File: Mathlib/Algebra/Group/Equiv/TypeTags.lean

full_name: AddEquiv.toMultiplicative_apply_symm_apply
code: /-- Reinterpret `G ≃+ H` as `Multiplicative G ≃* Multiplicative H`. -/
@[simps]
def AddEquiv.toMultiplicative [AddZeroClass G] [AddZeroClass H] :
    G ≃+ H ≃ (Multiplicative G ≃* Multiplicative H) 
goal: (AddEquiv.toMultiplicative f).symm a = (AddMonoidHom.toMultiplicative f.symm.toAddMonoidHom) a

full_name: AddEquiv.toMultiplicative
code: /-- Reinterpret `G ≃+ H` as `Multiplicative G ≃* Multiplicative H`. -/
@[simps]
def AddEquiv.toMultiplicative [AddZeroClass G] [AddZeroClass H] :
    G ≃+ H ≃ (Multiplicative G ≃* Multiplicative H) 
goal: G ≃+ H ≃ (Multiplicative G ≃* Multiplicative H)

full_name: AddEquiv.toMultiplicative_symm_apply_symm_apply
code: /-- Reinterpret `G ≃+ H` as `Multiplicative G ≃* Multiplicative H`. -/
@[simps]
def AddEquiv.toMultiplicative [AddZeroClass G] [AddZeroClass H] :
    G ≃+ H ≃ (Multiplicative G ≃* Multiplicative H) 
goal: (AddEquiv.toMultiplicative.symm f).symm a = (AddMonoidHom.toMultiplicative.symm f.symm.toMonoidHom) a

full_name: AddEquiv.toMultiplicative_symm_apply_apply
code: /-- Reinterpret `G ≃+ H` as `Multiplicative G ≃* Multiplicative H`. -/
@[simps]
def AddEquiv.toMultiplicative [AddZeroClass G] [AddZeroClass H] :
    G ≃+ H ≃ (Multiplicative G ≃* Multiplicative H) 
goal: (AddEquiv.toMultiplicative.symm f) a = (AddMonoidHom.toMultiplicative.symm f.toMonoidHom) a

full_name: AddEquiv.toMultiplicative_apply_apply
code: /-- Reinterpret `G ≃+ H` as `Multiplicative G ≃* Multiplicative H`. -/
@[simps]
def AddEquiv.toMultiplicative [AddZeroClass G] [AddZeroClass H] :
    G ≃+ H ≃ (Multiplicative G ≃* Multiplicative H) 
goal: (AddEquiv.toMultiplicative f) a = (AddMonoidHom.toMultiplicative f.toAddMonoidHom) a

full_name: MulEquiv.toAdditive_symm_apply_apply
code: /-- Reinterpret `G ≃* H` as `Additive G ≃+ Additive H`. -/
@[simps]
def MulEquiv.toAdditive [MulOneClass G] [MulOneClass H] :
    G ≃* H ≃ (Additive G ≃+ Additive H) 
goal: (MulEquiv.toAdditive.symm f) a = (MonoidHom.toAdditive.symm f.toAddMonoidHom) a

full_name: MulEquiv.toAdditive_apply_apply
code: /-- Reinterpret `G ≃* H` as `Additive G ≃+ Additive H`. -/
@[simps]
def MulEquiv.toAdditive [MulOneClass G] [MulOneClass H] :
    G ≃* H ≃ (Additive G ≃+ Additive H) 
goal: (MulEquiv.toAdditive f) a = (MonoidHom.toAdditive f.toMonoidHom) a

full_name: MulEquiv.toAdditive_apply_symm_apply
code: /-- Reinterpret `G ≃* H` as `Additive G ≃+ Additive H`. -/
@[simps]
def MulEquiv.toAdditive [MulOneClass G] [MulOneClass H] :
    G ≃* H ≃ (Additive G ≃+ Additive H) 
goal: (MulEquiv.toAdditive f).symm a = (MonoidHom.toAdditive f.symm.toMonoidHom) a

