Theorems · Theorem · logic and foundations

FirstOrder.Language.DirectLimit.equiv_lift_of

∀ (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)]
  (H_commuting : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) ((f i j hij) x) = (f' i j hij) ((g i) x)) {i : ι} (x : G i),
  (FirstOrder.Language.DirectLimit.equiv_lift L ι G f G' f' g H_commuting)
      ((FirstOrder.Language.DirectLimit.of L ι G f i) x) =
    (FirstOrder.Language.DirectLimit.of L ι G' f' i) ((g i) x)

Defined in

Mathlib.ModelTheory.DirectLimit

Cited by

0 results in Mathlib

Foundations

Depth 98 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

PreorderFirstOrder.Language.StructureIsDirectedOrderDirectedSystemNonemptyFirstOrder.Language.StructureDirectedSystem