Not top of Nontrivial Totally Ordered Real Vector Space
HarderNarasimhan.impl.not_top_of_Nontrivial_TotallyOrderedRealVectorSpace
Project documentation
In a nontrivial totally ordered real vector space, the coercion of any vector is strictly below ⊤ in the Dedekind–MacNeille completion. This lemma is used to derive contradictions when an equality forces a coerced value to be ⊤.
Exact Lean statement
lemma not_top_of_Nontrivial_TotallyOrderedRealVectorSpace
{V : Type*} [TotallyOrderedRealVectorSpace V] [hnt : Nontrivial V] :
∀ v : V, OrderTheory.coe' v < (⊤ : OrderTheory.DedekindMacNeilleCompletion V)Formal artifact
Lean source
lemma not_top_of_Nontrivial_TotallyOrderedRealVectorSpace{V : Type*} [TotallyOrderedRealVectorSpace V] [hnt : Nontrivial V] :∀ v : V, OrderTheory.coe' v < (⊤ : OrderTheory.DedekindMacNeilleCompletion V) := by intro v rcases hnt.exists_pair_ne with ⟨v₁, v₂, hne⟩ let v₀ := if v₁ < v₂ then v₂ - v₁ else v₁ - v₂ have hpos : v₀ > 0 := by by_cases h : v₁ < v₂ · simp only [h, ↓reduceIte, gt_iff_lt, sub_pos, v₀] · simp only [h, ↓reduceIte, gt_iff_lt, sub_pos, v₀] exact (eq_or_gt_of_not_lt h).resolve_left hne by_contra! exact not_top_lt <| top_le_iff.1 this ▸ (OrderTheory.coe'.lt_iff_lt.2 <| lt_add_of_pos_right v hpos)- Project
- Harder-Narasimhan
- License
- Apache-2.0
- Commit
- 20220e90b72c
- Source
- HarderNarasimhan/SlopeLike/Impl.lean:108-121
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Related declarations
Associated Primes ker mk Linear Map eq
HarderNarasimhan.CommutativeAlgebra.associatedPrimes_ker_mkLinearMap_eq
Project documentation
Associated primes of the kernel of the localization map. This identifies the associated primes of ker (LocalizedModule.mkLinearMap S M) : Submodule R M with the associated primes of M that do meet the multiplicative set S. Equivalently, these are the associated primes of M after removing those disjoint from S. This lemma is used as the “kernel...
Source project: Harder-Narasimhan
Person-level attribution pending.
Associated Primes ker mk Linear Map subset
HarderNarasimhan.CommutativeAlgebra.associatedPrimes_ker_mkLinearMap_subset
Plain-language statement
One-sided inclusion for associated primes of the kernel of the localization map. If p ∈ associatedPrimes R (ker (mkLinearMap S M)), then p is an associated prime of M and p is not disjoint from the multiplicative set S. This is one direction of associatedPrimes_ker_mkLinearMap_eq.
Source project: Harder-Narasimhan
Person-level attribution pending.
Associated Primes localized Module subset disjoint
HarderNarasimhan.CommutativeAlgebra.associatedPrimes_localizedModule_subset_disjoint
Plain-language statement
Associated primes of a localized module are disjoint from the multiplicative set. More precisely, if p ∈ associatedPrimes R (LocalizedModule S M), then p.carrier ∩ S = ∅. This is a standard fact in commutative algebra: an element of S becomes a unit after localization, so no associated prime of the localized module can contain an element of S.
Source project: Harder-Narasimhan
Person-level attribution pending.