exp_mul_exp_neg_eq_one — Mathlib

English

For any natural k, exp A raised to the k-th power equals the rescaled exponential: exp A^k = rescale(k, exp A).

Русский

Для любого натурального k справедливо exp A^k = rescale(k, exp A).

LaTeX

$$$$ \exp A^k = \operatorname{rescale}(k, \exp A) $$$$

Lean4

/-- Shows that $e^{x} * e^{-x} = 1$ -/
theorem exp_mul_exp_neg_eq_one [Algebra ℚ A] : exp A * evalNegHom (exp A) = 1 := by
  convert exp_mul_exp_eq_exp_add (1 : A) (-1) <;> simp

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