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Add constantCoeff_smul to RingTheory.PowerSeries.Basic (#12616)
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```
example : (constantCoeff ℝ) ((2:ℝ)•X) = 0 := by simp
```
simp doesn't solve the above because there is no lemma constantCoeff_smul tagged simp. This PR adds it to RingTheory.PowerSeries.Basic.



Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr>
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kcaze and sgouezel committed May 5, 2024
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Expand Up @@ -355,6 +355,11 @@ theorem coeff_smul {S : Type*} [Semiring S] [Module R S] (n : ℕ) (φ : PowerSe
rfl
#align power_series.coeff_smul PowerSeries.coeff_smul

@[simp]
theorem constantCoeff_smul {S : Type*} [Semiring S] [Module R S] (φ : PowerSeries S) (a : R) :
constantCoeff S (a • φ) = a • constantCoeff S φ :=
rfl

theorem smul_eq_C_mul (f : R⟦X⟧) (a : R) : a • f = C R a * f := by
ext
simp
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