ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- SemilinearIsometryClass.diam_image
theorem diam_image [SemilinearIsometryClass 𝓕 σ₁₂ E E₂] (f : 𝓕) (s : Set E) :
    Metric.diam (f '' s) = Metric.diam s 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} {𝓕 : Type u_10} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] [FunLike 𝓕 E E₂] [SemilinearIsometryClass 𝓕 σ₁₂ E E₂] (f : 𝓕) (s : Set E)
Goal: Metric.diam (⇑f '' s) = Metric.diam s
-- SemilinearIsometryClass.diam_range
theorem diam_range [SemilinearIsometryClass 𝓕 σ₁₂ E E₂] (f : 𝓕) :
    Metric.diam (range f) = Metric.diam (univ : Set E) 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} {𝓕 : Type u_10} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] [FunLike 𝓕 E E₂] [SemilinearIsometryClass 𝓕 σ₁₂ E E₂] (f : 𝓕)
Goal: Metric.diam (Set.range ⇑f) = Metric.diam Set.univ
-- LinearIsometry.toLinearMap_injective
theorem toLinearMap_injective : Injective (toLinearMap : (E →ₛₗᵢ[σ₁₂] E₂) → E →ₛₗ[σ₁₂] E₂)
  
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂]
Goal: Function.Injective LinearIsometry.toLinearMap
-- LinearIsometry.toLinearMap_inj
@[simp]
theorem toLinearMap_inj {f g : E →ₛₗᵢ[σ₁₂] E₂} : f.toLinearMap = g.toLinearMap ↔ f = g 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] {f g : E →ₛₗᵢ[σ₁₂] E₂}
Goal: f.toLinearMap = g.toLinearMap ↔ f = g
-- LinearIsometry.coe_toLinearMap
@[simp]
theorem coe_toLinearMap : ⇑f.toLinearMap = f 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂)
Goal: ⇑f.toLinearMap = ⇑f
-- LinearIsometry.coe_mk
@[simp]
theorem coe_mk (f : E →ₛₗ[σ₁₂] E₂) (hf) : ⇑(mk f hf) = f 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗ[σ₁₂] E₂) (hf : ∀ (x : E), ‖f x‖ = ‖x‖)
Goal: ⇑{ toLinearMap := f, norm_map' := hf } = ⇑f
-- LinearIsometry.coe_injective
theorem coe_injective : @Injective (E →ₛₗᵢ[σ₁₂] E₂) (E → E₂) (fun f => f) 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂]
Goal: Function.Injective fun f => ⇑f
-- LinearIsometry.ext
@[ext]
theorem ext {f g : E →ₛₗᵢ[σ₁₂] E₂} (h : ∀ x, f x = g x) : f = g 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] {f g : E →ₛₗᵢ[σ₁₂] E₂} (h : ∀ (x : E), f x = g x)
Goal: f = g
-- LinearIsometry.congr_arg
protected theorem congr_arg {f : 𝓕} :
    ∀ {x x' : E}, x = x' → f x = f x'
  
Arguments: {E : Type u_5} {E₂ : Type u_6} {𝓕 : Type u_10} [FunLike 𝓕 E E₂] {f : 𝓕} {x x' : E}
Goal: x = x' → f x = f x'
-- LinearIsometry.congr_fun
protected theorem congr_fun {f g : 𝓕} (h : f = g) (x : E) :
    f x = g x 
Arguments: {E : Type u_5} {E₂ : Type u_6} {𝓕 : Type u_10} [FunLike 𝓕 E E₂] {f g : 𝓕} (h : f = g) (x : E)
Goal: f x = g x
-- LinearIsometry.map_zero
protected theorem map_zero : f 0 = 0 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂)
Goal: f 0 = 0
-- LinearIsometry.map_add
protected theorem map_add (x y : E) : f (x + y) = f x + f y 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) (x y : E)
