ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- LinearIsometryEquiv.refl_mul
@[simp]
theorem refl_mul (e : E ≃ₗᵢ[R] E) : refl _ _ * e = e 
Arguments: {R : Type u_1} {E : Type u_5} [Semiring R] [SeminormedAddCommGroup E] [Module R E] (e : E ≃ₗᵢ[R] E)
Goal: LinearIsometryEquiv.refl R E * e = e
-- LinearIsometryEquiv.mul_refl
@[simp]
theorem mul_refl (e : E ≃ₗᵢ[R] E) : e * refl _ _ = e 
Arguments: {R : Type u_1} {E : Type u_5} [Semiring R] [SeminormedAddCommGroup E] [Module R E] (e : E ≃ₗᵢ[R] E)
Goal: e * LinearIsometryEquiv.refl R E = e
-- LinearIsometryEquiv.coe_coe
@[simp]
theorem coe_coe : ⇑(e : E ≃SL[σ₁₂] E₂) = 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} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂)
Goal: ⇑{ toLinearEquiv := e.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ } = ⇑e
-- LinearIsometryEquiv.coe_coe''
@[simp]
theorem coe_coe'' : ⇑(e : E →SL[σ₁₂] E₂) = 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} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂)
Goal: ⇑↑{ toLinearEquiv := e.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ } = ⇑e
-- LinearIsometryEquiv.map_zero
theorem map_zero : e 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} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂)
Goal: e 0 = 0
-- LinearIsometryEquiv.map_add
theorem map_add (x y : E) : e (x + y) = e x + e 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} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) (x y : E)
Goal: e (x + y) = e x + e y
-- LinearIsometryEquiv.map_sub
theorem map_sub (x y : E) : e (x - y) = e x - e 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} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) (x y : E)
Goal: e (x - y) = e x - e y
-- LinearIsometryEquiv.map_smulₛₗ
theorem map_smulₛₗ (c : R) (x : E) : e (c • x) = σ₁₂ c • e 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} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) (c : R) (x : E)
Goal: e (c • x) = σ₁₂ c • e x
-- LinearIsometryEquiv.map_smul
theorem map_smul [Module R E₂] {e : E ≃ₗᵢ[R] E₂} (c : R) (x : E) : e (c • x) = c • e 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] {e : E ≃ₗᵢ[R] E₂} (c : R) (x : E)
Goal: e (c • x) = c • e x
-- LinearIsometryEquiv.nnnorm_map
@[simp] -- Should be replaced with `SemilinearIsometryClass.nnorm_map` when `lean4#3107` is fixed.
theorem nnnorm_map (x : E) : ‖e 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} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) (x : E)
Goal: ‖e x‖₊ = ‖x‖₊
-- LinearIsometryEquiv.dist_map
@[simp]
theorem dist_map (x y : E) : dist (e x) (e 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} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) (x y : E)
Goal: dist (e x) (e y) = dist x y
-- LinearIsometryEquiv.edist_map
@[simp]
theorem edist_map (x y : E) : edist (e x) (e 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} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) (x y : E)
Goal: edist (e x) (e y) = edist x y
-- LinearIsometryEquiv.bijective
protected theorem bijective : Bijective 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} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂)
Goal: Function.Bijective ⇑e
-- LinearIsometryEquiv.injective
protected theorem injective : Injective 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} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂)
Goal: Function.Injective ⇑e
-- LinearIsometryEquiv.surjective
protected theorem surjective : Surjective 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} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂)
