Theorems · Definition · logic and foundations

FirstOrder.Language.DirectLimit.equiv_lift

(L : FirstOrder.Language) →
  (ι : Type v) →
    [inst : Preorder ι] →
      (G : ι → Type w) →
        [inst_1 : (i : ι) → L.Structure (G i)] →
          (f : (i j : ι) → i ≤ j → L.Embedding (G i) (G j)) →
            [inst_2 : IsDirectedOrder ι] →
              [inst_3 : DirectedSystem G fun i j h => ⇑(f i j h)] →
                [inst_4 : Nonempty ι] →
                  (G' : ι → Type w') →
                    [inst_5 : (i : ι) → L.Structure (G' i)] →
                      (f' : (i j : ι) → i ≤ j → L.Embedding (G' i) (G' j)) →
                        (g : (i : ι) → L.Equiv (G i) (G' i)) →
                          [inst_6 : DirectedSystem G' fun i j h => ⇑(f' i j h)] →
                            (∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) ((f i j hij) x) = (f' i j hij) ((g i) x)) →
                              L.Equiv (FirstOrder.Language.DirectLimit G f) (FirstOrder.Language.DirectLimit G' f')

The isomorphism between limits of isomorphic systems.

Defined in

Mathlib.ModelTheory.DirectLimit

Cited by

3 results in Mathlib

Foundations

Depth 97 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

PreorderFirstOrder.Language.StructureIsDirectedOrderDirectedSystemNonemptyFirstOrder.Language.StructureDirectedSystem