ext_elem_iff — Mathlib

English

For a basis B, b.repr = f if and only if for every i, f(b_i) = δ_i,1 (the i-th unit vector).

Русский

Для базы B равенство repr = f эквивалентно тому, что ∀ i, f(b_i) = δ_i,1.

LaTeX

$$$$ b.repr = f \iff \forall i, f(b_i) = Finsupp.single i 1 $$$$

Lean4

/-- Two elements are equal iff their coordinates are equal. -/
theorem ext_elem_iff {x y : M} : x = y ↔ ∀ i, b.repr x i = b.repr y i := by
  simp only [← DFunLike.ext_iff, EmbeddingLike.apply_eq_iff_eq]

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