Goal: f (x + y) = f x + f y
-- LinearIsometry.map_neg
protected theorem map_neg (x : E) : f (-x) = -f x 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) (x : E)
Goal: f (-x) = -f x
-- LinearIsometry.map_sub
protected theorem map_sub (x y : E) : f (x - y) = f x - f y 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) (x y : E)
Goal: f (x - y) = f x - f y
-- LinearIsometry.map_smulₛₗ
protected theorem map_smulₛₗ (c : R) (x : E) : f (c • x) = σ₁₂ c • f x 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) (c : R) (x : E)
Goal: f (c • x) = σ₁₂ c • f x
-- LinearIsometry.map_smul
protected theorem map_smul [Module R E₂] (f : E →ₗᵢ[R] E₂) (c : R) (x : E) : f (c • x) = c • f x 
Arguments: {R : Type u_1} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R E₂] (f : E →ₗᵢ[R] E₂) (c : R) (x : E)
Goal: f (c • x) = c • f x
-- LinearIsometry.norm_map
@[simp]
theorem norm_map (x : E) : ‖f x‖ = ‖x‖ 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) (x : E)
Goal: ‖f x‖ = ‖x‖
-- LinearIsometry.nnnorm_map
@[simp] -- Should be replaced with `SemilinearIsometryClass.nnorm_map` when `lean4#3107` is fixed.
theorem nnnorm_map (x : E) : ‖f x‖₊ = ‖x‖₊ 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) (x : E)
Goal: ‖f x‖₊ = ‖x‖₊
-- LinearIsometry.isometry
protected theorem isometry : Isometry f 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂)
Goal: Isometry ⇑f
-- LinearIsometry.isComplete_image_iff
theorem isComplete_image_iff [SemilinearIsometryClass 𝓕 σ₁₂ E E₂] (f : 𝓕) {s : Set E} :
    IsComplete (f '' s) ↔ IsComplete s 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} {𝓕 : Type u_10} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] [FunLike 𝓕 E E₂] [SemilinearIsometryClass 𝓕 σ₁₂ E E₂] (f : 𝓕) {s : Set E}
Goal: IsComplete (⇑f '' s) ↔ IsComplete s
-- LinearIsometry.isComplete_image_iff'
@[simp] -- Should be replaced with `LinearIsometry.isComplete_image_iff` when `lean4#3107` is fixed.
theorem isComplete_image_iff' (f : LinearIsometry σ₁₂ E E₂) {s : Set E} :
    IsComplete (f '' s) ↔ IsComplete s 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) {s : Set E}
Goal: IsComplete (⇑f '' s) ↔ IsComplete s
-- LinearIsometry.isComplete_map_iff
theorem isComplete_map_iff [RingHomSurjective σ₁₂] {p : Submodule R E} :
    IsComplete (p.map f.toLinearMap : Set E₂) ↔ IsComplete (p : Set E) 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) [RingHomSurjective σ₁₂] {p : Submodule R E}
Goal: IsComplete ↑(Submodule.map f.toLinearMap p) ↔ IsComplete ↑p
-- LinearIsometry.isComplete_map_iff'
theorem isComplete_map_iff' [SemilinearIsometryClass 𝓕 σ₁₂ E E₂] (f : 𝓕) [RingHomSurjective σ₁₂]
    {p : Submodule R E} : IsComplete (p.map f : Set E₂) ↔ IsComplete (p : Set E) 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} {𝓕 : Type u_10} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] [FunLike 𝓕 E E₂] [SemilinearIsometryClass 𝓕 σ₁₂ E E₂] (f : 𝓕) [RingHomSurjective σ₁₂] {p : Submodule R E}
