOnWith K C'
-- NormedAddGroupHom.SurjectiveOnWith.exists_pos
theorem SurjectiveOnWith.exists_pos {f : NormedAddGroupHom V₁ V₂} {K : AddSubgroup V₂} {C : ℝ}
    (h : f.SurjectiveOnWith K C) : ∃ C' > 0, f.SurjectiveOnWith K C' 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] {f : NormedAddGroupHom V₁ V₂} {K : AddSubgroup V₂} {C : ℝ} (h : f.SurjectiveOnWith K C)
Goal: ∃ C' > 0, f.SurjectiveOnWith K C'
-- NormedAddGroupHom.SurjectiveOnWith.surjOn
theorem SurjectiveOnWith.surjOn {f : NormedAddGroupHom V₁ V₂} {K : AddSubgroup V₂} {C : ℝ}
    (h : f.SurjectiveOnWith K C) : Set.SurjOn f Set.univ K 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] {f : NormedAddGroupHom V₁ V₂} {K : AddSubgroup V₂} {C : ℝ} (h : f.SurjectiveOnWith K C)
Goal: Set.SurjOn (⇑f) Set.univ ↑K
-- NormedAddGroupHom.norm_def
theorem norm_def : ‖f‖ = sInf { c | 0 ≤ c ∧ ∀ x, ‖f x‖ ≤ c * ‖x‖ } 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] (f : NormedAddGroupHom V₁ V₂)
Goal: ‖f‖ = sInf {c | 0 ≤ c ∧ ∀ (x : V₁), ‖f x‖ ≤ c * ‖x‖}
-- NormedAddGroupHom.bounds_nonempty
theorem bounds_nonempty {f : NormedAddGroupHom V₁ V₂} :
    ∃ c, c ∈ { c | 0 ≤ c ∧ ∀ x, ‖f x‖ ≤ c * ‖x‖ } 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] {f : NormedAddGroupHom V₁ V₂}
Goal: ∃ c, c ∈ {c | 0 ≤ c ∧ ∀ (x : V₁), ‖f x‖ ≤ c * ‖x‖}
-- NormedAddGroupHom.bounds_bddBelow
theorem bounds_bddBelow {f : NormedAddGroupHom V₁ V₂} :
    BddBelow { c | 0 ≤ c ∧ ∀ x, ‖f x‖ ≤ c * ‖x‖ } 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] {f : NormedAddGroupHom V₁ V₂}
Goal: BddBelow {c | 0 ≤ c ∧ ∀ (x : V₁), ‖f x‖ ≤ c * ‖x‖}
-- NormedAddGroupHom.opNorm_nonneg
theorem opNorm_nonneg : 0 ≤ ‖f‖ 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] (f : NormedAddGroupHom V₁ V₂)
Goal: 0 ≤ ‖f‖
-- NormedAddGroupHom.le_opNorm
theorem le_opNorm (x : V₁) : ‖f x‖ ≤ ‖f‖ * ‖x‖ 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] (f : NormedAddGroupHom V₁ V₂) (x : V₁)
Goal: ‖f x‖ ≤ ‖f‖ * ‖x‖
-- NormedAddGroupHom.le_opNorm_of_le
theorem le_opNorm_of_le {c : ℝ} {x} (h : ‖x‖ ≤ c) : ‖f x‖ ≤ ‖f‖ * c 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] (f : NormedAddGroupHom V₁ V₂) {c : ℝ} {x : V₁} (h : ‖x‖ ≤ c)
Goal: ‖f x‖ ≤ ‖f‖ * c
-- NormedAddGroupHom.le_of_opNorm_le
