exp_nonneg — Mathlib

English

For a real-linear field 𝕜 and a compact space α, the exponential of a continuous map f: α → 𝕜 equals the continuous map sending a ∈ α to exp(𝕜, f(a)).

Русский

Для поля 𝕜 и компактного пространства α экспонента от непрерывной карты f: α → 𝕜 равна непрерывной карте a ↦ exp(𝕜, f(a)).

LaTeX

$$$$ \mathrm{exp}_{𝕜} \circ f = \text{toFun} := (a \mapsto \exp_{𝕜}(f(a))) \in C(\alpha, 𝕜). $$$$

Lean4

@[aesop safe apply (rule_sets := [CStarAlgebra])]
theorem _root_.IsSelfAdjoint.exp_nonneg {𝕜 : Type*} [Field 𝕜] [Algebra 𝕜 A] [PartialOrder A] [StarOrderedRing A] {a : A}
    (ha : IsSelfAdjoint a) : 0 ≤ exp 𝕜 a :=
  by
  rw [exp_eq_exp 𝕜 ℝ, ← real_exp_eq_normedSpace_exp]
  exact cfc_nonneg fun x _ => Real.exp_nonneg x

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