ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContinuousLinearEquiv.coe_symm_comp_coe
@[simp high]
theorem coe_symm_comp_coe (e : M₁ ≃SL[σ₁₂] M₂) :
    (e.symm : M₂ →SL[σ₂₁] M₁).comp (e : M₁ →SL[σ₁₂] M₂) = ContinuousLinearMap.id R₁ M₁ 
Arguments: {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂)
Goal: (↑e.symm).comp ↑e = ContinuousLinearMap.id R₁ M₁
-- ContinuousLinearEquiv.symm_comp_self
@[simp]
theorem symm_comp_self (e : M₁ ≃SL[σ₁₂] M₂) : (e.symm : M₂ → M₁) ∘ (e : M₁ → M₂) = id 
Arguments: {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂)
Goal: ⇑e.symm ∘ ⇑e = id
-- ContinuousLinearEquiv.self_comp_symm
@[simp]
theorem self_comp_symm (e : M₁ ≃SL[σ₁₂] M₂) : (e : M₁ → M₂) ∘ (e.symm : M₂ → M₁) = id 
Arguments: {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂)
Goal: ⇑e ∘ ⇑e.symm = id
-- ContinuousLinearEquiv.symm_symm
@[simp]
theorem symm_symm (e : M₁ ≃SL[σ₁₂] M₂) : e.symm.symm = e 
Arguments: {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂)
Goal: e.symm.symm = e
-- ContinuousLinearEquiv.refl_symm
@[simp]
theorem refl_symm : (ContinuousLinearEquiv.refl R₁ M₁).symm = ContinuousLinearEquiv.refl R₁ M₁ 
Arguments: {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁]
Goal: (ContinuousLinearEquiv.refl R₁ M₁).symm = ContinuousLinearEquiv.refl R₁ M₁
-- ContinuousLinearEquiv.symm_symm_apply
theorem symm_symm_apply (e : M₁ ≃SL[σ₁₂] M₂) (x : M₁) : e.symm.symm x = e x 
Arguments: {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) (x : M₁)
Goal: e.symm.symm x = e x
-- ContinuousLinearEquiv.symm_apply_eq
theorem symm_apply_eq (e : M₁ ≃SL[σ₁₂] M₂) {x y} : e.symm x = y ↔ x = e y 
Arguments: {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) {x : M₂} {y : M₁}
Goal: e.symm x = y ↔ x = e y
-- ContinuousLinearEquiv.eq_symm_apply
theorem eq_symm_apply (e : M₁ ≃SL[σ₁₂] M₂) {x y} : y = e.symm x ↔ e y = x 
Arguments: {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) {x : M₂} {y : M₁}
Goal: y = e.symm x ↔ e y = x
-- ContinuousLinearEquiv.image_eq_preimage
protected theorem image_eq_preimage (e : M₁ ≃SL[σ₁₂] M₂) (s : Set M₁) : e '' s = e.symm ⁻¹' s 
Arguments: {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) (s : Set M₁)
Goal: ⇑e '' s = ⇑e.symm ⁻¹' s
-- ContinuousLinearEquiv.image_symm_eq_preimage
protected theorem image_symm_eq_preimage (e : M₁ ≃SL[σ₁₂] M₂) (s : Set M₂) :
    e.symm '' s = e ⁻¹' s 
Arguments: {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) (s : Set M₂)
Goal: ⇑e.symm '' s = ⇑e ⁻¹' s
-- ContinuousLinearEquiv.symm_preimage_preimage
@[simp]
protected theorem symm_preimage_preimage (e : M₁ ≃SL[σ₁₂] M₂) (s : Set M₂) :
    e.symm ⁻¹' (e ⁻¹' s) = s 
Arguments: {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) (s : Set M₂)
Goal: ⇑e.symm ⁻¹' (⇑e ⁻¹' s) = s
-- ContinuousLinearEquiv.preimage_symm_preimage
@[simp]