theorem le_of_opNorm_le {c : ℝ} (h : ‖f‖ ≤ c) (x : V₁) : ‖f x‖ ≤ c * ‖x‖ 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] (f : NormedAddGroupHom V₁ V₂) {c : ℝ} (h : ‖f‖ ≤ c) (x : V₁)
Goal: ‖f x‖ ≤ c * ‖x‖
-- NormedAddGroupHom.lipschitz
theorem lipschitz : LipschitzWith ⟨‖f‖, opNorm_nonneg f⟩ f 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] (f : NormedAddGroupHom V₁ V₂)
Goal: LipschitzWith ⟨‖f‖, ⋯⟩ ⇑f
-- NormedAddGroupHom.uniformContinuous
protected theorem uniformContinuous (f : NormedAddGroupHom V₁ V₂) : UniformContinuous f 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] (f : NormedAddGroupHom V₁ V₂)
Goal: UniformContinuous ⇑f
-- NormedAddGroupHom.continuous
@[continuity]
protected theorem continuous (f : NormedAddGroupHom V₁ V₂) : Continuous f 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] (f : NormedAddGroupHom V₁ V₂)
Goal: Continuous ⇑f
-- NormedAddGroupHom.ratio_le_opNorm
theorem ratio_le_opNorm (x : V₁) : ‖f x‖ / ‖x‖ ≤ ‖f‖ 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] (f : NormedAddGroupHom V₁ V₂) (x : V₁)
Goal: ‖f x‖ / ‖x‖ ≤ ‖f‖
-- NormedAddGroupHom.opNorm_le_bound
theorem opNorm_le_bound {M : ℝ} (hMp : 0 ≤ M) (hM : ∀ x, ‖f x‖ ≤ M * ‖x‖) : ‖f‖ ≤ M 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] (f : NormedAddGroupHom V₁ V₂) {M : ℝ} (hMp : 0 ≤ M) (hM : ∀ (x : V₁), ‖f x‖ ≤ M * ‖x‖)
Goal: ‖f‖ ≤ M
-- NormedAddGroupHom.opNorm_eq_of_bounds
theorem opNorm_eq_of_bounds {M : ℝ} (M_nonneg : 0 ≤ M) (h_above : ∀ x, ‖f x‖ ≤ M * ‖x‖)
    (h_below : ∀ N ≥ 0, (∀ x, ‖f x‖ ≤ N * ‖x‖) → M ≤ N) : ‖f‖ = M 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] (f : NormedAddGroupHom V₁ V₂) {M : ℝ} (M_nonneg : 0 ≤ M) (h_above : ∀ (x : V₁), ‖f x‖ ≤ M * ‖x‖) (h_below : ∀ N ≥ 0, (∀ (x : V₁), ‖f x‖ ≤ N * ‖x‖) → M ≤ N)
Goal: ‖f‖ = M
-- NormedAddGroupHom.opNorm_le_of_lipschitz
theorem opNorm_le_of_lipschitz {f : NormedAddGroupHom V₁ V₂} {K : ℝ≥0} (hf : LipschitzWith K f) :
    ‖f‖ ≤ K 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] {f : NormedAddGroupHom V₁ V₂} {K : ℝ≥0} (hf : LipschitzWith K ⇑f)
Goal: ‖f‖ ≤ ↑K
-- NormedAddGroupHom.mkNormedAddGroupHom_norm_le
theorem mkNormedAddGroupHom_norm_le (f : V₁ →+ V₂) {C : ℝ} (hC : 0 ≤ C) (h : ∀ x, ‖f x‖ ≤ C * ‖x‖) :
    ‖f.mkNormedAddGroupHom C h‖ ≤ C 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] (f : V₁ →+ V₂) {C : ℝ} (hC : 0 ≤ C) (h : ∀ (x : V₁), ‖f x‖ ≤ C * ‖x‖)
