English
Same as coe_liftLie_eq_lift_coe for another instance; the underlying correspondence matches the lift construction.
Русский
То же самое утверждение коэрциона для другого варианта подъемного контекста; соответствие сохраняется.
LaTeX
$$$\operatorname{coe}\Big(\mathrm{liftLie}\,R\,L\,M\,N\,P\,f\Big) = \mathrm{lift}\,R\,L\,M\,N\,P\,f.$$$
Lean4
/-- The geometric sequence `q^n` is a solution of `E` iff
`q` is a root of `E`'s characteristic polynomial. -/
theorem geom_sol_iff_root_charPoly (q : R) : (E.IsSolution fun n ↦ q ^ n) ↔ E.charPoly.IsRoot q :=
by
rw [charPoly, Polynomial.IsRoot.def, Polynomial.eval]
simp only [Polynomial.eval₂_finset_sum, one_mul, RingHom.id_apply, Polynomial.eval₂_monomial, Polynomial.eval₂_sub]
constructor
· intro h
simpa [sub_eq_zero] using h 0
· intro h n
simp only [pow_add, sub_eq_zero.mp h, mul_sum]
exact sum_congr rfl fun _ _ ↦ by ring