Goal: IsComplete ↑(Submodule.map f p) ↔ IsComplete ↑p
-- LinearIsometry.dist_map
@[simp]
theorem dist_map (x y : E) : dist (f x) (f y) = dist x y 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) (x y : E)
Goal: dist (f x) (f y) = dist x y
-- LinearIsometry.edist_map
@[simp]
theorem edist_map (x y : E) : edist (f x) (f y) = edist x y 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) (x y : E)
Goal: edist (f x) (f y) = edist x y
-- LinearIsometry.injective
protected theorem injective : Injective f₁ 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E₂ : Type u_6} {F : Type u_9} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E₂] [Module R₂ E₂] [NormedAddCommGroup F] [Module R F] (f₁ : F →ₛₗᵢ[σ₁₂] E₂)
Goal: Function.Injective ⇑f₁
-- LinearIsometry.map_eq_iff
@[simp]
theorem map_eq_iff {x y : F} : f₁ x = f₁ y ↔ x = y 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E₂ : Type u_6} {F : Type u_9} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E₂] [Module R₂ E₂] [NormedAddCommGroup F] [Module R F] (f₁ : F →ₛₗᵢ[σ₁₂] E₂) {x y : F}
Goal: f₁ x = f₁ y ↔ x = y
-- LinearIsometry.map_ne
theorem map_ne {x y : F} (h : x ≠ y) : f₁ x ≠ f₁ y 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E₂ : Type u_6} {F : Type u_9} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E₂] [Module R₂ E₂] [NormedAddCommGroup F] [Module R F] (f₁ : F →ₛₗᵢ[σ₁₂] E₂) {x y : F} (h : x ≠ y)
Goal: f₁ x ≠ f₁ y
-- LinearIsometry.lipschitz
protected theorem lipschitz : LipschitzWith 1 f 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂)
Goal: LipschitzWith 1 ⇑f
-- LinearIsometry.antilipschitz
protected theorem antilipschitz : AntilipschitzWith 1 f 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂)
Goal: AntilipschitzWith 1 ⇑f
-- LinearIsometry.continuous
@[continuity]
protected theorem continuous : Continuous f 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂)
Goal: Continuous ⇑f
-- LinearIsometry.preimage_ball
@[simp]
theorem preimage_ball (x : E) (r : ℝ) : f ⁻¹' Metric.ball (f x) r = Metric.ball x r 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) (x : E) (r : ℝ)
Goal: ⇑f ⁻¹' Metric.ball (f x) r = Metric.ball x r
-- LinearIsometry.preimage_sphere
@[simp]
theorem preimage_sphere (x : E) (r : ℝ) : f ⁻¹' Metric.sphere (f x) r = Metric.sphere x r 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) (x : E) (r : ℝ)
Goal: ⇑f ⁻¹' Metric.sphere (f x) r = Metric.sphere x r
-- LinearIsometry.preimage_closedBall
@[simp]
theorem preimage_closedBall (x : E) (r : ℝ) :
    f ⁻¹' Metric.closedBall (f x) r = Metric.closedBall x r 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) (x : E) (r : ℝ)
Goal: ⇑f ⁻¹' Metric.closedBall (f x) r = Metric.closedBall x r
-- LinearIsometry.ediam_image
theorem ediam_image (s : Set E) : EMetric.diam (f '' s) = EMetric.diam s 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) (s : Set E)