Goal: ‖f.mkNormedAddGroupHom C h‖ ≤ C
-- NormedAddGroupHom.ofLipschitz_norm_le
theorem ofLipschitz_norm_le (f : V₁ →+ V₂) {K : ℝ≥0} (h : LipschitzWith K f) :
    ‖ofLipschitz f h‖ ≤ K 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] (f : V₁ →+ V₂) {K : ℝ≥0} (h : LipschitzWith K ⇑f)
Goal: ‖NormedAddGroupHom.ofLipschitz f h‖ ≤ ↑K
-- NormedAddGroupHom.mkNormedAddGroupHom_norm_le'
theorem mkNormedAddGroupHom_norm_le' (f : V₁ →+ V₂) {C : ℝ} (h : ∀ x, ‖f x‖ ≤ C * ‖x‖) :
    ‖f.mkNormedAddGroupHom C h‖ ≤ max C 0 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] (f : V₁ →+ V₂) {C : ℝ} (h : ∀ (x : V₁), ‖f x‖ ≤ C * ‖x‖)
Goal: ‖f.mkNormedAddGroupHom C h‖ ≤ max C 0
-- NormedAddGroupHom.opNorm_add_le
theorem opNorm_add_le : ‖f + g‖ ≤ ‖f‖ + ‖g‖ 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] (f g : NormedAddGroupHom V₁ V₂)
Goal: ‖f + g‖ ≤ ‖f‖ + ‖g‖
-- NormedAddGroupHom.coe_add
@[simp]
theorem coe_add (f g : NormedAddGroupHom V₁ V₂) : ⇑(f + g) = f + g 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] (f g : NormedAddGroupHom V₁ V₂)
Goal: ⇑(f + g) = ⇑f + ⇑g
-- NormedAddGroupHom.add_apply
@[simp]
theorem add_apply (f g : NormedAddGroupHom V₁ V₂) (v : V₁) :
    (f + g) v = f v + g v 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] (f g : NormedAddGroupHom V₁ V₂) (v : V₁)
Goal: (f + g) v = f v + g v
-- NormedAddGroupHom.opNorm_zero
theorem opNorm_zero : ‖(0 : NormedAddGroupHom V₁ V₂)‖ = 0 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂]
Goal: ‖0‖ = 0
-- NormedAddGroupHom.opNorm_zero_iff
theorem opNorm_zero_iff {V₁ V₂ : Type*} [NormedAddCommGroup V₁] [NormedAddCommGroup V₂]
    {f : NormedAddGroupHom V₁ V₂} : ‖f‖ = 0 ↔ f = 0 
Arguments: {V₁ : Type u_5} {V₂ : Type u_6} [NormedAddCommGroup V₁] [NormedAddCommGroup V₂] {f : NormedAddGroupHom V₁ V₂}
Goal: ‖f‖ = 0 ↔ f = 0
-- NormedAddGroupHom.coe_zero
@[simp]
theorem coe_zero : ⇑(0 : NormedAddGroupHom V₁ V₂) = 0 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂]
Goal: ⇑0 = 0
-- NormedAddGroupHom.zero_apply
@[simp]
theorem zero_apply (v : V₁) : (0 : NormedAddGroupHom V₁ V₂) v = 0 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] (v : V₁)
Goal: 0 v = 0
-- NormedAddGroupHom.id_apply
@[simps!]
