feat(data/mv_polynomial/basic): lemmas about map (#10092)

This adds `map_alg_hom`, which fills the gap between `map` and `map_alg_equiv`.

The only new proof here is `map_surjective`; everything else is just a reworked existing proof.

Author: eric-wieser

src/data/mv_polynomial/basic.lean

@@ -846,6 +846,27 @@ begin
   exact hf (h m),
 end
 
+lemma map_surjective (hf : function.surjective f) :
+  function.surjective (map f : mv_polynomial σ R → mv_polynomial σ S₁) :=
+λ p, begin
+  induction p using mv_polynomial.induction_on' with i fr a b ha hb,
+  { obtain ⟨r, rfl⟩ := hf fr,
+    exact ⟨monomial i r, map_monomial _ _ _⟩, },
+  { obtain ⟨a, rfl⟩ := ha,
+    obtain ⟨b, rfl⟩ := hb,
+    exact ⟨a + b, ring_hom.map_add _ _ _⟩ },
+end
+
+/-- If `f` is a left-inverse of `g` then `map f` is a left-inverse of `map g`. -/
+lemma map_left_inverse {f : R →+* S₁} {g : S₁ →+* R} (hf : function.left_inverse f g) :
+  function.left_inverse (map f : mv_polynomial σ R → mv_polynomial σ S₁) (map g) :=
+λ x, by rw [map_map, (ring_hom.ext hf : f.comp g = ring_hom.id _), map_id]
+
+/-- If `f` is a right-inverse of `g` then `map f` is a right-inverse of `map g`. -/
+lemma map_right_inverse {f : R →+* S₁} {g : S₁ →+* R} (hf : function.right_inverse f g) :
+  function.right_inverse (map f : mv_polynomial σ R → mv_polynomial σ S₁) (map g) :=
+(map_left_inverse hf.left_inverse).right_inverse
+
 @[simp] lemma eval_map (f : R →+* S₁) (g : σ → S₁) (p : mv_polynomial σ R) :
   eval g (map f p) = eval₂ f g p :=
 by { apply mv_polynomial.induction_on p; { simp { contextual := tt } } }
@@ -913,6 +934,22 @@ begin
   refl
 end
 
+/-- If `f : S₁ →ₐ[R] S₂` is a morphism of `R`-algebras, then so is `mv_polynomial.map f`. -/
+@[simps]
+def map_alg_hom [comm_semiring S₂] [algebra R S₁] [algebra R S₂] (f : S₁ →ₐ[R] S₂) :
+  mv_polynomial σ S₁ →ₐ[R] mv_polynomial σ S₂ :=
+{ to_fun := map ↑f,
+  commutes' := λ r, begin
+    have h₁ : algebra_map R (mv_polynomial σ S₁) r = C (algebra_map R S₁ r) := rfl,
+    have h₂ : algebra_map R (mv_polynomial σ S₂) r = C (algebra_map R S₂ r) := rfl,
+    rw [h₁, h₂, map, eval₂_hom_C, ring_hom.comp_apply, alg_hom.coe_to_ring_hom, alg_hom.commutes],
+  end,
+  ..map ↑f }
+
+@[simp] lemma map_alg_hom_id [algebra R S₁] :
+  map_alg_hom (alg_hom.id R S₁) = alg_hom.id R (mv_polynomial σ S₁) :=
+alg_hom.ext map_id
+
 end map
 
 

src/data/mv_polynomial/equiv.lean

@@ -91,12 +91,8 @@ def map_equiv [comm_semiring S₁] [comm_semiring S₂] (e : S₁ ≃+* S₂) :
   mv_polynomial σ S₁ ≃+* mv_polynomial σ S₂ :=
 { to_fun    := map (e : S₁ →+* S₂),
   inv_fun   := map (e.symm : S₂ →+* S₁),
-  left_inv  := λ p,
-    have (e.symm : S₂ →+* S₁).comp ↑e = ring_hom.id _ := ring_hom.ext e.symm_apply_apply,
-    by rw [map_map, this, map_id],
-  right_inv := assume p,
-    have (e : S₁ →+* S₂).comp ↑e.symm = ring_hom.id _ := ring_hom.ext e.apply_symm_apply,
-    by rw [map_map, this, map_id],
+  left_inv  := map_left_inverse e.left_inv,
+  right_inv := map_right_inverse e.right_inv,
   ..map (e : S₁ →+* S₂) }
 
 @[simp] lemma map_equiv_refl :
@@ -115,19 +111,12 @@ variables {A₁ A₂ A₃ : Type*} [comm_semiring A₁] [comm_semiring A₂] [co
 variables [algebra R A₁] [algebra R A₂] [algebra R A₃]
 
 /-- If `e : A ≃ₐ[R] B` is an isomorphism of `R`-algebras, then so is `map e`. -/
+@[simps apply]
 def map_alg_equiv (e : A₁ ≃ₐ[R] A₂) :
   mv_polynomial σ A₁ ≃ₐ[R] mv_polynomial σ A₂ :=
-{ commutes' := λ r, begin
-    dsimp,
-    have h₁ : algebra_map R (mv_polynomial σ A₁) r = C (algebra_map R A₁ r) := rfl,
-    have h₂ : algebra_map R (mv_polynomial σ A₂) r = C (algebra_map R A₂ r) := rfl,
-    rw [h₁, h₂, map, eval₂_hom_C, ring_hom.comp_apply,
-      ring_equiv.coe_to_ring_hom, alg_equiv.coe_ring_equiv, alg_equiv.commutes],
-  end,
-  ..(map_equiv σ ↑e) }
-
-@[simp] lemma map_alg_equiv_apply (e : A₁ ≃ₐ[R] A₂) (x : mv_polynomial σ A₁) :
-  map_alg_equiv σ e x = map ↑e x := rfl
+{ to_fun := map (e : A₁ →+* A₂),
+  ..map_alg_hom (e : A₁ →ₐ[R] A₂),
+  ..map_equiv σ (e : A₁ ≃+* A₂) }
 
 @[simp] lemma map_alg_equiv_refl :
   map_alg_equiv σ (alg_equiv.refl : A₁ ≃ₐ[R] A₁) = alg_equiv.refl :=