protected theorem preimage_symm_preimage (e : M₁ ≃SL[σ₁₂] M₂) (s : Set M₁) :
    e ⁻¹' (e.symm ⁻¹' s) = s 
Arguments: {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) (s : Set M₁)
Goal: ⇑e ⁻¹' (⇑e.symm ⁻¹' s) = s
-- ContinuousLinearEquiv.uniformEmbedding
protected theorem uniformEmbedding {E₁ E₂ : Type*} [UniformSpace E₁] [UniformSpace E₂]
    [AddCommGroup E₁] [AddCommGroup E₂] [Module R₁ E₁] [Module R₂ E₂] [UniformAddGroup E₁]
    [UniformAddGroup E₂] (e : E₁ ≃SL[σ₁₂] E₂) : UniformEmbedding e 
Arguments: {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {E₁ : Type u_9} {E₂ : Type u_10} [UniformSpace E₁] [UniformSpace E₂] [AddCommGroup E₁] [AddCommGroup E₂] [Module R₁ E₁] [Module R₂ E₂] [UniformAddGroup E₁] [UniformAddGroup E₂] (e : E₁ ≃SL[σ₁₂] E₂)
Goal: UniformEmbedding ⇑e
-- LinearEquiv.uniformEmbedding
protected theorem _root_.LinearEquiv.uniformEmbedding {E₁ E₂ : Type*} [UniformSpace E₁]
    [UniformSpace E₂] [AddCommGroup E₁] [AddCommGroup E₂] [Module R₁ E₁] [Module R₂ E₂]
    [UniformAddGroup E₁] [UniformAddGroup E₂] (e : E₁ ≃ₛₗ[σ₁₂] E₂)
    (h₁ : Continuous e) (h₂ : Continuous e.symm) : UniformEmbedding e 
Arguments: {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {E₁ : Type u_9} {E₂ : Type u_10} [UniformSpace E₁] [UniformSpace E₂] [AddCommGroup E₁] [AddCommGroup E₂] [Module R₁ E₁] [Module R₂ E₂] [UniformAddGroup E₁] [UniformAddGroup E₂] (e : E₁ ≃ₛₗ[σ₁₂] E₂) (h₁ : Continuous ⇑e) (h₂ : Continuous ⇑e.symm)
Goal: UniformEmbedding ⇑e
-- ContinuousLinearEquiv.equivOfInverse_apply
@[simp]
theorem equivOfInverse_apply (f₁ : M₁ →SL[σ₁₂] M₂) (f₂ h₁ h₂ x) :
    equivOfInverse f₁ f₂ h₁ h₂ x = f₁ x 
Arguments: {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (f₁ : M₁ →SL[σ₁₂] M₂) (f₂ : M₂ →SL[σ₂₁] M₁) (h₁ : Function.LeftInverse ⇑f₂ ⇑f₁) (h₂ : Function.RightInverse ⇑f₂ ⇑f₁) (x : M₁)
Goal: (ContinuousLinearEquiv.equivOfInverse f₁ f₂ h₁ h₂) x = f₁ x
-- ContinuousLinearEquiv.symm_equivOfInverse
@[simp]
theorem symm_equivOfInverse (f₁ : M₁ →SL[σ₁₂] M₂) (f₂ h₁ h₂) :
    (equivOfInverse f₁ f₂ h₁ h₂).symm = equivOfInverse f₂ f₁ h₂ h₁ 
Arguments: {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (f₁ : M₁ →SL[σ₁₂] M₂) (f₂ : M₂ →SL[σ₂₁] M₁) (h₁ : Function.LeftInverse ⇑f₂ ⇑f₁) (h₂ : Function.RightInverse ⇑f₂ ⇑f₁)
Goal: (ContinuousLinearEquiv.equivOfInverse f₁ f₂ h₁ h₂).symm = ContinuousLinearEquiv.equivOfInverse f₂ f₁ h₂ h₁
-- ContinuousLinearEquiv.arrowCongrEquiv_symm_apply
@[simps]
def arrowCongrEquiv (e₁₂ : M₁ ≃SL[σ₁₂] M₂) (e₄₃ : M₄ ≃SL[σ₄₃] M₃) :
    (M₁ →SL[σ₁₄] M₄) ≃ (M₂ →SL[σ₂₃] M₃) 
Arguments: {R₁ : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} [Semiring R₁] [Semiring R₂] [Semiring R₃] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] {M₄ : Type u_8} [TopologicalSpace M₄] [AddCommMonoid M₄] [Module R₁ M₁] [Module R₂ M₂] [Module R₃ M₃] {R₄ : Type u_9} [Semiring R₄] [Module R₄ M₄] {σ₃₄ : R₃ →+* R₄} {σ₄₃ : R₄ →+* R₃} [RingHomInvPair σ₃₄ σ₄₃] [RingHomInvPair σ₄₃ σ₃₄] {σ₂₄ : R₂ →+* R₄} {σ₁₄ : R₁ →+* R₄} [RingHomCompTriple σ₂₁ σ₁₄ σ₂₄] [RingHomCompTriple σ₂₄ σ₄₃ σ₂₃] [RingHomCompTriple σ₁₃ σ₃₄ σ₁₄] (e₁₂ : M₁ ≃SL[σ₁₂] M₂) (e₄₃ : M₄ ≃SL[σ₄₃] M₃) (f : M₂ →SL[σ₂₃] M₃)