def id : NormedAddGroupHom V V 
Arguments: (V : Type u_1) [SeminormedAddCommGroup V]
Goal: ∀ (a : V), (NormedAddGroupHom.id V) a = a
-- NormedAddGroupHom.norm_id_le
theorem norm_id_le : ‖(id V : NormedAddGroupHom V V)‖ ≤ 1 
Arguments: (V : Type u_1) [SeminormedAddCommGroup V]
Goal: ‖NormedAddGroupHom.id V‖ ≤ 1
-- NormedAddGroupHom.norm_id_of_nontrivial_seminorm
theorem norm_id_of_nontrivial_seminorm (h : ∃ x : V, ‖x‖ ≠ 0) : ‖id V‖ = 1 
Arguments: (V : Type u_1) [SeminormedAddCommGroup V] (h : ∃ x, ‖x‖ ≠ 0)
Goal: ‖NormedAddGroupHom.id V‖ = 1
-- NormedAddGroupHom.norm_id
theorem norm_id {V : Type*} [NormedAddCommGroup V] [Nontrivial V] : ‖id V‖ = 1 
Arguments: {V : Type u_5} [NormedAddCommGroup V] [Nontrivial V]
Goal: ‖NormedAddGroupHom.id V‖ = 1
-- NormedAddGroupHom.coe_id
theorem coe_id : (NormedAddGroupHom.id V : V → V) = _root_.id 
Arguments: (V : Type u_1) [SeminormedAddCommGroup V]
Goal: ⇑(NormedAddGroupHom.id V) = id
-- NormedAddGroupHom.coe_neg
@[simp]
theorem coe_neg (f : NormedAddGroupHom V₁ V₂) : ⇑(-f) = -f 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] (f : NormedAddGroupHom V₁ V₂)
Goal: ⇑(-f) = -⇑f
-- NormedAddGroupHom.neg_apply
@[simp]
theorem neg_apply (f : NormedAddGroupHom V₁ V₂) (v : V₁) :
    (-f : NormedAddGroupHom V₁ V₂) v = -f v 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] (f : NormedAddGroupHom V₁ V₂) (v : V₁)
Goal: (-f) v = -f v
-- NormedAddGroupHom.opNorm_neg
theorem opNorm_neg (f : NormedAddGroupHom V₁ V₂) : ‖-f‖ = ‖f‖ 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] (f : NormedAddGroupHom V₁ V₂)
Goal: ‖-f‖ = ‖f‖
-- NormedAddGroupHom.coe_sub
@[simp]
theorem coe_sub (f g : NormedAddGroupHom V₁ V₂) : ⇑(f - g) = f - g 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] (f g : NormedAddGroupHom V₁ V₂)
Goal: ⇑(f - g) = ⇑f - ⇑g
-- NormedAddGroupHom.sub_apply
@[simp]
theorem sub_apply (f g : NormedAddGroupHom V₁ V₂) (v : V₁) :
    (f - g : NormedAddGroupHom V₁ V₂) v = f v - g v 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] (f g : NormedAddGroupHom V₁ V₂) (v : V₁)
Goal: (f - g) v = f v - g v
-- NormedAddGroupHom.coe_smul
@[simp]
theorem coe_smul (r : R) (f : NormedAddGroupHom V₁ V₂) : ⇑(r • f) = r • ⇑f 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] {R : Type u_5} [MonoidWithZero R] [DistribMulAction R V₂] [PseudoMetricSpace R] [BoundedSMul R V₂] (r : R) (f : NormedAddGroupHom V₁ V₂)
Goal: ⇑(r • f) = r • ⇑f
-- NormedAddGroupHom.smul_apply
@[simp]
theorem smul_apply (r : R) (f : NormedAddGroupHom V₁ V₂) (v : V₁) : (r • f) v = r • f v 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] {R : Type u_5} [MonoidWithZero R] [DistribMulAction R V₂] [PseudoMetricSpace R] [BoundedSMul R V₂] (r : R) (f : NormedAddGroupHom V₁ V₂) (v : V₁)
Goal: (r • f) v = r • f v
-- NormedAddGroupHom.coe_nsmul
@[simp]
theorem coe_nsmul (r : ℕ) (f : NormedAddGroupHom V₁ V₂) : ⇑(r • f) = r • ⇑f 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] (r : ℕ) (f : NormedAddGroupHom V₁ V₂)
