reps_zero_empty — Mathlib

English

The fourth application of S trivially returns the original matrix under the appropriate FixedDetMatrix context: S^4 · A = A.

Русский

Четвёртое применение S возвращает исходную матрицу: S^4·A = A.

LaTeX

$$$S^4 \cdot A = A$ for all $A \in FixedDetMatrix$ (in the given setting).$$

Lean4

@[simp]
theorem reps_zero_empty : reps 0 = ∅ :=
  by
  rw [reps, Set.eq_empty_iff_forall_notMem]
  rintro A ⟨h₁, h₂, -, h₄⟩
  suffices |A.1 0 1| < 0 by linarith [abs_nonneg (A.1 0 1)]
  have := A_c_eq_zero h₁
  simp_all [h₂.ne']

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