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Mathlib.LinearAlgebra.QuadraticForm.TensorProduct.Isometries

Linear equivalences of tensor products as isometries #

These results are separate from the definition of QuadraticForm.tmul as that file is very slow.

Main definitions #

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@[simp]
theorem QuadraticForm.tmul_comp_tensorMap {R : Type uR} {M₁ : Type uM₁} {M₂ : Type uM₂} {M₃ : Type uM₃} {M₄ : Type uM₄} [CommRing R] [AddCommGroup M₁] [AddCommGroup M₂] [AddCommGroup M₃] [AddCommGroup M₄] [Module R M₁] [Module R M₂] [Module R M₃] [Module R M₄] [Invertible 2] {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂} {Q₃ : QuadraticForm R M₃} {Q₄ : QuadraticForm R M₄} (f : Q₁ →qᵢ Q₂) (g : Q₃ →qᵢ Q₄) :
QuadraticForm.comp (QuadraticForm.tmul Q₂ Q₄) (TensorProduct.map f.toLinearMap g.toLinearMap) = QuadraticForm.tmul Q₁ Q₃
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@[simp]
theorem QuadraticForm.tmul_tensorMap_apply {R : Type uR} {M₁ : Type uM₁} {M₂ : Type uM₂} {M₃ : Type uM₃} {M₄ : Type uM₄} [CommRing R] [AddCommGroup M₁] [AddCommGroup M₂] [AddCommGroup M₃] [AddCommGroup M₄] [Module R M₁] [Module R M₂] [Module R M₃] [Module R M₄] [Invertible 2] {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂} {Q₃ : QuadraticForm R M₃} {Q₄ : QuadraticForm R M₄} (f : Q₁ →qᵢ Q₂) (g : Q₃ →qᵢ Q₄) (x : TensorProduct R M₁ M₃) :
(QuadraticForm.tmul Q₂ Q₄) ((TensorProduct.map f.toLinearMap g.toLinearMap) x) = (QuadraticForm.tmul Q₁ Q₃) x
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noncomputable def QuadraticForm.Isometry.tmul {R : Type uR} {M₁ : Type uM₁} {M₂ : Type uM₂} {M₃ : Type uM₃} {M₄ : Type uM₄} [CommRing R] [AddCommGroup M₁] [AddCommGroup M₂] [AddCommGroup M₃] [AddCommGroup M₄] [Module R M₁] [Module R M₂] [Module R M₃] [Module R M₄] [Invertible 2] {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂} {Q₃ : QuadraticForm R M₃} {Q₄ : QuadraticForm R M₄} (f : Q₁ →qᵢ Q₂) (g : Q₃ →qᵢ Q₄) :
QuadraticForm.tmul Q₁ Q₃ →qᵢ QuadraticForm.tmul Q₂ Q₄

TensorProduct.map for Quadraticform.Isometrys

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    @[simp]
    theorem QuadraticForm.Isometry.tmul_apply {R : Type uR} {M₁ : Type uM₁} {M₂ : Type uM₂} {M₃ : Type uM₃} {M₄ : Type uM₄} [CommRing R] [AddCommGroup M₁] [AddCommGroup M₂] [AddCommGroup M₃] [AddCommGroup M₄] [Module R M₁] [Module R M₂] [Module R M₃] [Module R M₄] [Invertible 2] {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂} {Q₃ : QuadraticForm R M₃} {Q₄ : QuadraticForm R M₄} (f : Q₁ →qᵢ Q₂) (g : Q₃ →qᵢ Q₄) (x : TensorProduct R M₁ M₃) :
    (QuadraticForm.Isometry.tmul f g) x = (TensorProduct.map f.toLinearMap g.toLinearMap) x
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    @[simp]
    theorem QuadraticForm.tmul_comp_tensorComm {R : Type uR} {M₁ : Type uM₁} {M₂ : Type uM₂} [CommRing R] [AddCommGroup M₁] [AddCommGroup M₂] [Module R M₁] [Module R M₂] [Invertible 2] (Q₁ : QuadraticForm R M₁) (Q₂ : QuadraticForm R M₂) :
    QuadraticForm.comp (QuadraticForm.tmul Q₂ Q₁) ↑(TensorProduct.comm R M₁ M₂) = QuadraticForm.tmul Q₁ Q₂
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    @[simp]
    theorem QuadraticForm.tmul_tensorComm_apply {R : Type uR} {M₁ : Type uM₁} {M₂ : Type uM₂} [CommRing R] [AddCommGroup M₁] [AddCommGroup M₂] [Module R M₁] [Module R M₂] [Invertible 2] (Q₁ : QuadraticForm R M₁) (Q₂ : QuadraticForm R M₂) (x : TensorProduct R M₁ M₂) :
    (QuadraticForm.tmul Q₂ Q₁) ((TensorProduct.comm R M₁ M₂) x) = (QuadraticForm.tmul Q₁ Q₂) x
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    noncomputable def QuadraticForm.tensorComm {R : Type uR} {M₁ : Type uM₁} {M₂ : Type uM₂} [CommRing R] [AddCommGroup M₁] [AddCommGroup M₂] [Module R M₁] [Module R M₂] [Invertible 2] (Q₁ : QuadraticForm R M₁) (Q₂ : QuadraticForm R M₂) :
    QuadraticForm.IsometryEquiv (QuadraticForm.tmul Q₁ Q₂) (QuadraticForm.tmul Q₂ Q₁)