Goal: ⇑(r • f) = r • ⇑f
-- NormedAddGroupHom.nsmul_apply
@[simp]
theorem nsmul_apply (r : ℕ) (f : NormedAddGroupHom V₁ V₂) (v : V₁) : (r • f) v = r • f v 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] (r : ℕ) (f : NormedAddGroupHom V₁ V₂) (v : V₁)
Goal: (r • f) v = r • f v
-- NormedAddGroupHom.coe_zsmul
@[simp]
theorem coe_zsmul (r : ℤ) (f : NormedAddGroupHom V₁ V₂) : ⇑(r • f) = r • ⇑f 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] (r : ℤ) (f : NormedAddGroupHom V₁ V₂)
Goal: ⇑(r • f) = r • ⇑f
-- NormedAddGroupHom.zsmul_apply
@[simp]
theorem zsmul_apply (r : ℤ) (f : NormedAddGroupHom V₁ V₂) (v : V₁) : (r • f) v = r • f v 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] (r : ℤ) (f : NormedAddGroupHom V₁ V₂) (v : V₁)
Goal: (r • f) v = r • f v
-- NormedAddGroupHom.coeAddHom_apply
@[simps]
def coeAddHom : NormedAddGroupHom V₁ V₂ →+ V₁ → V₂ 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂]
Goal: ∀ (a : NormedAddGroupHom V₁ V₂) (a_1 : V₁), NormedAddGroupHom.coeAddHom a a_1 = a a_1
-- NormedAddGroupHom.coe_sum
@[simp]
theorem coe_sum {ι : Type*} (s : Finset ι) (f : ι → NormedAddGroupHom V₁ V₂) :
    ⇑(∑ i ∈ s, f i) = ∑ i ∈ s, (f i : V₁ → V₂) 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] {ι : Type u_5} (s : Finset ι) (f : ι → NormedAddGroupHom V₁ V₂)
Goal: ⇑(∑ i ∈ s, f i) = ∑ i ∈ s, ⇑(f i)
-- NormedAddGroupHom.sum_apply
theorem sum_apply {ι : Type*} (s : Finset ι) (f : ι → NormedAddGroupHom V₁ V₂) (v : V₁) :
    (∑ i ∈ s, f i) v = ∑ i ∈ s, f i v 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] {ι : Type u_5} (s : Finset ι) (f : ι → NormedAddGroupHom V₁ V₂) (v : V₁)
Goal: (∑ i ∈ s, f i) v = ∑ i ∈ s, (f i) v
-- NormedAddGroupHom.comp_apply
@[simps!]
protected def comp (g : NormedAddGroupHom V₂ V₃) (f : NormedAddGroupHom V₁ V₂) :
    NormedAddGroupHom V₁ V₃ 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} {V₃ : Type u_4} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] [SeminormedAddCommGroup V₃] (g : NormedAddGroupHom V₂ V₃) (f : NormedAddGroupHom V₁ V₂)
Goal: ∀ (a : V₁), (g.comp f) a = g (f a)
-- NormedAddGroupHom.norm_comp_le
theorem norm_comp_le (g : NormedAddGroupHom V₂ V₃) (f : NormedAddGroupHom V₁ V₂) :
    ‖g.comp f‖ ≤ ‖g‖ * ‖f‖ 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} {V₃ : Type u_4} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] [SeminormedAddCommGroup V₃] (g : NormedAddGroupHom V₂ V₃) (f : NormedAddGroupHom V₁ V₂)
Goal: ‖g.comp f‖ ≤ ‖g‖ * ‖f‖
-- NormedAddGroupHom.norm_comp_le_of_le
theorem norm_comp_le_of_le {g : NormedAddGroupHom V₂ V₃} {C₁ C₂ : ℝ} (hg : ‖g‖ ≤ C₂)
    (hf : ‖f‖ ≤ C₁) : ‖g.comp f‖ ≤ C₂ * C₁ 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} {V₃ : Type u_4} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] [SeminormedAddCommGroup V₃] {f : NormedAddGroupHom V₁ V₂} {g : NormedAddGroupHom V₂ V₃} {C₁ C₂ : ℝ} (hg : ‖g‖ ≤ C₂) (hf : ‖f‖ ≤ C₁)