full_name: MulEquiv.toAdditive
code: /-- Reinterpret `G ≃* H` as `Additive G ≃+ Additive H`. -/
@[simps]
def MulEquiv.toAdditive [MulOneClass G] [MulOneClass H] :
    G ≃* H ≃ (Additive G ≃+ Additive H) 
goal: G ≃* H ≃ (Additive G ≃+ Additive H)

full_name: MulEquiv.toAdditive_symm_apply_symm_apply
code: /-- Reinterpret `G ≃* H` as `Additive G ≃+ Additive H`. -/
@[simps]
def MulEquiv.toAdditive [MulOneClass G] [MulOneClass H] :
    G ≃* H ≃ (Additive G ≃+ Additive H) 
goal: (MulEquiv.toAdditive.symm f).symm a = (MonoidHom.toAdditive.symm f.symm.toAddMonoidHom) a

full_name: AddEquiv.toMultiplicative'
code: /-- Reinterpret `Additive G ≃+ H` as `G ≃* Multiplicative H`. -/
@[simps]
def AddEquiv.toMultiplicative' [MulOneClass G] [AddZeroClass H] :
    Additive G ≃+ H ≃ (G ≃* Multiplicative H) 
goal: Additive G ≃+ H ≃ (G ≃* Multiplicative H)

full_name: AddEquiv.toMultiplicative'_apply_apply
code: /-- Reinterpret `Additive G ≃+ H` as `G ≃* Multiplicative H`. -/
@[simps]
def AddEquiv.toMultiplicative' [MulOneClass G] [AddZeroClass H] :
    Additive G ≃+ H ≃ (G ≃* Multiplicative H) 
goal: (AddEquiv.toMultiplicative' f) a = (AddMonoidHom.toMultiplicative' f.toAddMonoidHom) a

full_name: AddEquiv.toMultiplicative'_symm_apply_apply
code: /-- Reinterpret `Additive G ≃+ H` as `G ≃* Multiplicative H`. -/
@[simps]
def AddEquiv.toMultiplicative' [MulOneClass G] [AddZeroClass H] :
    Additive G ≃+ H ≃ (G ≃* Multiplicative H) 
goal: (AddEquiv.toMultiplicative'.symm f) a = (AddMonoidHom.toMultiplicative'.symm f.toMonoidHom) a

full_name: AddEquiv.toMultiplicative'_apply_symm_apply
code: /-- Reinterpret `Additive G ≃+ H` as `G ≃* Multiplicative H`. -/
@[simps]
def AddEquiv.toMultiplicative' [MulOneClass G] [AddZeroClass H] :
    Additive G ≃+ H ≃ (G ≃* Multiplicative H) 
goal: (AddEquiv.toMultiplicative' f).symm a = (AddMonoidHom.toMultiplicative'' f.symm.toAddMonoidHom) a

full_name: AddEquiv.toMultiplicative'_symm_apply_symm_apply
code: /-- Reinterpret `Additive G ≃+ H` as `G ≃* Multiplicative H`. -/
@[simps]
def AddEquiv.toMultiplicative' [MulOneClass G] [AddZeroClass H] :
    Additive G ≃+ H ≃ (G ≃* Multiplicative H) 
goal: (AddEquiv.toMultiplicative'.symm f).symm a = (AddMonoidHom.toMultiplicative''.symm f.symm.toMonoidHom) a

full_name: MulEquiv.toAdditive'
code: /-- Reinterpret `G ≃* Multiplicative H` as `Additive G ≃+ H` as. -/
abbrev MulEquiv.toAdditive' [MulOneClass G] [AddZeroClass H] :
    G ≃* Multiplicative H ≃ (Additive G ≃+ H) 
goal: G ≃* Multiplicative H ≃ (Additive G ≃+ H)