    TensorProduct.comm preserves tensor products of quadratic forms.

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      @[simp]
      theorem QuadraticForm.tensorComm_toLinearEquiv {R : Type uR} {M₁ : Type uM₁} {M₂ : Type uM₂} [CommRing R] [AddCommGroup M₁] [AddCommGroup M₂] [Module R M₁] [Module R M₂] [Invertible 2] (Q₁ : QuadraticForm R M₁) (Q₂ : QuadraticForm R M₂) :
      (QuadraticForm.tensorComm Q₁ Q₂).toLinearEquiv = TensorProduct.comm R M₁ M₂
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      @[simp]
      theorem QuadraticForm.tensorComm_apply {R : Type uR} {M₁ : Type uM₁} {M₂ : Type uM₂} [CommRing R] [AddCommGroup M₁] [AddCommGroup M₂] [Module R M₁] [Module R M₂] [Invertible 2] (Q₁ : QuadraticForm R M₁) (Q₂ : QuadraticForm R M₂) (x : TensorProduct R M₁ M₂) :
      (QuadraticForm.tensorComm Q₁ Q₂) x = (TensorProduct.comm R M₁ M₂) x
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      @[simp]
      theorem QuadraticForm.tensorComm_symm {R : Type uR} {M₁ : Type uM₁} {M₂ : Type uM₂} [CommRing R] [AddCommGroup M₁] [AddCommGroup M₂] [Module R M₁] [Module R M₂] [Invertible 2] (Q₁ : QuadraticForm R M₁) (Q₂ : QuadraticForm R M₂) :
      QuadraticForm.IsometryEquiv.symm (QuadraticForm.tensorComm Q₁ Q₂) = QuadraticForm.tensorComm Q₂ Q₁
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      @[simp]
      theorem QuadraticForm.tmul_comp_tensorAssoc {R : Type uR} {M₁ : Type uM₁} {M₂ : Type uM₂} {M₃ : Type uM₃} [CommRing R] [AddCommGroup M₁] [AddCommGroup M₂] [AddCommGroup M₃] [Module R M₁] [Module R M₂] [Module R M₃] [Invertible 2] (Q₁ : QuadraticForm R M₁) (Q₂ : QuadraticForm R M₂) (Q₃ : QuadraticForm R M₃) :
      QuadraticForm.comp (QuadraticForm.tmul Q₁ (QuadraticForm.tmul Q₂ Q₃)) ↑(TensorProduct.assoc R M₁ M₂ M₃) = QuadraticForm.tmul (QuadraticForm.tmul Q₁ Q₂) Q₃
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      @[simp]
      theorem QuadraticForm.tmul_tensorAssoc_apply {R : Type uR} {M₁ : Type uM₁} {M₂ : Type uM₂} {M₃ : Type uM₃} [CommRing R] [AddCommGroup M₁] [AddCommGroup M₂] [AddCommGroup M₃] [Module R M₁] [Module R M₂] [Module R M₃] [Invertible 2] (Q₁ : QuadraticForm R M₁) (Q₂ : QuadraticForm R M₂) (Q₃ : QuadraticForm R M₃) (x : TensorProduct R (TensorProduct R M₁ M₂) M₃) :
      (QuadraticForm.tmul Q₁ (QuadraticForm.tmul Q₂ Q₃)) ((TensorProduct.assoc R M₁ M₂ M₃) x) = (QuadraticForm.tmul (QuadraticForm.tmul Q₁ Q₂) Q₃) x
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      noncomputable def QuadraticForm.tensorAssoc {R : Type uR} {M₁ : Type uM₁} {M₂ : Type uM₂} {M₃ : Type uM₃} [CommRing R] [AddCommGroup M₁] [AddCommGroup M₂] [AddCommGroup M₃] [Module R M₁] [Module R M₂] [Module R M₃] [Invertible 2] (Q₁ : QuadraticForm R M₁) (Q₂ : QuadraticForm R M₂) (Q₃ : QuadraticForm R M₃) :
      QuadraticForm.IsometryEquiv (QuadraticForm.tmul (QuadraticForm.tmul Q₁ Q₂) Q₃) (QuadraticForm.tmul Q₁ (QuadraticForm.tmul Q₂ Q₃))