Goal: ‖g.comp f‖ ≤ C₂ * C₁
-- NormedAddGroupHom.norm_comp_le_of_le'
theorem norm_comp_le_of_le' {g : NormedAddGroupHom V₂ V₃} (C₁ C₂ C₃ : ℝ) (h : C₃ = C₂ * C₁)
    (hg : ‖g‖ ≤ C₂) (hf : ‖f‖ ≤ C₁) : ‖g.comp f‖ ≤ C₃ 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} {V₃ : Type u_4} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] [SeminormedAddCommGroup V₃] {f : NormedAddGroupHom V₁ V₂} {g : NormedAddGroupHom V₂ V₃} (C₁ C₂ C₃ : ℝ) (h : C₃ = C₂ * C₁) (hg : ‖g‖ ≤ C₂) (hf : ‖f‖ ≤ C₁)
Goal: ‖g.comp f‖ ≤ C₃
-- NormedAddGroupHom.comp_zero
@[simp]
theorem comp_zero (f : NormedAddGroupHom V₂ V₃) : f.comp (0 : NormedAddGroupHom V₁ V₂) = 0 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} {V₃ : Type u_4} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] [SeminormedAddCommGroup V₃] (f : NormedAddGroupHom V₂ V₃)
Goal: f.comp 0 = 0
-- NormedAddGroupHom.zero_comp
@[simp]
theorem zero_comp (f : NormedAddGroupHom V₁ V₂) : (0 : NormedAddGroupHom V₂ V₃).comp f = 0 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} {V₃ : Type u_4} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] [SeminormedAddCommGroup V₃] (f : NormedAddGroupHom V₁ V₂)
Goal: NormedAddGroupHom.comp 0 f = 0
-- NormedAddGroupHom.comp_assoc
theorem comp_assoc {V₄ : Type*} [SeminormedAddCommGroup V₄] (h : NormedAddGroupHom V₃ V₄)
    (g : NormedAddGroupHom V₂ V₃) (f : NormedAddGroupHom V₁ V₂) :
    (h.comp g).comp f = h.comp (g.comp f) 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} {V₃ : Type u_4} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] [SeminormedAddCommGroup V₃] {V₄ : Type u_5} [SeminormedAddCommGroup V₄] (h : NormedAddGroupHom V₃ V₄) (g : NormedAddGroupHom V₂ V₃) (f : NormedAddGroupHom V₁ V₂)
Goal: (h.comp g).comp f = h.comp (g.comp f)
-- NormedAddGroupHom.coe_comp
theorem coe_comp (f : NormedAddGroupHom V₁ V₂) (g : NormedAddGroupHom V₂ V₃) :
    (g.comp f : V₁ → V₃) = (g : V₂ → V₃) ∘ (f : V₁ → V₂) 
Arguments: {V₁ : Type u_2} {V₂ : Type u_3} {V₃ : Type u_4} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] [SeminormedAddCommGroup V₃] (f : NormedAddGroupHom V₁ V₂) (g : NormedAddGroupHom V₂ V₃)
Goal: ⇑(g.comp f) = ⇑g ∘ ⇑f
-- NormedAddGroupHom.incl_apply
@[simps!]
def incl (s : AddSubgroup V) : NormedAddGroupHom s V 
Arguments: {V : Type u_1} [SeminormedAddCommGroup V] (s : AddSubgroup V) (self : ↥s)
Goal: (NormedAddGroupHom.incl s) self = ↑self
-- NormedAddGroupHom.norm_incl
theorem norm_incl {V' : AddSubgroup V} (x : V') : ‖incl _ x‖ = ‖x‖ 
Arguments: {V : Type u_1} [SeminormedAddCommGroup V] {V' : AddSubgroup V} (x : ↥V')
Goal: ‖(NormedAddGroupHom.incl V') x‖ = ‖x‖
-- NormedAddGroupHom.mem_ker
theorem mem_ker (v : V₁) : v ∈ f.ker ↔ f v = 0 
Arguments: {V₁ : Type u_3} {V₂ : Type u_4} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] (f : NormedAddGroupHom V₁ V₂) (v : V₁)
Goal: v ∈ f.ker ↔ f v = 0
-- NormedAddGroupHom.ker.lift_apply_coe
@[simps]
def ker.lift (h : g.comp f = 0) : NormedAddGroupHom V₁ g.ker 