Goal: (e₁₂.arrowCongrEquiv e₄₃).symm f = (↑e₄₃.symm).comp (f.comp ↑e₁₂)
-- ContinuousLinearEquiv.arrowCongrEquiv_apply
@[simps]
def arrowCongrEquiv (e₁₂ : M₁ ≃SL[σ₁₂] M₂) (e₄₃ : M₄ ≃SL[σ₄₃] M₃) :
    (M₁ →SL[σ₁₄] M₄) ≃ (M₂ →SL[σ₂₃] M₃) 
Arguments: {R₁ : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} [Semiring R₁] [Semiring R₂] [Semiring R₃] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] {M₄ : Type u_8} [TopologicalSpace M₄] [AddCommMonoid M₄] [Module R₁ M₁] [Module R₂ M₂] [Module R₃ M₃] {R₄ : Type u_9} [Semiring R₄] [Module R₄ M₄] {σ₃₄ : R₃ →+* R₄} {σ₄₃ : R₄ →+* R₃} [RingHomInvPair σ₃₄ σ₄₃] [RingHomInvPair σ₄₃ σ₃₄] {σ₂₄ : R₂ →+* R₄} {σ₁₄ : R₁ →+* R₄} [RingHomCompTriple σ₂₁ σ₁₄ σ₂₄] [RingHomCompTriple σ₂₄ σ₄₃ σ₂₃] [RingHomCompTriple σ₁₃ σ₃₄ σ₁₄] (e₁₂ : M₁ ≃SL[σ₁₂] M₂) (e₄₃ : M₄ ≃SL[σ₄₃] M₃) (f : M₁ →SL[σ₁₄] M₄)
Goal: (e₁₂.arrowCongrEquiv e₄₃) f = (↑e₄₃).comp (f.comp ↑e₁₂.symm)
-- ContinuousLinearEquiv.skewProd_apply
@[simp]
theorem skewProd_apply (e : M ≃L[R] M₂) (e' : M₃ ≃L[R] M₄) (f : M →L[R] M₄) (x) :
    e.skewProd e' f x = (e x.1, e' x.2 + f x.1) 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_3} [TopologicalSpace M₂] [AddCommGroup M₂] {M₃ : Type u_4} [TopologicalSpace M₃] [AddCommGroup M₃] {M₄ : Type u_5} [TopologicalSpace M₄] [AddCommGroup M₄] [Module R M] [Module R M₂] [Module R M₃] [Module R M₄] [TopologicalAddGroup M₄] (e : M ≃L[R] M₂) (e' : M₃ ≃L[R] M₄) (f : M →L[R] M₄) (x : M × M₃)
Goal: (e.skewProd e' f) x = (e x.1, e' x.2 + f x.1)
-- ContinuousLinearEquiv.skewProd_symm_apply
@[simp]
theorem skewProd_symm_apply (e : M ≃L[R] M₂) (e' : M₃ ≃L[R] M₄) (f : M →L[R] M₄) (x) :
    (e.skewProd e' f).symm x = (e.symm x.1, e'.symm (x.2 - f (e.symm x.1))) 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_3} [TopologicalSpace M₂] [AddCommGroup M₂] {M₃ : Type u_4} [TopologicalSpace M₃] [AddCommGroup M₃] {M₄ : Type u_5} [TopologicalSpace M₄] [AddCommGroup M₄] [Module R M] [Module R M₂] [Module R M₃] [Module R M₄] [TopologicalAddGroup M₄] (e : M ≃L[R] M₂) (e' : M₃ ≃L[R] M₄) (f : M →L[R] M₄) (x : M₂ × M₄)
Goal: (e.skewProd e' f).symm x = (e.symm x.1, e'.symm (x.2 - f (e.symm x.1)))
-- ContinuousLinearEquiv.map_sub
theorem map_sub (e : M ≃SL[σ₁₂] M₂) (x y : M) : e (x - y) = e x - e y 
Arguments: {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_3} [TopologicalSpace M] [AddCommGroup M] [Module R M] {M₂ : Type u_4} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R₂ M₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] (e : M ≃SL[σ₁₂] M₂) (x y : M)
Goal: e (x - y) = e x - e y
