eq_iff_norm_eq_of_oangle_eq_zero — Mathlib

English

If norms are equal, then x = y is equivalent to o.oangle x y = 0.

Русский

Если нормa равны, то x = y эквивалентно o.oangle x y = 0.

LaTeX

$$Eq( x = y ) \Rightarrow (\|x\| = \|y\| \land o.oangle x y = 0) \,\land\, (\|x\| = \|y\| \land o.oangle x y = 0) \Rightarrow x = y$$

Lean4

/-- Two vectors with zero angle between them are equal if and only if they have equal norms. -/
theorem eq_iff_norm_eq_of_oangle_eq_zero {x y : V} (h : o.oangle x y = 0) : x = y ↔ ‖x‖ = ‖y‖ :=
  ⟨fun he => ((o.eq_iff_norm_eq_and_oangle_eq_zero x y).1 he).1, fun hn =>
    (o.eq_iff_norm_eq_and_oangle_eq_zero x y).2 ⟨hn, h⟩⟩

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