Goal: EMetric.diam (⇑f '' s) = EMetric.diam s
-- LinearIsometry.ediam_range
theorem ediam_range : EMetric.diam (range f) = EMetric.diam (univ : Set E) 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂)
Goal: EMetric.diam (Set.range ⇑f) = EMetric.diam Set.univ
-- LinearIsometry.diam_image
theorem diam_image (s : Set E) : Metric.diam (f '' s) = Metric.diam s 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) (s : Set E)
Goal: Metric.diam (⇑f '' s) = Metric.diam s
-- LinearIsometry.diam_range
theorem diam_range : Metric.diam (range f) = Metric.diam (univ : Set E) 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂)
Goal: Metric.diam (Set.range ⇑f) = Metric.diam Set.univ
-- LinearIsometry.toContinuousLinearMap_injective
theorem toContinuousLinearMap_injective :
    Function.Injective (toContinuousLinearMap : _ → E →SL[σ₁₂] E₂) 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂]
Goal: Function.Injective LinearIsometry.toContinuousLinearMap
-- LinearIsometry.toContinuousLinearMap_inj
@[simp]
theorem toContinuousLinearMap_inj {f g : E →ₛₗᵢ[σ₁₂] E₂} :
    f.toContinuousLinearMap = g.toContinuousLinearMap ↔ f = g 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] {f g : E →ₛₗᵢ[σ₁₂] E₂}
Goal: f.toContinuousLinearMap = g.toContinuousLinearMap ↔ f = g
-- LinearIsometry.coe_toContinuousLinearMap
@[simp]
theorem coe_toContinuousLinearMap : ⇑f.toContinuousLinearMap = f 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂)
Goal: ⇑f.toContinuousLinearMap = ⇑f
-- LinearIsometry.comp_continuous_iff
@[simp]
theorem comp_continuous_iff {α : Type*} [TopologicalSpace α] {g : α → E} :
    Continuous (f ∘ g) ↔ Continuous g 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) {α : Type u_11} [TopologicalSpace α] {g : α → E}
Goal: Continuous (⇑f ∘ g) ↔ Continuous g
-- LinearIsometry.coe_id
@[simp]
theorem coe_id : ((id : E →ₗᵢ[R] E) : E → E) = _root_.id 
Arguments: {R : Type u_1} {E : Type u_5} [Semiring R] [SeminormedAddCommGroup E] [Module R E]
Goal: ⇑LinearIsometry.id = id
-- LinearIsometry.id_apply
@[simp]
theorem id_apply (x : E) : (id : E →ₗᵢ[R] E) x = x 
Arguments: {R : Type u_1} {E : Type u_5} [Semiring R] [SeminormedAddCommGroup E] [Module R E] (x : E)
Goal: LinearIsometry.id x = x
-- LinearIsometry.id_toLinearMap
@[simp]
theorem id_toLinearMap : (id.toLinearMap : E →ₗ[R] E) = LinearMap.id 
Arguments: {R : Type u_1} {E : Type u_5} [Semiring R] [SeminormedAddCommGroup E] [Module R E]
Goal: LinearIsometry.id.toLinearMap = LinearMap.id
-- LinearIsometry.id_toContinuousLinearMap
@[simp]
theorem id_toContinuousLinearMap : id.toContinuousLinearMap = ContinuousLinearMap.id R E 
Arguments: {R : Type u_1} {E : Type u_5} [Semiring R] [SeminormedAddCommGroup E] [Module R E]
Goal: LinearIsometry.id.toContinuousLinearMap = ContinuousLinearMap.id R E
-- LinearIsometry.coe_comp
@[simp]
theorem coe_comp (g : E₂ →ₛₗᵢ[σ₂₃] E₃) (f : E →ₛₗᵢ[σ₁₂] E₂) : ⇑(g.comp f) = g ∘ f 