Goal: Function.Surjective ⇑e
-- LinearIsometryEquiv.map_eq_iff
theorem map_eq_iff {x y : E} : e x = e y ↔ 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} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) {x y : E}
Goal: e x = e y ↔ x = y
-- LinearIsometryEquiv.map_ne
theorem map_ne {x y : E} (h : x ≠ y) : e x ≠ e 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} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) {x y : E} (h : x ≠ y)
Goal: e x ≠ e y
-- LinearIsometryEquiv.lipschitz
protected theorem lipschitz : LipschitzWith 1 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} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂)
Goal: LipschitzWith 1 ⇑e
-- LinearIsometryEquiv.antilipschitz
protected theorem antilipschitz : AntilipschitzWith 1 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} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂)
Goal: AntilipschitzWith 1 ⇑e
-- LinearIsometryEquiv.image_eq_preimage
theorem image_eq_preimage (s : Set E) : e '' s = e.symm ⁻¹' 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} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) (s : Set E)
Goal: ⇑e '' s = ⇑e.symm ⁻¹' s
-- LinearIsometryEquiv.ediam_image
@[simp]
theorem ediam_image (s : Set E) : EMetric.diam (e '' 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} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) (s : Set E)
Goal: EMetric.diam (⇑e '' s) = EMetric.diam s
-- LinearIsometryEquiv.diam_image
@[simp]
theorem diam_image (s : Set E) : Metric.diam (e '' 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} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) (s : Set E)
Goal: Metric.diam (⇑e '' s) = Metric.diam s
-- LinearIsometryEquiv.preimage_ball
@[simp]
theorem preimage_ball (x : E₂) (r : ℝ) : e ⁻¹' Metric.ball x r = Metric.ball (e.symm 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} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) (x : E₂) (r : ℝ)
Goal: ⇑e ⁻¹' Metric.ball x r = Metric.ball (e.symm x) r
-- LinearIsometryEquiv.preimage_sphere
@[simp]
theorem preimage_sphere (x : E₂) (r : ℝ) : e ⁻¹' Metric.sphere x r = Metric.sphere (e.symm 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} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) (x : E₂) (r : ℝ)
Goal: ⇑e ⁻¹' Metric.sphere x r = Metric.sphere (e.symm x) r
-- LinearIsometryEquiv.preimage_closedBall
@[simp]
theorem preimage_closedBall (x : E₂) (r : ℝ) :
    e ⁻¹' Metric.closedBall x r = Metric.closedBall (e.symm 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} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) (x : E₂) (r : ℝ)
Goal: ⇑e ⁻¹' Metric.closedBall x r = Metric.closedBall (e.symm x) r
-- LinearIsometryEquiv.image_ball
@[simp]
theorem image_ball (x : E) (r : ℝ) : e '' Metric.ball x r = Metric.ball (e 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} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) (x : E) (r : ℝ)
Goal: ⇑e '' Metric.ball x r = Metric.ball (e x) r
-- LinearIsometryEquiv.image_sphere
@[simp]
theorem image_sphere (x : E) (r : ℝ) : e '' Metric.sphere x r = Metric.sphere (e 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} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) (x : E) (r : ℝ)
Goal: ⇑e '' Metric.sphere x r = Metric.sphere (e x) r
-- LinearIsometryEquiv.image_closedBall
@[simp]
theorem image_closedBall (x : E) (r : ℝ) : e '' Metric.closedBall x r = Metric.closedBall (e 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} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) (x : E) (r : ℝ)
