File: Mathlib/Algebra/Star/CentroidHom.lean

full_name: CentroidHom.instStar
code: instance : Star (CentroidHom α) 
goal: Star (CentroidHom α)

full_name: CentroidHom.star_apply
code: @[simp] lemma star_apply (f : CentroidHom α) (a : α) : (star f) a = star (f (star a)) 
goal: (star f) a = star (f (star a))

full_name: CentroidHom.instStarAddMonoid
code: instance instStarAddMonoid : StarAddMonoid (CentroidHom α) 
goal: StarAddMonoid (CentroidHom α)

full_name: CentroidHom.instStarSubtypeMemSubsemiringCenter
code: instance : Star (Subsemiring.center (CentroidHom α)) 
goal: Star ↥(Subsemiring.center (CentroidHom α))

full_name: CentroidHom.instStarAddMonoidCenter
code: instance instStarAddMonoidCenter : StarAddMonoid (Subsemiring.center (CentroidHom α)) 
goal: StarAddMonoid ↥(Subsemiring.center (CentroidHom α))

full_name: CentroidHom.instStarRingSubtypeMemSubsemiringCenter
code: instance : StarRing (Subsemiring.center (CentroidHom α)) 
goal: StarRing ↥(Subsemiring.center (CentroidHom α))

full_name: CentroidHom.centerStarEmbedding
code: /-- The canonical *-homomorphism embedding the center of `CentroidHom α` into `CentroidHom α`. -/
def centerStarEmbedding : Subsemiring.center (CentroidHom α) →⋆ₙ+* CentroidHom α 
goal: ↥(Subsemiring.center (CentroidHom α)) →⋆ₙ+* CentroidHom α

full_name: CentroidHom.star_centerToCentroidCenter
code: theorem star_centerToCentroidCenter (z : NonUnitalStarSubsemiring.center α) :
    star (centerToCentroidCenter z) =
      (centerToCentroidCenter (star z : NonUnitalStarSubsemiring.center α)) 
goal: star (CentroidHom.centerToCentroidCenter z) = CentroidHom.centerToCentroidCenter (star z)

full_name: CentroidHom.starCenterToCentroidCenter
code: /-- The canonical *-homomorphism from the center of a non-unital, non-associative *-semiring into
the center of its centroid. -/
def starCenterToCentroidCenter :
    NonUnitalStarSubsemiring.center α →⋆ₙ+* Subsemiring.center (CentroidHom α) 
goal: ↥(NonUnitalStarSubsemiring.center α) →⋆ₙ+* ↥(Subsemiring.center (CentroidHom α))

full_name: CentroidHom.starCenterToCentroid
code: /-- The canonical homomorphism from the center into the centroid -/
def starCenterToCentroid : NonUnitalStarSubsemiring.center α →⋆ₙ+* CentroidHom α 
goal: ↥(NonUnitalStarSubsemiring.center α) →⋆ₙ+* CentroidHom α

full_name: CentroidHom.starCenterToCentroid_apply
code: lemma starCenterToCentroid_apply (z : NonUnitalStarSubsemiring.center α) (a : α) :
    (starCenterToCentroid z) a = z * a 
goal: (CentroidHom.starCenterToCentroid z) a = ↑z * a

full_name: CentroidHom.starRingOfCommCentroidHom
code: /--
Let `α` be a star ring with commutative centroid. Then the centroid is a star ring.
-/
@[reducible]
def starRingOfCommCentroidHom (mul_comm : Std.Commutative (α := CentroidHom α) (· * ·)) :
    StarRing (CentroidHom α) 
goal: StarRing (CentroidHom α)

full_name: CentroidHom.starCenterIsoCentroid
code: /-- The canonical isomorphism from the center of a (non-associative) semiring onto its centroid. -/
def starCenterIsoCentroid : StarSubsemiring.center α ≃⋆+* CentroidHom α 
goal: ↥(StarSubsemiring.center α) ≃⋆+* CentroidHom α

full_name: CentroidHom.starCenterIsoCentroid_apply
code: @[simp]
lemma starCenterIsoCentroid_apply (a : ↥(NonUnitalStarSubsemiring.center α)) :
    CentroidHom.starCenterIsoCentroid a = CentroidHom.starCenterToCentroid a 
goal: CentroidHom.starCenterIsoCentroid a = CentroidHom.starCenterToCentroid a

full_name: CentroidHom.starCenterIsoCentroid_symm_apply_coe
code: @[simp]
lemma starCenterIsoCentroid_symm_apply_coe (T : CentroidHom α) :
    ↑(CentroidHom.starCenterIsoCentroid.symm T) = T 1 
goal: ↑(CentroidHom.starCenterIsoCentroid.symm T) = T 1