Mathlib.Algebra.Order.Pointwise._auxLemma.1

theorem Mathlib.Algebra.Order.Pointwise._auxLemma.1 : ∀ {α : Type u_2} {β : Type u_3} [inst : SMul α β] {t : Set β} {a : α} {x : β}, (x ∈ a • t) = ∃ y ∈ t, a • y = x

This theorem, `Mathlib.Algebra.Order.Pointwise._auxLemma.1`, is a statement about scalar multiplication of sets in the context of algebra. It states that for all types `α` and `β`, provided `α` has an operation of scalar multiplication (`SMul`) to `β`, and given a set `t` of elements of type `β`, a scalar `a` of type `α`, and an element `x` of type `β`, then `x` belongs to the scalar multiplication of `a` and `t` if and only if there exists an element `y` in the set `t` such that `x` equals the scalar multiplication of `a` and `y`. In mathematical terms, this theorem asserts that $x \in a \cdot t$ if and only if there exists a $y \in t$ such that $x = a \cdot y$, where $a \cdot t$ denotes the set obtained by scalar multiplication of each element of $t$ by $a$, and $a \cdot y$ represents the multiplication of $a$ and $y$.

More concisely: For any type `α` with scalar multiplication `SMul` to `β`, and set `t` of elements in `β`, an element `x` belongs to the scalar multiplication of `a` and `t` if and only if there exists an element `y` in `t` such that `x = a * y`.