of_injective — Mathlib

English

The Frattini-context variant asserts the order of G is odd under hypothesis.

Русский

Контекст фраттини — получаем нечетную кардинальность G; гипотеза аналогична ранее.

LaTeX

$$$Odd(|G|)$$$

Lean4

theorem of_injective : Function.Injective (of : α → FreeAbelianGroup α) := fun x y hoxy ↦
  Classical.by_contradiction fun hxy : x ≠ y ↦
    let f : FreeAbelianGroup α →+ ℤ := lift fun z ↦ if x = z then (1 : ℤ) else 0
    have hfx1 : f (of x) = 1 := (lift_apply_of _ _).trans <| if_pos rfl
    have hfy1 : f (of y) = 1 := hoxy ▸ hfx1
    have hfy0 : f (of y) = 0 := (lift_apply_of _ _).trans <| if_neg hxy
    one_ne_zero <| hfy1.symm.trans hfy0

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