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Theorems · Theorem · order theory

Order.height_eq_zero

∀ {α : Type u_1} [inst : Preorder α] {x : α}, Order.height x = 0 ↔ IsMin x

The elements of height zero are the minimal elements.

Defined in
Mathlib.Order.KrullDimension
Cited by
4 results in Mathlib
Foundations
Depth 63 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes
Preorder
Mathlib docsSourceLoogleLeanSearch

Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

Ideal.height_eq_zero_iff · cited by 5Ideal.height_eq_zero_iffOrder.coheight_eq_zero · cited by 2Order.coheight_eq_zeroOrder.height_ne_zero · cited by 0Order.height_ne_zeroOrder.IsMin.height_eq_zero · cited by 0IsMin.height_eq_zeroPreorder · cited by 7952PreorderENat · cited by 4985ENatNat.cast_zero · cited by 1870Nat.cast_zeroIsMin · cited by 277IsMinOrder.height · cited by 67Order.heightOrder.height_le_coe_iff · cited by 5Order.height_le_coe_iffOrder.height_eq_zeroCITED BYCITES

Cites6

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Cited by4

Results whose statement or proof uses this declaration.