chore(topology/algebra/order/basic): deduplicate (#16287)

Primed and non-primed versions of 2 lemmas were almost defeq. Merge them.

Author: urkud

src/analysis/calculus/extend_deriv.lean

@@ -131,7 +131,7 @@ begin
   have : has_deriv_within_at f e (Icc a b) a,
   { rw [has_deriv_within_at_iff_has_fderiv_within_at, ← t_closure],
     exact has_fderiv_at_boundary_of_tendsto_fderiv t_diff t_conv t_open t_cont t_diff' },
-  exact this.nhds_within (mem_nhds_within_Ici_iff_exists_Icc_subset.2 ⟨b, ab, subset.refl _⟩)
+  exact this.nhds_within (Icc_mem_nhds_within_Ici $ left_mem_Ico.2 ab)
 end
 
 /-- If a function is differentiable on the left of a point `a : ℝ`, continuous at `a`, and
@@ -170,7 +170,7 @@ begin
   have : has_deriv_within_at f e (Icc b a) a,
   { rw [has_deriv_within_at_iff_has_fderiv_within_at, ← t_closure],
     exact has_fderiv_at_boundary_of_tendsto_fderiv t_diff t_conv t_open t_cont t_diff' },
-  exact this.nhds_within (mem_nhds_within_Iic_iff_exists_Icc_subset.2 ⟨b, ba, subset.refl _⟩)
+  exact this.nhds_within (Icc_mem_nhds_within_Iic $ right_mem_Ioc.2 ba)
 end
 
 /-- If a real function `f` has a derivative `g` everywhere but at a point, and `f` and `g` are

src/topology/algebra/order/basic.lean

@@ -1527,8 +1527,8 @@ let ⟨u', hu'⟩ := exists_gt a in mem_nhds_within_Ici_iff_exists_Ico_subset' h
 
 /-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u]`
 with `a < u`. -/
-lemma mem_nhds_within_Ici_iff_exists_Icc_subset' [no_max_order α] [densely_ordered α]
-  {a : α} {s : set α} : s ∈ 𝓝[≥] a ↔ ∃u ∈ Ioi a, Icc a u ⊆ s :=
+lemma mem_nhds_within_Ici_iff_exists_Icc_subset [no_max_order α] [densely_ordered α]
+  {a : α} {s : set α} : s ∈ 𝓝[≥] a ↔ ∃ u, a < u ∧ Icc a u ⊆ s :=
 begin
   rw mem_nhds_within_Ici_iff_exists_Ico_subset,
   split,
@@ -1573,42 +1573,14 @@ let ⟨l', hl'⟩ := exists_lt a in mem_nhds_within_Iic_iff_exists_Ioc_subset' h
 
 /-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `[l, a]`
 with `l < a`. -/
-lemma mem_nhds_within_Iic_iff_exists_Icc_subset' [no_min_order α] [densely_ordered α]
-  {a : α} {s : set α} : s ∈ 𝓝[≤] a ↔ ∃l ∈ Iio a, Icc l a ⊆ s :=
+lemma mem_nhds_within_Iic_iff_exists_Icc_subset [no_min_order α] [densely_ordered α]
+  {a : α} {s : set α} : s ∈ 𝓝[≤] a ↔ ∃ l, l < a ∧ Icc l a ⊆ s :=
 begin
-  convert @mem_nhds_within_Ici_iff_exists_Icc_subset' αᵒᵈ _ _ _ _ _ _ _,
+  convert @mem_nhds_within_Ici_iff_exists_Icc_subset αᵒᵈ _ _ _ _ _ _ _,
   simp_rw (show ∀ u : αᵒᵈ, @Icc αᵒᵈ _ a u = @Icc α _ u a, from λ u, dual_Icc),
   refl,
 end
 
-/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u]`
-with `a < u`. -/
-lemma mem_nhds_within_Ici_iff_exists_Icc_subset [no_max_order α] [densely_ordered α]
-  {a : α} {s : set α} : s ∈ 𝓝[≥] a ↔ ∃u, a < u ∧ Icc a u ⊆ s :=
-begin
-  rw mem_nhds_within_Ici_iff_exists_Ico_subset,
-  split,
-  { rintros ⟨u, au, as⟩,
-    rcases exists_between au with ⟨v, hv⟩,
-    exact ⟨v, hv.1, λx hx, as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩ },
-  { rintros ⟨u, au, as⟩,
-    exact ⟨u, au, subset.trans Ico_subset_Icc_self as⟩ }
-end
-
-/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `[l, a]`
-with `l < a`. -/
-lemma mem_nhds_within_Iic_iff_exists_Icc_subset [no_min_order α] [densely_ordered α]
-  {a : α} {s : set α} : s ∈ 𝓝[≤] a ↔ ∃l, l < a ∧ Icc l a ⊆ s :=
-begin
-  rw mem_nhds_within_Iic_iff_exists_Ioc_subset,
-  split,
-  { rintros ⟨l, la, as⟩,
-    rcases exists_between la with ⟨v, hv⟩,
-    refine ⟨v, hv.2, λx hx, as ⟨lt_of_lt_of_le hv.1 hx.1, hx.2⟩⟩, },
-  { rintros ⟨l, la, as⟩,
-    exact ⟨l, la, subset.trans Ioc_subset_Icc_self as⟩ }
-end
-
 end order_topology
 
 end linear_order