      TensorProduct.assoc preserves tensor products of quadratic forms.

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        @[simp]
        theorem QuadraticForm.tensorAssoc_toLinearEquiv {R : Type uR} {M₁ : Type uM₁} {M₂ : Type uM₂} {M₃ : Type uM₃} [CommRing R] [AddCommGroup M₁] [AddCommGroup M₂] [AddCommGroup M₃] [Module R M₁] [Module R M₂] [Module R M₃] [Invertible 2] (Q₁ : QuadraticForm R M₁) (Q₂ : QuadraticForm R M₂) (Q₃ : QuadraticForm R M₃) :
        (QuadraticForm.tensorAssoc Q₁ Q₂ Q₃).toLinearEquiv = TensorProduct.assoc R M₁ M₂ M₃
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        @[simp]
        theorem QuadraticForm.tensorAssoc_apply {R : Type uR} {M₁ : Type uM₁} {M₂ : Type uM₂} {M₃ : Type uM₃} [CommRing R] [AddCommGroup M₁] [AddCommGroup M₂] [AddCommGroup M₃] [Module R M₁] [Module R M₂] [Module R M₃] [Invertible 2] (Q₁ : QuadraticForm R M₁) (Q₂ : QuadraticForm R M₂) (Q₃ : QuadraticForm R M₃) (x : TensorProduct R (TensorProduct R M₁ M₂) M₃) :
        (QuadraticForm.tensorAssoc Q₁ Q₂ Q₃) x = (TensorProduct.assoc R M₁ M₂ M₃) x
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        @[simp]
        theorem QuadraticForm.tensorAssoc_symm_apply {R : Type uR} {M₁ : Type uM₁} {M₂ : Type uM₂} {M₃ : Type uM₃} [CommRing R] [AddCommGroup M₁] [AddCommGroup M₂] [AddCommGroup M₃] [Module R M₁] [Module R M₂] [Module R M₃] [Invertible 2] (Q₁ : QuadraticForm R M₁) (Q₂ : QuadraticForm R M₂) (Q₃ : QuadraticForm R M₃) (x : TensorProduct R M₁ (TensorProduct R M₂ M₃)) :
        (QuadraticForm.IsometryEquiv.symm (QuadraticForm.tensorAssoc Q₁ Q₂ Q₃)) x = (LinearEquiv.symm (TensorProduct.assoc R M₁ M₂ M₃)) x
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        theorem QuadraticForm.comp_tensorRId_eq {R : Type uR} {M₁ : Type uM₁} [CommRing R] [AddCommGroup M₁] [Module R M₁] [Invertible 2] (Q₁ : QuadraticForm R M₁) :
        QuadraticForm.comp Q₁ ↑(TensorProduct.rid R M₁) = QuadraticForm.tmul Q₁ QuadraticForm.sq
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        @[simp]
        theorem QuadraticForm.tmul_tensorRId_apply {R : Type uR} {M₁ : Type uM₁} [CommRing R] [AddCommGroup M₁] [Module R M₁] [Invertible 2] (Q₁ : QuadraticForm R M₁) (x : TensorProduct R M₁ R) :
        Q₁ ((TensorProduct.rid R M₁) x) = (QuadraticForm.tmul Q₁ QuadraticForm.sq) x
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        noncomputable def QuadraticForm.tensorRId {R : Type uR} {M₁ : Type uM₁} [CommRing R] [AddCommGroup M₁] [Module R M₁] [Invertible 2] (Q₁ : QuadraticForm R M₁) :
        QuadraticForm.IsometryEquiv (QuadraticForm.tmul Q₁ QuadraticForm.sq) Q₁

        TensorProduct.rid preserves tensor products of quadratic forms.

