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chore(Algebra): shorten proofs where intermediate steps are not necessary #37888

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9 changes: 2 additions & 7 deletions Mathlib/Algebra/Star/CentroidHom.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -76,10 +76,7 @@ instance : StarRing (Subsemiring.center (CentroidHom α)) where
def centerStarEmbedding : Subsemiring.center (CentroidHom α) →⋆ₙ+* CentroidHom α where
toNonUnitalRingHom :=
(SubsemiringClass.subtype (Subsemiring.center (CentroidHom α))).toNonUnitalRingHom
map_star' f := by
simp only [RingHom.toMonoidHom_eq_coe, OneHom.toFun_eq_coe, MonoidHom.toOneHom_coe,
MonoidHom.coe_coe, SubsemiringClass.coe_subtype]
exact rfl
map_star' _ := rfl

theorem star_centerToCentroidCenter (z : NonUnitalStarSubsemiring.center α) :
star (centerToCentroidCenter z) =
Expand All @@ -97,9 +94,7 @@ the center of its centroid. -/
def starCenterToCentroidCenter :
NonUnitalStarSubsemiring.center α →⋆ₙ+* Subsemiring.center (CentroidHom α) where
toNonUnitalRingHom := centerToCentroidCenter
map_star' _ := by
simp only [MulHom.toFun_eq_coe, NonUnitalRingHom.coe_toMulHom]
exact (star_centerToCentroidCenter _).symm
map_star' _ := (star_centerToCentroidCenter _).symm

/-- The canonical homomorphism from the center into the centroid -/
def starCenterToCentroid : NonUnitalStarSubsemiring.center α →⋆ₙ+* CentroidHom α :=
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