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Mathlib.Algebra.Module.PointwisePi

Pointwise actions on sets in Pi types #

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

Tags #

set multiplication, set addition, pointwise addition, pointwise multiplication, pi

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