chore(linear_algebra/affine_space/affine_subspace) add set_like insta…

…nce (#18622)

Needed for the port.

src/linear_algebra/affine_space/affine_subspace.lean

@@ -178,9 +178,9 @@ variables (k : Type*) {V : Type*} (P : Type*) [ring k] [add_comm_group V] [modul
           [affine_space V P]
 include V
 
--- TODO Refactor to use `instance : set_like (affine_subspace k P) P :=` instead
-instance : has_coe (affine_subspace k P) (set P) := ⟨carrier⟩
-instance : has_mem P (affine_subspace k P) := ⟨λ p s, p ∈ (s : set P)⟩
+instance : set_like (affine_subspace k P) P :=
+{ coe := carrier,
+  coe_injective' := λ p q _, by cases p; cases q; congr' }
 
 /-- A point is in an affine subspace coerced to a set if and only if
 it is in that affine subspace. -/
@@ -354,17 +354,15 @@ begin
 end
 
 /-- Two affine subspaces are equal if they have the same points. -/
-@[ext] lemma coe_injective : function.injective (coe : affine_subspace k P → set P) :=
-λ s1 s2 h, begin
-  cases s1,
-  cases s2,
-  congr,
-  exact h
-end
+lemma coe_injective : function.injective (coe : affine_subspace k P → set P) :=
+set_like.coe_injective
+
+@[ext] theorem ext {p q : affine_subspace k P} (h : ∀ x, x ∈ p ↔ x ∈ q) : p = q :=
+set_like.ext h
 
 @[simp] lemma ext_iff (s₁ s₂ : affine_subspace k P) :
   (s₁ : set P) = s₂ ↔ s₁ = s₂ :=
-⟨λ h, coe_injective h, by tidy⟩
+set_like.ext'_iff.symm
 
 /-- Two affine subspaces with the same direction and nonempty
 intersection are equal. -/

src/linear_algebra/affine_space/pointwise.lean

@@ -49,7 +49,7 @@ lemma vadd_mem_pointwise_vadd_iff {v : V} {s : affine_subspace k P} {p : P} :
 vadd_mem_vadd_set_iff
 
 lemma pointwise_vadd_bot (v : V) : v +ᵥ (⊥ : affine_subspace k P) = ⊥ :=
-by { ext, simp, }
+by simp [set_like.ext'_iff]
 
 lemma pointwise_vadd_direction (v : V) (s : affine_subspace k P) :
   (v +ᵥ s).direction = s.direction :=