Theorems · Theorem · order theory

HahnEmbedding.Partial.exists_sub_mem_ball

∀ {K : Type u_1} [inst : DivisionRing K] [inst_1 : LinearOrder K] [inst_2 : IsOrderedRing K] [inst_3 : Archimedean K]
  {M : Type u_2} [inst_4 : AddCommGroup M] [inst_5 : LinearOrder M] [inst_6 : IsOrderedAddMonoid M]
  [inst_7 : Module K M] [inst_8 : IsOrderedModule K M] {R : Type u_3} [inst_9 : AddCommGroup R]
  [inst_10 : LinearOrder R] [inst_11 : Module K R] {seed : HahnEmbedding.Seed K M R} (f : HahnEmbedding.Partial seed)
  [IsOrderedAddMonoid R] [Archimedean R] {x : M} (hx : x ∉ (↑f).domain) (y : ↥(↑f).domain),
  ∃ z, ↑z - x ∈ FiniteArchimedeanClass.ball K (FiniteArchimedeanClass.mk (↑y - x) ⋯)

For x not in f's domain, there is an infinite chain of y from f's domain that keeps getting closer to x in terms of Archimedean classes.

Defined in

Mathlib.Algebra.Order.Module.HahnEmbedding

Cited by

1 results in Mathlib

Foundations

Depth 131 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

DivisionRingLinearOrderIsOrderedRingArchimedeanAddCommGroupLinearOrderIsOrderedAddMonoidModuleIsOrderedModuleAddCommGroupLinearOrderModuleIsOrderedAddMonoidArchimedean