mapsTo_emetric_ball — Mathlib

English

The ratio of the composition equals the product of the ratios.

Русский

Отношение композиции равно произведению отношений.

LaTeX

$$$\\operatorname{ratio}(g \\circ f) = \\operatorname{ratio}(g) \\cdot \\operatorname{ratio}(f)$$$

Lean4

/-- A dilation maps balls to balls and scales the radius by `ratio f`. -/
theorem mapsTo_emetric_ball (x : α) (r : ℝ≥0∞) :
    MapsTo (f : α → β) (EMetric.ball x r) (EMetric.ball (f x) (ratio f * r)) := fun y hy =>
  (edist_eq f y x).trans_lt <|
    (ENNReal.mul_lt_mul_left (ENNReal.coe_ne_zero.2 <| ratio_ne_zero f) ENNReal.coe_ne_top).2 hy

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