English
There is a constructive way to build an OfLocalizationSpan from given local data and a suitable hP, showing how to assemble global P from local checks.
Русский
Существует конструктивный способ построить OfLocalizationSpan из локальных данных и подходящего hP, показывающий, как собрать глобальное P из локальных проверок.
LaTeX
$$$\\exists (P\\text{ satisfies OfLocalizationSpan})\\;\\text{from }(hP, H)\\text{ via the given H}$$$
Lean4
theorem respectsIso (hP : StableUnderCompositionWithLocalizationAway P) : RespectsIso P
where
left {R S T} _ _ _ f e
hf := by
letI := e.toRingHom.toAlgebra
have : IsLocalization.Away (1 : S) T := IsLocalization.away_of_isUnit_of_bijective _ isUnit_one e.bijective
exact hP.right T (1 : S) f hf
right {R S T} _ _ _ f e
hf := by
letI := e.toRingHom.toAlgebra
have : IsLocalization.Away (1 : R) S := IsLocalization.away_of_isUnit_of_bijective _ isUnit_one e.bijective
exact hP.left S (1 : R) f hf
