Range unipotent Mul Diag U1

lemma range_unipotentMulDiagU1 :    Set.range (unipotentMulDiagU1 v p α hα) =      QuotientGroup.mk '' (GL2.localPTameLevel v p *        {diag (Units.mk0 α (by simpa))} : Set GL₂(v.adicCompletion F)) := by  rw [← Set.image_univ, ← Ideal.Quotient.mk_surjective.range_eq, ← Set.range_comp]  refine subset_antisymm (Set.range_subset_iff.mpr ?_) (Set.image_subset_iff.mpr ?_)  · exact fun a ↦ ⟨_, Set.mul_mem_mul (GL2.localTameLevel_le_localPTameLevel _ _      (unipotent_mem_localTameLevel _ a.2)) rfl, rfl⟩  · simp only [Set.mul_singleton, Set.image_mul_right, Set.subset_def, Set.mem_preimage]    intro x hx    have H : Valued.v (x.1 0 0) = Valued.v α.1 ∧ Valued.v (x.1 0 1) ≤ 1 ∧      Valued.v (x.1 1 0) < Valued.v α.1 ∧ Valued.v (x.1 1 1) = 1 := by      simpa [diag, mul_inv_eq_iff_eq_mul₀, hα, mul_inv_lt_iff₀, Valuation.pos_iff] using        GL2.mem_localIwahoriLevel_iff_v.mp (localPTameLevel_le_localIwahoriLevel _ _ hx)    refine ⟨⟨x 0 1 / x 1 1, by simp [mem_adicCompletionIntegers, H]⟩, QuotientGroup.eq.mpr ?_⟩    dsimp [unipotentMulDiag]    convert (GL2.localPTameLevel.conjBy_diag_mem_iff _ _ (mul_mem (inv_mem      (GL2.localTameLevel_le_localPTameLevel _ _ <|      unipotent_mem_localTameLevel (x 0 1 / x 1 1) (by simp [H]))) hx) α hα).mpr ?_ using 1    · group    have h' : x 1 1 ≠ 0 := fun h ↦ by simp [h] at H    simp only [← map_inv]    simpa [unipotent_inv, Matrix.mul_apply, -map_inv, diag,      unipotent_def, h', -dvd_zero] using! dvd_zero α