-- ContinuousLinearEquiv.map_neg
theorem map_neg (e : M ≃SL[σ₁₂] M₂) (x : M) : e (-x) = -e x 
Arguments: {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_3} [TopologicalSpace M] [AddCommGroup M] [Module R M] {M₂ : Type u_4} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R₂ M₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] (e : M ≃SL[σ₁₂] M₂) (x : M)
Goal: e (-x) = -e x
-- ContinuousLinearEquiv.unitsEquiv_apply
@[simp]
theorem unitsEquiv_apply (f : (M →L[R] M)ˣ) (x : M) : unitsEquiv R M f x = (f : M →L[R] M) x 
Arguments: (R : Type u_1) [Ring R] (M : Type u_3) [TopologicalSpace M] [AddCommGroup M] [Module R M] (f : (M →L[R] M)ˣ) (x : M)
Goal: ((ContinuousLinearEquiv.unitsEquiv R M) f) x = ↑f x
-- ContinuousLinearEquiv.unitsEquivAut_apply
@[simp]
theorem unitsEquivAut_apply (u : Rˣ) (x : R) : unitsEquivAut R u x = x * u 
Arguments: {R : Type u_1} [Ring R] [TopologicalSpace R] [ContinuousMul R] (u : Rˣ) (x : R)
Goal: ((ContinuousLinearEquiv.unitsEquivAut R) u) x = x * ↑u
-- ContinuousLinearEquiv.unitsEquivAut_apply_symm
@[simp]
theorem unitsEquivAut_apply_symm (u : Rˣ) (x : R) : (unitsEquivAut R u).symm x = x * ↑u⁻¹ 
Arguments: {R : Type u_1} [Ring R] [TopologicalSpace R] [ContinuousMul R] (u : Rˣ) (x : R)
Goal: ((ContinuousLinearEquiv.unitsEquivAut R) u).symm x = x * ↑u⁻¹
-- ContinuousLinearEquiv.unitsEquivAut_symm_apply
@[simp]
theorem unitsEquivAut_symm_apply (e : R ≃L[R] R) : ↑((unitsEquivAut R).symm e) = e 1 
Arguments: {R : Type u_1} [Ring R] [TopologicalSpace R] [ContinuousMul R] (e : R ≃L[R] R)
Goal: ↑((ContinuousLinearEquiv.unitsEquivAut R).symm e) = e 1
-- ContinuousLinearEquiv.fst_equivOfRightInverse
@[simp]
theorem fst_equivOfRightInverse (f₁ : M →L[R] M₂) (f₂ : M₂ →L[R] M)
    (h : Function.RightInverse f₂ f₁) (x : M) : (equivOfRightInverse f₁ f₂ h x).1 = f₁ x 
Arguments: {R : Type u_1} [Ring R] {M : Type u_3} [TopologicalSpace M] [AddCommGroup M] [Module R M] {M₂ : Type u_4} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M₂] [TopologicalAddGroup M] (f₁ : M →L[R] M₂) (f₂ : M₂ →L[R] M) (h : Function.RightInverse ⇑f₂ ⇑f₁) (x : M)
Goal: ((ContinuousLinearEquiv.equivOfRightInverse f₁ f₂ h) x).1 = f₁ x
-- ContinuousLinearEquiv.snd_equivOfRightInverse
@[simp]
theorem snd_equivOfRightInverse (f₁ : M →L[R] M₂) (f₂ : M₂ →L[R] M)
    (h : Function.RightInverse f₂ f₁) (x : M) :
    ((equivOfRightInverse f₁ f₂ h x).2 : M) = x - f₂ (f₁ x) 
Arguments: {R : Type u_1} [Ring R] {M : Type u_3} [TopologicalSpace M] [AddCommGroup M] [Module R M] {M₂ : Type u_4} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M₂] [TopologicalAddGroup M] (f₁ : M →L[R] M₂) (f₂ : M₂ →L[R] M) (h : Function.RightInverse ⇑f₂ ⇑f₁) (x : M)
Goal: ↑((ContinuousLinearEquiv.equivOfRightInverse f₁ f₂ h) x).2 = x - f₂ (f₁ x)
-- ContinuousLinearEquiv.equivOfRightInverse_symm_apply
@[simp]
