prod_self — Mathlib

English

If la has basis pa sa, then la × la has basis pa at sa(i) × sa(i).

Русский

Если у ла есть базис pa sa, то la × la имеет базис pa по sa(i) × sa(i).

LaTeX

$$If $\mathcal{L}$ has basis $p_i$ with sets $S_i$, then $(\mathcal{L} \times \mathcal{L})$ has basis $p_i$ with sets $S_i \times S_i$.$$

Lean4

theorem prod_self (hl : la.HasBasis pa sa) : (la ×ˢ la).HasBasis pa fun i => sa i ×ˢ sa i :=
  hl.prod_same_index hl fun {i j} hi hj => by
    simpa only [exists_prop, subset_inter_iff] using hl.mem_iff.1 (inter_mem (hl.mem_of_mem hi) (hl.mem_of_mem hj))

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