English
If two families of units u and v satisfy rank and inclusion conditions, then the ratio regOfFamily(u)/regOfFamily(v) equals the index of the subgroup closures with torsion.
Русский
Если две семьи единиц u и v удовлетворяют условиям ранга и включения, то отношение regOfFamily(u)/regOfFamily(v) равно индексу замыкания подгрупп с торсионностью.
LaTeX
$$regOfFamily u / regOfFamily v = (Subgroup.closure (Set.range u) ⊔ torsion K).relIndex (Subgroup.closure (Set.range v) ⊔ torsion K)$$
Lean4
/-- The absolute value of `p` is `p ^ (-t)` for some positive real number `t`. -/
theorem exists_pos_eq_pow_neg : ∃ t : ℝ, 0 < t ∧ f p = p ^ (-t) :=
by
have pprime := is_prime_of_minimal_nat_zero_lt_and_lt_one hp0 hp1 hmin
refine ⟨-logb p (f p), Left.neg_pos_iff.mpr <| logb_neg (mod_cast pprime.one_lt) hp0 hp1, ?_⟩
rw [neg_neg]
exact (rpow_logb (mod_cast pprime.pos) (mod_cast pprime.ne_one) hp0).symm
