English
A further naturality check for δ₂ faces with mk.app across Fin 3 holds, ensuring compatibility with higher faces.
Русский
further naturality check для δ₂ лицевых отображений с mk.app по Fin3 сохраняется, обеспечивая совместимость с высшими лицами.
LaTeX
$$$\forall i:\, fin(3),\; mk_naturality_δ1i(F,i)$$$
Lean4
theorem mk_naturality_σ1i (i : Fin 2) : toNerve₂.mk.naturalityProperty F (σ₂ i) :=
by
apply (cancel_mono (nerve₂.seagull _)).1
simp only [nerve₂.seagull, prod.comp_lift, assoc]
congr 1 <;> rw [← map_comp, ← op_comp]
· unfold δ2₂
rw [← toNerve₂.mk_naturality_δ1i F hyp, ← assoc, ← map_comp, ← op_comp]
change toNerve₂.mk.naturalityProperty F (δ₂ 2 ≫ σ₂ i)
fin_cases i
· dsimp only [Fin.zero_eta]
rw [δ₂_two_comp_σ₂_zero]
exact
(toNerve₂.mk.naturalityProperty F).comp_mem _ _ (toNerve₂.mk_naturality_σ00 F) (toNerve₂.mk_naturality_δ0i F _)
· dsimp only [Fin.mk_one]
rw [δ₂_two_comp_σ₂_one]
exact (toNerve₂.mk.naturalityProperty F).id_mem _
· unfold δ0₂
rw [← toNerve₂.mk_naturality_δ1i F hyp, ← assoc, ← map_comp, ← op_comp]
change toNerve₂.mk.naturalityProperty F (δ₂ 0 ≫ σ₂ i)
fin_cases i <;> dsimp only [Fin.zero_eta, Fin.isValue, Fin.mk_one]
· rw [δ₂_zero_comp_σ₂_zero]
exact (toNerve₂.mk.naturalityProperty F).id_mem _
· rw [δ₂_zero_comp_σ₂_one]
exact
(toNerve₂.mk.naturalityProperty F).comp_mem _ _ (toNerve₂.mk_naturality_σ00 F) (toNerve₂.mk_naturality_δ0i F _)
