orbit_eq_orbit_out — Mathlib

English

Two quotient-orbits determine equality of quotient elements; equality of orbits implies equality of the quotient classes.

Русский

Разделительная энумерация орбит квотирования: равенство орбит эквивалентно равенству эквивалентных классов.

LaTeX

$$$\text{orbit}_{\text{Quotient}} \;\text{injective}: x.orbit = y.orbit \iff x = y$$$

Lean4

/-- Note that `hφ = Quotient.out_eq'` is a useful choice here. -/
@[to_additive /-- Note that `hφ = Quotient.out_eq'` is a useful choice here. -/
    ]
theorem orbit_eq_orbit_out (x : orbitRel.Quotient G α) {φ : orbitRel.Quotient G α → α}
    (hφ :
      letI := orbitRel G α;
      RightInverse φ Quotient.mk') :
    orbitRel.Quotient.orbit x = MulAction.orbit G (φ x) :=
  by
  conv_lhs => rw [← hφ x]
  rfl

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