Goal: ⇑e '' Metric.closedBall x r = Metric.closedBall (e x) r
-- LinearIsometryEquiv.comp_continuousOn_iff
@[simp]
theorem comp_continuousOn_iff {f : α → E} {s : Set α} : ContinuousOn (e ∘ f) s ↔ ContinuousOn f 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} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) {α : Type u_11} [TopologicalSpace α] {f : α → E} {s : Set α}
Goal: ContinuousOn (⇑e ∘ f) s ↔ ContinuousOn f s
-- LinearIsometryEquiv.comp_continuous_iff
@[simp]
theorem comp_continuous_iff {f : α → E} : Continuous (e ∘ f) ↔ 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} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) {α : Type u_11} [TopologicalSpace α] {f : α → E}
Goal: Continuous (⇑e ∘ f) ↔ Continuous f
-- LinearIsometryEquiv.coe_ofSurjective
@[simp]
theorem coe_ofSurjective (f : F →ₛₗᵢ[σ₁₂] E₂) (hfr : Function.Surjective f) :
    ⇑(LinearIsometryEquiv.ofSurjective f hfr) = f 
Arguments: {R : Type u_1} {R₂ : Type u_2} {E₂ : Type u_6} {F : Type u_9} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E₂] [Module R₂ E₂] [NormedAddCommGroup F] [Module R F] (f : F →ₛₗᵢ[σ₁₂] E₂) (hfr : Function.Surjective ⇑f)
Goal: ⇑(LinearIsometryEquiv.ofSurjective f hfr) = ⇑f
-- LinearIsometryEquiv.coe_ofLinearIsometry
@[simp]
theorem coe_ofLinearIsometry (f : E →ₛₗᵢ[σ₁₂] E₂) (g : E₂ →ₛₗ[σ₂₁] E)
    (h₁ : f.toLinearMap.comp g = LinearMap.id) (h₂ : g.comp f.toLinearMap = LinearMap.id) :
    (ofLinearIsometry f g h₁ h₂ : E → E₂) = (f : 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] (f : E →ₛₗᵢ[σ₁₂] E₂) (g : E₂ →ₛₗ[σ₂₁] E) (h₁ : f.comp g = LinearMap.id) (h₂ : g.comp f.toLinearMap = LinearMap.id)
Goal: ⇑(LinearIsometryEquiv.ofLinearIsometry f g h₁ h₂) = ⇑f
-- LinearIsometryEquiv.coe_ofLinearIsometry_symm
@[simp]
theorem coe_ofLinearIsometry_symm (f : E →ₛₗᵢ[σ₁₂] E₂) (g : E₂ →ₛₗ[σ₂₁] E)
    (h₁ : f.toLinearMap.comp g = LinearMap.id) (h₂ : g.comp f.toLinearMap = LinearMap.id) :
    ((ofLinearIsometry f g h₁ h₂).symm : E₂ → E) = (g : E₂ → 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} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) (g : E₂ →ₛₗ[σ₂₁] E) (h₁ : f.comp g = LinearMap.id) (h₂ : g.comp f.toLinearMap = LinearMap.id)
Goal: ⇑(LinearIsometryEquiv.ofLinearIsometry f g h₁ h₂).symm = ⇑g
-- LinearIsometryEquiv.coe_neg
@[simp]
theorem coe_neg : (neg R : E → E) = fun x => -x 
Arguments: {R : Type u_1} {E : Type u_5} [Semiring R] [SeminormedAddCommGroup E] [Module R E]
Goal: ⇑(LinearIsometryEquiv.neg R) = fun x => -x
-- LinearIsometryEquiv.symm_neg
@[simp]
theorem symm_neg : (neg R : E ≃ₗᵢ[R] E).symm = neg R 
Arguments: {R : Type u_1} {E : Type u_5} [Semiring R] [SeminormedAddCommGroup E] [Module R E]
Goal: (LinearIsometryEquiv.neg R).symm = LinearIsometryEquiv.neg R
-- LinearIsometryEquiv.coe_prodAssoc
@[simp]
theorem coe_prodAssoc [Module R E₂] [Module R E₃] :
    (prodAssoc R E E₂ E₃ : (E × E₂) × E₃ → E × E₂ × E₃) = Equiv.prodAssoc E E₂ E₃ 
Arguments: (R : Type u_1) (E : Type u_5) (E₂ : Type u_6) (E₃ : Type u_7) [Semiring R] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [SeminormedAddCommGroup E₃] [Module R E] [Module R E₂] [Module R E₃]
Goal: ⇑(LinearIsometryEquiv.prodAssoc R E E₂ E₃) = ⇑(Equiv.prodAssoc E E₂ E₃)
-- LinearIsometryEquiv.coe_prodAssoc_symm
@[simp]
theorem coe_prodAssoc_symm [Module R E₂] [Module R E₃] :
    ((prodAssoc R E E₂ E₃).symm : E × E₂ × E₃ → (E × E₂) × E₃) = (Equiv.prodAssoc E E₂ E₃).symm 
Arguments: (R : Type u_1) (E : Type u_5) (E₂ : Type u_6) (E₃ : Type u_7) [Semiring R] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [SeminormedAddCommGroup E₃] [Module R E] [Module R E₂] [Module R E₃]