Arguments: {V₁ : Type u_3} {V₂ : Type u_4} {V₃ : Type u_5} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] [SeminormedAddCommGroup V₃] (f : NormedAddGroupHom V₁ V₂) (g : NormedAddGroupHom V₂ V₃) (h : g.comp f = 0) (v : V₁)
Goal: ↑((NormedAddGroupHom.ker.lift f g h) v) = f v
-- NormedAddGroupHom.ker.incl_comp_lift
@[simp]
theorem ker.incl_comp_lift (h : g.comp f = 0) : (incl g.ker).comp (ker.lift f g h) = f 
Arguments: {V₁ : Type u_3} {V₂ : Type u_4} {V₃ : Type u_5} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] [SeminormedAddCommGroup V₃] (f : NormedAddGroupHom V₁ V₂) (g : NormedAddGroupHom V₂ V₃) (h : g.comp f = 0)
Goal: (NormedAddGroupHom.incl g.ker).comp (NormedAddGroupHom.ker.lift f g h) = f
-- NormedAddGroupHom.ker_zero
@[simp]
theorem ker_zero : (0 : NormedAddGroupHom V₁ V₂).ker = ⊤ 
Arguments: {V₁ : Type u_3} {V₂ : Type u_4} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂]
Goal: NormedAddGroupHom.ker 0 = ⊤
-- NormedAddGroupHom.coe_ker
theorem coe_ker : (f.ker : Set V₁) = (f : V₁ → V₂) ⁻¹' {0} 
Arguments: {V₁ : Type u_3} {V₂ : Type u_4} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] (f : NormedAddGroupHom V₁ V₂)
Goal: ↑f.ker = ⇑f ⁻¹' {0}
-- NormedAddGroupHom.isClosed_ker
theorem isClosed_ker {V₂ : Type*} [NormedAddCommGroup V₂] (f : NormedAddGroupHom V₁ V₂) :
    IsClosed (f.ker : Set V₁) 
Arguments: {V₁ : Type u_3} [SeminormedAddCommGroup V₁] {V₂ : Type u_6} [NormedAddCommGroup V₂] (f : NormedAddGroupHom V₁ V₂)
Goal: IsClosed ↑f.ker
-- NormedAddGroupHom.mem_range
theorem mem_range (v : V₂) : v ∈ f.range ↔ ∃ w, f w = v 
Arguments: {V₁ : Type u_3} {V₂ : Type u_4} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] (f : NormedAddGroupHom V₁ V₂) (v : V₂)
Goal: v ∈ f.range ↔ ∃ w, f w = v
-- NormedAddGroupHom.mem_range_self
@[simp]
theorem mem_range_self (v : V₁) : f v ∈ f.range 
Arguments: {V₁ : Type u_3} {V₂ : Type u_4} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] (f : NormedAddGroupHom V₁ V₂) (v : V₁)
Goal: f v ∈ f.range
-- NormedAddGroupHom.comp_range
theorem comp_range : (g.comp f).range = AddSubgroup.map g.toAddMonoidHom f.range 
Arguments: {V₁ : Type u_3} {V₂ : Type u_4} {V₃ : Type u_5} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] [SeminormedAddCommGroup V₃] (f : NormedAddGroupHom V₁ V₂) (g : NormedAddGroupHom V₂ V₃)
Goal: (g.comp f).range = AddSubgroup.map g.toAddMonoidHom f.range
-- NormedAddGroupHom.incl_range
theorem incl_range (s : AddSubgroup V₁) : (incl s).range = s 
Arguments: {V₁ : Type u_3} [SeminormedAddCommGroup V₁] (s : AddSubgroup V₁)
Goal: (NormedAddGroupHom.incl s).range = s
-- NormedAddGroupHom.range_comp_incl_top
@[simp]
theorem range_comp_incl_top : (f.comp (incl (⊤ : AddSubgroup V₁))).range = f.range 
Arguments: {V₁ : Type u_3} {V₂ : Type u_4} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] (f : NormedAddGroupHom V₁ V₂)
Goal: (f.comp (NormedAddGroupHom.incl ⊤)).range = f.range