Arguments: {R : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} {E : Type u_5} {E₂ : Type u_6} {E₃ : Type u_7} [Semiring R] [Semiring R₂] [Semiring R₃] {σ₁₂ : R →+* R₂} {σ₁₃ : R →+* R₃} {σ₂₃ : R₂ →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [SeminormedAddCommGroup E₃] [Module R E] [Module R₂ E₂] [Module R₃ E₃] (g : E₂ →ₛₗᵢ[σ₂₃] E₃) (f : E →ₛₗᵢ[σ₁₂] E₂)
Goal: ⇑(g.comp f) = ⇑g ∘ ⇑f
-- LinearIsometry.id_comp
@[simp]
theorem id_comp : (id : E₂ →ₗᵢ[R₂] E₂).comp f = f 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂)
Goal: LinearIsometry.id.comp f = f
-- LinearIsometry.comp_id
@[simp]
theorem comp_id : f.comp id = f 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂)
Goal: f.comp LinearIsometry.id = f
-- LinearIsometry.comp_assoc
theorem comp_assoc (f : E₃ →ₛₗᵢ[σ₃₄] E₄) (g : E₂ →ₛₗᵢ[σ₂₃] E₃) (h : E →ₛₗᵢ[σ₁₂] E₂) :
    (f.comp g).comp h = f.comp (g.comp h) 
Arguments: {R : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} {R₄ : Type u_4} {E : Type u_5} {E₂ : Type u_6} {E₃ : Type u_7} {E₄ : Type u_8} [Semiring R] [Semiring R₂] [Semiring R₃] [Semiring R₄] {σ₁₂ : R →+* R₂} {σ₁₃ : R →+* R₃} {σ₁₄ : R →+* R₄} {σ₂₃ : R₂ →+* R₃} {σ₂₄ : R₂ →+* R₄} {σ₃₄ : R₃ →+* R₄} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [RingHomCompTriple σ₁₂ σ₂₄ σ₁₄] [RingHomCompTriple σ₂₃ σ₃₄ σ₂₄] [RingHomCompTriple σ₁₃ σ₃₄ σ₁₄] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [SeminormedAddCommGroup E₃] [SeminormedAddCommGroup E₄] [Module R E] [Module R₂ E₂] [Module R₃ E₃] [Module R₄ E₄] (f : E₃ →ₛₗᵢ[σ₃₄] E₄) (g : E₂ →ₛₗᵢ[σ₂₃] E₃) (h : E →ₛₗᵢ[σ₁₂] E₂)
Goal: (f.comp g).comp h = f.comp (g.comp h)
-- LinearIsometry.coe_one
@[simp]
theorem coe_one : ((1 : E →ₗᵢ[R] E) : E → E) = _root_.id 
Arguments: {R : Type u_1} {E : Type u_5} [Semiring R] [SeminormedAddCommGroup E] [Module R E]
Goal: ⇑1 = id
-- LinearIsometry.coe_mul
@[simp]
theorem coe_mul (f g : E →ₗᵢ[R] E) : ⇑(f * g) = f ∘ g 
Arguments: {R : Type u_1} {E : Type u_5} [Semiring R] [SeminormedAddCommGroup E] [Module R E] (f g : E →ₗᵢ[R] E)
Goal: ⇑(f * g) = ⇑f ∘ ⇑g
-- LinearIsometry.one_def
theorem one_def : (1 : E →ₗᵢ[R] E) = id 
Arguments: {R : Type u_1} {E : Type u_5} [Semiring R] [SeminormedAddCommGroup E] [Module R E]
Goal: 1 = LinearIsometry.id
-- LinearIsometry.mul_def
theorem mul_def (f g : E →ₗᵢ[R] E) : (f * g : E →ₗᵢ[R] E) = f.comp g 
Arguments: {R : Type u_1} {E : Type u_5} [Semiring R] [SeminormedAddCommGroup E] [Module R E] (f g : E →ₗᵢ[R] E)
Goal: f * g = f.comp g
-- LinearIsometry.coe_pow
theorem coe_pow (f : E →ₗᵢ[R] E) (n : ℕ) : ⇑(f ^ n) = f^[n] 
Arguments: {R : Type u_1} {E : Type u_5} [Semiring R] [SeminormedAddCommGroup E] [Module R E] (f : E →ₗᵢ[R] E) (n : ℕ)
Goal: ⇑(f ^ n) = (⇑f)^[n]
-- Submodule.coe_subtypeₗᵢ
@[simp]
theorem coe_subtypeₗᵢ : ⇑p.subtypeₗᵢ = p.subtype 
Arguments: {E : Type u_5} [SeminormedAddCommGroup E] {R' : Type u_11} [Ring R'] [Module R' E] (p : Submodule R' E)
