chore(order/filter): turn tendsto_id' into an iff lemma (#14791)

src/dynamics/omega_limit.lean

@@ -68,7 +68,7 @@ begin
 end
 
 lemma omega_limit_mono_left {f₁ f₂ : filter τ} (hf : f₁ ≤ f₂) : ω f₁ ϕ s ⊆ ω f₂ ϕ s :=
-omega_limit_subset_of_tendsto ϕ s (tendsto_id' hf)
+omega_limit_subset_of_tendsto ϕ s (tendsto_id'.2 hf)
 
 lemma omega_limit_mono_right {s₁ s₂ : set α} (hs : s₁ ⊆ s₂) : ω f ϕ s₁ ⊆ ω f ϕ s₂ :=
 Inter₂_mono $ λ u hu, closure_mono (image2_subset subset.rfl hs)

src/measure_theory/measure/haar_lebesgue.lean

@@ -445,7 +445,7 @@ begin
   have A : tendsto (λ (r : ℝ), ennreal.of_real (r ^ (finrank ℝ E)) * μ (closed_ball (0 : E) 1))
     (𝓝[<] 1) (𝓝 (ennreal.of_real (1 ^ (finrank ℝ E)) * μ (closed_ball (0 : E) 1))),
   { refine ennreal.tendsto.mul _ (by simp) tendsto_const_nhds (by simp),
-    exact ennreal.tendsto_of_real ((tendsto_id' nhds_within_le_nhds).pow _) },
+    exact ennreal.tendsto_of_real ((tendsto_id'.2 nhds_within_le_nhds).pow _) },
   simp only [one_pow, one_mul, ennreal.of_real_one] at A,
   refine le_of_tendsto A _,
   refine mem_nhds_within_Iio_iff_exists_Ioo_subset.2 ⟨(0 : ℝ), by simp, λ r hr, _⟩,

src/order/filter/basic.lean

@@ -2320,16 +2320,13 @@ theorem tendsto.congr {f₁ f₂ : α → β} {l₁ : filter α} {l₂ : filter
   (h : ∀ x, f₁ x = f₂ x) : tendsto f₁ l₁ l₂ → tendsto f₂ l₁ l₂ :=
 (tendsto_congr h).1
 
-lemma tendsto_id' {x y : filter α} : x ≤ y → tendsto id x y :=
-by simp only [tendsto, map_id, forall_true_iff] {contextual := tt}
+lemma tendsto_id' {x y : filter α} : tendsto id x y ↔ x ≤ y := iff.rfl
 
-lemma tendsto_id {x : filter α} : tendsto id x x := tendsto_id' $ le_refl x
+lemma tendsto_id {x : filter α} : tendsto id x x := le_refl x
 
 lemma tendsto.comp {f : α → β} {g : β → γ} {x : filter α} {y : filter β} {z : filter γ}
   (hg : tendsto g y z) (hf : tendsto f x y) : tendsto (g ∘ f) x z :=
-calc map (g ∘ f) x = map g (map f x) : by rw [map_map]
-  ... ≤ map g y : map_mono hf
-  ... ≤ z : hg
+λ s hs, hf (hg hs)
 
 lemma tendsto.mono_left {f : α → β} {x y : filter α} {z : filter β}
   (hx : tendsto f x z) (h : y ≤ x) : tendsto f y z :=

src/topology/instances/ennreal.lean

@@ -614,7 +614,7 @@ have Inf ((λb, ↑r - b) '' range b) = ↑r - (⨆i, b i),
   from is_glb.Inf_eq $ is_lub_supr.is_glb_of_tendsto
     (assume x _ y _, tsub_le_tsub (le_refl (r : ℝ≥0∞)))
     (range_nonempty _)
-    (ennreal.tendsto_coe_sub.comp (tendsto_id' inf_le_left)),
+    (ennreal.tendsto_coe_sub.comp (tendsto_id'.2 inf_le_left)),
 by rw [eq, ←this]; simp [Inf_image, infi_range, -mem_range]; exact le_rfl
 
 lemma exists_countable_dense_no_zero_top :

src/topology/order.lean

@@ -725,12 +725,11 @@ continuous_iff_le_induced.2 $ bot_le
 @[continuity] lemma continuous_top {t : tspace α} : cont t ⊤ f :=
 continuous_iff_coinduced_le.2 $ le_top
 
+lemma continuous_id_iff_le {t t' : tspace α} : cont t t' id ↔ t ≤ t' :=
+@continuous_def _ _ t t' id
+
 lemma continuous_id_of_le {t t' : tspace α} (h : t ≤ t') : cont t t' id :=
-begin
-  rw continuous_def,
-  assume u hu,
-  exact h u hu
-end
+continuous_id_iff_le.2 h
 
 /- 𝓝 in the induced topology -/
 

src/topology/separation.lean

@@ -587,7 +587,7 @@ lemma eq_of_tendsto_nhds [topological_space β] [t1_space β] {f : α → β} {a
   (h : tendsto f (𝓝 a) (𝓝 b)) : f a = b :=
 by_contra $ assume (hfa : f a ≠ b),
 have fact₁ : {f a}ᶜ ∈ 𝓝 b := compl_singleton_mem_nhds hfa.symm,
-have fact₂ : tendsto f (pure a) (𝓝 b) := h.comp (tendsto_id' $ pure_le_nhds a),
+have fact₂ : tendsto f (pure a) (𝓝 b) := h.comp (tendsto_id'.2 $ pure_le_nhds a),
 fact₂ fact₁ (eq.refl $ f a)
 
 /-- To prove a function to a `t1_space` is continuous at some point `a`, it suffices to prove that