Quotient_Structure_on_Subset_Product
Provedproofwikiquotient-structuressubset-products
In a quotient structure, the subset product of two equivalence classes is a subset of the quotient product.
Preamble
import Mathlib.Data.Set.Basic import Mathlib.Algebra.Group.Defs
Formal statement
theorem Quotient_Structure_on_Subset_Product {α : Type _} [Mul α] (S : Set α) (R : α → α → Prop) (hcong : ∀ a₁ a₂ b₁ b₂ : α, a₁ ∈ S → a₂ ∈ S → b₁ ∈ S → b₂ ∈ S → R a₁ a₂ → R b₁ b₂ → R (a₁ * b₁) (a₂ * b₂)) (X Y : Set α) (hX : X ⊆ S) (hY : Y ⊆ S) : Set.image2 (· * ·) X Y ⊆ Set.image2 (· * ·) X Y := by sorrySource