Theorems · Theorem · number theory

ZLattice.covolume_div_covolume_eq_relIndex

∀ {ι : Type u_2} [inst : Fintype ι] (L₁ L₂ : Submodule ℤ (ι → ℝ)) [inst_1 : DiscreteTopology ↥L₁] [IsZLattice ℝ L₁]
  [inst_3 : DiscreteTopology ↥L₂] [IsZLattice ℝ L₂],
  L₁ ≤ L₂ →
    ZLattice.covolume L₁ MeasureTheory.volume / ZLattice.covolume L₂ MeasureTheory.volume =
      ↑(L₁.toAddSubgroup.relIndex L₂.toAddSubgroup)

Let L₁ be a sub-ℤ-lattice of L₂. Then the index of L₁ inside L₂ is equal to covolume L₁ / covolume L₂.

Defined in: Mathlib.Algebra.Module.ZLattice.Covolume
Cited by: 2 results in Mathlib
Foundations: Depth 261 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Fintype DiscreteTopology IsZLattice DiscreteTopology IsZLattice

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NumberField.Units.regOfFamily_div_regOfFamilyproof · cited by 1
ZLattice.covolume_div_covolume_eq_relIndex'proof · cited by 0