chore: fix naming of coeff_C_ne_zero (#39632)

Split off from #39623

Author: justus-springer

Mathlib/Algebra/Polynomial/Basic.lean

@@ -640,7 +640,9 @@ theorem coeff_C : coeff (C a) n = ite (n = 0) a 0 := by
 theorem coeff_C_zero : coeff (C a) 0 = a :=
   coeff_monomial
 
-theorem coeff_C_ne_zero (h : n ≠ 0) : (C a).coeff n = 0 := by rw [coeff_C, if_neg h]
+theorem coeff_C_of_ne_zero (h : n ≠ 0) : (C a).coeff n = 0 := by rw [coeff_C, if_neg h]
+
+@[deprecated (since := "2026-05-20")] alias coeff_C_ne_zero := coeff_C_of_ne_zero
 
 @[simp]
 lemma coeff_C_succ {r : R} {n : ℕ} : coeff (C r) (n + 1) = 0 := by simp [coeff_C]

Mathlib/Algebra/Polynomial/Eval/Defs.lean

@@ -162,7 +162,7 @@ theorem eval₂_mul_C' (h : Commute (f a) x) : eval₂ f x (p * C a) = eval₂ f
   intro k
   by_cases hk : k = 0
   · simp only [hk, h, coeff_C_zero]
-  · simp only [coeff_C_ne_zero hk, map_zero, Commute.zero_left]
+  · simp only [coeff_C_of_ne_zero hk, map_zero, Commute.zero_left]
 
 theorem eval₂_list_prod_noncomm (ps : List R[X])
     (hf : ∀ p ∈ ps, ∀ (k), Commute (f <| coeff p k) x) :

Mathlib/RingTheory/HahnSeries/HEval.lean

@@ -192,7 +192,7 @@ theorem heval_C (r : R) : heval x (C r) = r • 1 := by
   ext g
   simp only [heval_apply, coeff_hsum, smulFamily_toFun, powers_toFun, HahnSeries.coeff_smul,
     HahnSeries.coeff_one, smul_eq_mul, mul_ite, mul_one, mul_zero]
-  rw [finsum_eq_single _ 0 (fun n hn ↦ by simp [coeff_ne_zero_C hn])]
+  rw [finsum_eq_single _ 0 (fun n hn ↦ by simp [coeff_C_of_ne_zero hn])]
   by_cases hg : g = 0 <;> simp
 
 theorem heval_X (hx : 0 < x.orderTop) : heval x X = x := by

Mathlib/RingTheory/Polynomial/Basic.lean

@@ -599,7 +599,7 @@ theorem isPrime_map_C_iff_isPrime (P : Ideal R) :
     constructor
     · rintro h (- | n)
       · rwa [coeff_C_zero]
-      · simp only [coeff_C_ne_zero (Nat.succ_ne_zero _), Submodule.zero_mem]
+      · simp only [coeff_C_of_ne_zero (Nat.succ_ne_zero _), Submodule.zero_mem]
     · intro h
       simpa only [coeff_C_zero] using h 0
   · intro h

Mathlib/RingTheory/PowerSeries/Basic.lean

@@ -185,12 +185,14 @@ theorem coeff_C (n : ℕ) (a : R) : coeff n (C a : R⟦X⟧) = if n = 0 then a e
 theorem coeff_zero_C (a : R) : coeff 0 (C a) = a := by
   rw [coeff_C, if_pos rfl]
 
-theorem coeff_ne_zero_C {a : R} {n : ℕ} (h : n ≠ 0) : coeff n (C a) = 0 := by
+theorem coeff_C_of_ne_zero {a : R} {n : ℕ} (h : n ≠ 0) : coeff n (C a) = 0 := by
   rw [coeff_C, if_neg h]
 
+@[deprecated (since := "2026-05-20")] alias coeff_ne_zero_C := coeff_C_of_ne_zero
+
 @[simp]
 theorem coeff_succ_C {a : R} {n : ℕ} : coeff (n + 1) (C a) = 0 :=
-  coeff_ne_zero_C n.succ_ne_zero
+  coeff_C_of_ne_zero n.succ_ne_zero
 
 @[grind inj]
 theorem C_injective : Function.Injective (C (R := R)) := MvPowerSeries.C_injective