full_name: AddEquiv.toMultiplicative''_apply_apply
code: /-- Reinterpret `G ≃+ Additive H` as `Multiplicative G ≃* H`. -/
@[simps]
def AddEquiv.toMultiplicative'' [AddZeroClass G] [MulOneClass H] :
    G ≃+ Additive H ≃ (Multiplicative G ≃* H) 
goal: (AddEquiv.toMultiplicative'' f) a = (AddMonoidHom.toMultiplicative'' f.toAddMonoidHom) a

full_name: AddEquiv.toMultiplicative''_apply_symm_apply
code: /-- Reinterpret `G ≃+ Additive H` as `Multiplicative G ≃* H`. -/
@[simps]
def AddEquiv.toMultiplicative'' [AddZeroClass G] [MulOneClass H] :
    G ≃+ Additive H ≃ (Multiplicative G ≃* H) 
goal: (AddEquiv.toMultiplicative'' f).symm a = (AddMonoidHom.toMultiplicative' f.symm.toAddMonoidHom) a

full_name: AddEquiv.toMultiplicative''
code: /-- Reinterpret `G ≃+ Additive H` as `Multiplicative G ≃* H`. -/
@[simps]
def AddEquiv.toMultiplicative'' [AddZeroClass G] [MulOneClass H] :
    G ≃+ Additive H ≃ (Multiplicative G ≃* H) 
goal: G ≃+ Additive H ≃ (Multiplicative G ≃* H)

full_name: AddEquiv.toMultiplicative''_symm_apply_symm_apply
code: /-- Reinterpret `G ≃+ Additive H` as `Multiplicative G ≃* H`. -/
@[simps]
def AddEquiv.toMultiplicative'' [AddZeroClass G] [MulOneClass H] :
    G ≃+ Additive H ≃ (Multiplicative G ≃* H) 
goal: (AddEquiv.toMultiplicative''.symm f).symm a = (AddMonoidHom.toMultiplicative'.symm f.symm.toMonoidHom) a

full_name: AddEquiv.toMultiplicative''_symm_apply_apply
code: /-- Reinterpret `G ≃+ Additive H` as `Multiplicative G ≃* H`. -/
@[simps]
def AddEquiv.toMultiplicative'' [AddZeroClass G] [MulOneClass H] :
    G ≃+ Additive H ≃ (Multiplicative G ≃* H) 
goal: (AddEquiv.toMultiplicative''.symm f) a = (AddMonoidHom.toMultiplicative''.symm f.toMonoidHom) a

full_name: MulEquiv.toAdditive''
code: /-- Reinterpret `Multiplicative G ≃* H` as `G ≃+ Additive H` as. -/
abbrev MulEquiv.toAdditive'' [AddZeroClass G] [MulOneClass H] :
    Multiplicative G ≃* H ≃ (G ≃+ Additive H) 
goal: Multiplicative G ≃* H ≃ (G ≃+ Additive H)

full_name: monoidEndToAdditive
code: /-- Multiplicative equivalence between multiplicative endomorphisms of a `MulOneClass` `M`
and additive endomorphisms of `Additive M`. -/
@[simps!] def monoidEndToAdditive (M : Type*) [MulOneClass M] :
    Monoid.End M ≃* AddMonoid.End (Additive M) 
goal: Monoid.End M ≃* AddMonoid.End (Additive M)

full_name: monoidEndToAdditive_apply_apply
code: /-- Multiplicative equivalence between multiplicative endomorphisms of a `MulOneClass` `M`
and additive endomorphisms of `Additive M`. -/
@[simps!] def monoidEndToAdditive (M : Type*) [MulOneClass M] :
    Monoid.End M ≃* AddMonoid.End (Additive M) 
goal: ((monoidEndToAdditive M) f) a = Additive.ofMul (f (Additive.toMul a))

full_name: monoidEndToAdditive_symm_apply_apply
code: /-- Multiplicative equivalence between multiplicative endomorphisms of a `MulOneClass` `M`
and additive endomorphisms of `Additive M`. -/
@[simps!] def monoidEndToAdditive (M : Type*) [MulOneClass M] :
    Monoid.End M ≃* AddMonoid.End (Additive M) 
goal: ((monoidEndToAdditive M).symm f) a = Additive.toMul (f (Additive.ofMul a))