theorem equivOfRightInverse_symm_apply (f₁ : M →L[R] M₂) (f₂ : M₂ →L[R] M)
    (h : Function.RightInverse f₂ f₁) (y : M₂ × ker f₁) :
    (equivOfRightInverse f₁ f₂ h).symm y = f₂ y.1 + y.2 
Arguments: {R : Type u_1} [Ring R] {M : Type u_3} [TopologicalSpace M] [AddCommGroup M] [Module R M] {M₂ : Type u_4} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M₂] [TopologicalAddGroup M] (f₁ : M →L[R] M₂) (f₂ : M₂ →L[R] M) (h : Function.RightInverse ⇑f₂ ⇑f₁) (y : M₂ × ↥(LinearMap.ker f₁))
Goal: (ContinuousLinearEquiv.equivOfRightInverse f₁ f₂ h).symm y = f₂ y.1 + ↑y.2
-- ContinuousLinearEquiv.coe_funUnique
@[simp]
theorem coe_funUnique : ⇑(funUnique ι R M) = Function.eval default 
Arguments: {ι : Type u_1} {R : Type u_2} {M : Type u_3} [Unique ι] [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M]
Goal: ⇑(ContinuousLinearEquiv.funUnique ι R M) = Function.eval default
-- ContinuousLinearEquiv.coe_funUnique_symm
@[simp]
theorem coe_funUnique_symm : ⇑(funUnique ι R M).symm = Function.const ι 
Arguments: {ι : Type u_1} {R : Type u_2} {M : Type u_3} [Unique ι] [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M]
Goal: ⇑(ContinuousLinearEquiv.funUnique ι R M).symm = Function.const ι
-- ContinuousLinearEquiv.piFinTwo_symm_apply
@[simps! (config := .asFn) apply symm_apply]
def piFinTwo (M : Fin 2 → Type*) [∀ i, AddCommMonoid (M i)] [∀ i, Module R (M i)]
    [∀ i, TopologicalSpace (M i)] : ((i : _) → M i) ≃L[R] M 0 × M 1 
Arguments: (R : Type u_2) [Semiring R] (M : Fin 2 → Type u_4) [(i : Fin 2) → AddCommMonoid (M i)] [(i : Fin 2) → Module R (M i)] [(i : Fin 2) → TopologicalSpace (M i)]
Goal: ⇑(ContinuousLinearEquiv.piFinTwo R M).symm = fun p => Fin.cons p.1 (Fin.cons p.2 finZeroElim)
-- ContinuousLinearEquiv.piFinTwo_apply
@[simps! (config := .asFn) apply symm_apply]
def piFinTwo (M : Fin 2 → Type*) [∀ i, AddCommMonoid (M i)] [∀ i, Module R (M i)]
    [∀ i, TopologicalSpace (M i)] : ((i : _) → M i) ≃L[R] M 0 × M 1 
Arguments: (R : Type u_2) [Semiring R] (M : Fin 2 → Type u_4) [(i : Fin 2) → AddCommMonoid (M i)] [(i : Fin 2) → Module R (M i)] [(i : Fin 2) → TopologicalSpace (M i)]
Goal: ⇑(ContinuousLinearEquiv.piFinTwo R M) = fun f => (f 0, f 1)
-- ContinuousLinearEquiv.finTwoArrow_apply
@[simps! (config := .asFn) apply symm_apply]
def finTwoArrow : (Fin 2 → M) ≃L[R] M × M 
Arguments: (R : Type u_2) (M : Type u_3) [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M]
Goal: ⇑(ContinuousLinearEquiv.finTwoArrow R M) = fun f => (f 0, f 1)
-- ContinuousLinearEquiv.finTwoArrow_symm_apply
@[simps! (config := .asFn) apply symm_apply]
def finTwoArrow : (Fin 2 → M) ≃L[R] M × M 
Arguments: (R : Type u_2) (M : Type u_3) [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M]
Goal: ⇑(ContinuousLinearEquiv.finTwoArrow R M).symm = fun x => ![x.1, x.2]
-- ContinuousLinearMap.inverse_equiv
@[simp]