Goal: ⇑(LinearIsometryEquiv.prodAssoc R E E₂ E₃).symm = ⇑(Equiv.prodAssoc E E₂ E₃).symm
-- LinearIsometryEquiv.ofTop_toLinearEquiv
@[simps! toLinearEquiv apply symm_apply_coe]
def ofTop {R : Type*} [Ring R] [Module R E] (p : Submodule R E) (hp : p = ⊤) : p ≃ₗᵢ[R] E 
Arguments: (E : Type u_5) [SeminormedAddCommGroup E] {R : Type u_12} [Ring R] [Module R E] (p : Submodule R E) (hp : p = ⊤)
Goal: (LinearIsometryEquiv.ofTop E p hp).toLinearEquiv = LinearEquiv.ofTop p hp
-- LinearIsometryEquiv.ofTop_symm_apply_coe
@[simps! toLinearEquiv apply symm_apply_coe]
def ofTop {R : Type*} [Ring R] [Module R E] (p : Submodule R E) (hp : p = ⊤) : p ≃ₗᵢ[R] E 
Arguments: (E : Type u_5) [SeminormedAddCommGroup E] {R : Type u_12} [Ring R] [Module R E] (p : Submodule R E) (hp : p = ⊤) (x : E)
Goal: ↑((LinearIsometryEquiv.ofTop E p hp).symm x) = x
-- LinearIsometryEquiv.ofTop_apply
@[simps! toLinearEquiv apply symm_apply_coe]
def ofTop {R : Type*} [Ring R] [Module R E] (p : Submodule R E) (hp : p = ⊤) : p ≃ₗᵢ[R] E 
Arguments: (E : Type u_5) [SeminormedAddCommGroup E] {R : Type u_12} [Ring R] [Module R E] (p : Submodule R E) (hp : p = ⊤) (self : ↥p)
Goal: (LinearIsometryEquiv.ofTop E p hp) self = ↑self
-- LinearIsometryEquiv.coe_ofEq_apply
@[simp]
theorem coe_ofEq_apply (h : p = q) (x : p) : (ofEq p q h x : E) = x 
Arguments: {E : Type u_5} [SeminormedAddCommGroup E] {R' : Type u_12} [Ring R'] [Module R' E] {p q : Submodule R' E} (h : p = q) (x : ↥p)
Goal: ↑((LinearIsometryEquiv.ofEq p q h) x) = ↑x
-- LinearIsometryEquiv.ofEq_symm
@[simp]
theorem ofEq_symm (h : p = q) : (ofEq p q h).symm = ofEq q p h.symm 
Arguments: {E : Type u_5} [SeminormedAddCommGroup E] {R' : Type u_12} [Ring R'] [Module R' E] {p q : Submodule R' E} (h : p = q)
Goal: (LinearIsometryEquiv.ofEq p q h).symm = LinearIsometryEquiv.ofEq q p ⋯
-- LinearIsometryEquiv.ofEq_rfl
@[simp]
theorem ofEq_rfl : ofEq p p rfl = LinearIsometryEquiv.refl R' p 
Arguments: {E : Type u_5} [SeminormedAddCommGroup E] {R' : Type u_12} [Ring R'] [Module R' E] {p : Submodule R' E}
Goal: LinearIsometryEquiv.ofEq p p ⋯ = LinearIsometryEquiv.refl R' ↥p
-- Basis.ext_linearIsometry
theorem Basis.ext_linearIsometry {ι : Type*} (b : Basis ι R E) {f₁ f₂ : E →ₛₗᵢ[σ₁₂] E₂}
    (h : ∀ i, f₁ (b i) = f₂ (b i)) : f₁ = f₂ 
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] {ι : Type u_11} (b : Basis ι R E) {f₁ f₂ : E →ₛₗᵢ[σ₁₂] E₂} (h : ∀ (i : ι), f₁ (b i) = f₂ (b i))
Goal: f₁ = f₂
-- Basis.ext_linearIsometryEquiv
theorem Basis.ext_linearIsometryEquiv {ι : Type*} (b : Basis ι R E) {f₁ f₂ : E ≃ₛₗᵢ[σ₁₂] E₂}
    (h : ∀ i, f₁ (b i) = f₂ (b i)) : f₁ = f₂ 
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] {ι : Type u_11} (b : Basis ι R E) {f₁ f₂ : E ≃ₛₗᵢ[σ₁₂] E₂} (h : ∀ (i : ι), f₁ (b i) = f₂ (b i))
Goal: f₁ = f₂
-- LinearIsometry.equivRange_apply_coe
@[simps! apply_coe] -- Porting note: `toLinearEquiv` projection does not simplify using itself
noncomputable def LinearIsometry.equivRange {R S : Type*} [Semiring R] [Ring S] [Module S E]
    [Module R F] {σ₁₂ : R →+* S} {σ₂₁ : S →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂]
    (f : F →ₛₗᵢ[σ₁₂] E) : F ≃ₛₗᵢ[σ₁₂] (LinearMap.range f.toLinearMap) 
Arguments: {E : Type u_5} {F : Type u_9} [SeminormedAddCommGroup E] [NormedAddCommGroup F] {R : Type u_11} {S : Type u_12} [Semiring R] [Ring S] [Module S E] [Module R F] {σ₁₂ : R →+* S} {σ₂₁ : S →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] (f : F →ₛₗᵢ[σ₁₂] E) (a : F)
