chore(algebra/category): remove duplicated proofs (#7349)

The results added in #7100 already exist. I moved them to the place where Scott added the duplicates. Hopefully that will make them more discoverable.

Author: TwoFX

src/algebra/category/Module/basic.lean

@@ -229,29 +229,6 @@ instance : preadditive (Module.{v} R) :=
 
 end preadditive
 
-section epi_mono
-variables {M N : Module.{v} R} (f : M ⟶ N)
-
-lemma ker_eq_bot_of_mono [mono f] : f.ker = ⊥ :=
-linear_map.ker_eq_bot_of_cancel $ λ u v, (@cancel_mono _ _ _ _ _ f _ ↟u ↟v).1
-
-lemma range_eq_top_of_epi [epi f] : f.range = ⊤ :=
-linear_map.range_eq_top_of_cancel $ λ u v, (@cancel_epi _ _ _ _ _ f _ ↟u ↟v).1
-
-lemma mono_of_ker_eq_bot (hf : f.ker = ⊥) : mono f :=
-concrete_category.mono_of_injective _ $ linear_map.ker_eq_bot.1 hf
-
-lemma epi_of_range_eq_top (hf : f.range = ⊤) : epi f :=
-concrete_category.epi_of_surjective _ $ linear_map.range_eq_top.1 hf
-
-instance mono_as_hom'_subtype (U : submodule R M) : mono ↾U.subtype :=
-mono_of_ker_eq_bot _ (submodule.ker_subtype U)
-
-instance epi_as_hom''_mkq (U : submodule R M) : epi ↿U.mkq :=
-epi_of_range_eq_top _ $ submodule.range_mkq _
-
-end epi_mono
-
 end Module
 
 instance (M : Type u) [add_comm_group M] [module R M] : has_coe (submodule R M) (Module R) :=

src/algebra/category/Module/epi_mono.lean

@@ -17,38 +17,36 @@ universes v u
 
 open category_theory
 open Module
+open_locale Module
 
 namespace Module
 
-variables {R : Type u} [ring R]
+variables {R : Type u} [ring R] {X Y : Module.{v} R} (f : X ⟶ Y)
 
-/--
-We could also give a direct proof via `linear_map.ker_eq_bot_of_cancel`.
-(This would allow generalising from `Module.{u}` to `Module.{v}`.)
--/
-lemma mono_iff_injective {X Y : Module.{u} R} (f : X ⟶ Y) : mono f ↔ function.injective f :=
-begin
-  rw ←category_theory.mono_iff_injective,
-  exact ⟨right_adjoint_preserves_mono (adj R), faithful_reflects_mono (forget (Module.{u} R))⟩,
-end
-
-lemma epi_iff_surjective {X Y : Module.{v} R} (f : X ⟶ Y) : epi f ↔ function.surjective f :=
-begin
-  fsplit,
-  { intro h,
-    rw ←linear_map.range_eq_top,
-    apply linear_map.range_eq_top_of_cancel,
-    -- Now we have to fight a bit with the difference between `Y` and `↥Y`.
-    intros u v w,
-    change Y ⟶ Module.of R (linear_map.range f).quotient at u,
-    change Y ⟶ Module.of R (linear_map.range f).quotient at v,
-    apply (cancel_epi (Module.of_self_iso Y).hom).mp,
-    apply h.left_cancellation,
-    cases X, cases Y, -- after this we can see `Module.of_self_iso` is just the identity.
-    convert w; { dsimp, erw category.id_comp, }, },
-  { rw ←category_theory.epi_iff_surjective,
-    exact faithful_reflects_epi (forget (Module.{v} R)), },
-end
+lemma ker_eq_bot_of_mono [mono f] : f.ker = ⊥ :=
+linear_map.ker_eq_bot_of_cancel $ λ u v, (@cancel_mono _ _ _ _ _ f _ ↟u ↟v).1
+
+lemma range_eq_top_of_epi [epi f] : f.range = ⊤ :=
+linear_map.range_eq_top_of_cancel $ λ u v, (@cancel_epi _ _ _ _ _ f _ ↟u ↟v).1
+
+lemma mono_iff_ker_eq_bot : mono f ↔ f.ker = ⊥ :=
+⟨λ hf, by exactI ker_eq_bot_of_mono _,
+ λ hf, concrete_category.mono_of_injective _ $ linear_map.ker_eq_bot.1 hf⟩
+
+lemma mono_iff_injective : mono f ↔ function.injective f :=
+by rw [mono_iff_ker_eq_bot, linear_map.ker_eq_bot]
+
+lemma epi_iff_range_eq_top : epi f ↔ f.range = ⊤ :=
+⟨λ hf, by exactI range_eq_top_of_epi _,
+ λ hf, concrete_category.epi_of_surjective _ $ linear_map.range_eq_top.1 hf⟩
+
+lemma epi_iff_surjective : epi f ↔ function.surjective f :=
+by rw [epi_iff_range_eq_top, linear_map.range_eq_top]
+
+instance mono_as_hom'_subtype (U : submodule R X) : mono ↾U.subtype :=
+(mono_iff_ker_eq_bot _).mpr (submodule.ker_subtype U)
 
+instance epi_as_hom''_mkq (U : submodule R X) : epi ↿U.mkq :=
+(epi_iff_range_eq_top _).mpr $ submodule.range_mkq _
 
 end Module

src/algebra/category/Module/kernels.lean

@@ -3,7 +3,7 @@ Copyright (c) 2020 Markus Himmel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Markus Himmel
 -/
-import algebra.category.Module.basic
+import algebra.category.Module.epi_mono
 
 /-!
 # The concrete (co)kernels in the category of modules are (co)kernels in the categorical sense.
@@ -48,7 +48,7 @@ cofork.is_colimit.mk _
   (λ s, f.range.liftq_mkq (cofork.π s) _)
   (λ s m h,
   begin
-    haveI : epi (as_hom f.range.mkq) := epi_of_range_eq_top _ (submodule.range_mkq _),
+    haveI : epi (as_hom f.range.mkq) := (epi_iff_range_eq_top _).mpr (submodule.range_mkq _),
     apply (cancel_epi (as_hom f.range.mkq)).1,
     convert h walking_parallel_pair.one,
     exact submodule.liftq_mkq _ _ _

src/algebra/category/Module/subobject.lean

@@ -3,7 +3,7 @@ Copyright (c) 2021 Markus Himmel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Markus Himmel
 -/
-import algebra.category.Module.basic
+import algebra.category.Module.epi_mono
 import category_theory.subobject.well_powered
 
 /-!