Update partialDeriv.lean

Nevanlinna/partialDeriv.lean

@@ -93,6 +93,25 @@ lemma partialDeriv_fderiv {f : E → F} (hf : ContDiff 𝕜 2 f) (z a b : E) :
   · simp
 
 
+section restrictScalars
+
+variable (𝕜 : Type*) [NontriviallyNormedField 𝕜]
+variable {𝕜' : Type*} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜']
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedSpace 𝕜' E]
+variable [IsScalarTower 𝕜 𝕜' E]
+variable {F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedSpace 𝕜' F]
+variable [IsScalarTower 𝕜 𝕜' F]
+--variable {f : E → F}
+
+theorem partialDeriv_restrictScalars {f : E → F} {v : E} :
+  Differentiable 𝕜' f → partialDeriv 𝕜 v f = partialDeriv 𝕜' v f := by
+  intro hf
+  unfold partialDeriv
+  funext x
+  rw [(hf x).fderiv_restrictScalars 𝕜]
+  simp
+
+
 theorem partialDeriv_comm
   {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
   {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]