feat(NumberTheory/NumberField/Completion/FinitePlace): add multiplicity lemmas

[MichaelStollBayreuth]

This adds some more API for multiplicities of prime ideals in factorizations (and some more basic API lemmas, too), in preparation for a proof of the Northcott property for heights on number fields.

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• depends on: feat(RingTheory.DedekindDomain.Factorization): add API for factorization #38654

[github-actions]

PR summary 4efb9eabf5

Import changes for modified files

Dependency changes

File	Base Count	Head Count	Change
Mathlib.RingTheory.DedekindDomain.Factorization	2130	2131	+1 (+0.05%)
Mathlib.NumberTheory.NumberField.Completion.FinitePlace	2998	2999	+1 (+0.03%)

Import changes for all files

Files

Import difference

19 files Mathlib.Algebra.Module.Torsion.PrimaryComponent Mathlib.AlgebraicGeometry.EllipticCurve.LFunction Mathlib.NumberTheory.FLT.Three Mathlib.NumberTheory.Height.NumberField Mathlib.NumberTheory.NumberField.AdeleRing Mathlib.NumberTheory.NumberField.Completion.FinitePlace Mathlib.NumberTheory.NumberField.Cyclotomic.Basic Mathlib.NumberTheory.NumberField.Cyclotomic.Galois Mathlib.NumberTheory.NumberField.Cyclotomic.Ideal Mathlib.NumberTheory.NumberField.Cyclotomic.PID Mathlib.NumberTheory.NumberField.Cyclotomic.Three Mathlib.NumberTheory.NumberField.Discriminant.Different Mathlib.NumberTheory.NumberField.FinitePlaces Mathlib.NumberTheory.NumberField.Ideal.Basic Mathlib.NumberTheory.NumberField.ProductFormula Mathlib.RingTheory.DedekindDomain.Factorization Mathlib.RingTheory.DedekindDomain.FiniteAdeleRing Mathlib.RingTheory.Ideal.Norm.RelNorm Mathlib.Topology.Algebra.Ring.Compact

1

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Declarations diff

+ Ideal.emultiplicity_bot
+ Ideal.finprod_heightOneSpectrum_pow_multiplicity
+ apply_mul_absNorm_pow_eq_one
+ asIdeal_maximalIdeal_injective
+ count_normalizedFactors_eq_multiplicity
+ emultiplicity_iSup
+ emultiplicity_sup
+ equivHeightOneSpectrum_apply
+ factorization_eq_multiplicity
+ finprod_finitePlace_pow_multiplicity
+ maxPowDividing_eq_pow_multiplicity
+ multiplicity_iSup
+ multiplicity_le_of_ideal_ge
+ multiplicity_sup
+ normalizedFactors_prod_inter_eq_inter
+ one_le_finrank_rat
+ prod_inter_normalizedFactors_ne_zero

You can run this locally as follows

## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci

## summary with just the declaration names:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.

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No changes to strong technical debt.

No changes to weak technical debt.

[mathlib-dependent-issues]

This PR/issue depends on:

• feat(RingTheory.DedekindDomain.Factorization): add API for factorization #38654
  By Dependent Issues (🤖). Happy coding!