Documentation

Mathlib.Algebra.Hom.Equiv.TypeTags

Additive and multiplicative equivalences associated to Multiplicative and Additive. #

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@[simp]
theorem AddEquiv.toMultiplicative_symm_apply_apply {G : Type u_1} {H : Type u_2} [AddZeroClass G] [AddZeroClass H] (f : Multiplicative G ≃* Multiplicative H) (a : G) :
↑(↑AddEquiv.toMultiplicative.symm f) a = ↑(↑AddMonoidHom.toMultiplicative.symm (MulEquiv.toMonoidHom f)) a
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@[simp]
theorem AddEquiv.toMultiplicative_symm_apply_symm_apply {G : Type u_1} {H : Type u_2} [AddZeroClass G] [AddZeroClass H] (f : Multiplicative G ≃* Multiplicative H) (a : H) :
↑(AddEquiv.symm (↑AddEquiv.toMultiplicative.symm f)) a = ↑(↑AddMonoidHom.toMultiplicative.symm (MulEquiv.toMonoidHom (MulEquiv.symm f))) a
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@[simp]
theorem AddEquiv.toMultiplicative_apply_symm_apply {G : Type u_1} {H : Type u_2} [AddZeroClass G] [AddZeroClass H] (f : G ≃+ H) (a : Multiplicative H) :
↑(MulEquiv.symm (↑AddEquiv.toMultiplicative f)) a = ↑(↑AddMonoidHom.toMultiplicative (AddEquiv.toAddMonoidHom (AddEquiv.symm f))) a
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@[simp]
theorem AddEquiv.toMultiplicative_apply_apply {G : Type u_1} {H : Type u_2} [AddZeroClass G] [AddZeroClass H] (f : G ≃+ H) (a : Multiplicative G) :
↑(↑AddEquiv.toMultiplicative f) a = ↑(↑AddMonoidHom.toMultiplicative (AddEquiv.toAddMonoidHom f)) a
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def AddEquiv.toMultiplicative {G : Type u_1} {H : Type u_2} [AddZeroClass G] [AddZeroClass H] :
G ≃+ H ≃ (Multiplicative G ≃* Multiplicative H)

Reinterpret G ≃+ H as Multiplicative G ≃* Multiplicative H.

Instances For
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    @[simp]
    theorem MulEquiv.toAdditive_apply_symm_apply {G : Type u_1} {H : Type u_2} [MulOneClass G] [MulOneClass H] (f : G ≃* H) (a : Additive H) :
    ↑(AddEquiv.symm (↑MulEquiv.toAdditive f)) a = ↑(↑MonoidHom.toAdditive (MulEquiv.toMonoidHom (MulEquiv.symm f))) a
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    @[simp]
    theorem MulEquiv.toAdditive_apply_apply {G : Type u_1} {H : Type u_2} [MulOneClass G] [MulOneClass H] (f : G ≃* H) (a : Additive G) :
    ↑(↑MulEquiv.toAdditive f) a = ↑(↑MonoidHom.toAdditive (MulEquiv.toMonoidHom f)) a
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    @[simp]
    theorem MulEquiv.toAdditive_symm_apply_symm_apply {G : Type u_1} {H : Type u_2} [MulOneClass G] [MulOneClass H] (f : Additive G ≃+ Additive H) (a : H) :
    ↑(MulEquiv.symm (↑MulEquiv.toAdditive.symm f)) a = ↑(↑MonoidHom.toAdditive.symm (AddEquiv.toAddMonoidHom (AddEquiv.symm f))) a
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    @[simp]
    theorem MulEquiv.toAdditive_symm_apply_apply {G : Type u_1} {H : Type u_2} [MulOneClass G] [MulOneClass H] (f : Additive G ≃+ Additive H) (a : G) :
    ↑(↑MulEquiv.toAdditive.symm f) a = ↑(↑MonoidHom.toAdditive.symm (AddEquiv.toAddMonoidHom f)) a
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    def MulEquiv.toAdditive {G : Type u_1} {H : Type u_2} [MulOneClass G] [MulOneClass H] :
    G ≃* H ≃ (Additive G ≃+ Additive H)

    Reinterpret G ≃* H as Additive G ≃+ Additive H.

    Instances For
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      @[simp]
      theorem AddEquiv.toMultiplicative'_symm_apply_symm_apply {G : Type u_1} {H : Type u_2} [MulOneClass G] [AddZeroClass H] (f : G ≃* Multiplicative H) (a : H) :
      ↑(AddEquiv.symm (↑AddEquiv.toMultiplicative'.symm f)) a = ↑(↑AddMonoidHom.toMultiplicative''.symm (MulEquiv.toMonoidHom (MulEquiv.symm f))) a
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      @[simp]
      theorem AddEquiv.toMultiplicative'_symm_apply_apply {G : Type u_1} {H : Type u_2} [MulOneClass G] [AddZeroClass H] (f : G ≃* Multiplicative H) (a : Additive G) :
      ↑(↑AddEquiv.toMultiplicative'.symm f) a = ↑(↑AddMonoidHom.toMultiplicative'.symm (MulEquiv.toMonoidHom f)) a
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      @[simp]
      theorem AddEquiv.toMultiplicative'_apply_symm_apply {G : Type u_1} {H : Type u_2} [MulOneClass G] [AddZeroClass H] (f : Additive G ≃+ H) (a : Multiplicative H) :
      ↑(MulEquiv.symm (↑AddEquiv.toMultiplicative' f)) a = ↑(↑AddMonoidHom.toMultiplicative'' (AddEquiv.toAddMonoidHom (AddEquiv.symm f))) a
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      @[simp]
      theorem AddEquiv.toMultiplicative'_apply_apply {G : Type u_1} {H : Type u_2} [MulOneClass G] [AddZeroClass H] (f : Additive G ≃+ H) (a : G) :
      ↑(↑AddEquiv.toMultiplicative' f) a = ↑(↑AddMonoidHom.toMultiplicative' (AddEquiv.toAddMonoidHom f)) a
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      def AddEquiv.toMultiplicative' {G : Type u_1} {H : Type u_2} [MulOneClass G] [AddZeroClass H] :
      Additive G ≃+ H ≃ (G ≃* Multiplicative H)

