expect_eq_zero_iff_ne_zero — Mathlib

English

For a finite group G and AddChar ψ, the average is nonzero if and only if ψ is the trivial character; equivalently, 𝔼 a, ψ a ≠ 0 ⇔ ψ = 0.

Русский

Среднее не ноль тогда, когда ψ тривиальна; эквивалентно 𝔼 a, ψ a ≠ 0 ⇔ ψ = 0.

LaTeX

$$$\\mathbb{E}_{a}[\\psi(a)] \\neq 0 \\quad \\Leftrightarrow \\quad \\psi = 0.$$$

Lean4

theorem expect_eq_zero_iff_ne_zero : 𝔼 x, ψ x = 0 ↔ ψ ≠ 0 := by rw [expect_eq_ite, one_ne_zero.ite_eq_right_iff]

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