feat(topology/algebra/equicontinuity): a family of group homomorphism…

…s is equicontinuous iff it is equicontinuous at `1` (#17128)

src/topology/algebra/equicontinuity.lean

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+/-
+Copyright (c) 2022 Anatole Dedecker. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Anatole Dedecker
+-/
+import topology.algebra.uniform_convergence
+
+/-!
+# Algebra-related equicontinuity criterions
+-/
+
+open function
+open_locale uniform_convergence
+
+@[to_additive] lemma equicontinuous_of_equicontinuous_at_one {ι G M hom : Type*}
+  [topological_space G] [uniform_space M] [group G] [group M] [topological_group G]
+  [uniform_group M] [monoid_hom_class hom G M] (F : ι → hom)
+  (hf : equicontinuous_at (coe_fn ∘ F) (1 : G)) :
+  equicontinuous (coe_fn ∘ F) :=
+begin
+  letI : has_coe_to_fun hom (λ _, G → M) := fun_like.has_coe_to_fun,
+  rw equicontinuous_iff_continuous,
+  rw equicontinuous_at_iff_continuous_at at hf,
+  let φ : G →* (ι → M) :=
+  { to_fun := swap (coe_fn ∘ F),
+    map_one' := by ext; exact map_one _,
+    map_mul' := λ a b, by ext; exact map_mul _ _ _ },
+  exact continuous_of_continuous_at_one φ hf
+end
+
+@[to_additive] lemma uniform_equicontinuous_of_equicontinuous_at_one {ι G M hom : Type*}
+  [uniform_space G] [uniform_space M] [group G] [group M] [uniform_group G] [uniform_group M]
+  [monoid_hom_class hom G M] (F : ι → hom) (hf : equicontinuous_at (coe_fn ∘ F) (1 : G)) :
+  uniform_equicontinuous (coe_fn ∘ F) :=
+begin
+  letI : has_coe_to_fun hom (λ _, G → M) := fun_like.has_coe_to_fun,
+  rw uniform_equicontinuous_iff_uniform_continuous,
+  rw equicontinuous_at_iff_continuous_at at hf,
+  let φ : G →* (ι → M) :=
+  { to_fun := swap (coe_fn ∘ F),
+    map_one' := by ext; exact map_one _,
+    map_mul' := λ a b, by ext; exact map_mul _ _ _ },
+  exact uniform_continuous_of_continuous_at_one φ hf
+end