# Documentation

Mathlib.Algebra.Module.PointwisePi

# Pointwise actions on sets in Pi types #

This file contains lemmas about pointwise actions on sets in Pi types.

## Tags #

theorem vadd_pi_subset {K : Type u_1} {ι : Type u_2} {R : ιType u_3} [(i : ι) → VAdd K (R i)] (r : K) (s : Set ι) (t : (i : ι) → Set (R i)) :
r +ᵥ Set.pi s t Set.pi s (r +ᵥ t)
theorem smul_pi_subset {K : Type u_1} {ι : Type u_2} {R : ιType u_3} [(i : ι) → SMul K (R i)] (r : K) (s : Set ι) (t : (i : ι) → Set (R i)) :
r Set.pi s t Set.pi s (r t)
theorem vadd_univ_pi {K : Type u_1} {ι : Type u_2} {R : ιType u_3} [(i : ι) → VAdd K (R i)] (r : K) (t : (i : ι) → Set (R i)) :
r +ᵥ Set.pi Set.univ t = Set.pi Set.univ (r +ᵥ t)
theorem smul_univ_pi {K : Type u_1} {ι : Type u_2} {R : ιType u_3} [(i : ι) → SMul K (R i)] (r : K) (t : (i : ι) → Set (R i)) :
r Set.pi Set.univ t = Set.pi Set.univ (r t)
theorem vadd_pi {K : Type u_1} {ι : Type u_2} {R : ιType u_3} [] [(i : ι) → AddAction K (R i)] (r : K) (S : Set ι) (t : (i : ι) → Set (R i)) :
r +ᵥ Set.pi S t = Set.pi S (r +ᵥ t)
theorem smul_pi {K : Type u_1} {ι : Type u_2} {R : ιType u_3} [] [(i : ι) → MulAction K (R i)] (r : K) (S : Set ι) (t : (i : ι) → Set (R i)) :
r Set.pi S t = Set.pi S (r t)
theorem smul_pi₀ {K : Type u_1} {ι : Type u_2} {R : ιType u_3} [] [(i : ι) → MulAction K (R i)] {r : K} (S : Set ι) (t : (i : ι) → Set (R i)) (hr : r 0) :
r Set.pi S t = Set.pi S (r t)