English
Continuation: equivalence of finite coordinate representations preserves the affine image.
Русский
Продолжение эквивалентности — сохранение афинного изображения.
LaTeX
$$$ (P \approx Q) \Rightarrow toAffine W P = toAffine W Q $$$
Lean4
theorem toAffine_smul (P : Fin 3 → F) {u : F} (hu : IsUnit u) : toAffine W (u • P) = toAffine W P :=
by
by_cases hP : W.Nonsingular P
· by_cases hPz : P z = 0
· rw [toAffine_of_Z_eq_zero <| mul_eq_zero_of_right u hPz, toAffine_of_Z_eq_zero hPz]
· rw [toAffine_of_Z_ne_zero ((nonsingular_smul P hu).mpr hP) <| mul_ne_zero hu.ne_zero hPz,
toAffine_of_Z_ne_zero hP hPz, Affine.Point.some.injEq]
simp only [smul_fin3_ext, mul_div_mul_left _ _ hu.ne_zero, and_self]
· rw [toAffine_of_singular <| hP.comp (nonsingular_smul P hu).mp, toAffine_of_singular hP]