full_name: addMonoidEndToMultiplicative
code: /-- Multiplicative equivalence between additive endomorphisms of an `AddZeroClass` `A`
and multiplicative endomorphisms of `Multiplicative A`. -/
@[simps!] def addMonoidEndToMultiplicative (A : Type*) [AddZeroClass A] :
    AddMonoid.End A ≃* Monoid.End (Multiplicative A) 
goal: AddMonoid.End A ≃* Monoid.End (Multiplicative A)

full_name: addMonoidEndToMultiplicative_symm_apply_apply
code: /-- Multiplicative equivalence between additive endomorphisms of an `AddZeroClass` `A`
and multiplicative endomorphisms of `Multiplicative A`. -/
@[simps!] def addMonoidEndToMultiplicative (A : Type*) [AddZeroClass A] :
    AddMonoid.End A ≃* Monoid.End (Multiplicative A) 
goal: ((addMonoidEndToMultiplicative A).symm f) a = Multiplicative.toAdd (f (Multiplicative.ofAdd a))

full_name: addMonoidEndToMultiplicative_apply_apply
code: /-- Multiplicative equivalence between additive endomorphisms of an `AddZeroClass` `A`
and multiplicative endomorphisms of `Multiplicative A`. -/
@[simps!] def addMonoidEndToMultiplicative (A : Type*) [AddZeroClass A] :
    AddMonoid.End A ≃* Monoid.End (Multiplicative A) 
goal: ((addMonoidEndToMultiplicative A) f) a = Multiplicative.ofAdd (f (Multiplicative.toAdd a))

full_name: AddEquiv.additiveMultiplicative_apply
code: /-- `Additive (Multiplicative G)` is just `G`. -/
@[simps!]
def AddEquiv.additiveMultiplicative [AddZeroClass G] : Additive (Multiplicative G) ≃+ G 
goal: (AddEquiv.additiveMultiplicative G) a = Multiplicative.toAdd (Additive.toMul a)

full_name: AddEquiv.additiveMultiplicative_symm_apply
code: /-- `Additive (Multiplicative G)` is just `G`. -/
@[simps!]
def AddEquiv.additiveMultiplicative [AddZeroClass G] : Additive (Multiplicative G) ≃+ G 
goal: (AddEquiv.additiveMultiplicative G).symm a = Additive.ofMul (Multiplicative.ofAdd a)

full_name: AddEquiv.additiveMultiplicative
code: /-- `Additive (Multiplicative G)` is just `G`. -/
@[simps!]
def AddEquiv.additiveMultiplicative [AddZeroClass G] : Additive (Multiplicative G) ≃+ G 
goal: Additive (Multiplicative G) ≃+ G

full_name: MulEquiv.multiplicativeAdditive_apply
code: /-- `Multiplicative (Additive H)` is just `H`. -/
@[simps!]
def MulEquiv.multiplicativeAdditive [MulOneClass H] : Multiplicative (Additive H) ≃* H 
goal: (MulEquiv.multiplicativeAdditive H) a = Additive.toMul (Multiplicative.toAdd a)

full_name: MulEquiv.multiplicativeAdditive_symm_apply
code: /-- `Multiplicative (Additive H)` is just `H`. -/
@[simps!]
def MulEquiv.multiplicativeAdditive [MulOneClass H] : Multiplicative (Additive H) ≃* H 
goal: (MulEquiv.multiplicativeAdditive H).symm a = Multiplicative.ofAdd (Additive.ofMul a)

full_name: MulEquiv.multiplicativeAdditive
code: /-- `Multiplicative (Additive H)` is just `H`. -/
@[simps!]
def MulEquiv.multiplicativeAdditive [MulOneClass H] : Multiplicative (Additive H) ≃* H 
goal: Multiplicative (Additive H) ≃* H