Goal: ⇑p.subtypeₗᵢ = ⇑p.subtype
-- Submodule.subtypeₗᵢ_toLinearMap
@[simp]
theorem subtypeₗᵢ_toLinearMap : p.subtypeₗᵢ.toLinearMap = p.subtype 
Arguments: {E : Type u_5} [SeminormedAddCommGroup E] {R' : Type u_11} [Ring R'] [Module R' E] (p : Submodule R' E)
Goal: p.subtypeₗᵢ.toLinearMap = p.subtype
-- Submodule.subtypeₗᵢ_toContinuousLinearMap
@[simp]
theorem subtypeₗᵢ_toContinuousLinearMap : p.subtypeₗᵢ.toContinuousLinearMap = p.subtypeL 
Arguments: {E : Type u_5} [SeminormedAddCommGroup E] {R' : Type u_11} [Ring R'] [Module R' E] (p : Submodule R' E)
Goal: p.subtypeₗᵢ.toContinuousLinearMap = p.subtypeL
-- LinearIsometryEquiv.norm_map'
structure LinearIsometryEquiv (σ₁₂ : R →+* R₂) {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁]
  [RingHomInvPair σ₂₁ σ₁₂] (E E₂ : Type*) [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂]
  [Module R E] [Module R₂ E₂] extends E ≃ₛₗ[σ₁₂] E₂ where
  norm_map' : ∀ x, ‖toLinearEquiv x‖ = ‖x‖
Arguments: {R : Type u_1} {R₂ : Type u_2} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {E : Type u_11} {E₂ : Type u_12} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (self : E ≃ₛₗᵢ[σ₁₂] E₂) (x : E)
Goal: ‖self.toLinearEquiv x‖ = ‖x‖
-- SemilinearIsometryEquivClass.norm_map
class SemilinearIsometryEquivClass (𝓕 : Type*) {R R₂ : outParam Type*} [Semiring R]
  [Semiring R₂] (σ₁₂ : outParam <| R →+* R₂) {σ₂₁ : outParam <| R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁]
  [RingHomInvPair σ₂₁ σ₁₂] (E E₂ : outParam Type*) [SeminormedAddCommGroup E]
  [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] [EquivLike 𝓕 E E₂]
  extends SemilinearEquivClass 𝓕 σ₁₂ E E₂ : Prop where
  norm_map : ∀ (f : 𝓕) (x : E), ‖f x‖ = ‖x‖
Arguments: {𝓕 : Type u_11} {R : outParam (Type u_12)} {R₂ : outParam (Type u_13)} [Semiring R] [Semiring R₂] {σ₁₂ : outParam (R →+* R₂)} {σ₂₁ : outParam (R₂ →+* R)} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {E : outParam (Type u_14)} {E₂ : outParam (Type u_15)} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] [EquivLike 𝓕 E E₂] [self : SemilinearIsometryEquivClass 𝓕 σ₁₂ E E₂] (f : 𝓕) (x : E)
Goal: ‖f x‖ = ‖x‖
-- LinearIsometryEquiv.toLinearEquiv_injective
theorem toLinearEquiv_injective : Injective (toLinearEquiv : (E ≃ₛₗᵢ[σ₁₂] E₂) → E ≃ₛₗ[σ₁₂] E₂)
  
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂]
Goal: Function.Injective LinearIsometryEquiv.toLinearEquiv
-- LinearIsometryEquiv.toLinearEquiv_inj
@[simp]
theorem toLinearEquiv_inj {f g : E ≃ₛₗᵢ[σ₁₂] E₂} : f.toLinearEquiv = g.toLinearEquiv ↔ f = g 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] {f g : E ≃ₛₗᵢ[σ₁₂] E₂}
Goal: f.toLinearEquiv = g.toLinearEquiv ↔ f = g
-- LinearIsometryEquiv.coe_injective
theorem coe_injective : @Function.Injective (E ≃ₛₗᵢ[σ₁₂] E₂) (E → E₂) (↑) 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂]
Goal: Function.Injective DFunLike.coe