English
The set W.Point of nonsingular projective points carries a natural additive abelian group structure transported from the affine side by the affine-additive equivalence.
Русский
Множество ненонсигурных проективных точек W образует естественную структуру абелевой группы, перенесённую из аффинной стороны через эквивалентность сложения.
LaTeX
$$$\\text{AddGroup}(W.Point) \\cong \\text{AddGroup}(W.toAffine.Point)$ via the affine addEquiv$$
Lean4
noncomputable instance : AddCommGroup W.Point where
nsmul := nsmulRec
zsmul := zsmulRec
zero_add
_ := by
classical
apply (toAffineAddEquiv W).injective
simp only [map_add, toAffineAddEquiv_apply, toAffineLift_zero, zero_add]
add_zero
_ := by
classical
apply (toAffineAddEquiv W).injective
simp only [map_add, toAffineAddEquiv_apply, toAffineLift_zero, add_zero]
neg_add_cancel
P := by
classical
apply (toAffineAddEquiv W).injective
simp only [map_add, toAffineAddEquiv_apply, toAffineLift_neg, neg_add_cancel, toAffineLift_zero]
add_comm _
_ := by
classical
apply (toAffineAddEquiv W).injective
simp only [map_add, add_comm]
add_assoc _ _
_ := by
classical
apply (toAffineAddEquiv W).injective
simp only [map_add, add_assoc]
