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_3} {R : ι → Type u_2} [inst : (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)

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theorem smul_pi_subset {K : Type u_1} {ι : Type u_3} {R : ι → Type u_2} [inst : (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)

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theorem vadd_univ_pi {K : Type u_1} {ι : Type u_3} {R : ι → Type u_2} [inst : (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)

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theorem smul_univ_pi {K : Type u_1} {ι : Type u_3} {R : ι → Type u_2} [inst : (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)

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theorem vadd_pi {K : Type u_1} {ι : Type u_3} {R : ι → Type u_2} [inst : AddGroup K] [inst : (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)

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theorem smul_pi {K : Type u_1} {ι : Type u_3} {R : ι → Type u_2} [inst : Group K] [inst : (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)

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theorem smul_pi₀ {K : Type u_1} {ι : Type u_3} {R : ι → Type u_2} [inst : GroupWithZero K] [inst : (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)