      Reinterpret Additive G ≃+ H as G ≃* Multiplicative H.

      Instances For
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        @[inline, reducible]
        abbrev MulEquiv.toAdditive' {G : Type u_1} {H : Type u_2} [MulOneClass G] [AddZeroClass H] :
        G ≃* Multiplicative H ≃ (Additive G ≃+ H)

        Reinterpret G ≃* Multiplicative H as Additive G ≃+ H as.

        Instances For
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          @[simp]
          theorem AddEquiv.toMultiplicative''_apply_apply {G : Type u_1} {H : Type u_2} [AddZeroClass G] [MulOneClass H] (f : G ≃+ Additive H) (a : Multiplicative G) :
          ↑(↑AddEquiv.toMultiplicative'' f) a = ↑(↑AddMonoidHom.toMultiplicative'' (AddEquiv.toAddMonoidHom f)) a
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          @[simp]
          theorem AddEquiv.toMultiplicative''_apply_symm_apply {G : Type u_1} {H : Type u_2} [AddZeroClass G] [MulOneClass H] (f : G ≃+ Additive H) (a : H) :
          ↑(MulEquiv.symm (↑AddEquiv.toMultiplicative'' f)) a = ↑(↑AddMonoidHom.toMultiplicative' (AddEquiv.toAddMonoidHom (AddEquiv.symm f))) a
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          @[simp]
          theorem AddEquiv.toMultiplicative''_symm_apply_apply {G : Type u_1} {H : Type u_2} [AddZeroClass G] [MulOneClass H] (f : Multiplicative G ≃* H) (a : G) :
          ↑(↑AddEquiv.toMultiplicative''.symm f) a = ↑(↑AddMonoidHom.toMultiplicative''.symm (MulEquiv.toMonoidHom f)) a
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          @[simp]
          theorem AddEquiv.toMultiplicative''_symm_apply_symm_apply {G : Type u_1} {H : Type u_2} [AddZeroClass G] [MulOneClass H] (f : Multiplicative G ≃* H) (a : Additive H) :
          ↑(AddEquiv.symm (↑AddEquiv.toMultiplicative''.symm f)) a = ↑(↑AddMonoidHom.toMultiplicative'.symm (MulEquiv.toMonoidHom (MulEquiv.symm f))) a
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          def AddEquiv.toMultiplicative'' {G : Type u_1} {H : Type u_2} [AddZeroClass G] [MulOneClass H] :
          G ≃+ Additive H ≃ (Multiplicative G ≃* H)

          Reinterpret G ≃+ Additive H as Multiplicative G ≃* H.

          Instances For
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            @[inline, reducible]
            abbrev MulEquiv.toAdditive'' {G : Type u_1} {H : Type u_2} [AddZeroClass G] [MulOneClass H] :
            Multiplicative G ≃* H ≃ (G ≃+ Additive H)

            Reinterpret Multiplicative G ≃* H as G ≃+ Additive H as.

            Instances For
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              @[simp]
              theorem AddEquiv.additiveMultiplicative_symm_apply (G : Type u_1) [AddZeroClass G] (a : G) :
              ↑(AddEquiv.symm (AddEquiv.additiveMultiplicative G)) a = ↑Additive.ofMul (↑Multiplicative.ofAdd a)
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              @[simp]
              theorem AddEquiv.additiveMultiplicative_apply (G : Type u_1) [AddZeroClass G] (a : Additive (Multiplicative G)) :
              ↑(AddEquiv.additiveMultiplicative G) a = ↑Multiplicative.toAdd (↑Additive.toMul a)
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              def AddEquiv.additiveMultiplicative (G : Type u_1) [AddZeroClass G] :
              Additive (Multiplicative G) ≃+ G

              Additive (Multiplicative G) is just G.

              Instances For
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                @[simp]
                theorem MulEquiv.multiplicativeAdditive_symm_apply (H : Type u_2) [MulOneClass H] (a : H) :
                ↑(MulEquiv.symm (MulEquiv.multiplicativeAdditive H)) a = ↑Multiplicative.ofAdd (↑Additive.ofMul a)
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                @[simp]
                theorem MulEquiv.multiplicativeAdditive_apply (H : Type u_2) [MulOneClass H] (a : Multiplicative (Additive H)) :
                ↑(MulEquiv.multiplicativeAdditive H) a = ↑Additive.toMul (↑Multiplicative.toAdd a)
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                def MulEquiv.multiplicativeAdditive (H : Type u_2) [MulOneClass H] :
                Multiplicative (Additive H) ≃* H

                Multiplicative (Additive H) is just H.

                Instances For