feat(Algebra/IsScalarTower): add transitivity results (#27866)

Add the following: 
```
Let `Q / P / N / M` be a scalar tower. If `P / N / M`, `Q / P / M` and `Q / P / N` are
scalar towers, then `Q / N / M` is also a scalar tower.
```
and variations.

Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>



Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>

Author: xroblot

Mathlib/Algebra/Group/Action/Defs.lean

@@ -488,7 +488,6 @@ lemma isScalarTower_iff_smulCommClass_of_commMonoid (R₁ R : Type*)
     SMulCommClass R₁ R R ↔ IsScalarTower R₁ R R :=
   ⟨fun _ ↦ IsScalarTower.of_commMonoid R₁ R, fun _ ↦ SMulCommClass.of_commMonoid R₁ R R⟩
 
-
 end
 
 section CompatibleScalar
@@ -516,6 +515,34 @@ lemma SMulCommClass.of_mul_smul_one {M N} [Monoid N] [SMul M N]
     (H : ∀ (x : M) (y : N), y * x • (1 : N) = x • y) : SMulCommClass M N N :=
   ⟨fun x y z ↦ by rw [← H x z, smul_eq_mul, ← H, smul_eq_mul, mul_assoc]⟩
 
+/--
+Let `Q / P / N / M` be a tower. If `P / N / M`, `Q / P / M` and `Q / P / N` are
+scalar towers, then `Q / N / M` is also a scalar tower.
+-/
+@[to_additive] lemma IsScalarTower.to₁₂₄ (M N P Q)
+    [SMul M N] [SMul M P] [SMul M Q] [SMul N P] [SMul N Q] [Monoid P] [MulAction P Q]
+    [IsScalarTower M N P] [IsScalarTower M P Q] [IsScalarTower N P Q] : IsScalarTower M N Q where
+  smul_assoc m n q := by rw [← smul_one_smul P, smul_assoc m, smul_assoc, smul_one_smul]
+
+/--
+Let `Q / P / N / M` be a tower. If `P / N / M`, `Q / N / M` and `Q / P / N` are
+scalar towers, then `Q / P / M` is also a scalar tower.
+-/
+@[to_additive] lemma IsScalarTower.to₁₃₄ (M N P Q)
+    [SMul M N] [SMul M P] [SMul M Q] [SMul P Q] [Monoid N] [MulAction N P] [MulAction N Q]
+    [IsScalarTower M N P] [IsScalarTower M N Q] [IsScalarTower N P Q] : IsScalarTower M P Q where
+  smul_assoc m p q := by rw [← smul_one_smul N m, smul_assoc, smul_one_smul]
+
+/--
+Let `Q / P / N / M` be a tower. If `P / N / M`, `Q / N / M` and `Q / P / M` are
+scalar towers, then `Q / P / N` is also a scalar tower.
+-/
+@[to_additive] lemma IsScalarTower.to₂₃₄ (M N P Q)
+    [SMul M N] [SMul M P] [SMul M Q] [SMul P Q] [Monoid N] [MulAction N P] [MulAction N Q]
+    [IsScalarTower M N P] [IsScalarTower M N Q] [IsScalarTower M P Q]
+    (h : Function.Surjective fun m : M ↦ m • (1 : N)) : IsScalarTower N P Q where
+  smul_assoc n p q := by obtain ⟨m, rfl⟩ := h n; simp_rw [smul_one_smul, smul_assoc]
+
 end CompatibleScalar
 
 /-- Typeclass for multiplicative actions on multiplicative structures.

Mathlib/Algebra/Group/Action/Faithful.lean

@@ -70,3 +70,12 @@ lemma faithfulSMul_iff_injective_smul_one (R A : Type*)
   · simp only at hr
     rw [← one_mul a, ← smul_mul_assoc, ← smul_mul_assoc, hr]
   · simpa using hr 1
+
+/--
+Let `Q / P / N / M` be a tower. If `Q / N / M`, `Q / P / M` and `Q / P / N` are
+scalar towers, then `P / N / M` is also a scalar tower.
+-/
+@[to_additive] lemma IsScalarTower.to₁₂₃ (M N P Q)
+    [SMul M N] [SMul M P] [SMul M Q] [SMul N P] [SMul N Q] [SMul P Q] [FaithfulSMul P Q]
+    [IsScalarTower M N Q] [IsScalarTower M P Q] [IsScalarTower N P Q] : IsScalarTower M N P where
+  smul_assoc m n p := by simp_rw [← (smul_left_injective' (α := Q)).eq_iff, smul_assoc]