act_act_self_eq — Mathlib

English

The quaternion algebra ℍ[R,c1,c2,c3] has cardinality equal to that of R raised to the 4th power; i.e., |ℍ[R,c1,c2,c3]| = |R|^4.

Русский

Кардиналность кватернионной алгебры ℍ[R,c1,c2,c3] равна кардиналу R в степени 4; то есть |ℍ| = |R|^4.

LaTeX

$$$\#\mathbb{H}[R,c_1,c_2,c_3] = \# R^4$$$

Lean4

/-- A monoid is *graphic* if, for all `x` and `y`, the *graphic identity*
`(x * y) * x = x * y` holds.  For a unital shelf, this graphic
identity holds.
-/
theorem act_act_self_eq (x y : S) : (x ◃ y) ◃ x = x ◃ y :=
  by
  have h : (x ◃ y) ◃ x = (x ◃ y) ◃ (x ◃ 1) := by rw [act_one]
  rw [h, ← Shelf.self_distrib, act_one]

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