rel_bot_eq_right_group_rel — Mathlib

English

rel_bot_eq_right_group_rel: the left-right setoid relation equals the right quotient relation by H.

Русский

rel_bot_eq_right_group_rel: левая-правой сетей равна правой факторной relation по H.

LaTeX

$$$ \\text{rel}_{\\bot,\\!H} = \\text{QuotientGroup.rightRel } H $$$

Lean4

theorem rel_bot_eq_right_group_rel (H : Subgroup G) : ⇑(setoid ↑H ↑(⊥ : Subgroup G)) = ⇑(QuotientGroup.rightRel H) :=
  by
  ext a b
  rw [rel_iff, QuotientGroup.rightRel_apply]
  constructor
  · rintro ⟨b, hb, a, rfl : a = 1, rfl⟩
    rwa [mul_one, mul_inv_cancel_right]
  · rintro (h : b * a⁻¹ ∈ H)
    exact ⟨b * a⁻¹, h, 1, rfl, by rw [mul_one, inv_mul_cancel_right]⟩

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