Theorems · Theorem · order theory

HahnEmbedding.Partial.eval_ne

∀ {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)
  [inst_12 : IsOrderedAddMonoid R] [inst_13 : Archimedean R] {x : M},
  x ∉ (↑f).domain → ∀ (y : ↥(↑f).domain), f.eval x ≠ ↑↑f y

If x isn't in f's domain, f.eval x produces a brand new value not in f's range.

Defined in

Mathlib.Algebra.Order.Module.HahnEmbedding

Cited by

1 results in Mathlib

Foundations

Depth 130 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

DivisionRingLinearOrderIsOrderedRingArchimedeanAddCommGroupLinearOrderIsOrderedAddMonoidModuleIsOrderedModuleAddCommGroupLinearOrderModuleIsOrderedAddMonoidArchimedean