Theorems · Definition · logic and foundations
FirstOrder.Language.Substructure.IsElementary
{L : FirstOrder.Language} → {M : Type u_1} → [inst : L.Structure M] → L.Substructure M → PropA substructure is elementary when every formula applied to a tuple in the substructure agrees with its value in the overall structure.
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 22 from the axioms · uses Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites5
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- FirstOrder.Languagestatement and proof · cited by 1,084
- FirstOrder.Language.Structurestatement and proof · cited by 775
- FirstOrder.Language.Substructurestatement and proof · cited by 242
- FirstOrder.Language.Formulaproof · cited by 93
- FirstOrder.Language.Formula.Realizeproof · cited by 81
Cited by13
Results whose statement or proof uses this declaration.
- FirstOrder.Language.Substructure.isElementary_of_existsstatement · cited by 2
- FirstOrder.Language.ElementarySubstructure.isElementary'statement · cited by 1
- FirstOrder.Language.ElementarySubstructure.mk.injstatement and proof · cited by 1
- FirstOrder.Language.ElementarySubstructure.mk.noConfusionstatement and proof · cited by 1
- FirstOrder.Language.ElementarySubstructure.isElementarystatement · cited by 0
- FirstOrder.Language.ElementarySubstructure.noConfusionproof · cited by 0
- FirstOrder.Language.ElementarySubstructure.noConfusionTypeproof · cited by 0
- FirstOrder.Language.ElementarySubstructure.recOnstatement and proof · cited by 0
- FirstOrder.Language.Substructure.skolem₁_reduct_isElementarystatement · cited by 0
- FirstOrder.Language.ElementarySubstructure.mk.injEqstatement and proof · cited by 0
- FirstOrder.Language.ElementarySubstructure.mk.sizeOf_specstatement and proof · cited by 0
- FirstOrder.Language.MeetsDefinable.isElementary_closurestatement · cited by 0