theorem inverse_equiv (e : M ≃L[R] M₂) : inverse (e : M →L[R] M₂) = e.symm 
Arguments: {R : Type u_1} {M : Type u_2} {M₂ : Type u_3} [TopologicalSpace M] [TopologicalSpace M₂] [Semiring R] [AddCommMonoid M₂] [Module R M₂] [AddCommMonoid M] [Module R M] (e : M ≃L[R] M₂)
Goal: (↑e).inverse = ↑e.symm
-- ContinuousLinearMap.inverse_non_equiv
@[simp]
theorem inverse_non_equiv (f : M →L[R] M₂) (h : ¬∃ e' : M ≃L[R] M₂, ↑e' = f) : inverse f = 0 
Arguments: {R : Type u_1} {M : Type u_2} {M₂ : Type u_3} [TopologicalSpace M] [TopologicalSpace M₂] [Semiring R] [AddCommMonoid M₂] [Module R M₂] [AddCommMonoid M] [Module R M] (f : M →L[R] M₂) (h : ¬∃ e', ↑e' = f)
Goal: f.inverse = 0
-- ContinuousLinearMap.ring_inverse_equiv
@[simp]
theorem ring_inverse_equiv (e : M ≃L[R] M) : Ring.inverse ↑e = inverse (e : M →L[R] M) 
Arguments: {R : Type u_1} {M : Type u_2} [TopologicalSpace M] [Ring R] [AddCommGroup M] [Module R M] (e : M ≃L[R] M)
Goal: Ring.inverse ↑e = (↑e).inverse
-- ContinuousLinearMap.to_ring_inverse
theorem to_ring_inverse (e : M ≃L[R] M₂) (f : M →L[R] M₂) :
    inverse f = Ring.inverse ((e.symm : M₂ →L[R] M).comp f) ∘L e.symm 
Arguments: {R : Type u_1} {M : Type u_2} {M₂ : Type u_3} [TopologicalSpace M] [TopologicalSpace M₂] [Ring R] [AddCommGroup M] [Module R M] [AddCommGroup M₂] [Module R M₂] (e : M ≃L[R] M₂) (f : M →L[R] M₂)
Goal: f.inverse = (Ring.inverse ((↑e.symm).comp f)).comp ↑e.symm
-- ContinuousLinearMap.ring_inverse_eq_map_inverse
theorem ring_inverse_eq_map_inverse : Ring.inverse = @inverse R M M _ _ _ _ _ _ _ 
Arguments: {R : Type u_1} {M : Type u_2} [TopologicalSpace M] [Ring R] [AddCommGroup M] [Module R M]
Goal: Ring.inverse = ContinuousLinearMap.inverse
-- Submodule.ClosedComplemented.exists_isClosed_isCompl
theorem ClosedComplemented.exists_isClosed_isCompl {p : Submodule R M} [T1Space p]
    (h : ClosedComplemented p) :
    ∃ q : Submodule R M, IsClosed (q : Set M) ∧ IsCompl p q 
Arguments: {R : Type u_1} [Ring R] {M : Type u_2} [TopologicalSpace M] [AddCommGroup M] [Module R M] {p : Submodule R M} [T1Space ↥p] (h : p.ClosedComplemented)
Goal: ∃ q, IsClosed ↑q ∧ IsCompl p q
-- Submodule.ClosedComplemented.isClosed
protected theorem ClosedComplemented.isClosed [TopologicalAddGroup M] [T1Space M]
    {p : Submodule R M} (h : ClosedComplemented p) : IsClosed (p : Set M) 
Arguments: {R : Type u_1} [Ring R] {M : Type u_2} [TopologicalSpace M] [AddCommGroup M] [Module R M] [TopologicalAddGroup M] [T1Space M] {p : Submodule R M} (h : p.ClosedComplemented)
Goal: IsClosed ↑p
-- Submodule.closedComplemented_bot
@[simp]
theorem closedComplemented_bot : ClosedComplemented (⊥ : Submodule R M) 
Arguments: {R : Type u_1} [Ring R] {M : Type u_2} [TopologicalSpace M] [AddCommGroup M] [Module R M]
Goal: ⊥.ClosedComplemented
-- Submodule.closedComplemented_top
@[simp]
theorem closedComplemented_top : ClosedComplemented (⊤ : Submodule R M) 
