expand — Mathlib

English

In a group G, if a x y describes a semiconjugacy h, then order(x) = order(y).

Русский

В группе G, если x и y связаны через полуприведение, то ord(x) = ord(y).

LaTeX

$$order(x) = order(y)$$

Lean4

/-- A fraction `r /ₒ s` is equal to its expansion by an arbitrary factor `t` if `t * s ∈ S`. -/
@[to_additive /-- A difference `r -ₒ s` is equal to its expansion by an
    arbitrary translation `t` if `t + s ∈ S`. -/
    ]
protected theorem expand (r : X) (s : S) (t : R) (hst : t * (s : R) ∈ S) : r /ₒ s = t • r /ₒ ⟨t * s, hst⟩ :=
  by
  apply Quotient.sound
  exact ⟨s, s * t, by rw [mul_smul, Submonoid.smul_def], by rw [← mul_assoc]⟩

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