edist_zero_of_eq_zero — Mathlib

English

If variationOnFromTo f s a b = 0, then the distance edist (f a) (f b) must also be 0, provided locally bounded variation on f.

Русский

Если вариация на отрезке равна нулю, то ев distância между f a и f b равна нулю при условии локальной ограниченности вариации.

LaTeX

$$edist (f a) (f b) = 0$$

Lean4

protected theorem edist_zero_of_eq_zero (hf : LocallyBoundedVariationOn f s) {a b : α} (ha : a ∈ s) (hb : b ∈ s)
    (h : variationOnFromTo f s a b = 0) : edist (f a) (f b) = 0 :=
  by
  wlog h' : a ≤ b
  · rw [edist_comm]
    apply this hf hb ha _ (le_of_not_ge h')
    rw [variationOnFromTo.eq_neg_swap, h, neg_zero]
  · apply le_antisymm _ (zero_le _)
    rw [← ENNReal.ofReal_zero, ← h, variationOnFromTo.eq_of_le f s h', ENNReal.ofReal_toReal (hf a b ha hb)]
    apply eVariationOn.edist_le
    exacts [⟨ha, ⟨le_rfl, h'⟩⟩, ⟨hb, ⟨h', le_rfl⟩⟩]

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