Arguments: {R : Type u_1} [Ring R] {M : Type u_2} [TopologicalSpace M] [AddCommGroup M] [Module R M]
Goal: ⊤.ClosedComplemented
-- Submodule.ClosedComplemented.exists_submodule_equiv_prod
lemma ClosedComplemented.exists_submodule_equiv_prod [TopologicalAddGroup M]
    {p : Submodule R M} (hp : p.ClosedComplemented) :
    ∃ (q : Submodule R M) (e : M ≃L[R] (p × q)),
      (∀ x : p, e x = (x, 0)) ∧ (∀ y : q, e y = (0, y)) ∧ (∀ x, e.symm x = x.1 + x.2) 
Arguments: {R : Type u_1} [Ring R] {M : Type u_2} [TopologicalSpace M] [AddCommGroup M] [Module R M] [TopologicalAddGroup M] {p : Submodule R M} (hp : p.ClosedComplemented)
Goal: ∃ q e, (∀ (x : ↥p), e ↑x = (x, 0)) ∧ (∀ (y : ↥q), e ↑y = (0, y)) ∧ ∀ (x : ↥p × ↥q), e.symm x = ↑x.1 + ↑x.2
-- ContinuousLinearMap.closedComplemented_ker_of_rightInverse
theorem ContinuousLinearMap.closedComplemented_ker_of_rightInverse {R : Type*} [Ring R]
    {M : Type*} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type*} [TopologicalSpace M₂]
    [AddCommGroup M₂] [Module R M] [Module R M₂] [TopologicalAddGroup M] (f₁ : M →L[R] M₂)
    (f₂ : M₂ →L[R] M) (h : Function.RightInverse f₂ f₁) : (ker f₁).ClosedComplemented 
Arguments: {R : Type u_1} [Ring R] {M : Type u_2} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_3} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M] [Module R M₂] [TopologicalAddGroup M] (f₁ : M →L[R] M₂) (f₂ : M₂ →L[R] M) (h : Function.RightInverse ⇑f₂ ⇑f₁)
Goal: (LinearMap.ker f₁).ClosedComplemented
-- Submodule.isOpenMap_mkQ
theorem isOpenMap_mkQ [TopologicalAddGroup M] : IsOpenMap S.mkQ 
Arguments: {R : Type u_1} {M : Type u_2} [Ring R] [AddCommGroup M] [Module R M] [TopologicalSpace M] (S : Submodule R M) [TopologicalAddGroup M]
Goal: IsOpenMap ⇑S.mkQ
-- _private.Mathlib.Topology.MetricSpace.GromovHausdorff.0.GromovHausdorff.equivalence_isometryRel
private theorem equivalence_isometryRel : Equivalence IsometryRel 
Goal: Equivalence GromovHausdorff.IsometryRel
-- GromovHausdorff.eq_toGHSpace_iff
theorem eq_toGHSpace_iff {X : Type u} [MetricSpace X] [CompactSpace X] [Nonempty X]
    {p : NonemptyCompacts ℓ_infty_ℝ} :
    ⟦p⟧ = toGHSpace X ↔ ∃ Ψ : X → ℓ_infty_ℝ, Isometry Ψ ∧ range Ψ = p 
Arguments: {X : Type u} [MetricSpace X] [CompactSpace X] [Nonempty X] {p : TopologicalSpace.NonemptyCompacts ↥(lp (fun n => ℝ) ⊤)}
Goal: ⟦p⟧ = GromovHausdorff.toGHSpace X ↔ ∃ Ψ, Isometry Ψ ∧ Set.range Ψ = ↑p
-- GromovHausdorff.eq_toGHSpace
theorem eq_toGHSpace {p : NonemptyCompacts ℓ_infty_ℝ} : ⟦p⟧ = toGHSpace p 
Arguments: {p : TopologicalSpace.NonemptyCompacts ↥(lp (fun n => ℝ) ⊤)}
Goal: ⟦p⟧ = GromovHausdorff.toGHSpace ↥p
-- GromovHausdorff.GHSpace.toGHSpace_rep
theorem GHSpace.toGHSpace_rep (p : GHSpace) : toGHSpace p.Rep = p 
Arguments: (p : GromovHausdorff.GHSpace)
Goal: GromovHausdorff.toGHSpace p.Rep = p