setOf_prime_and_eq_mod_infinite — Mathlib

English

A basic algebraic identity about negation and multiplication used in manipulation of series.

Русский

Базовая алгебраическая тождественность относительно отрицания и умножения, применяемая в работе с рядами.

LaTeX

$$$-(a\\,b) = a(-b)$$$

Lean4

/-- **Dirichlet's Theorem** on primes in arithmetic progression: if `q` is a positive
integer and `a : ZMod q` is a unit, then there are infinitely many prime numbers `p`
such that `(p : ZMod q) = a`. -/
theorem setOf_prime_and_eq_mod_infinite (ha : IsUnit a) : {p : ℕ | p.Prime ∧ (p : ZMod q) = a}.Infinite :=
  by
  by_contra H
  rw [Set.not_infinite] at H
  exact not_summable_residueClass_prime_div ha <| summable_of_finite_support <| support_residueClass_prime_div a ▸ H

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