Goal: ↑(f.equivRange a) = f a
-- spectrum.SpectralRadius.of_subsingleton
@[simp]
theorem SpectralRadius.of_subsingleton [Subsingleton A] (a : A) : spectralRadius 𝕜 a = 0 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [Subsingleton A] (a : A)
Goal: spectralRadius 𝕜 a = 0
-- spectrum.spectralRadius_zero
@[simp]
theorem spectralRadius_zero : spectralRadius 𝕜 (0 : A) = 0 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A]
Goal: spectralRadius 𝕜 0 = 0
-- spectrum.mem_resolventSet_of_spectralRadius_lt
theorem mem_resolventSet_of_spectralRadius_lt {a : A} {k : 𝕜} (h : spectralRadius 𝕜 a < ‖k‖₊) :
    k ∈ ρ a 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] {a : A} {k : 𝕜} (h : spectralRadius 𝕜 a < ↑‖k‖₊)
Goal: k ∈ resolventSet 𝕜 a
-- spectrum.isOpen_resolventSet
theorem isOpen_resolventSet (a : A) : IsOpen (ρ a) 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] (a : A)
Goal: IsOpen (resolventSet 𝕜 a)
-- spectrum.isClosed
protected theorem isClosed (a : A) : IsClosed (σ a) 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] (a : A)
Goal: IsClosed (spectrum 𝕜 a)
-- spectrum.mem_resolventSet_of_norm_lt_mul
theorem mem_resolventSet_of_norm_lt_mul {a : A} {k : 𝕜} (h : ‖a‖ * ‖(1 : A)‖ < ‖k‖) : k ∈ ρ a 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] {a : A} {k : 𝕜} (h : ‖a‖ * ‖1‖ < ‖k‖)
Goal: k ∈ resolventSet 𝕜 a
-- spectrum.mem_resolventSet_of_norm_lt
theorem mem_resolventSet_of_norm_lt [NormOneClass A] {a : A} {k : 𝕜} (h : ‖a‖ < ‖k‖) : k ∈ ρ a 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] [NormOneClass A] {a : A} {k : 𝕜} (h : ‖a‖ < ‖k‖)
Goal: k ∈ resolventSet 𝕜 a
-- spectrum.norm_le_norm_mul_of_mem
theorem norm_le_norm_mul_of_mem {a : A} {k : 𝕜} (hk : k ∈ σ a) : ‖k‖ ≤ ‖a‖ * ‖(1 : A)‖ 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] {a : A} {k : 𝕜} (hk : k ∈ spectrum 𝕜 a)
Goal: ‖k‖ ≤ ‖a‖ * ‖1‖
-- spectrum.norm_le_norm_of_mem
theorem norm_le_norm_of_mem [NormOneClass A] {a : A} {k : 𝕜} (hk : k ∈ σ a) : ‖k‖ ≤ ‖a‖ 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] [NormOneClass A] {a : A} {k : 𝕜} (hk : k ∈ spectrum 𝕜 a)
Goal: ‖k‖ ≤ ‖a‖
-- spectrum.subset_closedBall_norm_mul
theorem subset_closedBall_norm_mul (a : A) : σ a ⊆ Metric.closedBall (0 : 𝕜) (‖a‖ * ‖(1 : A)‖) 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] (a : A)
Goal: spectrum 𝕜 a ⊆ Metric.closedBall 0 (‖a‖ * ‖1‖)
-- spectrum.subset_closedBall_norm
theorem subset_closedBall_norm [NormOneClass A] (a : A) : σ a ⊆ Metric.closedBall (0 : 𝕜) ‖a‖ 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] [NormOneClass A] (a : A)
Goal: spectrum 𝕜 a ⊆ Metric.closedBall 0 ‖a‖
-- spectrum.isBounded
theorem isBounded (a : A) : Bornology.IsBounded (σ a) 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] (a : A)
Goal: Bornology.IsBounded (spectrum 𝕜 a)
-- spectrum.isCompact
protected theorem isCompact [ProperSpace 𝕜] (a : A) : IsCompact (σ a) 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] [ProperSpace 𝕜] (a : A)
Goal: IsCompact (spectrum 𝕜 a)
-- quasispectrum.isCompact
theorem _root_.quasispectrum.isCompact (a : B) : IsCompact (quasispectrum 𝕜 a) 
Arguments: {𝕜 : Type u_1} [NormedField 𝕜] {B : Type u_3} [NonUnitalNormedRing B] [NormedSpace 𝕜 B] [CompleteSpace B] [IsScalarTower 𝕜 B B] [SMulCommClass 𝕜 B B] [ProperSpace 𝕜] (a : B)
Goal: IsCompact (quasispectrum 𝕜 a)
-- spectrum.spectralRadius_le_nnnorm
theorem spectralRadius_le_nnnorm [NormOneClass A] (a : A) : spectralRadius 𝕜 a ≤ ‖a‖₊ 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] [NormOneClass A] (a : A)
Goal: spectralRadius 𝕜 a ≤ ↑‖a‖₊