congr_symm_apply_of — Mathlib

English

The composition of two successive maps between directed limits equals the map induced by the composition of the component maps.

Русский

Сумма отображений между пределами равна отображению, порождённому композицией отображений компонентов.

LaTeX

$$$\\text{map\_comp}(g_1,g_2,hg_1,hg_2) = \\text{map}(g_2\\circ g_1, hg_1,hg_2)$$$

Lean4

theorem congr_symm_apply_of (e : (i : ι) → G i ≃ₗ[R] G' i) (he : ∀ i j h, e j ∘ₗ f i j h = f' i j h ∘ₗ e i) {i : ι}
    (g : G' i) : (congr e he).symm (of _ _ G' f' i g) = of _ _ G f i ((e i).symm g) :=
  map_apply_of _
    (fun i j h ↦ by
      rw [toLinearMap_symm_comp_eq, ← comp_assoc, he i, comp_assoc, comp_coe, symm_trans_self, refl_toLinearMap,
        comp_id])
    _

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