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          @[simp]
          theorem QuadraticForm.tensorRId_toLinearEquiv {R : Type uR} {M₁ : Type uM₁} [CommRing R] [AddCommGroup M₁] [Module R M₁] [Invertible 2] (Q₁ : QuadraticForm R M₁) :
          (QuadraticForm.tensorRId Q₁).toLinearEquiv = TensorProduct.rid R M₁
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          @[simp]
          theorem QuadraticForm.tensorRId_apply {R : Type uR} {M₁ : Type uM₁} [CommRing R] [AddCommGroup M₁] [Module R M₁] [Invertible 2] (Q₁ : QuadraticForm R M₁) (x : TensorProduct R M₁ R) :
          (QuadraticForm.tensorRId Q₁) x = (TensorProduct.rid R M₁) x
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          @[simp]
          theorem QuadraticForm.tensorRId_symm_apply {R : Type uR} {M₁ : Type uM₁} [CommRing R] [AddCommGroup M₁] [Module R M₁] [Invertible 2] (Q₁ : QuadraticForm R M₁) (x : M₁) :
          (QuadraticForm.IsometryEquiv.symm (QuadraticForm.tensorRId Q₁)) x = (LinearEquiv.symm (TensorProduct.rid R M₁)) x
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          theorem QuadraticForm.comp_tensorLId_eq {R : Type uR} {M₂ : Type uM₂} [CommRing R] [AddCommGroup M₂] [Module R M₂] [Invertible 2] (Q₂ : QuadraticForm R M₂) :
          QuadraticForm.comp Q₂ ↑(TensorProduct.lid R M₂) = QuadraticForm.tmul QuadraticForm.sq Q₂
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          @[simp]
          theorem QuadraticForm.tmul_tensorLId_apply {R : Type uR} {M₂ : Type uM₂} [CommRing R] [AddCommGroup M₂] [Module R M₂] [Invertible 2] (Q₂ : QuadraticForm R M₂) (x : TensorProduct R R M₂) :
          Q₂ ((TensorProduct.lid R M₂) x) = (QuadraticForm.tmul QuadraticForm.sq Q₂) x
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          noncomputable def QuadraticForm.tensorLId {R : Type uR} {M₂ : Type uM₂} [CommRing R] [AddCommGroup M₂] [Module R M₂] [Invertible 2] (Q₂ : QuadraticForm R M₂) :
          QuadraticForm.IsometryEquiv (QuadraticForm.tmul QuadraticForm.sq Q₂) Q₂

          TensorProduct.lid preserves tensor products of quadratic forms.

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            @[simp]
            theorem QuadraticForm.tensorLId_toLinearEquiv {R : Type uR} {M₂ : Type uM₂} [CommRing R] [AddCommGroup M₂] [Module R M₂] [Invertible 2] (Q₂ : QuadraticForm R M₂) :
            (QuadraticForm.tensorLId Q₂).toLinearEquiv = TensorProduct.lid R M₂
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            @[simp]
            theorem QuadraticForm.tensorLId_apply {R : Type uR} {M₂ : Type uM₂} [CommRing R] [AddCommGroup M₂] [Module R M₂] [Invertible 2] (Q₂ : QuadraticForm R M₂) (x : TensorProduct R R M₂) :
            (QuadraticForm.tensorLId Q₂) x = (TensorProduct.lid R M₂) x
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            @[simp]
            theorem QuadraticForm.tensorLId_symm_apply {R : Type uR} {M₂ : Type uM₂} [CommRing R] [AddCommGroup M₂] [Module R M₂] [Invertible 2] (Q₂ : QuadraticForm R M₂) (x : M₂) :
            (QuadraticForm.IsometryEquiv.symm (QuadraticForm.tensorLId Q₂)) x = (LinearEquiv.symm (TensorProduct.lid R M₂)) x